hypothesis vs a thesis

Thesis Vs Hypothesis: Understanding The Basis And The Key Differences

Hypothesis vs. thesis: They sound similar and seem to discuss the same thing. However, these terms have vastly different meanings and purposes. You may have encountered these concepts in school or research, but understanding them is key to executing quality work. 

In this article, I’ll discuss hypothesis vs. thesis, break down their differences, and show you how to apply this knowledge to create quality written works. Let’s get to it!

Thesis vs. Hypothesis: Understanding the Basis

The power of a thesis.

A thesis statement is typically found at the end of the introduction in an essay or research paper, succinctly summarizing the overarching theme.

Crafting a strong thesis

Hypothesis: the scientific proposition.

In contrast, a hypothesis is a tentative proposition or educated guess. It is the initial step in the scientific method, where researchers formulate a hunch to test their assumptions and theories. 

Formulating a hypothesis

Key differences between thesis vs. hypothesis, 1. nature of statement, 3. testability, 4. research stage, 6. examples.

These differences highlight the distinct roles that the thesis and hypothesis play in academic writing and scientific research, with one providing a point of argumentation and the other guiding the scientific inquiry process.

Can a hypothesis become a thesis?

Do all research papers require a thesis, can a thesis be proven wrong.

Yes. The purpose of a thesis is not only to prove but also to encourage critical analysis. It can be proven wrong with compelling counterarguments and evidence.

How long should a thesis statement be?

Is a hypothesis only used in scientific research, can a hypothesis be vague.

No. When creating a hypothesis, it’s important to make it clear and able to be tested. Developing experiments and making conclusions based on the results can be difficult if the hypothesis needs clarification.

Final Thoughts

In conclusion, understanding the differences between a hypothesis and a thesis is vital to crafting successful research projects and academic papers. While they may seem interchangeable at first glance, these two concepts serve distinct purposes in the research process. 

So, the next time you embark on a research project, take the time to ensure that you understand the fundamental difference between a hypothesis and a thesis. Doing so can lead to more focused, meaningful research that advances knowledge and understanding in your field.

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Home » Education » Difference Between Thesis and Hypothesis

Difference Between Thesis and Hypothesis

Main difference –  thesis vs hypothesis                           .

Thesis and hypothesis are two common terms that are often found in research studies. Hypothesis is a logical proposition that is based on existing knowledge that serves as the starting point of an investigation. A thesis is a statement that is put forward as a premise to be maintained or proved. The main difference between thesis and hypothesis is that thesis is found in all research studies whereas a hypothesis is mainly found in experimental quantitative research studies.

This article explains,

1. What is a Thesis?      – Definition, Features, Function

2. What is a Hypothesis?      – Definition, Features, Function

What is a Thesis

The word thesis has two meanings in a research study. Thesis can either refer to a dissertation or a thesis statement. Thesis or dissertation is the long essay or document that consists of the research study.  Thesis can also refer to a theory or statement that is used as a premise to be maintained or proved.

The thesis statement in a research article is a sentence found at the beginning of the paper that presents the main argument of the paper. The rest of the document will gather, organize and present evidence to support this argument. The thesis statement will basically present the topic of the paper and indicate what position the researcher is going to take in relation to this topic. A thesis statement can generally be found at the end of the first paragraph (introductory paragraph) of the paper.

What is a Hypothesis

A hypothesis is a logical assumption based on available evidence. Hypothesis is defined as “a supposition or proposed explanation made on the basis of limited evidence as a starting point for further investigation” in the Oxford dictionary and as “an idea or theory that is not proven but that leads to further study or discussion” in the Merriam-Webster dictionary. In simple words, it is an educated guess that is not proven with concrete scientific evidence. Once it is scientifically tested and proven, it becomes a theory. However, it is important to note that a hypothesis can be accurate or inaccurate.

Hypotheses are mostly used in experiments and research studies. However, hypotheses are not used in every research study. They are mostly used in quantitative research studies  that deal with experiments. Hypotheses are often used to test a specific model or theory . They can be used only when the researcher has sufficient knowledge about the subject since hypothesis are always based on the existing knowledge. Once the hypothesis is built, the researcher can find and analyze data and use them to prove or disprove the hypothesis.

Thesis: A thesis is a “statement or theory that is put forward as a premise to be maintained or proved” or a “long essay or dissertation involving personal research, written by a candidate for a university degree” (Oxford dictionary).

Hypothesis: A hypothesis is “a supposition or proposed explanation made on the basis of limited evidence as a starting point for further investigation” (Oxford dictionary).

Thesis: Thesis statement can be found in all research papers.

Hypothesis: Hypotheses are usually found in experimental quantitative research studies.

Thesis: Thesis statement may explain the hypothesis and how the researcher intends to support it.

Hypothesis: Hypothesis is an educated guess based on the existing knowledge.

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Answered By: APUS Librarians Last Updated: Apr 15, 2022     Views: 129275

Both the hypothesis statement and the thesis statement answer a research question. 

• A hypothesis is a statement that can be proved or disproved. It is typically used in quantitative research and predicts the relationship between variables.  
• A thesis statement is a short, direct sentence that summarizes the main point or claim of an essay or research paper. It is seen in quantitative, qualitative, and mixed methods research. A thesis statement is developed, supported, and explained in the body of the essay or research report by means of examples and evidence.

Every research study should contain a concise and well-written thesis statement. If the intent of the study is to prove/disprove something, that research report will also contain a hypothesis statement.

NOTE: In some disciplines, the hypothesis is referred to as a thesis statement! This is not accurate but within those disciplines it is understood that "a short, direct sentence that summarizes the main point" will be included.

For more information, see The Research Question and Hypothesis (PDF file from the English Language Support, Department of Student Services, Ryerson University).

How do I write a good thesis statement?

How do I write a good hypothesis statement?

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The Real Differences Between Thesis and Hypothesis (With table)

A thesis and a hypothesis are two very different things, but they are often confused with one another. In this blog post, we will explain the differences between these two terms, and help you understand when to use which one in a research project.

As a whole, the main difference between a thesis and a hypothesis is that a thesis is an assertion that can be proven or disproven, while a hypothesis is a statement that can be tested by scientific research. 

We probably need to expand a bit on this topic to make things clearer for you, let’s start with definitions and examples.

Definitions

As always, let’s start with the definition of each term before going further.

A thesis is a statement or theory that is put forward as a premise to be maintained or proved. A thesis statement is usually one sentence, and it states your position on the topic at hand.

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The best way to understand the slight difference between those terms, is to give you an example for each of them.

If you are writing a paper about the effects of climate change on the environment, your thesis might be “Climate change is causing irreparable damage to our planet, and we must take action to prevent further damage”.

If your hypothesis is correct, then further research should be able to confirm it. However, if your hypothesis is incorrect, research will disprove it. Either way, a hypothesis is an important part of the scientific process.

The word “hypothesis” comes from the Greek words “hupo,” meaning “under”, and “thesis” that we just explained.

Argumentation vs idea

A thesis is usually the result of extensive research and contemplation, and seeks to prove a point or theory.

A hypothesis is only a statement that need to be tested by observation or experimentation.

5 mains differences between thesis and hypothesis

Thesis and hypothesis are different in several ways, here are the 5 keys differences between those terms:

So, in short, a thesis is an argument, while a hypothesis is a prediction. A thesis is more detailed and longer than a hypothesis, and it is based on research. Finally, a thesis must be proven, while a hypothesis does not need to be proven.

Is there a difference between a thesis and a claim?

Is a hypothesis a prediction.

No, a hypothesis is not a prediction. A prediction is a statement about what you think will happen in the future, whereas a hypothesis is a statement about what you think is causing a particular phenomenon.

What’s the difference between thesis and dissertation?

A thesis is usually shorter and more focused than a dissertation, and it is typically achieved in order to earn a bachelor’s degree. A dissertation is usually longer and more comprehensive, and it is typically completed in order to earn a master’s or doctorate degree.

What is a good thesis statement?

I am very curious and I love to learn about all types of subjects. Thanks to my experience on the web, I share my discoveries with you on this site :)

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Hypothesis vs. Thesis

What's the difference.

A hypothesis is a proposed explanation for a phenomenon that can be tested through research and experimentation. It is a tentative statement that serves as the basis for further investigation. On the other hand, a thesis is a statement or theory that is put forward as a premise to be maintained or proved. It is typically a longer, more detailed argument that is supported by evidence and analysis. While a hypothesis is more focused on predicting outcomes and guiding research, a thesis is more comprehensive and aims to persuade the reader of a particular perspective or argument.

Further Detail

A hypothesis is a proposed explanation for a phenomenon or a scientific question that can be tested through experimentation or observation. It is a tentative assumption made in order to draw out and test its logical or empirical consequences. On the other hand, a thesis is a statement or theory that is put forward as a premise to be maintained or proved. It is typically used in academic writing to present an argument or claim that will be supported with evidence and analysis.

Hypotheses are used in scientific research to guide the investigation and testing of a specific question or problem. They help researchers to make predictions about the outcome of experiments and observations. In contrast, a thesis is used in academic writing to present a central argument or claim that the author will support with evidence and analysis. It serves as the main point that the author is trying to prove or persuade the reader to accept.

A hypothesis is typically narrower in scope compared to a thesis. It focuses on a specific question or problem and proposes a possible explanation or solution. In contrast, a thesis is broader in scope as it presents an overarching argument or claim that encompasses the entire paper or essay. It provides a roadmap for the reader to understand the main point of the work and how it will be supported.

Both hypotheses and theses rely on evidence to support their claims. In scientific research, hypotheses are tested through experimentation and observation to gather data that either confirms or refutes the proposed explanation. In academic writing, theses are supported with evidence from sources such as research studies, scholarly articles, and other relevant sources. The quality and relevance of the evidence used can strengthen the credibility of both hypotheses and theses.

Flexibility

Hypotheses are more flexible compared to theses. If a hypothesis is not supported by the data or experiments, researchers can revise or refine it based on the new information gathered. This allows for the hypothesis to evolve as more evidence is collected. On the other hand, a thesis is typically more fixed in academic writing. While it can be revised during the writing process, the central argument or claim remains constant throughout the paper or essay.

In conclusion, while both hypotheses and theses play important roles in scientific research and academic writing, they differ in terms of definition, function, scope, evidence, and flexibility. Understanding the differences between the two can help researchers and writers effectively formulate and support their arguments. Whether testing a scientific question or presenting a central claim in an academic paper, both hypotheses and theses are essential tools for advancing knowledge and understanding in their respective fields.

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Thesis vs. Hypothesis: What's the Difference?

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What Is A Research (Scientific) Hypothesis? A plain-language explainer + examples

By:  Derek Jansen (MBA)  | Reviewed By: Dr Eunice Rautenbach | June 2020

If you’re new to the world of research, or it’s your first time writing a dissertation or thesis, you’re probably noticing that the words “research hypothesis” and “scientific hypothesis” are used quite a bit, and you’re wondering what they mean in a research context .

“Hypothesis” is one of those words that people use loosely, thinking they understand what it means. However, it has a very specific meaning within academic research. So, it’s important to understand the exact meaning before you start hypothesizing. 

Research Hypothesis 101

• What is a hypothesis ?
• What is a research hypothesis (scientific hypothesis)?
• Requirements for a research hypothesis
• Definition of a research hypothesis
• The null hypothesis

What is a hypothesis?

Let’s start with the general definition of a hypothesis (not a research hypothesis or scientific hypothesis), according to the Cambridge Dictionary:

Hypothesis: an idea or explanation for something that is based on known facts but has not yet been proved.

In other words, it’s a statement that provides an explanation for why or how something works, based on facts (or some reasonable assumptions), but that has not yet been specifically tested . For example, a hypothesis might look something like this:

Hypothesis: sleep impacts academic performance.

This statement predicts that academic performance will be influenced by the amount and/or quality of sleep a student engages in – sounds reasonable, right? It’s based on reasonable assumptions , underpinned by what we currently know about sleep and health (from the existing literature). So, loosely speaking, we could call it a hypothesis, at least by the dictionary definition.

But that’s not good enough…

Unfortunately, that’s not quite sophisticated enough to describe a research hypothesis (also sometimes called a scientific hypothesis), and it wouldn’t be acceptable in a dissertation, thesis or research paper . In the world of academic research, a statement needs a few more criteria to constitute a true research hypothesis .

What is a research hypothesis?

A research hypothesis (also called a scientific hypothesis) is a statement about the expected outcome of a study (for example, a dissertation or thesis). To constitute a quality hypothesis, the statement needs to have three attributes – specificity , clarity and testability .

Let’s take a look at these more closely.

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Hypothesis Essential #1: Specificity & Clarity

A good research hypothesis needs to be extremely clear and articulate about both what’ s being assessed (who or what variables are involved ) and the expected outcome (for example, a difference between groups, a relationship between variables, etc.).

Let’s stick with our sleepy students example and look at how this statement could be more specific and clear.

Hypothesis: Students who sleep at least 8 hours per night will, on average, achieve higher grades in standardised tests than students who sleep less than 8 hours a night.

As you can see, the statement is very specific as it identifies the variables involved (sleep hours and test grades), the parties involved (two groups of students), as well as the predicted relationship type (a positive relationship). There’s no ambiguity or uncertainty about who or what is involved in the statement, and the expected outcome is clear.

Contrast that to the original hypothesis we looked at – “Sleep impacts academic performance” – and you can see the difference. “Sleep” and “academic performance” are both comparatively vague , and there’s no indication of what the expected relationship direction is (more sleep or less sleep). As you can see, specificity and clarity are key.

Hypothesis Essential #2: Testability (Provability)

A statement must be testable to qualify as a research hypothesis. In other words, there needs to be a way to prove (or disprove) the statement. If it’s not testable, it’s not a hypothesis – simple as that.

For example, consider the hypothesis we mentioned earlier:

Hypothesis: Students who sleep at least 8 hours per night will, on average, achieve higher grades in standardised tests than students who sleep less than 8 hours a night.  

We could test this statement by undertaking a quantitative study involving two groups of students, one that gets 8 or more hours of sleep per night for a fixed period, and one that gets less. We could then compare the standardised test results for both groups to see if there’s a statistically significant difference. 

Again, if you compare this to the original hypothesis we looked at – “Sleep impacts academic performance” – you can see that it would be quite difficult to test that statement, primarily because it isn’t specific enough. How much sleep? By who? What type of academic performance?

So, remember the mantra – if you can’t test it, it’s not a hypothesis 🙂

Defining A Research Hypothesis

You’re still with us? Great! Let’s recap and pin down a clear definition of a hypothesis.

A research hypothesis (or scientific hypothesis) is a statement about an expected relationship between variables, or explanation of an occurrence, that is clear, specific and testable.

So, when you write up hypotheses for your dissertation or thesis, make sure that they meet all these criteria. If you do, you’ll not only have rock-solid hypotheses but you’ll also ensure a clear focus for your entire research project.

What about the null hypothesis?

You may have also heard the terms null hypothesis , alternative hypothesis, or H-zero thrown around. At a simple level, the null hypothesis is the counter-proposal to the original hypothesis.

For example, if the hypothesis predicts that there is a relationship between two variables (for example, sleep and academic performance), the null hypothesis would predict that there is no relationship between those variables.

At a more technical level, the null hypothesis proposes that no statistical significance exists in a set of given observations and that any differences are due to chance alone.

And there you have it – hypotheses in a nutshell. 

If you have any questions, be sure to leave a comment below and we’ll do our best to help you. If you need hands-on help developing and testing your hypotheses, consider our private coaching service , where we hold your hand through the research journey.

17 Comments

Very useful information. I benefit more from getting more information in this regard.

Very great insight,educative and informative. Please give meet deep critics on many research data of public international Law like human rights, environment, natural resources, law of the sea etc

In a book I read a distinction is made between null, research, and alternative hypothesis. As far as I understand, alternative and research hypotheses are the same. Can you please elaborate? Best Afshin

This is a self explanatory, easy going site. I will recommend this to my friends and colleagues.

Very good definition. How can I cite your definition in my thesis? Thank you. Is nul hypothesis compulsory in a research?

It’s a counter-proposal to be proven as a rejection

Please what is the difference between alternate hypothesis and research hypothesis?

It is a very good explanation. However, it limits hypotheses to statistically tasteable ideas. What about for qualitative researches or other researches that involve quantitative data that don’t need statistical tests?

In qualitative research, one typically uses propositions, not hypotheses.

could you please elaborate it more

I’ve benefited greatly from these notes, thank you.

This is very helpful

well articulated ideas are presented here, thank you for being reliable sources of information

Excellent. Thanks for being clear and sound about the research methodology and hypothesis (quantitative research)

I have only a simple question regarding the null hypothesis. – Is the null hypothesis (Ho) known as the reversible hypothesis of the alternative hypothesis (H1? – How to test it in academic research?

this is very important note help me much more

Hi” best wishes to you and your very nice blog” 

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B oth the hypothesis statement and the thesis statement answer the research question of the study.  When the statement is one that can be proved or disproved, it is an hypothesis statement.  If, instead, the statement specifically shows the intentions/objectives/position of the researcher, it is a thesis statement.

A hypothesis is a statement that can be proved or disproved.  It is typically used in quantitative research and predicts the relationship between variables.

A thesis statement is a short, direct sentence that summarizes the main point or claim of an essay or research paper. It is seen in quantitative, qualitative, and mixed methods research.  A thesis statement is developed, supported, and explained in the body of the essay or research report by means of examples and evidence.

Every research study should contain a concise and well-written thesis statement. If the intent of the study is to prove/disprove something, that research report will also contain an hypothesis statement.

Jablonski , Judith. What is the difference between a thesis statement and an hypothesis statement? Online Library. American Public University System. Jun 16, 2014. Web.   http://apus.libanswers.com/faq/2374

Let’s say you are interested in the conflict in Darfur, and you conclude that the issues you wish to address include the nature, causes, and effects of the conflict, and the international response. While you could address the issue of international response first, it makes the most sense to start with a description of the conflict, followed by an exploration of the causes, effects, and then to discuss the international response and what more could/should be done.

This hypothetical example may lead to the following title, introduction, and statement of questions:

Conflict in Darfur: Causes, Consequences, and International Response       This paper examines the conflict in Darfur, Sudan. It is organized around the following questions: (1) What is the nature of the conflict in Darfur? (2) What are the causes and effects of the conflict? (3) What has the international community done to address it, and what more could/should it do?

Following the section that presents your questions and background, you will offer a set of responses/answers/(hypo)theses. They should follow the order of the questions. This might look something like this, “The paper argues/contends/ maintains/seeks to develop the position that...etc.” The most important thing you can do in this section is to present as clearly as possible your best thinking on the subject matter guided by course material and research. As you proceed through the research process, your thinking about the issues/questions will become more nuanced, complex, and refined. The statement of your theses will reflect this as you move forward in the research process.

So, looking to our hypothetical example on Darfur:

The current conflict in Darfur goes back more than a decade and consists of fighting between government-supported troops and residents of Darfur. The causes of the conflict include x, y, and z. The effects of the conflict have been a, b, and c. The international community has done 0, and it should do 1, 2, and 3.

Once you have setup your thesis you will be ready to begin amassing supporting evidence for you claims. This is a very important part of the research paper, as you will provide the substance to defend your thesis.

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Hypothesis or Thesis

Looking for the author's thesis or hypothesis.

The image below shows the part of the scholarly article that shows where the authors are making their argument. 

(click on image to enlarge)

• The first few paragraphs of a journal article serve to introduce the topic, to provide the author's hypothesis or thesis, and to indicate why the research was done.  
• A thesis or hypothesis is not always clearly labeled; you may need to read through the introductory paragraphs to determine what the authors are proposing.
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What is the difference between hypothesis, thesis statement and research goal?

Can someone explain the difference between hypothesis, thesis statement and research goal based on an example?

• terminology

• 1 You should mention which subject you are in. 'Hypothesis' has opposite meanings in maths and physics. –  Jessica B Commented May 31, 2018 at 11:22

2 Answers 2

I had this same question recently and did some research on it. The definitions I found weren't consistent, but from them I derived the following.

Thesis statement -- A definitive statement about the way the world (or your system of interest) works, especially what is most important in causing or influencing the behavior of the system.

"Family expectations has primary significance on the performance in college for Latino girls in the Western US" is an example of a thesis statement.

Research goal -- Expresses what you hope to learn or shed light on in your research. Specifically, the goal should specify what type of results you are hoping to achieve. It contextualizes your work in relation to other research, especially theory. It also feeds into your choice of method.

"My research goal is to develop a theoretical model of cultural influence on college performance, contextualized by gender and ethnicity" is an example of a research goal.

Hypotheses -- What specific conditions or relations do you aim to test or evaluate in your research. Any research that does not include a method for hypothesis testing should not claim to test hypotheses. A hypothesis statement must be specific enough that it is testable by the methods you choose, and also it should be falsifiable -- i.e. it is clear what evidence might prove the hypothesis false, and such evidence should be plausible and possible.

"Low family expectations has a detrimental effect on the college completion rate and time-to-complete for high-achieving Latino girls" is an example of a hypothesis statement.

Notice how there are specific, testable conditions and metrics -- "college completion rates" and "time-to-complete". These conditions should appear as metrics in your research methods -- i.e. instruments and analysis methods.

A thesis statement usually helps guide the research paper. It is a short sentence or summary containing the central idea of the research paper. It helps a reader have a clear glimpse of what the paper is about.

The Hypothesis statement comes in different format but with the intent to help prove or disprove a phenomenon. The hypothesis can help defend, support, explain or disprove, argue against the thesis statement.Usually the hypothesis measures specific issues or variables-two or more and therefore should be testable. The thesis statement creates a background while the hypothesis creates a means to measure the interrelationship.

The research goal takes a look into the future of your study or research paper. |It tries to help you state what the outcomes you seek to achieve by the research work. With a research goal you can set specific milestones to accomplish at the end of the research work.

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HOW TO: Use Articles for Research: Introduction: Hypothesis/Thesis

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Hypothesis or Thesis

The first few paragraphs of a journal article serve to introduce the topic, to provide the author's hypothesis or thesis, and to indicate why the research was done.  A thesis or hypothesis is not always clearly labled; you may need to read through the introductory paragraphs to determine what the authors are proposing.

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While Sandel argues that pursuing perfection through genetic engineering would decrease our sense of humility, he claims that the sense of solidarity we would lose is also important.

This thesis summarizes several points in Sandel’s argument, but it does not make a claim about how we should understand his argument. A reader who read Sandel’s argument would not also need to read an essay based on this descriptive thesis.  

Broad thesis (arguable, but difficult to support with evidence) 

Michael Sandel’s arguments about genetic engineering do not take into consideration all the relevant issues.

This is an arguable claim because it would be possible to argue against it by saying that Michael Sandel’s arguments do take all of the relevant issues into consideration. But the claim is too broad. Because the thesis does not specify which “issues” it is focused on—or why it matters if they are considered—readers won’t know what the rest of the essay will argue, and the writer won’t know what to focus on. If there is a particular issue that Sandel does not address, then a more specific version of the thesis would include that issue—hand an explanation of why it is important.  

Arguable thesis with analytical claim 

While Sandel argues persuasively that our instinct to “remake” (54) ourselves into something ever more perfect is a problem, his belief that we can always draw a line between what is medically necessary and what makes us simply “better than well” (51) is less convincing.

This is an arguable analytical claim. To argue for this claim, the essay writer will need to show how evidence from the article itself points to this interpretation. It’s also a reasonable scope for a thesis because it can be supported with evidence available in the text and is neither too broad nor too narrow.  

Arguable thesis with normative claim 

Given Sandel’s argument against genetic enhancement, we should not allow parents to decide on using Human Growth Hormone for their children.

This thesis tells us what we should do about a particular issue discussed in Sandel’s article, but it does not tell us how we should understand Sandel’s argument.  

Questions to ask about your thesis 

• Is the thesis truly arguable? Does it speak to a genuine dilemma in the source, or would most readers automatically agree with it?  
• Is the thesis too obvious? Again, would most or all readers agree with it without needing to see your argument?  
• Is the thesis complex enough to require a whole essay's worth of argument?  
• Is the thesis supportable with evidence from the text rather than with generalizations or outside research?  
• Would anyone want to read a paper in which this thesis was developed? That is, can you explain what this paper is adding to our understanding of a problem, question, or topic?
• picture_as_pdf Thesis

Thesis vs. Hypothesis — What's the Difference?

Difference Between Thesis and Hypothesis

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• How to Write a Strong Hypothesis | Steps & Examples

How to Write a Strong Hypothesis | Steps & Examples

Published on May 6, 2022 by Shona McCombes . Revised on November 20, 2023.

A hypothesis is a statement that can be tested by scientific research. If you want to test a relationship between two or more variables, you need to write hypotheses before you start your experiment or data collection .

Example: Hypothesis

Daily apple consumption leads to fewer doctor’s visits.

Table of contents

What is a hypothesis, developing a hypothesis (with example), hypothesis examples, other interesting articles, frequently asked questions about writing hypotheses.

A hypothesis states your predictions about what your research will find. It is a tentative answer to your research question that has not yet been tested. For some research projects, you might have to write several hypotheses that address different aspects of your research question.

A hypothesis is not just a guess – it should be based on existing theories and knowledge. It also has to be testable, which means you can support or refute it through scientific research methods (such as experiments, observations and statistical analysis of data).

Variables in hypotheses

Hypotheses propose a relationship between two or more types of variables .

• An independent variable is something the researcher changes or controls.
• A dependent variable is something the researcher observes and measures.

If there are any control variables , extraneous variables , or confounding variables , be sure to jot those down as you go to minimize the chances that research bias  will affect your results.

In this example, the independent variable is exposure to the sun – the assumed cause . The dependent variable is the level of happiness – the assumed effect .

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Step 1. Ask a question

Writing a hypothesis begins with a research question that you want to answer. The question should be focused, specific, and researchable within the constraints of your project.

Step 2. Do some preliminary research

Your initial answer to the question should be based on what is already known about the topic. Look for theories and previous studies to help you form educated assumptions about what your research will find.

At this stage, you might construct a conceptual framework to ensure that you’re embarking on a relevant topic . This can also help you identify which variables you will study and what you think the relationships are between them. Sometimes, you’ll have to operationalize more complex constructs.

Step 3. Formulate your hypothesis

Now you should have some idea of what you expect to find. Write your initial answer to the question in a clear, concise sentence.

4. Refine your hypothesis

You need to make sure your hypothesis is specific and testable. There are various ways of phrasing a hypothesis, but all the terms you use should have clear definitions, and the hypothesis should contain:

• The relevant variables
• The specific group being studied
• The predicted outcome of the experiment or analysis

5. Phrase your hypothesis in three ways

To identify the variables, you can write a simple prediction in  if…then form. The first part of the sentence states the independent variable and the second part states the dependent variable.

In academic research, hypotheses are more commonly phrased in terms of correlations or effects, where you directly state the predicted relationship between variables.

If you are comparing two groups, the hypothesis can state what difference you expect to find between them.

6. Write a null hypothesis

If your research involves statistical hypothesis testing , you will also have to write a null hypothesis . The null hypothesis is the default position that there is no association between the variables. The null hypothesis is written as H 0 , while the alternative hypothesis is H 1 or H a .

• H 0 : The number of lectures attended by first-year students has no effect on their final exam scores.
• H 1 : The number of lectures attended by first-year students has a positive effect on their final exam scores.

If you want to know more about the research process , methodology , research bias , or statistics , make sure to check out some of our other articles with explanations and examples.

• Sampling methods
• Simple random sampling
• Stratified sampling
• Cluster sampling
• Likert scales
• Reproducibility

 Statistics

• Null hypothesis
• Statistical power
• Probability distribution
• Effect size
• Poisson distribution

Research bias

• Optimism bias
• Cognitive bias
• Implicit bias
• Hawthorne effect
• Anchoring bias
• Explicit bias

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A hypothesis is not just a guess — it should be based on existing theories and knowledge. It also has to be testable, which means you can support or refute it through scientific research methods (such as experiments, observations and statistical analysis of data).

Null and alternative hypotheses are used in statistical hypothesis testing . The null hypothesis of a test always predicts no effect or no relationship between variables, while the alternative hypothesis states your research prediction of an effect or relationship.

Hypothesis testing is a formal procedure for investigating our ideas about the world using statistics. It is used by scientists to test specific predictions, called hypotheses , by calculating how likely it is that a pattern or relationship between variables could have arisen by chance.

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Thesis and Purpose Statements

Use the guidelines below to learn the differences between thesis and purpose statements.

In the first stages of writing, thesis or purpose statements are usually rough or ill-formed and are useful primarily as planning tools.

A thesis statement or purpose statement will emerge as you think and write about a topic. The statement can be restricted or clarified and eventually worked into an introduction.

As you revise your paper, try to phrase your thesis or purpose statement in a precise way so that it matches the content and organization of your paper.

Thesis statements

A thesis statement is a sentence that makes an assertion about a topic and predicts how the topic will be developed. It does not simply announce a topic: it says something about the topic.

Good: X has made a significant impact on the teenage population due to its . . . Bad: In this paper, I will discuss X.

A thesis statement makes a promise to the reader about the scope, purpose, and direction of the paper. It summarizes the conclusions that the writer has reached about the topic.

A thesis statement is generally located near the end of the introduction. Sometimes in a long paper, the thesis will be expressed in several sentences or an entire paragraph.

A thesis statement is focused and specific enough to be proven within the boundaries of the paper. Key words (nouns and verbs) should be specific, accurate, and indicative of the range of research, thrust of the argument or analysis, and the organization of supporting information.

Purpose statements

A purpose statement announces the purpose, scope, and direction of the paper. It tells the reader what to expect in a paper and what the specific focus will be.

Common beginnings include:

“This paper examines . . .,” “The aim of this paper is to . . .,” and “The purpose of this essay is to . . .”

A purpose statement makes a promise to the reader about the development of the argument but does not preview the particular conclusions that the writer has drawn.

A purpose statement usually appears toward the end of the introduction. The purpose statement may be expressed in several sentences or even an entire paragraph.

A purpose statement is specific enough to satisfy the requirements of the assignment. Purpose statements are common in research papers in some academic disciplines, while in other disciplines they are considered too blunt or direct. If you are unsure about using a purpose statement, ask your instructor.

This paper will examine the ecological destruction of the Sahel preceding the drought and the causes of this disintegration of the land. The focus will be on the economic, political, and social relationships which brought about the environmental problems in the Sahel.

Sample purpose and thesis statements

The following example combines a purpose statement and a thesis statement (bold).

The goal of this paper is to examine the effects of Chile’s agrarian reform on the lives of rural peasants. The nature of the topic dictates the use of both a chronological and a comparative analysis of peasant lives at various points during the reform period. . . The Chilean reform example provides evidence that land distribution is an essential component of both the improvement of peasant conditions and the development of a democratic society. More extensive and enduring reforms would likely have allowed Chile the opportunity to further expand these horizons.

For more tips about writing thesis statements, take a look at our new handout on Developing a Thesis Statement.

Writing Process and Structure

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Getting Started with Your Paper

Interpreting Writing Assignments from Your Courses

Generating Ideas for Your Paper

Creating an Argument

Thesis vs. Purpose Statements

Developing a Thesis Statement

Architecture of Arguments

Working with Sources

Quoting and Paraphrasing Sources

Using Literary Quotations

Citing Sources in Your Paper

Drafting Your Paper

Introductions

Paragraphing

Developing Strategic Transitions

Conclusions

Revising Your Paper

Peer Reviews

Reverse Outlines

Revising an Argumentative Paper

Revision Strategies for Longer Projects

Finishing Your Paper

Twelve Common Errors: An Editing Checklist

How to Proofread your Paper

Writing Collaboratively

Collaborative and Group Writing

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How to Write a Great Hypothesis

Hypothesis Definition, Format, Examples, and Tips

Verywell / Alex Dos Diaz

• The Scientific Method

Hypothesis Format

Falsifiability of a hypothesis.

• Operationalization

Hypothesis Types

Hypotheses examples.

• Collecting Data

A hypothesis is a tentative statement about the relationship between two or more variables. It is a specific, testable prediction about what you expect to happen in a study. It is a preliminary answer to your question that helps guide the research process.

Consider a study designed to examine the relationship between sleep deprivation and test performance. The hypothesis might be: "This study is designed to assess the hypothesis that sleep-deprived people will perform worse on a test than individuals who are not sleep-deprived."

At a Glance

A hypothesis is crucial to scientific research because it offers a clear direction for what the researchers are looking to find. This allows them to design experiments to test their predictions and add to our scientific knowledge about the world. This article explores how a hypothesis is used in psychology research, how to write a good hypothesis, and the different types of hypotheses you might use.

The Hypothesis in the Scientific Method

In the scientific method , whether it involves research in psychology, biology, or some other area, a hypothesis represents what the researchers think will happen in an experiment. The scientific method involves the following steps:

• Forming a question
• Performing background research
• Creating a hypothesis
• Designing an experiment
• Collecting data
• Analyzing the results
• Drawing conclusions
• Communicating the results

The hypothesis is a prediction, but it involves more than a guess. Most of the time, the hypothesis begins with a question which is then explored through background research. At this point, researchers then begin to develop a testable hypothesis.

Unless you are creating an exploratory study, your hypothesis should always explain what you  expect  to happen.

In a study exploring the effects of a particular drug, the hypothesis might be that researchers expect the drug to have some type of effect on the symptoms of a specific illness. In psychology, the hypothesis might focus on how a certain aspect of the environment might influence a particular behavior.

Remember, a hypothesis does not have to be correct. While the hypothesis predicts what the researchers expect to see, the goal of the research is to determine whether this guess is right or wrong. When conducting an experiment, researchers might explore numerous factors to determine which ones might contribute to the ultimate outcome.

In many cases, researchers may find that the results of an experiment  do not  support the original hypothesis. When writing up these results, the researchers might suggest other options that should be explored in future studies.

In many cases, researchers might draw a hypothesis from a specific theory or build on previous research. For example, prior research has shown that stress can impact the immune system. So a researcher might hypothesize: "People with high-stress levels will be more likely to contract a common cold after being exposed to the virus than people who have low-stress levels."

In other instances, researchers might look at commonly held beliefs or folk wisdom. "Birds of a feather flock together" is one example of folk adage that a psychologist might try to investigate. The researcher might pose a specific hypothesis that "People tend to select romantic partners who are similar to them in interests and educational level."

Elements of a Good Hypothesis

So how do you write a good hypothesis? When trying to come up with a hypothesis for your research or experiments, ask yourself the following questions:

• Is your hypothesis based on your research on a topic?
• Can your hypothesis be tested?
• Does your hypothesis include independent and dependent variables?

Before you come up with a specific hypothesis, spend some time doing background research. Once you have completed a literature review, start thinking about potential questions you still have. Pay attention to the discussion section in the  journal articles you read . Many authors will suggest questions that still need to be explored.

How to Formulate a Good Hypothesis

To form a hypothesis, you should take these steps:

• Collect as many observations about a topic or problem as you can.
• Evaluate these observations and look for possible causes of the problem.
• Create a list of possible explanations that you might want to explore.
• After you have developed some possible hypotheses, think of ways that you could confirm or disprove each hypothesis through experimentation. This is known as falsifiability.

In the scientific method ,  falsifiability is an important part of any valid hypothesis. In order to test a claim scientifically, it must be possible that the claim could be proven false.

Students sometimes confuse the idea of falsifiability with the idea that it means that something is false, which is not the case. What falsifiability means is that  if  something was false, then it is possible to demonstrate that it is false.

One of the hallmarks of pseudoscience is that it makes claims that cannot be refuted or proven false.

The Importance of Operational Definitions

A variable is a factor or element that can be changed and manipulated in ways that are observable and measurable. However, the researcher must also define how the variable will be manipulated and measured in the study.

Operational definitions are specific definitions for all relevant factors in a study. This process helps make vague or ambiguous concepts detailed and measurable.

For example, a researcher might operationally define the variable " test anxiety " as the results of a self-report measure of anxiety experienced during an exam. A "study habits" variable might be defined by the amount of studying that actually occurs as measured by time.

These precise descriptions are important because many things can be measured in various ways. Clearly defining these variables and how they are measured helps ensure that other researchers can replicate your results.

Replicability

One of the basic principles of any type of scientific research is that the results must be replicable.

Replication means repeating an experiment in the same way to produce the same results. By clearly detailing the specifics of how the variables were measured and manipulated, other researchers can better understand the results and repeat the study if needed.

Some variables are more difficult than others to define. For example, how would you operationally define a variable such as aggression ? For obvious ethical reasons, researchers cannot create a situation in which a person behaves aggressively toward others.

To measure this variable, the researcher must devise a measurement that assesses aggressive behavior without harming others. The researcher might utilize a simulated task to measure aggressiveness in this situation.

Hypothesis Checklist

• Does your hypothesis focus on something that you can actually test?
• Does your hypothesis include both an independent and dependent variable?
• Can you manipulate the variables?
• Can your hypothesis be tested without violating ethical standards?

The hypothesis you use will depend on what you are investigating and hoping to find. Some of the main types of hypotheses that you might use include:

• Simple hypothesis : This type of hypothesis suggests there is a relationship between one independent variable and one dependent variable.
• Complex hypothesis : This type suggests a relationship between three or more variables, such as two independent and dependent variables.
• Null hypothesis : This hypothesis suggests no relationship exists between two or more variables.
• Alternative hypothesis : This hypothesis states the opposite of the null hypothesis.
• Statistical hypothesis : This hypothesis uses statistical analysis to evaluate a representative population sample and then generalizes the findings to the larger group.
• Logical hypothesis : This hypothesis assumes a relationship between variables without collecting data or evidence.

A hypothesis often follows a basic format of "If {this happens} then {this will happen}." One way to structure your hypothesis is to describe what will happen to the  dependent variable  if you change the  independent variable .

The basic format might be: "If {these changes are made to a certain independent variable}, then we will observe {a change in a specific dependent variable}."

A few examples of simple hypotheses:

• "Students who eat breakfast will perform better on a math exam than students who do not eat breakfast."
• "Students who experience test anxiety before an English exam will get lower scores than students who do not experience test anxiety."​
• "Motorists who talk on the phone while driving will be more likely to make errors on a driving course than those who do not talk on the phone."
• "Children who receive a new reading intervention will have higher reading scores than students who do not receive the intervention."

Examples of a complex hypothesis include:

• "People with high-sugar diets and sedentary activity levels are more likely to develop depression."
• "Younger people who are regularly exposed to green, outdoor areas have better subjective well-being than older adults who have limited exposure to green spaces."

Examples of a null hypothesis include:

• "There is no difference in anxiety levels between people who take St. John's wort supplements and those who do not."
• "There is no difference in scores on a memory recall task between children and adults."
• "There is no difference in aggression levels between children who play first-person shooter games and those who do not."

Examples of an alternative hypothesis:

• "People who take St. John's wort supplements will have less anxiety than those who do not."
• "Adults will perform better on a memory task than children."
• "Children who play first-person shooter games will show higher levels of aggression than children who do not." 

Collecting Data on Your Hypothesis

Once a researcher has formed a testable hypothesis, the next step is to select a research design and start collecting data. The research method depends largely on exactly what they are studying. There are two basic types of research methods: descriptive research and experimental research.

Descriptive Research Methods

Descriptive research such as  case studies ,  naturalistic observations , and surveys are often used when  conducting an experiment is difficult or impossible. These methods are best used to describe different aspects of a behavior or psychological phenomenon.

Once a researcher has collected data using descriptive methods, a  correlational study  can examine how the variables are related. This research method might be used to investigate a hypothesis that is difficult to test experimentally.

Experimental Research Methods

Experimental methods  are used to demonstrate causal relationships between variables. In an experiment, the researcher systematically manipulates a variable of interest (known as the independent variable) and measures the effect on another variable (known as the dependent variable).

Unlike correlational studies, which can only be used to determine if there is a relationship between two variables, experimental methods can be used to determine the actual nature of the relationship—whether changes in one variable actually  cause  another to change.

The hypothesis is a critical part of any scientific exploration. It represents what researchers expect to find in a study or experiment. In situations where the hypothesis is unsupported by the research, the research still has value. Such research helps us better understand how different aspects of the natural world relate to one another. It also helps us develop new hypotheses that can then be tested in the future.

Thompson WH, Skau S. On the scope of scientific hypotheses .  R Soc Open Sci . 2023;10(8):230607. doi:10.1098/rsos.230607

Taran S, Adhikari NKJ, Fan E. Falsifiability in medicine: what clinicians can learn from Karl Popper [published correction appears in Intensive Care Med. 2021 Jun 17;:].  Intensive Care Med . 2021;47(9):1054-1056. doi:10.1007/s00134-021-06432-z

Eyler AA. Research Methods for Public Health . 1st ed. Springer Publishing Company; 2020. doi:10.1891/9780826182067.0004

Nosek BA, Errington TM. What is replication ?  PLoS Biol . 2020;18(3):e3000691. doi:10.1371/journal.pbio.3000691

Aggarwal R, Ranganathan P. Study designs: Part 2 - Descriptive studies .  Perspect Clin Res . 2019;10(1):34-36. doi:10.4103/picr.PICR_154_18

Nevid J. Psychology: Concepts and Applications. Wadworth, 2013.

By Kendra Cherry, MSEd Kendra Cherry, MS, is a psychosocial rehabilitation specialist, psychology educator, and author of the "Everything Psychology Book."

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Home » What is a Hypothesis – Types, Examples and Writing Guide

What is a Hypothesis – Types, Examples and Writing Guide

Table of Contents

Definition:

Hypothesis is an educated guess or proposed explanation for a phenomenon, based on some initial observations or data. It is a tentative statement that can be tested and potentially proven or disproven through further investigation and experimentation.

Hypothesis is often used in scientific research to guide the design of experiments and the collection and analysis of data. It is an essential element of the scientific method, as it allows researchers to make predictions about the outcome of their experiments and to test those predictions to determine their accuracy.

Types of Hypothesis

Types of Hypothesis are as follows:

Research Hypothesis

A research hypothesis is a statement that predicts a relationship between variables. It is usually formulated as a specific statement that can be tested through research, and it is often used in scientific research to guide the design of experiments.

Null Hypothesis

The null hypothesis is a statement that assumes there is no significant difference or relationship between variables. It is often used as a starting point for testing the research hypothesis, and if the results of the study reject the null hypothesis, it suggests that there is a significant difference or relationship between variables.

Alternative Hypothesis

An alternative hypothesis is a statement that assumes there is a significant difference or relationship between variables. It is often used as an alternative to the null hypothesis and is tested against the null hypothesis to determine which statement is more accurate.

Directional Hypothesis

A directional hypothesis is a statement that predicts the direction of the relationship between variables. For example, a researcher might predict that increasing the amount of exercise will result in a decrease in body weight.

Non-directional Hypothesis

A non-directional hypothesis is a statement that predicts the relationship between variables but does not specify the direction. For example, a researcher might predict that there is a relationship between the amount of exercise and body weight, but they do not specify whether increasing or decreasing exercise will affect body weight.

Statistical Hypothesis

A statistical hypothesis is a statement that assumes a particular statistical model or distribution for the data. It is often used in statistical analysis to test the significance of a particular result.

Composite Hypothesis

A composite hypothesis is a statement that assumes more than one condition or outcome. It can be divided into several sub-hypotheses, each of which represents a different possible outcome.

Empirical Hypothesis

An empirical hypothesis is a statement that is based on observed phenomena or data. It is often used in scientific research to develop theories or models that explain the observed phenomena.

Simple Hypothesis

A simple hypothesis is a statement that assumes only one outcome or condition. It is often used in scientific research to test a single variable or factor.

Complex Hypothesis

A complex hypothesis is a statement that assumes multiple outcomes or conditions. It is often used in scientific research to test the effects of multiple variables or factors on a particular outcome.

Applications of Hypothesis

Hypotheses are used in various fields to guide research and make predictions about the outcomes of experiments or observations. Here are some examples of how hypotheses are applied in different fields:

• Science : In scientific research, hypotheses are used to test the validity of theories and models that explain natural phenomena. For example, a hypothesis might be formulated to test the effects of a particular variable on a natural system, such as the effects of climate change on an ecosystem.
• Medicine : In medical research, hypotheses are used to test the effectiveness of treatments and therapies for specific conditions. For example, a hypothesis might be formulated to test the effects of a new drug on a particular disease.
• Psychology : In psychology, hypotheses are used to test theories and models of human behavior and cognition. For example, a hypothesis might be formulated to test the effects of a particular stimulus on the brain or behavior.
• Sociology : In sociology, hypotheses are used to test theories and models of social phenomena, such as the effects of social structures or institutions on human behavior. For example, a hypothesis might be formulated to test the effects of income inequality on crime rates.
• Business : In business research, hypotheses are used to test the validity of theories and models that explain business phenomena, such as consumer behavior or market trends. For example, a hypothesis might be formulated to test the effects of a new marketing campaign on consumer buying behavior.
• Engineering : In engineering, hypotheses are used to test the effectiveness of new technologies or designs. For example, a hypothesis might be formulated to test the efficiency of a new solar panel design.

How to write a Hypothesis

Here are the steps to follow when writing a hypothesis:

Identify the Research Question

The first step is to identify the research question that you want to answer through your study. This question should be clear, specific, and focused. It should be something that can be investigated empirically and that has some relevance or significance in the field.

Conduct a Literature Review

Before writing your hypothesis, it’s essential to conduct a thorough literature review to understand what is already known about the topic. This will help you to identify the research gap and formulate a hypothesis that builds on existing knowledge.

Determine the Variables

The next step is to identify the variables involved in the research question. A variable is any characteristic or factor that can vary or change. There are two types of variables: independent and dependent. The independent variable is the one that is manipulated or changed by the researcher, while the dependent variable is the one that is measured or observed as a result of the independent variable.

Formulate the Hypothesis

Based on the research question and the variables involved, you can now formulate your hypothesis. A hypothesis should be a clear and concise statement that predicts the relationship between the variables. It should be testable through empirical research and based on existing theory or evidence.

Write the Null Hypothesis

The null hypothesis is the opposite of the alternative hypothesis, which is the hypothesis that you are testing. The null hypothesis states that there is no significant difference or relationship between the variables. It is important to write the null hypothesis because it allows you to compare your results with what would be expected by chance.

Refine the Hypothesis

After formulating the hypothesis, it’s important to refine it and make it more precise. This may involve clarifying the variables, specifying the direction of the relationship, or making the hypothesis more testable.

Examples of Hypothesis

Here are a few examples of hypotheses in different fields:

• Psychology : “Increased exposure to violent video games leads to increased aggressive behavior in adolescents.”
• Biology : “Higher levels of carbon dioxide in the atmosphere will lead to increased plant growth.”
• Sociology : “Individuals who grow up in households with higher socioeconomic status will have higher levels of education and income as adults.”
• Education : “Implementing a new teaching method will result in higher student achievement scores.”
• Marketing : “Customers who receive a personalized email will be more likely to make a purchase than those who receive a generic email.”
• Physics : “An increase in temperature will cause an increase in the volume of a gas, assuming all other variables remain constant.”
• Medicine : “Consuming a diet high in saturated fats will increase the risk of developing heart disease.”

Purpose of Hypothesis

The purpose of a hypothesis is to provide a testable explanation for an observed phenomenon or a prediction of a future outcome based on existing knowledge or theories. A hypothesis is an essential part of the scientific method and helps to guide the research process by providing a clear focus for investigation. It enables scientists to design experiments or studies to gather evidence and data that can support or refute the proposed explanation or prediction.

The formulation of a hypothesis is based on existing knowledge, observations, and theories, and it should be specific, testable, and falsifiable. A specific hypothesis helps to define the research question, which is important in the research process as it guides the selection of an appropriate research design and methodology. Testability of the hypothesis means that it can be proven or disproven through empirical data collection and analysis. Falsifiability means that the hypothesis should be formulated in such a way that it can be proven wrong if it is incorrect.

In addition to guiding the research process, the testing of hypotheses can lead to new discoveries and advancements in scientific knowledge. When a hypothesis is supported by the data, it can be used to develop new theories or models to explain the observed phenomenon. When a hypothesis is not supported by the data, it can help to refine existing theories or prompt the development of new hypotheses to explain the phenomenon.

When to use Hypothesis

Here are some common situations in which hypotheses are used:

• In scientific research , hypotheses are used to guide the design of experiments and to help researchers make predictions about the outcomes of those experiments.
• In social science research , hypotheses are used to test theories about human behavior, social relationships, and other phenomena.
• I n business , hypotheses can be used to guide decisions about marketing, product development, and other areas. For example, a hypothesis might be that a new product will sell well in a particular market, and this hypothesis can be tested through market research.

Characteristics of Hypothesis

Here are some common characteristics of a hypothesis:

• Testable : A hypothesis must be able to be tested through observation or experimentation. This means that it must be possible to collect data that will either support or refute the hypothesis.
• Falsifiable : A hypothesis must be able to be proven false if it is not supported by the data. If a hypothesis cannot be falsified, then it is not a scientific hypothesis.
• Clear and concise : A hypothesis should be stated in a clear and concise manner so that it can be easily understood and tested.
• Based on existing knowledge : A hypothesis should be based on existing knowledge and research in the field. It should not be based on personal beliefs or opinions.
• Specific : A hypothesis should be specific in terms of the variables being tested and the predicted outcome. This will help to ensure that the research is focused and well-designed.
• Tentative: A hypothesis is a tentative statement or assumption that requires further testing and evidence to be confirmed or refuted. It is not a final conclusion or assertion.
• Relevant : A hypothesis should be relevant to the research question or problem being studied. It should address a gap in knowledge or provide a new perspective on the issue.

Advantages of Hypothesis

Hypotheses have several advantages in scientific research and experimentation:

• Guides research: A hypothesis provides a clear and specific direction for research. It helps to focus the research question, select appropriate methods and variables, and interpret the results.
• Predictive powe r: A hypothesis makes predictions about the outcome of research, which can be tested through experimentation. This allows researchers to evaluate the validity of the hypothesis and make new discoveries.
• Facilitates communication: A hypothesis provides a common language and framework for scientists to communicate with one another about their research. This helps to facilitate the exchange of ideas and promotes collaboration.
• Efficient use of resources: A hypothesis helps researchers to use their time, resources, and funding efficiently by directing them towards specific research questions and methods that are most likely to yield results.
• Provides a basis for further research: A hypothesis that is supported by data provides a basis for further research and exploration. It can lead to new hypotheses, theories, and discoveries.
• Increases objectivity: A hypothesis can help to increase objectivity in research by providing a clear and specific framework for testing and interpreting results. This can reduce bias and increase the reliability of research findings.

Limitations of Hypothesis

Some Limitations of the Hypothesis are as follows:

• Limited to observable phenomena: Hypotheses are limited to observable phenomena and cannot account for unobservable or intangible factors. This means that some research questions may not be amenable to hypothesis testing.
• May be inaccurate or incomplete: Hypotheses are based on existing knowledge and research, which may be incomplete or inaccurate. This can lead to flawed hypotheses and erroneous conclusions.
• May be biased: Hypotheses may be biased by the researcher’s own beliefs, values, or assumptions. This can lead to selective interpretation of data and a lack of objectivity in research.
• Cannot prove causation: A hypothesis can only show a correlation between variables, but it cannot prove causation. This requires further experimentation and analysis.
• Limited to specific contexts: Hypotheses are limited to specific contexts and may not be generalizable to other situations or populations. This means that results may not be applicable in other contexts or may require further testing.
• May be affected by chance : Hypotheses may be affected by chance or random variation, which can obscure or distort the true relationship between variables.

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This is the Difference Between a Hypothesis and a Theory

What to Know A hypothesis is an assumption made before any research has been done. It is formed so that it can be tested to see if it might be true. A theory is a principle formed to explain the things already shown in data. Because of the rigors of experiment and control, it is much more likely that a theory will be true than a hypothesis.

As anyone who has worked in a laboratory or out in the field can tell you, science is about process: that of observing, making inferences about those observations, and then performing tests to see if the truth value of those inferences holds up. The scientific method is designed to be a rigorous procedure for acquiring knowledge about the world around us.

In scientific reasoning, a hypothesis is constructed before any applicable research has been done. A theory, on the other hand, is supported by evidence: it's a principle formed as an attempt to explain things that have already been substantiated by data.

Toward that end, science employs a particular vocabulary for describing how ideas are proposed, tested, and supported or disproven. And that's where we see the difference between a hypothesis and a theory .

A hypothesis is an assumption, something proposed for the sake of argument so that it can be tested to see if it might be true.

In the scientific method, the hypothesis is constructed before any applicable research has been done, apart from a basic background review. You ask a question, read up on what has been studied before, and then form a hypothesis.

What is a Hypothesis?

A hypothesis is usually tentative, an assumption or suggestion made strictly for the objective of being tested.

When a character which has been lost in a breed, reappears after a great number of generations, the most probable hypothesis is, not that the offspring suddenly takes after an ancestor some hundred generations distant, but that in each successive generation there has been a tendency to reproduce the character in question, which at last, under unknown favourable conditions, gains an ascendancy. Charles Darwin, On the Origin of Species , 1859 According to one widely reported hypothesis , cell-phone transmissions were disrupting the bees' navigational abilities. (Few experts took the cell-phone conjecture seriously; as one scientist said to me, "If that were the case, Dave Hackenberg's hives would have been dead a long time ago.") Elizabeth Kolbert, The New Yorker , 6 Aug. 2007

What is a Theory?

A theory , in contrast, is a principle that has been formed as an attempt to explain things that have already been substantiated by data. It is used in the names of a number of principles accepted in the scientific community, such as the Big Bang Theory . Because of the rigors of experimentation and control, its likelihood as truth is much higher than that of a hypothesis.

It is evident, on our theory , that coasts merely fringed by reefs cannot have subsided to any perceptible amount; and therefore they must, since the growth of their corals, either have remained stationary or have been upheaved. Now, it is remarkable how generally it can be shown, by the presence of upraised organic remains, that the fringed islands have been elevated: and so far, this is indirect evidence in favour of our theory . Charles Darwin, The Voyage of the Beagle , 1839 An example of a fundamental principle in physics, first proposed by Galileo in 1632 and extended by Einstein in 1905, is the following: All observers traveling at constant velocity relative to one another, should witness identical laws of nature. From this principle, Einstein derived his theory of special relativity. Alan Lightman, Harper's , December 2011

Non-Scientific Use

In non-scientific use, however, hypothesis and theory are often used interchangeably to mean simply an idea, speculation, or hunch (though theory is more common in this regard):

The theory of the teacher with all these immigrant kids was that if you spoke English loudly enough they would eventually understand. E. L. Doctorow, Loon Lake , 1979 Chicago is famous for asking questions for which there can be no boilerplate answers. Example: given the probability that the federal tax code, nondairy creamer, Dennis Rodman and the art of mime all came from outer space, name something else that has extraterrestrial origins and defend your hypothesis . John McCormick, Newsweek , 5 Apr. 1999 In his mind's eye, Miller saw his case suddenly taking form: Richard Bailey had Helen Brach killed because she was threatening to sue him over the horses she had purchased. It was, he realized, only a theory , but it was one he felt certain he could, in time, prove. Full of urgency, a man with a mission now that he had a hypothesis to guide him, he issued new orders to his troops: Find out everything you can about Richard Bailey and his crowd. Howard Blum, Vanity Fair , January 1995

And sometimes one term is used as a genus, or a means for defining the other:

Laplace's popular version of his astronomy, the Système du monde , was famous for introducing what came to be known as the nebular hypothesis , the theory that the solar system was formed by the condensation, through gradual cooling, of the gaseous atmosphere (the nebulae) surrounding the sun. Louis Menand, The Metaphysical Club , 2001 Researchers use this information to support the gateway drug theory — the hypothesis that using one intoxicating substance leads to future use of another. Jordy Byrd, The Pacific Northwest Inlander , 6 May 2015 Fox, the business and economics columnist for Time magazine, tells the story of the professors who enabled those abuses under the banner of the financial theory known as the efficient market hypothesis . Paul Krugman, The New York Times Book Review , 9 Aug. 2009

Incorrect Interpretations of "Theory"

Since this casual use does away with the distinctions upheld by the scientific community, hypothesis and theory are prone to being wrongly interpreted even when they are encountered in scientific contexts—or at least, contexts that allude to scientific study without making the critical distinction that scientists employ when weighing hypotheses and theories.

The most common occurrence is when theory is interpreted—and sometimes even gleefully seized upon—to mean something having less truth value than other scientific principles. (The word law applies to principles so firmly established that they are almost never questioned, such as the law of gravity.)

This mistake is one of projection: since we use theory in general use to mean something lightly speculated, then it's implied that scientists must be talking about the same level of uncertainty when they use theory to refer to their well-tested and reasoned principles.

The distinction has come to the forefront particularly on occasions when the content of science curricula in schools has been challenged—notably, when a school board in Georgia put stickers on textbooks stating that evolution was "a theory, not a fact, regarding the origin of living things." As Kenneth R. Miller, a cell biologist at Brown University, has said , a theory "doesn’t mean a hunch or a guess. A theory is a system of explanations that ties together a whole bunch of facts. It not only explains those facts, but predicts what you ought to find from other observations and experiments.”

While theories are never completely infallible, they form the basis of scientific reasoning because, as Miller said "to the best of our ability, we’ve tested them, and they’ve held up."

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Thesis vs Hypothesis vs Theory: the Differences and examples

thesis hypothesis and theory

Many students may have a hard time understanding the differences between a thesis, a hypothesis, and a theory. It is important to understand their differences. Such an understanding will be instrumental.

More so, when writing complex research papers that require a thesis that has a hypothesis and utilizes theories. We have gathered from responses of our college writing service that the difference between the three is confusing.

That being said, this article is meant to explain the differences between a thesis, a hypothesis, and a theory. 

Difference between Hypothesis and Thesis

There are major differences between hypothesis and thesis. While they seem to be related on the face, their differences are huge both in concept and practice.

A hypothesis is a proposed explanation of something or a phenomenon. A scientific hypothesis uses a scientific method that requires any hypothesis to be tested. As such, scientists and researchers base their hypothesis on observations that have been previously made and that which cannot be explained by the available or prevailing scientific theories.

From the definition of a hypothesis, you can see that theories must be included in any scientific method. This is the reason why this article tries to differentiate a thesis, a hypothesis, and a theory. 

Moving forward, a thesis can be defined as a written piece of academic work that is submitted by students to attain a university degree. However, on a smaller scale, there is something that is referred to as a thesis statement.

This is written at the introduction of a research paper or essay that is supported by a credible argument. The link between a hypothesis and thesis is that a thesis is a distinction or an affirmation of the hypothesis.

What this means is that whenever a research paper contains a hypothesis, there should be a thesis that validates it. 

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What is a Hypothesis?

A hypothesis can be defined as the proposed or suggested explanation for an occurrence, something, or a phenomenon. It should be testable through scientific methods. The reason why scholarly works should have a hypothesis is that the observed phenomena could not be explained using the prevailing scientific theories hence the reason why it should be tested. 

Testing the hypothesis may result in the development of new or improved scientific theories that are beneficial to the discipline and society in general. 

What is a Thesis?

A thesis is a written piece of academic work that is submitted by students to attain a university degree. When a thesis is used as a stand-alone word, it denotes academic papers written by university students. It is mostly written by those pursuing postgraduate degrees, at the end of their courses. They demonstrate their proficiency in their disciplines and the topics they have selected for research. 

However, when a thesis is used to refer to a statement, it denotes the statement that is written at the introduction of a research paper or essay. A thesis is supported by a credible argument.

Every research paper must have a thesis statement that acts as a guide to what the research will be all about. It is possible to receive very poor grades or even score a zero if your research paper lacks the thesis statement. 

What is a Theory?

A theory can be defined as a rational form of abstract perspectives or thinking concerning the results of such thinking or a phenomenon. The process of rational and contemplative thinking is mostly associated with processes such as research or observational study.

As such, a theory can be considered to belong to both scientific and non-scientific disciplines. Theories can also belong to no discipline.

From a modernistic scientific approach, a theory can mean scientific theories that have been well confirmed to explain nature and that are created in such a way that they are consistent with the standard scientific method. A theory should fulfill all the criteria required by modern-day science. 

A theory should be described in a way that scientific tests that have been conducted can provide empirical support or contradiction to the theory.

Because of the nature by which scientific theories are developed, they tend to be the most rigorous, reliable, and comprehensive when it comes to describing and supporting scientific knowledge. 

The connection between a theory and a hypothesis is that when a theory has not yet been proven, it can be referred to as a hypothesis.

The thing about theories is that they are not meant to help the scientist or researcher reach a particular goal. Rather, a theory is meant to guide the process of finding facts about a phenomenon or an observation. 

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Difference between a Theory and Thesis

A theory is a rational form of abstract perspectives or thinking concerning the results of such thinking or a phenomenon. The process of rational and contemplative thinking is mostly associated with processes such as research or observational study. On the other hand, a thesis is a written piece of academic work that is submitted by students to attain a university degree.

It denotes academic papers that are written by students in the university, especially those pursuing postgraduate degrees, at the end of their courses to demonstrate their proficiency in their disciplines and the topics they have selected for research. 

To understand the application of these, read our guide on the difference between a research paper and a thesis proposal to get a wider view.

How to write a Good Hypothesis

1. asking a question.

Asking a question is the first step in the scientific method and the question should be based on  who, what, where, when, why,  and  how . The question should be focused, specific, and researchable.

2. Gathering preliminary research 

This is the process of collecting relevant data. It can be done by researching academic journals, conducting case studies, observing phenomena, and conducting experiments. 

3. Formulating an answer

When the research is completed, you should think of how best to answer the question and defend your position. The answer to your question should be objective. 

4. Writing the hypothesis

When your answer is ready, you can move to the next step of formulating the hypothesis. A good hypothesis should contain relevant variables, predicted outcomes, and a study group that can include non-human things. The hypothesis should not be a question but a complete statement. 

5. Refining the hypothesis

Though you may skip this step, it is advisable to include it because your study may involve two groups or be a correlational study. Refining the hypothesis will ensure that you have stated the difference or relationship you expect to find. 

6. Creating a null and alternative hypotheses

A null hypothesis (H0) will postulate that there is no evidence to support the difference. On the other hand, an alternative hypothesis (H1) posits that there is evidence in support of the difference. 

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Frequently Asked Questions

Difference between thesis and hypothesis example.

Thesis:  High levels of alcohol consumption have detrimental effects on your health, such as weight gain, heart disease, and liver complications.

Hypothesis:  The people who consume high levels of alcohol experience detrimental effects on their health such as weight gain, heart disease, and liver complications. 

What is the difference between a summary and a thesis statement?

A summary is a brief account or statement of the main points from the researches. A thesis statement is a statement that is written at the end of the introduction of a research paper or essay that summarizes the main claims of the paper. 

Difference between hypothesis and statement of the problem

A hypothesis can be defined as the proposed or suggested explanation for an occurrence, something, or a phenomenon. The same should be testable through scientific methods. Conversely, a statement of a problem is a concise description of the issue to be addressed on how it can be improved. 

When not handling complex essays and academic writing tasks, Josh is busy advising students on how to pass assignments. In spare time, he loves playing football or walking with his dog around the park.

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Step-by-step guide to hypothesis testing in statistics

Hypothesis testing in statistics helps us use data to make informed decisions. It starts with an assumption or guess about a group or population—something we believe might be true. We then collect sample data to check if there is enough evidence to support or reject that guess. This method is useful in many fields, like science, business, and healthcare, where decisions need to be based on facts.

Learning how to do hypothesis testing in statistics step-by-step can help you better understand data and make smarter choices, even when things are uncertain. This guide will take you through each step, from creating your hypothesis to making sense of the results, so you can see how it works in practical situations.

What is Hypothesis Testing?

Table of Contents

Hypothesis testing is a method for determining whether data supports a certain idea or assumption about a larger group. It starts by making a guess, like an average or a proportion, and then uses a small sample of data to see if that guess seems true or not.

For example, if a company wants to know if its new product is more popular than its old one, it can use hypothesis testing. They start with a statement like “The new product is not more popular than the old one” (this is the null hypothesis) and compare it with “The new product is more popular” (this is the alternative hypothesis). Then, they look at customer feedback to see if there’s enough evidence to reject the first statement and support the second one.

Simply put, hypothesis testing is a way to use data to help make decisions and understand what the data is really telling us, even when we don’t have all the answers.

Importance Of Hypothesis Testing In Decision-Making And Data Analysis

Hypothesis testing is important because it helps us make smart choices and understand data better. Here’s why it’s useful:

• Reduces Guesswork : It helps us see if our guesses or ideas are likely correct, even when we don’t have all the details.
• Uses Real Data : Instead of just guessing, it checks if our ideas match up with real data, which makes our decisions more reliable.
• Avoids Errors : It helps us avoid mistakes by carefully checking if our ideas are right so we don’t make costly errors.
• Shows What to Do Next : It tells us if our ideas work or not, helping us decide whether to keep, change, or drop something. For example, a company might test a new ad and decide what to do based on the results.
• Confirms Research Findings : It makes sure that research results are accurate and not just random chance so that we can trust the findings.

Here’s a simple guide to understanding hypothesis testing, with an example:

1. Set Up Your Hypotheses

Explanation: Start by defining two statements:

• Null Hypothesis (H0): This is the idea that there is no change or effect. It’s what you assume is true.
• Alternative Hypothesis (H1): This is what you want to test. It suggests there is a change or effect.

Example: Suppose a company says their new batteries last an average of 500 hours. To check this:

• Null Hypothesis (H0): The average battery life is 500 hours.
• Alternative Hypothesis (H1): The average battery life is not 500 hours.

2. Choose the Test

Explanation: Pick a statistical test that fits your data and your hypotheses. Different tests are used for various kinds of data.

Example: Since you’re comparing the average battery life, you use a one-sample t-test .

3. Set the Significance Level

Explanation: Decide how much risk you’re willing to take if you make a wrong decision. This is called the significance level, often set at 0.05 or 5%.

Example: You choose a significance level of 0.05, meaning you’re okay with a 5% chance of being wrong.

4. Gather and Analyze Data

Explanation: Collect your data and perform the test. Calculate the test statistic to see how far your sample result is from what you assumed.

Example: You test 30 batteries and find they last an average of 485 hours. You then calculate how this average compares to the claimed 500 hours using the t-test.

5. Find the p-Value

Explanation: The p-value tells you the probability of getting a result as extreme as yours if the null hypothesis is true.

Example: You find a p-value of 0.0001. This means there’s a very small chance (0.01%) of getting an average battery life of 485 hours or less if the true average is 500 hours.

6. Make Your Decision

Explanation: Compare the p-value to your significance level. If the p-value is smaller, you reject the null hypothesis. If it’s larger, you do not reject it.

Example: Since 0.0001 is much less than 0.05, you reject the null hypothesis. This means the data suggests the average battery life is different from 500 hours.

7. Report Your Findings

Explanation: Summarize what the results mean. State whether you rejected the null hypothesis and what that implies.

Example: You conclude that the average battery life is likely different from 500 hours. This suggests the company’s claim might not be accurate.

Hypothesis testing is a way to use data to check if your guesses or assumptions are likely true. By following these steps—setting up your hypotheses, choosing the right test, deciding on a significance level, analyzing your data, finding the p-value, making a decision, and reporting results—you can determine if your data supports or challenges your initial idea.

Understanding Hypothesis Testing: A Simple Explanation

Hypothesis testing is a way to use data to make decisions. Here’s a straightforward guide:

1. What is the Null and Alternative Hypotheses?

• Null Hypothesis (H0): This is your starting assumption. It says that nothing has changed or that there is no effect. It’s what you assume to be true until your data shows otherwise. Example: If a company says their batteries last 500 hours, the null hypothesis is: “The average battery life is 500 hours.” This means you think the claim is correct unless you find evidence to prove otherwise.
• Alternative Hypothesis (H1): This is what you want to find out. It suggests that there is an effect or a difference. It’s what you are testing to see if it might be true. Example: To test the company’s claim, you might say: “The average battery life is not 500 hours.” This means you think the average battery life might be different from what the company says.

2. One-Tailed vs. Two-Tailed Tests

• One-Tailed Test: This test checks for an effect in only one direction. You use it when you’re only interested in finding out if something is either more or less than a specific value. Example: If you think the battery lasts longer than 500 hours, you would use a one-tailed test to see if the battery life is significantly more than 500 hours.
• Two-Tailed Test: This test checks for an effect in both directions. Use this when you want to see if something is different from a specific value, whether it’s more or less. Example: If you want to see if the battery life is different from 500 hours, whether it’s more or less, you would use a two-tailed test. This checks for any significant difference, regardless of the direction.

3. Common Misunderstandings

• Clarification: Hypothesis testing doesn’t prove that the null hypothesis is true. It just helps you decide if you should reject it. If there isn’t enough evidence against it, you don’t reject it, but that doesn’t mean it’s definitely true.
• Clarification: A small p-value shows that your data is unlikely if the null hypothesis is true. It suggests that the alternative hypothesis might be right, but it doesn’t prove the null hypothesis is false.
• Clarification: The significance level (alpha) is a set threshold, like 0.05, that helps you decide how much risk you’re willing to take for making a wrong decision. It should be chosen carefully, not randomly.
• Clarification: Hypothesis testing helps you make decisions based on data, but it doesn’t guarantee your results are correct. The quality of your data and the right choice of test affect how reliable your results are.

Benefits and Limitations of Hypothesis Testing

• Clear Decisions: Hypothesis testing helps you make clear decisions based on data. It shows whether the evidence supports or goes against your initial idea.
• Objective Analysis: It relies on data rather than personal opinions, so your decisions are based on facts rather than feelings.
• Concrete Numbers: You get specific numbers, like p-values, to understand how strong the evidence is against your idea.
• Control Risk: You can set a risk level (alpha level) to manage the chance of making an error, which helps avoid incorrect conclusions.
• Widely Used: It can be used in many areas, from science and business to social studies and engineering, making it a versatile tool.

Limitations

• Sample Size Matters: The results can be affected by the size of the sample. Small samples might give unreliable results, while large samples might find differences that aren’t meaningful in real life.
• Risk of Misinterpretation: A small p-value means the results are unlikely if the null hypothesis is true, but it doesn’t show how important the effect is.
• Needs Assumptions: Hypothesis testing requires certain conditions, like data being normally distributed . If these aren’t met, the results might not be accurate.
• Simple Decisions: It often results in a basic yes or no decision without giving detailed information about the size or impact of the effect.
• Can Be Misused: Sometimes, people misuse hypothesis testing, tweaking data to get a desired result or focusing only on whether the result is statistically significant.
• No Absolute Proof: Hypothesis testing doesn’t prove that your hypothesis is true. It only helps you decide if there’s enough evidence to reject the null hypothesis, so the conclusions are based on likelihood, not certainty.

Final Thoughts 

Hypothesis testing helps you make decisions based on data. It involves setting up your initial idea, picking a significance level, doing the test, and looking at the results. By following these steps, you can make sure your conclusions are based on solid information, not just guesses.

This approach lets you see if the evidence supports or contradicts your initial idea, helping you make better decisions. But remember that hypothesis testing isn’t perfect. Things like sample size and assumptions can affect the results, so it’s important to be aware of these limitations.

In simple terms, using a step-by-step guide for hypothesis testing is a great way to better understand your data. Follow the steps carefully and keep in mind the method’s limits.

What is the difference between one-tailed and two-tailed tests?

 A one-tailed test assesses the probability of the observed data in one direction (either greater than or less than a certain value). In contrast, a two-tailed test looks at both directions (greater than and less than) to detect any significant deviation from the null hypothesis.

How do you choose the appropriate test for hypothesis testing?

The choice of test depends on the type of data you have and the hypotheses you are testing. Common tests include t-tests, chi-square tests, and ANOVA. You get more details about ANOVA, you may read Complete Details on What is ANOVA in Statistics ?  It’s important to match the test to the data characteristics and the research question.

What is the role of sample size in hypothesis testing?  

Sample size affects the reliability of hypothesis testing. Larger samples provide more reliable estimates and can detect smaller effects, while smaller samples may lead to less accurate results and reduced power.

Can hypothesis testing prove that a hypothesis is true?  

Hypothesis testing cannot prove that a hypothesis is true. It can only provide evidence to support or reject the null hypothesis. A result can indicate whether the data is consistent with the null hypothesis or not, but it does not prove the alternative hypothesis with certainty.

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• Published: 06 September 2024

Dissociative and prioritized modeling of behaviorally relevant neural dynamics using recurrent neural networks

• Omid G. Sani   ORCID: orcid.org/0000-0003-3032-5669 1 ,
• Bijan Pesaran   ORCID: orcid.org/0000-0003-4116-0038 2 &
• Maryam M. Shanechi   ORCID: orcid.org/0000-0002-0544-7720 1 , 3 , 4 , 5  

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Understanding the dynamical transformation of neural activity to behavior requires new capabilities to nonlinearly model, dissociate and prioritize behaviorally relevant neural dynamics and test hypotheses about the origin of nonlinearity. We present dissociative prioritized analysis of dynamics (DPAD), a nonlinear dynamical modeling approach that enables these capabilities with a multisection neural network architecture and training approach. Analyzing cortical spiking and local field potential activity across four movement tasks, we demonstrate five use-cases. DPAD enabled more accurate neural–behavioral prediction. It identified nonlinear dynamical transformations of local field potentials that were more behavior predictive than traditional power features. Further, DPAD achieved behavior-predictive nonlinear neural dimensionality reduction. It enabled hypothesis testing regarding nonlinearities in neural–behavioral transformation, revealing that, in our datasets, nonlinearities could largely be isolated to the mapping from latent cortical dynamics to behavior. Finally, DPAD extended across continuous, intermittently sampled and categorical behaviors. DPAD provides a powerful tool for nonlinear dynamical modeling and investigation of neural–behavioral data.

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High resolution behavioral and neural activity representation using a geometrical approach

Understanding how neural population dynamics give rise to behavior is a major goal in neuroscience. Many methods that relate neural activity to behavior use static mappings or embeddings, which do not describe the temporal structure in how neural population activity evolves over time 1 . In comparison, dynamical models can describe these temporal structures in terms of low-dimensional latent states embedded in the high-dimensional space of neural recordings. Prior dynamical models have often been linear or generalized linear 1 , 2 , 3 , 4 , 5 , 6 , 7 , thus motivating recent work to develop support for piece-wise linear 8 , locally linear 9 , switching linear 10 , 11 , 12 , 13 or nonlinear 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 models of neural dynamics, especially in applications such as single-trial smoothing of neural population activity 9 , 14 , 15 , 16 , 17 , 18 , 19 and decoding behavior 20 , 21 , 22 , 23 , 24 , 26 . Once trained, the latent states of these models can subsequently be mapped to behavior 1 , 25 to learn an overall dynamical transformation from neural activity to behavior. However, multiple challenges hinder the dynamical modeling and interpretation of neural–behavioral transformations.

First, the neural–behavioral transformation can exhibit nonlinearities, which the dynamical model should capture. Moreover, these nonlinearities can be in one or more different elements within the dynamical model, for example, in the dynamics of the latent state or in its embedding. Enabling hypothesis testing regarding the origin of nonlinearity (that is, where the nonlinearity can be isolated to within the model) is important for interpreting neural computations and developing neurotechnology but remains largely unaddressed in current nonlinear models. Second, neural dynamics related to a given behavior often constitute a minority of the total neural variance 28 , 29 , 30 , 31 , 32 , 33 . To avoid missing or confounding these dynamics, nonlinear dynamical models need to dissociate behaviorally relevant neural dynamics from other neural dynamics and prioritize the learning of the former, which is currently not possible. Indeed, existing nonlinear methods for modeling neural activity either do not explicitly model temporal dynamics 34 , 35 , 36 or do not prioritize behaviorally relevant dynamics 16 , 37 , 38 , or have a mixed objective 18 that may mix behaviorally relevant and other neural dynamics in the same latent states ( Discussion and Extended Data Table 1 ). Our prior method, termed PSID 6 , has enabled prioritized dissociation of behaviorally relevant neural dynamics but for linear dynamical models. Third, for broad applicability, in addition to continuous behaviors, dynamical models should admit categorical (for example, choices) or intermittently sampled behaviors (for example, mood reports), which are not supported by existing dynamical methods with a mixed objective 18 or by PSID. To date, learning nonlinear dynamical models of neural population activity that can address the above challenges has not been achieved.

Here, we develop dissociative prioritized analysis of dynamics (DPAD), a nonlinear dynamical modeling framework using recurrent neural networks (RNNs) that addresses all the above challenges. DPAD models both behaviorally relevant and other neural dynamics but dissociates them into separate latent states and prioritizes the learning of the former. To do so, we formulate a two-section RNN as the DPAD nonlinear dynamical model and develop a four-step optimization algorithm to train it. The first RNN section learns the behaviorally relevant latent states with priority, and the second section learns any remaining neural dynamics (Fig. 1a and Supplementary Fig. 1 ). Moreover, DPAD adjusts these optimization steps as needed to admit continuous-valued, categorical or intermittently sampled data ( Methods ). Furthermore, to capture nonlinearity in the neural–behavioral transformation and enable hypothesis testing regarding its origins, DPAD decomposes this transformation into the following four interpretable elements and allows each element to become linear or nonlinear (Fig. 1a,b ): the mapping from neural activity to the latent space (neural input), the latent state dynamics within this space (recursion) and the mappings of the state to neural activity and behavior (neural and behavior readouts). Finally, we formulate the DPAD model in predictor form such that the learned model can be directly used for inference, enabling causal and computationally efficient decoding for data, whether with or without a fixed-length trial structure ( Methods ).

a , DPAD decomposes the neural–behavioral transformation into four interpretable mapping elements. It learns the mapping of neural activity ( y k ) to latent states ( x k ), termed neural input in the model; learns the dynamics or temporal structure of the latent states, termed recursion in the model; dissociates the behaviorally relevant latent states ( \({x}_{k}^{\left(1\right)}\) ) that are relevant to a measured behavior ( z k ) from other states ( \({x}_{k}^{\left(2\right)}\) ); learns the mapping of the latent states to behavior and to neural activity, termed behavior and neural readouts in the model; and allows flexible linear or nonlinear mappings in any of its elements. DPAD additionally prioritizes the learning of behaviorally relevant neural dynamics to learn them accurately. b , Computation graph of the DPAD model consists of a two-section RNN whose input is neural activity at the current time step and whose outputs are the predicted behavior and neural activity in the next time step ( Methods ). This graph assumes that computations are Markovian, that is, with a high enough dimension, latent states can summarize the information from past neural data that is useful for predicting future neural–behavioral data. Each of the four mapping elements from a has a corresponding parameter in each section of the RNN model, indicated by the same colors and termed as introduced in a . c , We developed a four-step optimization method to learn all the model parameters from training neural–behavioral data (Supplementary Fig. 1a ). Further, each model parameter can be specified via the ‘nonlinearity setting’ to be linear or nonlinear with various options to implement the nonlinearity (Supplementary Fig. 1b,c ). After a model is learned, only past neural activity is used to decode behavior and predict neural activity using the computation graph in b . d , DPAD also has the option of automatically selecting the ‘nonlinearity setting’ for the data by fitting candidate models and comparing them in terms of both behavior decoding and neural self-prediction accuracy ( Methods ). In this work, we chose among 90 candidate models with various nonlinearity settings ( Methods ). We refer to this automatic selection of nonlinearity as ‘DPAD with flexible nonlinearity’.

To show its broad utility, we demonstrate five distinct use-cases for DPAD across four diverse nonhuman primate (NHP) datasets consisting of both population spiking activity and local field potentials (LFPs). First, DPAD more accurately models the overall neural–behavioral data than alternative nonlinear and linear methods. This is due both to DPAD’s prioritized and dynamical modeling of behaviorally relevant neural dynamics and to its nonlinearity. Second, DPAD can automatically uncover nonlinear dynamical transformations of raw LFP that are more predictive of behavior than traditional LFP power band features and in some datasets can even outperform population spiking activity in terms of behavior prediction. Further, DPAD reveals that among the neural modalities, the degree of nonlinearity is greatest for the raw LFP. Third, DPAD enables nonlinear and dynamical neural dimensionality reduction while preserving behavior information, thus extracting lower-dimensional yet more behavior-predictive latent states from past neural activity. Fourth, DPAD enables hypothesis testing regarding the origin of nonlinearity in the neural–behavioral transformation. Consistently across our movement-related datasets, doing so revealed that summarizing the nonlinearities just in the behavior readout from the latent state is largely sufficient for predicting the neural–behavioral data (see Discussion ). Fifth, DPAD extends to categorical and intermittently observed behaviors, which is important for cognitive neuroscience 11 , 39 and neuropsychiatry 40 , 41 , 42 . Together, these results highlight DPAD’s broad utility as a dynamical modeling tool to investigate the nonlinear and dynamical transformation of neural activity to specific behaviors across various domains of neuroscience.

Overview of DPAD

Formulation.

We model neural activity and behavior jointly and nonlinearly ( Methods ) as

where k is the time index, \({y}_{k}\in {{\mathbb{R}}}^{{n}_{y}}\) and \({z}_{k}\in {{\mathbb{R}}}^{{n}_{z}}\) denote the neural activity and behavior time series, respectively, \({x}_{k}\in {{\mathbb{R}}}^{{n}_{x}}\) is the latent state, and e k and \({{\epsilon }}_{k}\) denote neural and behavior dynamics that are unpredictable from past neural activity. Multi-input–multi-output functions A ′ (recursion), K (neural input), C y (neural readout) and C z (behavior readout) are parameters that fully specify the model and have interpretable descriptions ( Methods , Supplementary Note 1 and Fig. 1a,b ). The adjusted formulation for intermittently sampled and noncontinuous-valued (for example, categorical) data is provided in Methods . DPAD supports both linear and nonlinear modeling, which will be termed linear DPAD and nonlinear DPAD (or just DPAD), respectively.

Dissociative and prioritized learning

We further expand the model in Eq. ( 1 ) in two sections, as depicted in Fig. 1b (Eq. ( 2 ) in Methods and Supplementary Note 2 ). The first and second sections describe the behaviorally relevant neural dynamics and the other neural dynamics with latent states \({x}_{k}^{(1)}\in {{\mathbb{R}}}^{{n}_{1}}\) and \({x}_{k}^{(2)}\in {{\mathbb{R}}}^{{n}_{x}-{n}_{1}}\) , respectively. We specify the parameters of the two RNN sections with superscripts (for example, K (1) and K (2) ) and learn them all sequentially via a four-step optimization ( Methods , Supplementary Fig. 1a and Fig. 1b ). The first two steps exclusively learn neural dynamics that are behaviorally relevant with the objective of behavior prediction, whereas the optional last two steps learn any remaining neural dynamics with the objective of residual neural prediction ( Methods and Supplementary Fig. 1 ). We implement DPAD in Tensorflow and use an ADAM 43 optimizer ( Methods ).

Comparison baselines

As a baseline, we compare DPAD with standard nonlinear RNNs fitted to maximize neural prediction, unsupervised with respect to behavior. We refer to this baseline as nonlinear neural dynamical modeling (NDM) 6 or as linear NDM if all RNN parameters are linear. NDM is nondissociative and nonprioritized, so comparisons with NDM show the benefit of DPAD’s prioritized dissociation of behaviorally relevant neural dynamics. We also compare DPAD with latent factor analysis via dynamical systems (LFADS) 16 and with two concurrently 44 developed methods with DPAD named targeted neural dynamical modeling (TNDM) 18 and consistent embeddings of high-dimensional recordings using auxiliary variables (CEBRA) 36 in terms of neural–behavioral prediction; however, as summarized in Extended Data Table 1 , these and other existing methods differ from DPAD in key goals and capabilities and do not enable some of DPAD’s use-cases (see Discussion ).

Decoding using past neural data

Given DPAD’s learned parameters, the latent states can be causally extracted from neural activity by iterating through the RNN in Eq. ( 1 ) ( Methods and Supplementary Note 1 ). Note that this decoding always only uses neural activity without seeing the behavior data.

Flexible control of nonlinearities

We allow each model parameter (for example, C z ) to be an arbitrary multilayer neural network (Supplementary Fig. 1c ), which can universally approximate any smooth nonlinear function or implement linear matrix multiplications ( Methods and Supplementary Fig. 1b ). Users can manually specify which parameters will be learned as nonlinear and with what architecture (Fig. 1c ; see application in use-case 4). Alternatively, DPAD can automatically determine the best nonlinearity setting for the data by conducting a search over nonlinearity options (Fig. 1d and Methods ), a process that we refer to as flexible nonlinearity. For a fair comparison, we also implement this flexible nonlinearity for NDM. To show the benefits of nonlinearity, we also compare with linear DPAD, where all parameters are set to be linear, in which case Eq. ( 1 ) formulates a standard linear state-space model in predictor form ( Methods ).

Evaluation metrics

We evaluate how well the models can use the past neural activity to predict the next sample of behavior (termed ‘decoding’) or the next sample of neural activity itself (termed ‘neural self-prediction’ or simply ‘self-prediction’). Thus, decoding and self-prediction assess the one-step-ahead prediction accuracies and reflect the learning of behaviorally relevant and overall neural dynamics, respectively. Both performance measures are always computed with cross-validation ( Methods ).

Our primary interest is to find models that simultaneously reach both accurate behavior decoding and accurate neural self-prediction. But in some applications, only one of these metrics may be of interest. Thus, we use the term ‘performance frontier’ to refer to the range of performances achievable by those models that compared to every other model are better in neural self-prediction and/or behavior decoding or are similar in terms of both metrics ( Methods ).

Diverse neural–behavioral datasets

We used DPAD to study the behaviorally relevant neural dynamics in four NHPs performing four different tasks (Fig. 2 and Methods ). In the first task, the animal made naturalistic three-dimensional (3D) reach, grasp and return movements to diverse locations while the joint angles in the arm, elbow, wrist and fingers were tracked as the behavior (Fig. 2a ) 6 , 45 . In the second task, the animal made saccadic eye movements to one of eight possible targets on a screen, with the two-dimensional (2D) eye position tracked as the behavior (Fig. 2d ) 6 , 46 . In the third task, the animal made sequential 2D reaches on a screen using a cursor controlled with a manipulandum while the 2D cursor position and velocity were tracked as the behavior (Fig. 2g ) 47 , 48 . In the fourth task, the animal made 2D reaches to random targets in a virtual-reality-presented grid via a cursor that mirrored the animal’s fingertip movements, for which the 2D position and velocity were tracked as the behavior (Fig. 2i ) 49 . In tasks 1 and 4, primary motor cortical activity was modeled. For tasks 2 and 3, prefrontal cortex and dorsal premotor cortical activities were modeled, respectively.

a , The 3D reach task, along with example true and decoded behavior dimensions, decoded from spiking activity using DPAD, with more example trajectories for all modalities shown in Supplementary Fig. 3 . b , Cross-validated decoding accuracy correlation coefficient (CC) achieved by linear and nonlinear DPAD. Results are shown for spiking activity, raw LFP activity and LFP band power activity ( Methods ). For nonlinear DPAD, the nonlinearities are selected automatically based on the training data to maximize behavior decoding accuracy (that is, flexible nonlinearity). The latent state dimension in each session and fold is chosen (among powers of 2 up to 128) as the smallest that reaches peak decoding in the training data among all state dimensions ( Methods ). Bars show the mean, whiskers show the s.e.m., and dots show all data points ( N  = 35 session-folds). Asterisks (*) show significance level for a one-sided Wilcoxon signed-rank test (* P  < 0.05, ** P  < 0.005 and *** P  < 0.0005); NS, not significant. c , The difference between the nonlinear and linear results from b shown with the same notations. d – f , Same as a – c for the second dataset with saccadic eye movements ( N  = 35 session-folds). g , h , Same as a and b for the third dataset, which did not include LFP data, with sequential cursor reaches controlled via a 2D manipulandum ( N  = 15 session-folds). Behavior consists of the 2D position and velocity of the cursor, denoted as ‘hand kinematics’ in the figure. i – k , Same as a – c for the fourth dataset, with random grid virtual reality cursor reaches controlled via fingertip movement ( N  = 35 session-folds). For all DPAD variations, only the first two optimization steps were used in this figure (that is, n 1  =  n x ) to only focus on learning behaviorally relevant neural dynamics.

Source data

In all datasets, we modeled the Gaussian smoothed spike counts as the main neural modality ( Methods ). In three datasets that had LFP, we also modeled the following two additional modalities: (1) raw LFP, downsampled to the sampling rate of behavior (that is, 50-ms time steps), which in the motor cortex is known as the local motor potential 50 , 51 , 52 and has been used to decode behavior 6 , 50 , 51 , 52 , 53 ; and (2) LFP power in standard frequency bands from delta (0.1–4 Hz) to high gamma (130–170 Hz (refs. 5 , 6 , 40 ); Methods ). Similar results held for all three modalities.

Numerical simulations validate DPAD

We first validate DPAD with linear simulations here (Extended Data Fig. 1 ) and then present nonlinear simulations under use-case 4 below (Extended Data Fig. 2 and Supplementary Fig. 2 ). We simulated general random linear models (not emulating any real data) in which only a subset of state dimensions contributed to generating behavior and thus were behaviorally relevant ( Methods ). We found that with a state dimension equal to that of the true model, DPAD achieved ideal cross-validated prediction (that is, similar to the true model) for both behavior and neural signals (Extended Data Fig. 1b,d ). Moreover, even given a minimal state dimension equal to the true behaviorally relevant state dimension, DPAD still achieved ideal prediction for behavior (Extended Data Fig. 1c ). Finally, across various regimens of training samples, linear DPAD performed similarly to the linear-algebraic-based PSID 6 from our prior work (Extended Data Fig. 1 ). Thus, hereafter, we use linear DPAD as our linear modeling benchmark.

Use-case 1: DPAD enables nonlinear neural–behavioral modeling across modalities

Dpad captures nonlinearity in behaviorally relevant dynamics.

We modeled each neural modality (spiking, raw LFP or LFP power) along with behavior using linear and nonlinear DPAD and compared their cross-validated behavior decoding (Fig. 2b,e,h,j and Supplementary Fig. 3 ). Across all neural modalities in all datasets, nonlinear DPAD achieved significantly higher decoding accuracy than linear DPAD. This result suggests that there is nonlinearity in the dynamical neural–behavioral transformation, which DPAD successfully captures (Fig. 2b,e,h,j ).

DPAD better predicts the neural–behavioral data

Across all datasets and modalities, compared to nonlinear NDM or linear DPAD, nonlinear DPAD reached higher behavior decoding accuracy while also being as accurate or better in terms of neural self-prediction (Fig. 3 , Extended Data Fig. 3 and Supplementary Fig. 4 ). Indeed, compared to these, DPAD was always on the best performance frontier for predicting the neural–behavioral data (Fig. 3 and Extended Data Fig. 3 ). Additionally, DPAD was always on the best performance frontier for predicting the neural–behavioral data compared to long short-term memory (LSTM) networks as well as a concurrently 44 developed method with DPAD termed CEBRA 36 on our four datasets (Fig. 4a–h ) in addition to a fifth movement dataset 54 analyzed in the CEBRA paper (Fig. 4i,j ). These results suggest that DPAD provides a more accurate description for neural–behavioral data.

a , The 3D reach task. b , Cross-validated neural self-prediction accuracy (CC) achieved by each method shown on the horizontal axis versus the corresponding behavior decoding accuracy on the vertical axis for modeling spiking activity. Latent state dimension for each method in each session, and fold is chosen (among powers of 2 up to 128) as the smallest that reaches peak neural self-prediction in training data or reaches peak decoding in training data, whichever is larger ( Methods ). The plus on the plot shows the mean self-prediction and decoding accuracy across sessions and folds ( N  = 35 session-folds), and the horizontal and vertical whiskers show the s.e.m. for these two measures, respectively. Capital letter annotations denote the methods according to the legend to make the plots more accessible. Models whose self-prediction and decoding accuracy measures lead to values toward the top-rightmost corner of the plot lie on the best performance frontier (indicated by red arrows) as they have better performance in both measures and thus better explain the neural–behavioral data ( Methods ). c , d , Same as a and b for the second dataset with saccadic eye movements ( N  = 35 session-folds). e , f , Same as a and b for the third dataset, with sequential cursor reaches controlled via a 2D manipulandum ( N  = 15 session-folds). g , h , Same as a and b for the fourth dataset with random grid virtual reality cursor reaches controlled via fingertip position ( N  = 35 session-folds). For all DPAD variations, the first 16 latent state dimensions are learned using the first two optimization steps, and the remaining dimensions are learned using the last two optimization steps (that is, n 1  = 16). For nonlinear DPAD/NDM, we fit models with different combinations of nonlinearities and then select a final model among these fitted models based on either decoding or self-prediction accuracy in the training data and report both sets of results (Supplementary Fig. 1 and Methods ). DPAD with nonlinearity selected based on neural self-prediction was better than all other methods overall ( b , d , f and h ).

a – h , Figure content is parallel to Fig. 3 (with pluses and whiskers defined in the same way) but instead of NDM shows CEBRA and LSTM networks as baselines ( Methods ). i , j , Here, we also add a fifth dataset 54 ( Methods ), where in each trial an NHP moves a cursor from a center point to one of eight peripheral targets ( i ). In this fifth dataset ( N  = 5 folds), we use the exact CEBRA hyperparameters that were used for this dataset from the paper introducing CEBRA 36 . In the other four datasets ( N  = 35 session-folds in b , d and h and N  = 15 session-folds in f ), we also show CEBRA results for when hyperparameters are picked based on an extensive search ( Methods ). Two types of LSTM networks are shown, one fitted to decode behavior from neural activity and another fitted to predict the next time step of neural activity (self-prediction). We also show the results for DPAD when only using the first two optimization steps. Note that CEBRA-Behavior (denoted by D and F), LSTM for behavior decoding (denoted by H) and DPAD when only using the first two optimization steps (denoted by G) dedicate all their latent states to behavior-related objectives (for example, prediction or contrastive loss), whereas other methods dedicate some or all latent states to neural self-prediction. As in Fig. 3 , the final latent dimension for each method in each session and fold is chosen (among powers of 2 up to 128) as the smallest that reaches peak neural self-prediction in training data or reaches peak decoding in training data, whichever is larger ( Methods ). Across all datasets, DPAD outperforms baseline methods in terms of cross-validated neural–behavioral prediction and lies on the best performance frontier. For a summary of the fundamental differences in goals and capabilities of these methods, see Extended Data Table 1 .

Beyond one-step-ahead predictions, we next evaluated DPAD in terms of multistep-ahead prediction of neural–behavioral data, also known as forecasting. To do this, starting with one-step-ahead predictions (that is, m  = 1), we pass m -step-ahead predictions of neural data using the learned models as the neural observation in the next time step to obtain ( m  + 1)-step-ahead predictions ( Methods ). Nonlinear DPAD was consistently better than nonlinear NDM and linear dynamical systems (LDS) modeling in multistep-ahead forecasting of behavior (Extended Data Fig. 4 ). For neural self-prediction, we used a naive predictor as a conservative forecasting baseline, which reflects how easy it is to predict the future in a model-free way purely based on the smoothness of neural data. DPAD significantly outperformed this baseline in terms of one-step-ahead and multistep-ahead neural self-predictions (Supplementary Fig. 5 ).

Use-case 2: DPAD extracts behavior-predictive nonlinear transformations from raw LFP

We next used DPAD to compare the amount of nonlinearity in the neural–behavioral transformation across different neural modalities (Fig. 2 and Supplementary Fig. 3 ). To do so, we compared the gain in behavior decoding accuracy when going from linear to nonlinear DPAD modeling in each modality. In all datasets, raw LFP activity had the highest gain from nonlinearity in behavior decoding accuracy (Fig. 2c,f,k ). Notably, using nonlinear DPAD, raw LFP reached more accurate behavior decoding than traditional LFP band powers in all tasks (Fig. 2b,e,j ). In one dataset, raw LFP even significantly surpassed spiking activity in terms of behavior decoding accuracy (Fig. 2e ). Note that computing LFP powers involves a prespecified nonreversible nonlinear transformation of raw LFP, which may be discarding important behaviorally relevant information that DPAD can uncover directly from raw LFP. Interestingly, linear dynamical modeling did worse for raw LFP than LFP powers in most tasks (compare linear DPAD for raw LFP versus LFP powers), suggesting that nonlinearity, captured by DPAD, was required for uncovering the extra behaviorally relevant information in raw LFP.

We next examined the spatial pattern of behaviorally relevant information across recording channels. For different channels, we compared the neural self-prediction of DPAD’s low-dimensional behaviorally relevant latent states (Extended Data Fig. 5 ). We computed the coefficient of variation (defined as standard deviation divided by mean) of the self-prediction over recording channels and found that the spatial distribution of behaviorally relevant information was less variable in raw LFP than spiking activity ( P  ≤ 0.00071, one-sided signed-rank test, N  = 35 for all three datasets with LFP). This could suggest that raw LFPs reflect large-scale network-level behaviorally relevant computations, which are thus less variable within the same spatial brain area than spiking, which represents local, smaller-scale computations 55 .

Use-case 3: DPAD enables behavior-predictive nonlinear dynamical dimensionality reduction

We next found that DPAD extracted latent states that were lower dimensional yet more behavior predictive than both nonlinear NDM and linear DPAD (Fig. 5 ). Specifically, we inspected the dimension required for nonlinear DPAD to reach almost (within 5% of) peak behavior decoding accuracy in each dataset (Fig. 5b,g,l,o ). At this low latent state dimension, linear DPAD and nonlinear and linear NDM all achieved much lower behavior decoding accuracy than nonlinear DPAD across all neural modalities (Fig. 5c–e,h–j,m,p–r ). The lower decoding accuracy of nonlinear NDM suggests that the dominant dynamics in spiking and LFP modalities can be unrelated to the modeled behavior. Thus, behaviorally relevant dynamics can be missed or confounded unless they are prioritized during nonlinear learning, as is done by DPAD. Moreover, we visualized the 2D latent state trajectories learned by each method (Extended Data Fig. 6 ). Consistent with the above results, DPAD extracted latent states from neural activity that were clearly different for different behavior/movement conditions (Extended Data Fig. 6b,e,h,k ). In comparison, NDM extracted latent states that did not as clearly dissociate different conditions (Extended Data Fig. 6c,f,i,l ). These results highlight the capability of DPAD for nonlinear dynamical dimensionality reduction in neural data while preserving behaviorally relevant neural dynamics.

a , The 3D reach task. b , Cross-validated decoding accuracy (CC) achieved by variations of linear/nonlinear DPAD/NDM for different latent state dimensions. For nonlinear DPAD/NDM, the nonlinearities are selected automatically based on the training data to maximize behavior decoding accuracy (flexible nonlinearity). Solid lines show the average across sessions and folds ( N  = 35 session-folds), and the shaded areas show the s.e.m.; Low-dim., low-dimensional. c , Decoding accuracy of nonlinear DPAD versus linear DPAD and nonlinear/linear NDM at the latent state dimension for which DPAD reaches within 5% of its peak decoding accuracy in the training data across all latent state dimensions. Bars, whiskers, dots and asterisks are defined as in Fig. 2b ( N  = 35 session-folds). d , Same as c for modeling of raw LFP ( N  = 35 session-folds). e , Same as c for modeling of LFP band power activity ( N  = 35 session-folds). f – j , Same as a – e for the second dataset with saccadic eye movements ( N  = 35 session-folds). k – m , Same as a – c for the third dataset, which did not include LFP data, with sequential cursor reaches controlled via a 2D manipulandum ( N  = 15 session-folds). n – r , Same as a – e for the fourth dataset, with random grid virtual reality cursor reaches controlled via fingertip position ( N  = 35 session-folds). For all DPAD variations, only the first two optimization steps were used in this figure (that is, n 1  =  n x ) to only focus on learning behaviorally relevant neural dynamics in the dimensionality reduction regimen.

Next, we found that at low dimensions, nonlinearity could improve the accuracy of both behavior decoding (Fig. 5b,g,l,o ) and neural self-prediction (Extended Data Fig. 7 ). However, as the state dimension was increased, linear methods reached similar neural self-prediction performance as nonlinear methods across modalities (Fig. 3 and Extended Data Fig. 3 ). This was in contrast to behavior decoding, which benefited from nonlinearity regardless of how high the dimension was (Figs. 2 and 3 ).

Use-case 4: DPAD localizes the nonlinearity in the neural–behavioral transformation

Numerical simulations validate dpad’s localization.

To demonstrate that DPAD can correctly find the origin of nonlinearity in the neural–behavioral transformation (Extended Data Fig. 2 and Supplementary Fig. 2 ), we simulated random models where only one of the parameters was set to a random nonlinear function ( Methods ). DPAD identifies a parameter as the origin if models with nonlinearity only in that parameter are on the best performance frontier when compared to alternative models, that is, models with nonlinearity in other parameters, models with flexible/full nonlinearity and fully linear models (Fig. 6a ). DPAD enables this assessment due to (1) its flexible control over nonlinearities to train alternative models and (2) its simultaneous neural–behavioral modeling and evaluation ( Methods ). In all simulations, DPAD identified that the model with the correct nonlinearity origin was on the best performance frontier compared to alternative nonlinear models (Extended Data Fig. 2 and Supplementary Fig. 2 ), thus correctly revealing the origin of nonlinearity.

a , The process of determining the origin of nonlinearity via hypothesis testing shown with an example simulation. Simulation results are taken from Extended Data Fig. 2b , and the origin is correctly identified as K . Pluses and whiskers are defined as in Fig. 3 ( N  = 20 random models). b , The 3D reach task. c , DPAD’s hypothesis testing. Cross-validated neural self-prediction accuracy (CC) for each nonlinearity and the corresponding decoding accuracy. DPAD variations that have only one nonlinear parameter (for example, C z ) use a nonlinear neural network for that parameter and keep all other parameters linear. Linear and flexible nonlinear results are as in Fig. 3 . Latent state dimension in each session and fold is chosen (among powers of 2 up to 128) as the smallest that reaches peak neural self-prediction in training data or reaches peak decoding in training data, whichever is larger ( Methods ). Pluses and whiskers are defined as in Fig. 3 ( N  = 35 session-folds). Annotated arrows indicate any individual nonlinearities that are on the best performance frontier compared to all other models. Results are shown for spiking activity here and for raw LFP and LFP power activity in Supplementary Fig. 6 . d , e , Same as b and c for the second dataset with saccadic eye movements ( N  = 35 session-folds). f , g , Same as b and c for the third dataset, with sequential cursor reaches controlled via a 2D manipulandum ( N  = 15 session-folds). h , i , Same as b and c for the fourth dataset, with random grid virtual reality cursor reaches controlled via fingertip position ( N  = 35 session-folds). For all DPAD variations, the first 16 latent state dimensions are learned using the first two optimization steps, and the remaining dimensions are learned using the last two optimization steps (that is, n 1  = 16).

DPAD consistently localized nonlinearities in the behavior readout

Having validated the localization of nonlinearity in simulations, we used DPAD to find where in the model nonlinearities could be isolated to in our real datasets. We found that having the nonlinearity only in the behavior readout parameter C z was largely sufficient for achieving high behavior decoding and neural self-prediction accuracies across all our datasets and modalities (Fig. 6b–i and Supplementary Fig. 6 ). First, for spiking activity, models with nonlinearity only in the behavior readout parameter C z reached the best behavior decoding accuracy compared to models with other individual nonlinearities (Fig. 6c,e,i ) while reaching almost the same decoding accuracy as fully nonlinear models (Fig. 6c,e,g,i ). Second, these models with nonlinearity only in the behavior readout also reached a self-prediction accuracy that was unmatched by other types of individual nonlinearity (Fig. 6c,e,g,i ). Overall, this meant that models with nonlinearity only in the behavior readout parameter C z were always on the best performance frontier when compared to all other linear or nonlinear models (Fig. 6c,e,g,i ). This result interestingly also held for both LFP modalities (Supplementary Fig. 6 ).

Consistent with the above localization results, DPAD with flexible nonlinearity also, very frequently, automatically selected models with nonlinearity in the behavior readout parameter (Supplementary Fig. 7 ). However, critically, this observation on its own cannot conclude that nonlinearities can be isolated in the behavior readout parameter. This is because in the flexible nonlinearity approach, parameters may be selected as nonlinear as long as this nonlinearity does not hurt the prediction accuracies, which does not imply that such nonlinearities are necessary ( Methods ); this is why we need the hypothesis testing procedure above (Fig. 6a ). Of note, using an LSTM for the recursion parameter A ′ is one of the nonlinearity options that is automatically considered in DPAD (Extended Data Fig. 3 ), but we found that LSTM was rarely selected in our datasets as the recursion dynamics in the flexible search over nonlinearities (Supplementary Fig. 7 ). Finally, note that fitting models with a nonlinear behavior readout via a post hoc nonlinear refitting of linear DPAD models (1) cannot identify the origin of nonlinearity in general (for example, other brain regions or tasks) and (2) even in our datasets resulted in significantly worse decoding than the same models being fitted end-to-end as done by nonlinear DPAD ( P  ≤ 0.0027, one-sided signed-rank test, N  ≥ 15).

Together, these results highlight the application of DPAD in enabling investigations of nonlinear processing in neural computations underlying specific behaviors. DPAD’s machinery can not only fit fully nonlinear models but also provide evidence for the location in the model where the nonlinearity can be isolated ( Discussion ).

Use-case 5: DPAD extends to noncontinuous and intermittent data

Dpad extends to intermittently sampled behavior observations.

DPAD also supports intermittently sampled behaviors ( Methods ) 56 , that is, when behavior is measured only during a subset of time steps. We first confirmed in numerical simulations with random models that DPAD correctly learns the model with intermittently sampled behavioral data (Supplementary Fig. 8 ). Next, in each of our neural datasets, we emulated intermittent sampling by randomly discarding up to 90% of behavior samples during learning. DPAD learned accurate nonlinear models even in this case (Extended Data Fig. 8 ). This capability is important, for example, in affective neuroscience or neuropsychiatry applications where the behavior consists of sparsely sampled momentary ecological assessments of mental states such as mood 40 . We next simulated a mood decoding application and found that with as low as one behavioral (for example, mood survey) sample per day, DPAD still outperformed NDM even when NDM had access to continuous behavior samples (Extended Data Fig. 9 ). These results suggest the potential utility of DPAD in such applications, although substantial future validation in data is needed 7 , 40 , 41 , 42 .

DPAD extends to noncontinuous-valued observations

DPAD also extends to modeling of noncontinuous-valued (for example, categorical) behaviors ( Methods ). To demonstrate this, we modeled the transformation from neural activity to the momentary phase of the task in the 3D reach task: reach, hold, return or rest (Fig. 7 ). Compared to nonlinear NDM (which is dynamic) or nonlinear nondynamic methods such as support vector machines, DPAD more accurately predicted the task phase at each point in time (Fig. 7 ). This capability can extend the utility of DPAD to categorical behaviors such as decision choices in cognitive neuroscience 39 .

a , In the 3D reach dataset, we model spiking activity along with the epoch of the task as discrete behavioral data ( Methods and Fig. 2a ). The epochs/classes are (1) reaching toward the target, (2) holding the target, (3) returning to resting position and (4) resting until the next reach. b , DPAD’s predicted probability for each class is shown in a continuous segment of the test data. Most of the time, DPAD predicts the highest probability for the correct class. c , The cross-validated behavior classification performance, quantified as the area under curve (AUC) for the four-class classification, is shown for different methods at different latent state dimensions. Solid lines and shaded areas are defined as in Fig. 5b ( N  = 35 session-folds). AUC of 1 and 0.5 indicate perfect and chance-level classification, respectively. We include three nondynamic/static classification methods that map neural activity for a given time step to class label at the same time step (Extended Data Table 1 ): (1) multilayer neural network, (2) nonlinear support vector machine (SVM) and (3) linear discriminant analysis (LDA). d , Cross-validated behavior classification performance (AUC) achieved by each method when choosing the state dimension in each session and fold as the smallest that reaches peak classification performance in the training data among all state dimensions with that method ( Methods ). Bars, whiskers, dots and asterisks are defined as in Fig. 2b ( N  = 35 session-folds). e , Same as d when all methods use the same latent state dimension as DPAD (best nonlinearity for decoding) does in d ( N  = 35 session-folds). c and e show DPAD’s benefit for dimensionality reduction. f , Cross-validated neural self-prediction accuracy achieved by each method versus the corresponding behavior classification performance. Here, the latent state dimension for each method in each session and fold is chosen (among powers of 2 up to 128) as the smallest that reaches peak neural self-prediction in training data or reaches peak decoding in training data, whichever is larger ( Methods ). Pluses and whiskers are defined as in Fig. 3 ( N  = 35 session-folds).

Finally, we applied DPAD to nonsmoothed spike counts, where we compared the results with two noncausal sequential autoencoder methods, termed LFADS 16 and TNDM 18 (Supplementary Fig. 9 ), both of which have Poisson observations that model nonsmoothed spike counts 16 , 18 . TNDM 18 , which was developed after LFADS 16 and concurrently with our work 44 , 56 , adds behavioral terms to the objective function for a subset of latents but unlike DPAD does so with a mixed objective and thus does not completely dissociate or prioritize behaviorally relevant dynamics (Extended Data Table 1 and Supplementary Note 3 ). Compared to both LFADS and TNDM, DPAD remained on the best performance frontier for predicting the neural–behavioral data (Supplementary Fig. 9a ) and more accurately predicted behavior using low-dimensional latent states (Supplementary Fig. 9b ). Beyond this, TNDM and LFADS also have fundamental differences with DPAD and do not address some of DPAD’s use-cases ( Discussion and Extended Data Table 1 ).

We developed DPAD for nonlinear dynamical modeling and investigation of neural dynamics underlying behavior. DPAD can dissociate the behaviorally relevant neural dynamics and prioritize their learning over other neural dynamics, enable hypothesis testing regarding the origin of nonlinearity in the neural–behavioral transformation and achieve causal decoding. DPAD enables prioritized dynamical dimensionality reduction by extracting lower-dimensional yet more behavior-predictive latent states from neural population activity and supports modeling noncontinuous-valued (for example, categorical) and intermittently sampled behavioral data. These attributes make DPAD suitable for diverse use-cases across neuroscience and neurotechnology, some of which we demonstrated here.

We found similar results for three neural modalities: spiking activity, LFP band powers and raw LFP. For all modalities, nonlinear DPAD more accurately learned the behaviorally relevant neural dynamics than linear DPAD and linear/nonlinear NDM as reflected in its better decoding while also reaching the best performance frontier when considering both behavior decoding and neural self-prediction. Notably, the raw LFP activity benefited the most from nonlinear modeling using DPAD and outperformed LFP powers in all tasks in terms of decoding. This suggests that automatic learning of nonlinear models from raw LFP using DPAD reveals behaviorally relevant information that may be discarded when extracting traditionally used features such as LFP band powers. Also, nonlinearity was necessary to recover the extra information in raw LFP, as, unlike DPAD modeling, linear dynamical modeling of raw LFP did not outperform that of LFP powers in most datasets. These results highlight another use-case of DPAD for automatic dynamic feature extraction from LFP data.

As another use-case, DPAD enabled an investigation of which element in the neural–behavioral transformation was nonlinear. Interestingly, consistently across our four movement-related datasets, DPAD models with nonlinearity only in the behavior readout performed similarly to fully nonlinear models, reaching the best performance frontier for predicting future behavior and neural data using past neural data. The consistency of this result across our datasets is interesting because, as demonstrated in simulations (Extended Data Fig. 2 , Supplementary Fig. 2 and Fig. 6a ), the detected origin of nonlinearity could have technically been in any one (or more) of the following four elements (Fig. 1a,b ): neural input, recurrent dynamics and neural or behavior readouts, all of which were correctly localized in simulations (Extended Data Fig. 2 and Supplementary Fig. 2 ). Thus, the consistent localization results on our neural datasets provide evidence that across these four tasks, neural dynamics in these recorded cortical areas may be largely describable with linear dynamics of sufficiently high dimension, with additional nonlinearities introduced somewhere between the neural state and behavior. This finding may be consistent with (1) introduction of nonlinear processing along the downstream neuromuscular pathway that goes from the recorded cortical area to the measured behavior or any of the convergent inputs along this pathway 57 , 58 , 59 or (2) cognition intervening nonlinearly between these latent neural states and behavior, for example, by implementing context-dependent computations 60 . This result illustrates how DPAD can provide new hypotheses and the machinery to test them in future experiments that would record from multiple additional brain regions (for example, both motor and cognitive regions) and use DPAD to model them together. Such analyses may narrow down or revise the origin of nonlinearity for the wider neural–behavioral measurement set; for example, the state dynamics may be found to be nonlinear once additional brain regions are added. Localization of nonlinearity could also guide the design of competitive deep learning architectures that are more flexible or easier to implement in neurotechnologies such as brain–computer interfaces 61 .

Interestingly, the behavior decoding aspect of the localization finding here is consistent with a prior study 22 that explored the mapping of the motor cortex to an electromyogram (EMG) during a one-dimensional movement task with varying forces and found that a fully linear model was worse than a nonlinear EMG readout in decoding the EMG 22 . However, as our simulations show (Extended Data Fig. 2b and Fig. 6a ), comparing a linear model to a model that has nonlinear behavior readout is not sufficient to conclude the origin of nonlinearity, and a stronger test is needed (see Fig. 6a for a counter example and details in Methods ). Further, this previous study 22 used a specific condition-dependent nonlinearity for behavior readout rather than a universal nonlinear function approximator that DPAD enables. Finally, to conclude localization, the model with that specific nonlinearity should perform similarly to fully nonlinear models; however, unlike our results, a fully nonlinear LSTM model in some cases appears to outperform models with nonlinear readout in this prior study (see Fig. 7a,b in ref. 22 versus Fig. 9c in ref. 22 ); it is unclear if this result is due to this prior study’s specific readout nonlinearity being suboptimal or to the nonlinear origin being different in its dataset 22 . DPAD can address such questions by (1) allowing for training and comparison of alternative models with different nonlinear origins and (2) enabling a general (versus specific) nonlinearity in model parameters.

When hypothesis testing about where in the model nonlinearity can be isolated to, it may be possible to equivalently explain the same data with multiple types of nonlinearities (for example, with either a nonlinear neural input or a nonlinear readout). Such nonidentifiability is a common limitation for latent models. However, when such equivalence exists, we expect all equivalent nonlinear models to have similar performance and thus lie on the best performance frontier. But this was not the case in our datasets. Instead, we found that the nonlinear behavior readout was in most cases the only individual nonlinear parameter on the best performance frontier, providing evidence that no other individual nonlinear parameter was as suitable in our datasets. Alternatively, the best model describing the data may require two or more of the four parameters to be nonlinear. But in our datasets, models with nonlinearity only in the behavior readout were always on the best performance frontier and could not be considerably outperformed by models with more than one nonlinearity (Fig. 6 ). Nevertheless, we note that ultimately our analysis simply provides evidence for one location of nonlinearity resulting in a better fit to data with a parsimonious model, but it does not rule out other possibilities for explaining the data. For example, one could reformulate a nonlinear readout model by adding latent states and representing the readout nonlinearity as a recursion nonlinearity for the additional states, although such an equivalent but less parsimonious model may need more data to be learned as accurately. Finally, we also note that our conclusions were based on the datasets and family of nonlinear models (recursive RNNs) considered here, and thus we cannot rule out different conclusions in other scenarios and/or brain regions. Nevertheless, by providing evidence for a nonlinearity configuration, DPAD can provide testable hypotheses for future experiments that record from more brain regions.

Sequential autoencoders, spearheaded by LFADS 16 , have been used to smooth single-trial neural activity 16 without considering relevance to behavior, which is a distinct goal as we showed in comparison to PSID in our prior work 6 . Notably, another sequential autoencoder, termed TNDM, has been developed concurrently with our work 44 , 56 that adds a behavior term to the optimization objective 18 . However, these approaches do not enable several of the use-cases of DPAD here. First, unlike DPAD’s four-step learning approach, TNDM and LFADS use a single learning step with a neural-only objective (LFADS) 16 or a mixed neural–behavioral objective (TNDM) 18 that does not fully prioritize the behaviorally relevant neural dynamics (Extended Data Table 1 and Supplementary Note 3 ). DPAD’s prioritization is important for accurate learning of behaviorally relevant neural dynamics and for preserving them in dimensionality reduction, as our results comparing DPAD to TNDM/LFADS suggest (Supplementary Fig. 9 ). Second, TNDM and LFADS 16 , 18 , like other prior works 16 , 18 , 20 , 23 , 24 , 26 , 61 , do not provide flexible nonlinearity or explore hypotheses regarding the origin of nonlinearities because they use fixed nonlinear network structures (use-case 4). Third, TNDM considers spiking activity and continuous behaviors 18 , whereas DPAD extends across diverse neural and behavioral modalities: spiking, raw LFP and LFP powers and continuous, categorical or intermittent behavioral modalities. Fourth, in contrast to these noncausal sequential autoencoders 16 , 18 and some other nonlinear methods 8 , 14 , DPAD can process the test data causally and without expensive computations such as iterative expectation maximization 8 , 14 or sampling and averaging 16 , 18 . This causal efficient processing is also important for real-time closed-loop brain–computer interfaces 62 , 63 . Of note, noncausal processing is also implemented in the DPAD code library as an option ( Methods ), although it is not shown in this work. Finally, unlike these prior methods 14 , 16 , 18 , DPAD does not require fixed-length trials or trial structure, making it suitable for modeling naturalistic behaviors 5 and neural dynamics with trial-to-trial variability in the alignment to task events 64 .

Several methods can in some ways prioritize behaviorally relevant information while extracting latent embeddings from neural data but are distinct from DPAD in terms of goals and capabilities. One group includes nondynamic/static methods that do not explicitly model temporal dynamics 1 . These methods build linear maps (for example, as in demixed principal component analysis (dPCA) 34 ) or nonlinear maps, such as convolutional maps in a concurrently 44 developed method with DPAD named CEBRA 36 , to extract latent embeddings that can be guided by behavior either as a trial condition 34 or indirectly as a contrastive loss 36 . These nondynamic mappings only use a single sample or a small fixed window around each sample of neural data to extract latent embeddings (Extended Data Table 1 ). By contrast, DPAD can recursively aggregate information from all past neural data by explicitly learning a model of temporal dynamics (recursion), which also enables forecasting unlike in static/nondynamic methods. These differences may be one reason why DPAD outperformed CEBRA in terms of neural–behavioral prediction (Fig. 4 ). Another approach is used by task aligned manifold estimation (TAME-GP) 9 , which uses a Gaussian process prior (as in Gaussian process factor analysis (GPFA) 14 ) to expand the window of neural activity used for extracting the embedding into a complete trial. Unlike DPAD, methods with a Gaussian process prior have limited support for nonlinearity, often do not have closed-forms for inference and thus necessitate numerical optimization even for inference 9 and often operate noncausally 9 . Finally, the above methods do not provide flexible nonlinearity or hypothesis testing to localize the nonlinearity.

Other prior works have used RNNs either causally 20 , 22 , 23 , 24 , 26 or noncausally 16 , 18 , for example, for causal decoding of behavior from neural activity 20 , 22 , 23 , 24 , 26 . These works 20 , 22 , 23 , 24 , 26 have similarities to the first step of DPAD’s four-step optimization (Supplementary Fig. 1a ) in that the RNNs in these works learn dynamical models by solely optimizing behavior prediction. However, these works do not learn the mapping from the RNN latent states to neural activity, which is done in DPAD’s second optimization step to enable neural self-prediction (Supplementary Fig. 1a ). In addition, unlike what the last two optimization steps in DPAD enable, these prior works do not model additional neural dynamics beyond those that decode behavior and thus do not dissociate the two types of neural dynamics (Extended Data Table 1 ). Finally, as noted earlier, these prior works 9 , 20 , 23 , 24 , 26 , 36 , 61 , similar to prior sequential autoencoders 16 , 18 , have fixed nonlinear network structures and thus cannot explore hypotheses regarding the origin of nonlinearities or flexibly learn the best nonlinear structure for the training data (Fig. 1c,d and Extended Data Table 1 ).

DPAD’s optimization objective functions are not convex, similar to most nonlinear deep learning methods. Thus, as usual with nonconvex optimizations, convergence to a global optimum is not guaranteed. Moreover, as with any method, quality and neural–behavioral prediction of the learned models depend on dataset properties such as signal-to-noise ratio. Thus, we compare alternative methods within each dataset, suggesting that (for example, Fig. 4 ) across the multiple datasets here, DPAD learns more accurate models of neural–behavioral data. However, models in other datasets/scenarios may not be as accurate.

Here, we focused on using DPAD to model the transformation of neural activity to behavior. DPAD can also be used to study the transformation between other signals. For example, when modeling data from multiple brain regions, one region can be taken as the primary signal ( y k ) and another as the secondary signal ( z k ) to dissociate their shared versus distinct dynamics. Alternatively, when modeling the brain response to electrical 7 , 41 , 42 or sensory 41 , 65 , 66 stimulation, one could take the primary signal ( y k ) to be the stimulation and the secondary signal ( z k ) to be neural activity to dissociate and predict neural dynamics that are driven by stimulation. Finally, one may apply DPAD to simultaneously recorded brain activity from two subjects as primary and secondary signals to find shared intersubject dynamics during social interactions.

Model formulation

Equation ( 1 ) simplifies the DPAD model by showing both of its RNN sections as one, but the general two-section form of the model is as follows:

This equation separates the latent states of Eq. ( 1 ) into the following two parts: \({x}_{k}^{\left(1\right)}\in {{\mathbb{R}}}^{{n}_{1}}\) denotes the latent states of the first RNN section that summarize the behaviorally relevant dynamics, and \({x}_{k}^{\left(2\right)}\in {{\mathbb{R}}}^{{n}_{2}}\) , with \({n}_{2}={n}_{x}-{n}_{1}\) , denotes those of the second RNN section that represent the other neural dynamics (Supplementary Fig. 1a ). Here, A ′(1) , A ′(2) , K (1) , K (2) , \({C}_{y}^{\,\left(1\right)}\) , \({C}_{y}^{\,\left(2\right)}\) , \({C}_{z}^{\,\left(1\right)}\) and \({C}_{z}^{\,\left(2\right)}\) are multi-input–multi-output functions that parameterize the model, which we learn using a four-step numerical optimization formulation expanded on in the next section (Supplementary Fig. 1a ). DPAD also supports learning the initial value of the latent states at time 0 (that is, \({x}_{0}^{\left(1\right)}\) and \({x}_{0}^{\left(2\right)}\) ) as a parameter, but in all analyses in this paper, the initial states are simply set to 0 given their minimal impact when modeling long data sequences. Each pair of superscripted parameters (for example, A ′(1) and A ′(2) ) in Eq. ( 2 ) is a dissociated version of the corresponding nonsuperscripted parameter in Eq. ( 1 ) (for example, A ′). The computation graph for Eq. ( 2 ) is provided in Fig. 1b (and Supplementary Fig. 1a ). In Eq. ( 2 ), the recursions for computing \({x}_{k}^{\left(1\right)}\) are not dependent on \({x}_{k}^{\left(2\right)}\) , thus allowing the former to be computed without the latter. By contrast, \({x}_{k}^{\left(2\right)}\) can depend on \({x}_{k}^{\left(1\right)}\) , and this dependence is modeled via K (2) (see Supplementary Note 2 ). Note that such dependence of \({x}_{k}^{\left(2\right)}\) on \({x}_{k}^{\left(1\right)}\) via K (2) does not introduce new dynamics to \({x}_{k}^{\left(2\right)}\) because it does not involve the recursion parameter A ′(2) , which describes the dynamics of \({x}_{k}^{\left(2\right)}\) . This two-section RNN formulation is mathematically motivated by equivalent representations of a dynamical system model in different bases and by the relation between the predictor and stochastic forms of dynamical systems (Supplementary Notes 1 and 2 ).

For the RNN formulated in Eq. ( 1 ) or ( 2 ), neural activity y k constitutes the input, and predictions of neural and behavioral signals are the outputs (Fig. 1b ) given by

Note that each x k is estimated purely using all past y k (that is, y 1 , …, y k   –  1 ), so the predictions in Eq. ( 3 ) are one-step-ahead predictions of y k and z k using past neural observations (Supplementary Note 1 ). Once the model parameters are learned, the extraction of latent states x k involves iteratively applying the first line from Eq. ( 2 ), and predicting behavior or neural activity involves applying Eq. ( 3 ) to the extracted x k . As such, by writing the nonlinear model in predictor form 67 , 68 (Supplementary Note 1 ), we enable causal and computationally efficient prediction.

Learning: four-step numerical optimization approach

Unlike nondynamic models 1 , 34 , 35 , 36 , 69 , dynamical models explicitly model temporal evolution in time series data. Recent dynamical models have gone beyond linear or generalized linear dynamical models 2 , 3 , 4 , 5 , 6 , 7 , 70 , 71 , 72 , 73 , 74 , 75 , 76 , 77 , 78 , 79 , 80 , 81 to incorporate switching linear 10 , 11 , 12 , 13 , locally linear 37 or nonlinear 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 23 , 24 , 26 , 27 , 38 , 61 , 82 , 83 , 84 , 85 , 86 , 87 , 88 , 89 , 90 dynamics, often using deep learning methods 25 , 91 , 92 , 93 , 94 . But these recent nonlinear/switching works do not aim to localize nonlinearity or allow for flexible nonlinearity and do not enable fully prioritized dissociation of behaviorally relevant neural dynamics because they either do not consider behavior in their learning objective at all 14 , 16 , 37 , 38 , 61 , 95 , 96 or incorporate it with a mixed neural–behavioral objective 9 , 18 , 35 , 61 (Extended Data Table 1 ).

In DPAD, we develop a four-step learning method for training our two-section RNN in Eq. ( 1 ) and extracting the latent states that (1) enables dissociation and prioritized learning of the behaviorally relevant neural dynamics in the nonlinear model, (2) allows for flexible modeling and localization of nonlinearities, (3) extends to data with diverse distributions and (4) does all this while also achieving causal decoding and being applicable to data both with and without a trial structure. DPAD is for nonlinear modeling, and its multistep learning approach, in each step, uses numerical optimization tools that are rooted in deep learning. Thus, DPAD is mathematically distinct from our prior PSID work for linear models, which is an analytical and linear technique. PSID is based on analytical linear algebraic projections rooted in control theory 6 , which are thus not extendable to nonlinear modeling or to non-Gaussian, noncontinuous or intermittently sampled data. Thus, even when we restrict DPAD to linear modeling as a special case, it is still mathematically different from PSID 6 .

To dissociate and prioritize the behaviorally relevant neural dynamics, we devise a four-step optimization approach for learning the two-section RNN model parameters (Supplementary Fig. 1a ). This approach prioritizes the extraction and learning of the behaviorally relevant dynamics in the first two steps with states \({x}_{k}^{\left(1\right)}\in {{\mathbb{R}}}^{{n}_{1}}\) while also learning the rest of the neural dynamics in the last two steps with states \({x}_{k}^{\left(2\right)}\in {{\mathbb{R}}}^{{n}_{2}}\) and dissociating the two subtypes of dynamics. This prioritization is important for accurate learning of behaviorally relevant neural dynamics and is achieved because of the multistep learning approach; the earlier steps learn the behaviorally relevant dynamics first, that is, with priority, and then the subsequent steps learn the other neural dynamics later so that they do not mask or confound the behaviorally relevant dynamics. Importantly, each optimization step is independent of subsequent steps so all steps can be performed in order, with no need to iteratively repeat any step. We define the neural and behavioral prediction losses that are used in the optimization steps based on the negative log-likelihoods (NLLs) associated with the neural and behavior distributions, respectively. This approach benefits from the statistical foundation of maximum likelihood estimation and facilitates generalizability across behavioral distributions. We now expand on each of the four optimization steps for RNN training.

Optimization step 1

In the first two optimization steps (Supplementary Fig. 1a ), the objective is to learn the behaviorally relevant latent states \({x}_{k}^{\left(1\right)}\) and their associated parameters. In the first optimization step, we learn the parameters A ′(1) , \({C}_{z}^{\,\left(1\right)}\) and K (1) of the RNN

and estimate its latent state \({x}_{k}^{\left(1\right)}\) while minimizing the NLL of the behavior z k given by \({x}_{k}^{\left(1\right)}\) . For continuous-valued (Gaussian) behavioral data, we minimize the following sum of squared prediction error 69 , 97 given by

where the sum is over all available samples of behavior z k , and \({\Vert .\Vert }_{2}\) indicates the two-norm operator. This objective, which is typically used when fitting models to continuous-valued data 69 , 97 , is proportional to the Gaussian NLL if we assume isotropic Gaussian residuals (that is, ∑ 𝜖  = σ 𝜖 I ) 69 , 97 . If desired, a general nonisotropic residual covariance ∑ 𝜖 can be empirically computed from model residuals after the above optimization is solved (see Learning noise statistics ), although having ∑ 𝜖 is mainly useful for simulating new data and is not needed when using the learned model for inference. Similarly, in the subsequent optimization steps detailed later, the same points hold regarding how the appropriate mean squared error used for continuous-valued data is proportional to the Gaussian NLL if we assume isotropic Gaussian residuals and how the residual covariance can be computed empirically after the optimization if desired.

Optimization step 2

The second optimization step uses the extracted latent state \({x}_{k}^{\left(1\right)}\) from the RNN and fits the parameter \({C}_{y}^{\left(1\right)}\) in

while minimizing the NLL of the neural activity y k given by \({x}_{k}^{(1)}\) . For continuous-valued (Gaussian) neural activity y k , we minimize the following sum of squared prediction error 69 :

where the sum is over all available samples of y k . Optimization steps 1 and 2 conclude the prioritized extraction and modeling of behaviorally relevant latent states \({x}_{k}^{(1)}\) (Fig. 1b ) and the learning of the first section of the RNN model (Supplementary Fig. 1a ).

Optimization step 3

In optimization steps 3 and 4 (Supplementary Fig. 1a ), the objective is to learn any additional dynamics in neural activity that are not learned in the first two optimization steps, that is, \({x}_{k}^{\left(2\right)}\) and the associated parameters. To do so, in the third optimization step, we learn the parameters A ′(2) , \({C}_{y}^{\,\left(2\right)}\) and K (2) of the RNN

and estimate its latent state \({x}_{k}^{\left(2\right)}\) while minimizing the aggregate NLL of y k given both latent states, that is, by also taking into account the NLL obtained from step 2 via the \({C}_{y}^{\,\left(1\right)}\left({x}_{k}^{\left(1\right)}\right)\) term in Eq. ( 6 ). The notations \({y}_{k}^{{\prime} }\) and \({e}_{k}^{{\prime} }\) in the second line of Eq. ( 8 ) signify the fact that it is not y k that is predicted by the RNN of Eq. ( 8 ), rather it is the yet unpredicted parts of y k (that is, unpredicted after extracting \({x}_{k}^{(1)}\) ) that are being predicted. In the case of continuous-valued (Gaussian) neural activity y k , we minimize the following loss:

where the sum is over all available samples of y k . Note that in the continuous-valued (Gaussian) case, this loss is equivalent to minimizing the error in predicting the residual neural activity given by \({y}_{k}-{C}_{y}^{\,\left(1\right)}\left({x}_{k}^{\left(1\right)}\right)\) and is computed using the previously learned parameter \({C}_{y}^{\,\left(1\right)}\) and the previously extracted states \({x}_{k}^{\left(1\right)}\) in steps 1 and 2. Also, the input to the RNN in Eq. ( 8 ) includes both y k and the extracted \({x}_{k+1}^{\left(1\right)}\) from optimization step 1. The above shows how the optimization steps are appropriately linked together to compute the aggregate likelihoods.

Optimization step 4

If we assume that the second set of states \({x}_{k}^{\left(2\right)}\) do not contain any information about behavior, we could stop the modeling. However, this may not be the case if the dimension of the states extracted in the first optimization step (that is, n 1 ) is selected to be very small such that some behaviorally relevant neural dynamics are not learned in the first step. To be robust to such selections of n 1 , we can use another final numerical optimization to determine based on the data whether and how \({x}_{k}^{\left(2\right)}\) should affect behavior prediction. Thus, a fourth optimization step uses the extracted latent state in optimization steps 1 and 3 and fits C z in

while minimizing the negative log-likelihood of behavior given both latent states. In the case of continuous-valued (Gaussian) behavior z k , we minimize the following loss:

The parameter C z that is learned in this optimization step will replace both \({C}_{z}^{\,\left(1\right)}\) and \({C}_{z}^{\,\left(2\right)}\) in Eq. ( 2 ). Optionally, in a final optimization step, a similar nonlinear mapping from \({x}_{k}^{\left(1\right)}\) and \({x}_{k}^{\left(2\right)}\) can also be learned, this time to predict y k , which allows DPAD to support nonlinear interactions of \({x}_{k}^{\left(1\right)}\) and \({x}_{k}^{\left(2\right)}\) in predicting neural activity. In this case, the resulting learned C y parameter will replace both \({C}_{y}^{\,\left(1\right)}\) and \({C}_{y}^{\,\left(2\right)}\) in Eq. ( 2 ). This concludes the learning of both model sections (Supplementary Fig. 1a ) and all model parameters in Eq. ( 2 ).

In this work, when optimization steps 1 and 3 are both used to extract the latent states (that is, when 0 <  n 1  <  n x ), we do not perform the additional fourth optimization step in Eq. ( 10 ), and the prediction of behavior is done solely using the \({x}_{k}^{\left(1\right)}\) states extracted in the first optimization step. Note that DPAD can also cover NDM as a special case if we only use the third optimization step to extract the states (that is, n 1  = 0, in which case the first two steps are not needed). In this case, we use the fourth optimization step to learn C z , which is the mapping from the latent states to behavior. Also, in this case, we simply have a unified state x k as there is no dissociation in NDM, and the only goal is to extract states that predict neural activity accurately.

Additional generalizations of state dynamics

Finally, the first lines of Eqs. ( 4 ) and ( 8 ) can also be written more generally as

where instead of an additive relation between the two terms of the righthand side, both terms are combined in nonlinear functions \({{A}^{{\prime} {\prime} }}^{\left(1\right)}\) and \({{A}^{{\prime} {\prime} }}^{\left(2\right)}\) , which as a special case can still learn the additive relation in Eqs. ( 4 ) and ( 8 ). Whenever both the state recursion A and neural input K parameters (with the appropriate superscripts) are specified to be nonlinear, we use the more general architecture in Eqs. ( 12 ) and ( 13 ), and if any one of A or K or both are linear, we use Eqs. ( 4 ) and ( 8 ).

As another option, both RNN sections can be made bidirectional, which enables noncausal prediction for DPAD by using future data in addition to past data, with the goal of improving prediction, especially in datasets with stereotypical trials. Although this option is not reported in this work, it is implemented and available for use in DPAD’s public code library.

Learning noise statistics

Once the learning is complete, we also compute the covariances of the neural and behavior residual time series e k and 𝜖 k as ∑ e and ∑ 𝜖 , respectively. This allows the learned model in Eq. ( 1 ) to be usable for generating new simulated data. This application is not the focus of this work, but an explanation of it is provided in Numerical simulations .

Regularization

Adding norm 1 or norm 2 regularization for any set of parameters and the option to automatically select the regularization weight with inner cross-validation is implemented in the DPAD code. However, we did not use regularization in any of the analyses presented here.

Forecasting

DPAD also enables the capability to predict neural–behavioral data more than one time step into the future. To obtain two-step-ahead prediction, we pass the one-step-ahead neural predictions of the model as neural observations into it. This allows us to perform one state update iteration, that is, line 1 of Eq. ( 2 ), with y k being replaced with \({\hat{y}}_{k}\) from Eq. ( 3 ). Repeating this procedure m times gives the ( m  + 1)-step-ahead prediction of the latent state and neural–behavioral data.

Extending to intermittently measured behaviors

We also extend DPAD to modeling intermittently measured behavior time series (Extended Data Figs. 8 and 9 and Supplementary Fig. 8 ). To do so, when forming the behavior loss (Eqs. ( 5 ) and ( 11 )), we only compute the loss on samples where the behavior is measured and solve the optimization with this loss.

Extending to noncontinuous-valued data observations

We can also extend DPAD to noncontinuous-valued (non-Gaussian) observations by devising modified loss functions and observation models. Here, we demonstrate this extension for categorical behavioral observations, for example, discrete choices or epochs/phases during a task (Fig. 7 ). A similar approach could be used in the future to model other non-Gaussian behaviors and non-Gaussian (for example, Poisson) neural modalities, as shown in a thesis 56 .

To model categorical behaviors, we devise a new behavior observation model for DPAD by making three changes. First, we change the behavior loss (Eqs. ( 5 ) and ( 11 )) to the NLL of a categorical distribution, which we implement using the dedicated class in the TensorFlow library (that is, tf.keras.losses.CategoricalCrossentropy). Second, we change the behavior readout parameter C z to have an output dimension of n z  ×  n c instead of n z , where n c denotes the number of behavior categories or classes. Third, we apply Softmax normalization (Eq. ( 14 )) to the output of the behavior readout parameter C z to ensure that for each of the n z behavior dimensions, the predicted probabilities for all the n c classes add up to 1 so that they represent valid probability mass functions. Softmax normalization can be written as

where \({l}_{k}\in {{\mathbb{R}}}^{{n}_{z}\times {n}_{c}}\) is the output of C z at time k , and the superscript ( m , n ) denotes the element of l k associated with the behavior dimension m and the class/category number n . With these changes, we obtain a new RNN architecture with categorical behavioral outputs. We then learn this new RNN architecture with DPAD’s four-step prioritized optimization approach as before but now incorporating the modified NLL losses for categorical data. Together, with these changes, DPAD extends to modeling categorical behavioral measurements.

Behavior decoding and neural self-prediction metrics and performance frontier

Cross-validation.

To evaluate the learning, we perform a cross-validation with five folds (unless otherwise noted). We cut the data from the recording session into five equal continuous segments, leave these segments out one by one as the test data and train the model only using the data in the remaining segments. Once the model is trained using the neural and behavior training data, we pass the neural test data to the model to get the latent states in the test data using the first line of Eq. ( 1 ) (or Eq. ( 2 ), equivalently). We then pass the extracted latent states to Eq. ( 3 ) to get the one-step-ahead prediction of the behavior and neural test data, which we refer to as behavior decoding and neural self-prediction, respectively. Note that only past neural data are used to get the behavior and neural predictions. Also, the behavior test data are never used in predictions. Given the predicted behavior and neural time series, we compute the CC between each dimension of these time series and the actual behavior and neural test time series. We then take the mean of CC across dimensions of behavior and neural data to get one final cross-validated CC value for behavior decoding and one final CC value for neural self-prediction in each cross-validation fold.

Selection of the latent state dimension

We often need to select a latent state dimension to report an overall behavior decoding and/or neural self-prediction accuracy for each model/method (for example, Figs. 2 – 7 ). By latent state dimension, we always refer to the total latent state dimension of the model, that is, n x . For DPAD, unless otherwise noted, we always used n 1  = 16 to extract the first 16 latent state dimensions (or all latent state dimensions when n x  ≤ 16) using steps 1 and 2 and any remaining dimensions using steps 3 and 4. We chose n 1  = 16 because dedicating more, even all, latent state dimensions to behavior prediction only minimally improved it across datasets and neural modalities. For all methods, to select a state dimension n x , in each cross-validation fold, we fit models with latent state dimensions 1, 2, 4, 16,…and 128 (powers of 2 from 1 to 128) and select one of these models based on their decoding and neural self-prediction accuracies within the training data of that fold. We then report the decoding/self-prediction of this selected model computed in the test data of that fold. Our goal is often to select a model that simultaneously explains behavior and neural data well. For this goal, we pick the state dimension that reaches the peak neural self-prediction in the training data or the state dimension that reaches the peak behavior decoding in the training data, whichever is larger; we then report both the neural self-prediction and the corresponding behavior decoding accuracy of the same model with the selected state dimension in the test data (Figs. 3 – 4 , 6 and 7f , Extended Data Figs. 3 and 4 and Supplementary Figs. 4 – 7 and 9 ). Alternatively, for all methods, when our goal is to find models that solely aim to optimize behavior prediction, we report the cross-validated prediction performances for the smallest state dimension that reaches peak behavior decoding in training data (Figs. 2 , 5 and 7d , Extended Data Fig. 8 and Supplementary Fig. 3 ). We emphasize that in all cases, the reported performances are always computed in the test data of the cross-validation fold, which is not used for any other purpose such as model fitting or selection of the state dimension.

Performance frontier

When comparing a group of alternative models, we use the term ‘performance frontier’ to describe the best performances reached by models that in every comparison with any alternative model are in some sense better than or at least comparable to the alternative model. More precisely, when comparing a group \({\mathcal{M}}\) of models, model \({\mathcal{A}}\in {\mathcal{M}}\) will be described as reaching the best performance frontier when compared to every other model \({\mathcal{B}}{\mathscr{\in }}{\mathcal{M}}\) , \({\mathcal{A}}\) is significantly better than \({\mathcal{B}}\) in behavior decoding or in neural self-prediction or is comparable to \({\mathcal{B}}\) in both. Note that \({\mathcal{A}}\) may be better than some model \({{\mathcal{B}}}_{1}\in {\mathcal{M}}\) in decoding while being better than another model \({{\mathcal{B}}}_{2}\in {\mathcal{M}}\) in self-prediction; nevertheless \({\mathcal{A}}\) will be on the frontier as long as in every comparison one of the following conditions hold: (1) there is at least one measure for which \({\mathcal{A}}\) is more performant and (2) \({\mathcal{A}}\) is at least equally performant in both measures. To avoid exclusion of models from the best performance frontier due to very minimal performance differences, in this analysis, we only declare a difference in performance significant if in addition to resulting in P  ≤ 0.05 in a one-sided signed-rank test there is also at least 1% relative difference in the mean performance measures.

DPAD with flexible nonlinearity: automatic determination of appropriate nonlinearity

Fine-grained control over nonlinearities.

Each parameter in the DPAD model represents an operation in the computation graph of DPAD (Fig. 1b and Supplementary Fig. 1a ). We solve the numerical optimizations involved in model learning in each step of our multistep learning via standard stochastic gradient descent 43 , which remains applicable for any modification of the computation graph that remains acyclic. Thus, the operation associated with each model parameter (for example, A ′, K , C y and C z ) can be replaced with any multilayer neural network with an arbitrary number of hidden units and layers (Supplementary Fig. 1c ), and the model remains trainable with the same approach. Having no hidden layers implements the special case of a linear mapping (Supplementary Fig. 1b ). Of course, given that the training data are finite, the typical trade-off between model capacity and generalization error remains 69 . Given that neural networks can approximate any continuous function (with a compact domain) 98 , replacing model parameters with neural networks should have the capacity to learn any nonlinear function in their place 99 , 100 , 101 . The resulting RNN in Eq. ( 1 ) can in turn approximate any state-space dynamics (under mild conditions) 102 . In this work, for nonlinear parameters, we use multilayer feed-forward networks with one or two hidden layers, each with 64 or 128 units. For all hidden layers, we always use a rectified linear unit (ReLU) nonlinear activation (Supplementary Fig. 1c ). Finally, when making a parameter (for example, C z ) nonlinear, we always do so for that parameter in both sections of the RNN (for example, both \({C}_{z}^{\,\left(1\right)}\) and \({C}_{z}^{\,\left(2\right)}\) ; see Supplementary Fig. 1a ) and using the same feed-forward network structure. Given that no existing RNN implementation allowed individual RNN elements to be independently set to arbitrary multilayer neural networks, we developed a custom TensorFlow RNN cell to implement the RNNs in DPAD (Eqs. ( 4 ) and ( 8 )). We used the Adam optimizer to implement gradient descent for all optimization steps 43 . We continued each optimization for up to 2,500 epochs but stopped earlier if the objective function did not improve in three consecutive epochs (convergence criteria).

Automatic selection of nonlinearity settings

We devise a procedure for automatically determining the most suitable combination of nonlinearities for the data, which we refer to as DPAD with flexible nonlinearity. In this procedure, for each cross-validation fold in each recording session of each dataset, we try a series of nonlinearities within the training data and select one based on an inner cross-validation within the training data (Fig. 1d ). Specifically, we consider the following options for the nonlinearity. First, each of the four main parameters (that is, A ′, K , C y and C z ) can be linear or nonlinear, resulting in 16 cases (that is, 2 4 ). In cases with nonlinearity, we consider four network structures for the parameters, that is, having one or two hidden layers and having 64 or 128 units in each hidden layer (Supplementary Fig. 1c ), resulting in 61 cases (that is, 15 × 4 + 1, where 1 is for the fully linear model) overall. Finally, specifically for the recursion parameter A ′, we also consider modeling it as an LSTM, with the other parameters still having the same nonlinearity options as before, resulting in another 29 cases for when this LSTM recursion is used (that is, 7 × 4 + 1, where 1 is for the case where the other three model parameters are all linear), bringing the total number of considered cases to 90. For each of these 90 considered linear or nonlinear architectures, we perform a twofold inner cross-validation within the training data to compute an estimate of the behavior decoding and neural self-prediction of each architecture using the training data. Note that although this process for automatic selection of nonlinearities is computationally expensive, it is parallelizable because each candidate model can be fitted independently on a different processor. Once all candidate architectures are fitted and evaluated within the training data, we select one final architecture purely based on training data to be used for that cross-validation fold based on one of the following two criteria: (1) decoding focused: pick the architecture with the best neural self-prediction in training data among all those that reach within 1 s.e.m. of the best behavior decoding; or (2) self-prediction focused: pick the architecture with the best behavior decoding in training data among all those that reach within 1 s.e.m. of the best neural self-prediction. The first criterion prioritizes good behavior decoding in the selection, and the second criterion prioritizes good neural self-prediction. Note that these two criteria are used when selecting among different already-learned models with different nonlinearities and thus are independent of the four internal objective functions used in learning the parameters for a given model with the four-step optimization approach (Supplementary Fig. 1a ). For example, in the first optimization step of DPAD, model parameters are always learned to optimize behavior decoding (Eq. ( 5 )). But once the four-step optimization is concluded and different models (with different combinations of nonlinearities) are learned, we can then select among these already-learned models based on either neural self-prediction or behavior decoding. Thus, whenever neural self-prediction is also of interest, we report the results for flexible nonlinearity based on both model selection criteria (for example, Figs. 3 , 4 and 6 ).

Localization of nonlinearities

DPAD enables an inspection of where nonlinearities can be localized to by providing two capabilities, without either of which the origin of nonlinearities may be incorrectly found. As the first capability, DPAD can train alternative models with different individual nonlinearities and then compare these alternative nonlinear models not only with a fully linear model but also with each other and with fully nonlinear models (that is, flexible nonlinearity). Indeed, our simulations showed that simply comparing a linear model to a model with nonlinearity in a given parameter may incorrectly identify the origin of nonlinearity (Extended Data Fig. 2b and Fig. 6a ). For example, in Fig. 6a , although the nonlinearity is just in the neural input parameter, a linear model does worse than a model with a nonlinear behavior readout parameter. Thus, just a comparison of the latter model to a linear model would incorrectly find the origin of nonlinearity to be the behavior readout. This issue is avoided in DPAD because it can also train a model with the neural input being nonlinear, thus finding it to be more predictive than models with any other individual nonlinearity and as predictive as a fully nonlinear model (Fig. 6a ). As the second capability, DPAD can compare alternative nonlinear models in terms of overall neural–behavioral prediction rather than either behavior decoding or neural prediction alone. Indeed, our simulations showed that comparing the models in terms of just behavior decoding (Extended Data Fig. 2d,f ) or just neural self-prediction (Extended Data Fig. 2d,h ) may lead to incorrect conclusions about the origin of nonlinearities; this is because a model with the incorrect origin may be equivalent in one of these metrics to the one with the correct origin. DPAD avoids this problem by jointly evaluating both neural–behavioral metrics. Here, when comparing models with nonlinearity in different individual parameters for localization purposes (for example, Fig. 6 ), we only consider one network architecture for the nonlinearity, that is, having one hidden layer with 64 units.

Numerical simulations

To validate DPAD in numerical simulations, we perform two sets of simulations. One set validates linear modeling to show the correctness of the four-step numerical optimization for learning. The other set validates nonlinear modeling. In the linear simulation, we randomly generate 100 linear models with various dimensionality and noise statistics, as described in our prior work 6 . Briefly, the neural and behavior dimensions are selected from 5 ≤  n y , n z  ≤ 10 randomly with uniform probability. The state dimension is selected as n x  = 16, of which n 1  = 4 latent state dimensions are selected to drive behavior. Eigenvalues of the state transition matrix are selected randomly as complex conjugate pairs with uniform probability within the unit disk. Each element in the behavior and neural readout matrices is generated as a random Gaussian variable. State and neural observation noise covariances are generated as random positive definite matrices and scaled randomly with a number between 0.003 and 0.3 or between 0.01 and 100, respectively, to obtain a wide range of relative noises across random models. A separate random linear state-space model with four latent state dimensions is generated to produce the behavior readout noise 𝜖 k , representing the behavior dynamics that are not encoded in the recorded neural activity. Finally, the behavior readout matrix is scaled to set the ratio of the signal standard deviation to noise standard deviation in each behavior dimension to a random number from 0.5 to 50. We perform model learning and evaluation with twofold cross-validation (Extended Data Fig. 1 ).

In the nonlinear simulations that are used to validate both DPAD and the hypothesis testing procedure it enables to find the origin of nonlinearity, we start by generating 20 random linear models ( n y  =  n z  = 1) either with n x  =  n z  =  n y (Extended Data Fig. 2 ) or n x  = 2 latent states, only one of which drives behavior (Supplementary Fig. 2 ). We then introduce nonlinearity in one of the four model parameters (that is, A ′, K , C y or C z ) by replacing that parameter with a nonlinear trigonometric function, such that roughly one period of the trigonometric function is visited by the model (while keeping the rest of the parameters linear). To do this, we first scale each latent state in the initial random linear model to find a similarity transform for it where the latent state has a 95% confidence interval range of 2 π . We then add a sine function to the original parameter that is to be changed to nonlinear and scale the amplitude of the sine such that its output reaches roughly 0.25 of the range of the outputs from the original linear parameter. This was done to reduce the chance of generating unrealistic unstable nonlinear models that produce outputs with infinite energy, which is likely when A ′ is nonlinear. Changing one parameter to nonlinear can change the range of the statistics of the latent states in the model; thus, we generate some simulated data from the model and redo the scaling of the nonlinearity until ratio conditions are met.

To generate data from any nonlinear model in Eq. ( 1 ), we first generate a neural noise time series e k based on its covariance ∑ e in the model and initialize the state as x 0  = 0. We then iteratively apply the second and first lines of Eq. ( 1 ) to get the simulated neural activity y k from line 2 and then the next state \({x}_{k+1}\) from line 1, respectively. Finally, once the state time series is produced, we generate a behavior noise time series 𝜖 k based on its covariance ∑ 𝜖 in the model and apply the third line of Eq. ( 1 ) to get the simulated behavior z k . Similar to linear simulations, we perform the modeling and evaluation of nonlinear simulations with twofold cross-validation (Extended Data Fig. 2 and Supplementary Fig. 2 ).

Neural datasets and behavioral tasks

We evaluate DPAD in five datasets with different behavioral tasks, brain regions and neural recording modalities to show the generality of our conclusions. For each dataset, all animal procedures were performed in compliance with the National Research Council Guide for Care and Use of Laboratory Animals and were approved by the Institutional Animal Care and Use Committee at the respective institution, namely New York University (datasets 1 and 2) 6 , 45 , 46 , Northwestern University (datasets 3 and 5) 47 , 48 , 54 and University of California San Francisco (dataset 4) 21 , 49 .

Across all four main datasets (datasets 1 to 4), the spiking activity was binned with 10-ms nonoverlapping bins, smoothed with a Gaussian kernel with standard deviation of 50 ms (refs. 6 , 14 , 34 , 103 , 104 ) and downsampled to 50 ms to be used as the neural signal to be modeled. The behavior time series was also downsampled to a matching 50 ms before modeling. In the three datasets where LFP activity was also available, we also studied two types of features extracted from LFP. As the first LFP feature, we considered raw LFP activity itself, which was high-pass filtered above 0.5 Hz to remove the baseline, low-pass filtered below 10 Hz (that is, antialiasing) and downsampled to the behavior sampling rate of a 50-ms time step (that is, 20 Hz). Note that in the context of the motor cortex, low-pass-filtered raw LFP is also referred to as the local motor potential 50 , 51 , 52 , 105 , 106 and has been used to decode behavior 6 , 50 , 51 , 52 , 53 , 105 , 106 , 107 . As the second feature, we computed the LFP log-powers 5 , 6 , 7 , 40 , 77 , 79 , 106 , 108 , 109 in eight standard frequency bands (delta: 0.1–4 Hz, theta: 4–8 Hz, alpha: 8–12 Hz, low beta: 12–24 Hz, mid-beta: 24–34 Hz, high beta: 34–55 Hz, low gamma: 65–95 Hz and high gamma: 130–170 Hz) in sliding 300-ms windows at a time step of 50 ms using Welch’s method (using eight subwindows with 50% overlap) 6 . The median analyzed data length for each session across the datasets ranged from 4.6 to 9.9 min.

First dataset: 3D reaches to random targets

In the first dataset, the animal (named J) performed reaches to a target randomly positioned in 3D space within the reach of the animal, grasped the target and returned its hand to resting position 6 , 45 . Kinematic data were acquired using the Cortex software package (version 5.3) to track retroreflective markers in 3D (Motion Analysis) 6 , 45 . Joint angles were solved from the 3D marker data using a Rhesus macaque musculoskeletal model via the SIMM toolkit (version 4.0, MusculoGraphics) 6 , 45 . Angles of 27 joints in the shoulder, elbow, wrist and fingers in the active hand (right hand) were taken as the behavior signal 6 , 45 . Neural activity was recorded with a 137-electrode microdrive (Gray Matter Research), of which 28 electrodes were in the contralateral primary motor cortex M1. The multiunit spiking activity in these M1 electrodes was used as the neural signal. For LFP analyses, LFP features were also extracted from the same M1 electrodes. We analyzed the data from seven recording sessions.

To visualize the low-dimensional latent state trajectories for each behavioral condition (Extended Data Fig. 6 ), we determined the periods of reach and return movements in the data (Fig. 7a ), resampled them to have similar number of time samples and averaged the latent states across those resampled trials. Given the redundancy in latent descriptions (that is, any scaling, rotation and so on on the latent states still gives an equivalent model), before averaging trials across cross-validation folds and sessions, we devised the following procedure to standardize the latent states for each fold in the case of 2D latent states (Extended Data Fig. 6 ). (1) We z score all state dimensions to have zero mean and unit variance. (2) We rotate the 2D latent states such that the average 2D state trajectory for the first condition (here, the reach epochs) starts from an angle of 0. (3) We estimate the direction of the rotation for the average 2D state trajectory of the first condition, and if it is not counterclockwise, we multiply the second state dimension by –1 to make it so. Note that in each step, the same mapping is applied to the latent states during the whole test data, regardless of condition, so this procedure does not alter the relative differences in the state trajectory across different conditions. The procedure also does not change the learned model and simply corresponds to a similarity transform that changes the basis of the model. This procedure only removes the redundancies for describing a 2D latent state-space model and standardizes the extracted latent states so that trials across different test sets can be averaged together.

Second dataset: saccadic eye movements

In the second dataset, the animal (named A) performed saccadic eye movements to one of eight targets on a display 6 , 46 . The visual stimuli in the task with saccadic eye movements were controlled via custom LabVIEW (version 9.0, National Instruments) software executed on a real-time embedded system (NI PXI-8184, National Instruments) 46 . The 2D position of the eye was tracked and taken as the behavior signal. Neural activity was recorded with a 32-electrode microdrive (Gray Matter Research) covering the prefrontal cortex 6 , 46 . Single-unit activity from these electrodes, ranging from 34 to 43 units across different recording sessions, was used as the neural signal. For LFP analyses, LFP features were also extracted from the same 32 electrodes. We analyzed the data from the first 7 days of recordings. We only included data from successful trials where the animal performed the task correctly by making a saccadic eye movement to the specified target. To visualize the low-dimensional latent state trajectories for each behavioral condition (Extended Data Fig. 6 ), we grouped the trials based on their target position. Standardization across folds before averaging was done as in the first dataset.

Third dataset: sequential reaches with a 2D cursor controlled with a manipulandum

In the third dataset, which was collected and made publicly available by the laboratory of L. E. Miller 47 , 48 , the animal (named T) controlled a cursor on a 2D screen using a manipulandum and performed a sequential reach task 47 , 48 . The 2D cursor position and velocity were taken as the behavior signal. Neural activity was recorded using a 100-electrode microelectrode array (Blackrock Microsystems) in the dorsal premotor cortex 47 , 48 . Single-unit activity, recorded from 37 to 49 units across recording sessions, was used as the neural signal. This dataset did not include any LFP recordings, so LFP features could not be considered. We analyzed the data from all three recording sessions. To visualize the low-dimensional latent state trajectories for each behavioral condition (Extended Data Fig. 6 ), we grouped the trials into eight different conditions based on the angle of the direction of movement (that is, end position minus starting position) during the trial, with each condition covering movement directions within a 45° (that is, 360/8) range. Standardization across folds before averaging was performed as in the first dataset.

Fourth dataset: virtual reality random reaches with a 2D cursor controlled with the fingertip

In the fourth dataset, which was collected and made publicly available by the laboratory of P. N. Sabes 49 , the animal (named I) controlled a cursor based on the fingertip position on a 2D surface within a 3D virtual reality environment 21 , 49 . The 2D cursor position and velocity were taken as the behavior signal. Neural activity was recorded with a 96-electrode microelectrode array (Blackrock Microsystems) 21 , 49 covering M1. We selected a random subset of 32 of these electrodes, which had 77 to 99 single units across the recording sessions, as the neural signal. LFP features were also extracted from the same 32 electrodes. We analyzed the data for the first seven sessions for which the wideband activity was also available (sessions 20160622/01 to 20160921/01). Grouping into conditions for visualization of low-dimensional latent state trajectories (Extended Data Fig. 6 ) was done as in the third dataset. Standardization across folds before averaging was done as in the first dataset.

Fifth dataset: center-out cursor control reaching task

In the fifth dataset, which was collected and made publicly available by the laboratory of L. E. Miller 54 , the animal (named H) controlled a cursor on a 2D screen using a manipulandum and performed reaches from a center point to one of eight peripheral targets (Fig. 4i ). The 2D cursor position was taken as the behavior signal. Neural activity was recorded with a 96-electrode microelectrode array (Blackrock Microsystems) covering area 2 of the somatosensory cortex 54 . Preprocessing for this dataset was done as in ref. 36 . Specifically, the spiking activity was binned with 1-ms nonoverlapping bins and smoothed with a Gaussian kernel with a standard deviation of 40 ms (ref. 110 ), with the behavior also being sampled with the same 1-ms sampling rate. Trials were also aligned as in the same prior work 110 with data from –100 to 500 ms around movement onset of each trial being used for modeling 36 .

Additional details for baseline methods

For the fifth dataset, which has been analyzed in ref. 36 and introduces CEBRA, we used the exact same CEBRA hyperparameters as those reported in ref. 36 (Fig. 4i,j ). For each of the other four datasets (Fig. 4a–h ), when learning a CEBRA-Behavior or CEBRA-Time model for each session, fold and latent dimension, we also performed an extensive search over CEBRA hyperparameters and picked the best value with the same inner cross-validation approach as we use for the automatic selection of nonlinearities in DPAD. We considered 30 different sets of hyperparameters: 3 options for the ‘time-offset’ hyperparameter (1, 2 or 10) and 10 options for the ‘temperature’ hyperparameter (from 0.0001 to 0.01), which were designed to include all sets of hyperparameters reported for primate data in ref. 36 . We swept the CEBRA latent dimension over the same values as DPAD, that is, powers of 2 up to 128. In all cases, we used a k -nearest neighbors regression to map the CEBRA-extracted latent embeddings to behavior and neural data as done in ref. 36 because CEBRA itself does not learn a reconstruction model 36 (Extended Data Table 1 ).

It is important to note that CEBRA and DPAD have fundamentally different architectures and goals (Extended Data Table 1 ). CEBRA uses a small ten-sample window (when ‘model_architecture’ is ‘offset10-model’) around each datapoint to extract a latent embedding via a series of convolutions. By contrast, DPAD learns a dynamical model that recursively aggregates all past neural data to extract an embedding. Also, in contrast to CEBRA-Behavior, DPAD’s embedding includes and dissociates both behaviorally relevant neural dimensions and other neural dimensions to predict not only the behavior but also the neural data well. Finally, CEBRA does not automatically map its latent embeddings back to neural data or to behavior during learning but does so post hoc, whereas DPAD learns these mappings for all its latent states. Given these differences, several use-cases of DPAD are not targeted by CEBRA, including explicit dynamical modeling of neural–behavioral data (use-case 1), flexible nonlinearity, hypothesis testing regarding the origin of nonlinearity (use-case 4) and forecasting.

We used the Wilcoxon signed-rank test for all paired statistical tests.

Reporting summary

Further information on research design is available in the Nature Portfolio Reporting Summary linked to this article.

Data availability

Three of the datasets used in this work are publicly available 47 , 48 , 49 , 54 . The other two datasets used to support the results are available upon reasonable request from the corresponding author. Source data are provided with this paper.

Code availability

The code for DPAD is available at https://github.com/ShanechiLab/DPAD .

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Acknowledgements

This work was supported, in part, by the following organizations and grants: the Office of Naval Research (ONR) Young Investigator Program under contract N00014-19-1-2128, National Institutes of Health (NIH) Director’s New Innovator Award DP2-MH126378, NIH R01MH123770, NIH BRAIN Initiative R61MH135407 and the Army Research Office (ARO) under contract W911NF-16-1-0368 as part of the collaboration between the US DOD, the UK MOD and the UK Engineering and Physical Research Council (EPSRC) under the Multidisciplinary University Research Initiative (MURI) (to M.M.S.) and a University of Southern California Annenberg Fellowship (to O.G.S.).

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Ming Hsieh Department of Electrical and Computer Engineering, Viterbi School of Engineering, University of Southern California, Los Angeles, CA, USA

Omid G. Sani & Maryam M. Shanechi

Perelman School of Medicine, University of Pennsylvania, Philadelphia, PA, USA

Bijan Pesaran

Thomas Lord Department of Computer Science, University of Southern California, Los Angeles, CA, USA

Maryam M. Shanechi

Neuroscience Graduate Program, University of Southern California, Los Angeles, CA, USA

Alfred E. Mann Department of Biomedical Engineering, University of Southern California, Los Angeles, CA, USA

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Contributions

O.G.S. and M.M.S. conceived the study, developed the DPAD algorithm and wrote the manuscript, and O.G.S. performed all the analyses. B.P. designed and performed the experiments for two of the NHP datasets and provided feedback on the manuscript. M.M.S. supervised the work.

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Correspondence to Maryam M. Shanechi .

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University of Southern California has a patent related to modeling and decoding of shared dynamics between signals in which M.M.S. and O.G.S. are inventors. The other author declares no competing interests.

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Extended data

Extended data fig. 1 dpad dissociates and prioritizes the behaviorally relevant neural dynamics while also learning the other neural dynamics in numerical simulations of linear models..

a , Example data generated from one of 100 random models ( Methods ). These random models do not emulate real data but for terminological consistency, we still refer to the primary signal (that is, y k in Eq. ( 1 )) as the ‘neural activity’ and to the secondary signal (that is, z k in Eq. ( 1 )) as the ‘behavior’. b , Cross-validated behavior decoding accuracy (correlation coefficient, CC) for each method as a function of the number of training samples when we use a state dimension equal to the total state dimension of the true model. The performance measures for each random model are normalized by their ideal values that were achieved by the true model itself. Performance for the true model is shown in black. Solid lines and shaded areas are defined as in Fig. 5b ( N  = 100 random models). c , Same as b but when learned models have low-dimensional latent states with enough dimensions just for the behaviorally relevant latent states (that is, n x  =  n 1 ). d - e , Same as b - c showing the cross-validated normalized neural self-prediction accuracy. Linear NDM, which learns the parameters using a numerical optimization, performs similarly to a linear algebraic subspace-based implementation of linear NDM 67 , thus validating NDM’s numerical optimization implementation. Linear DPAD, just like PSID 6 , achieves almost ideal behavior decoding even with low-dimensional latent states ( c ); this shows that DPAD correctly dissociates and prioritizes behaviorally relevant dynamics, as opposed to aiming to simply explain the most neural variance as non-prioritized methods such as NDM do. For this reason, with a low-dimensional state, non-prioritized NDM methods can explain neural activity well ( e ) but prioritized methods can explain behavior much better ( c ). Nevertheless, using the second stage of PSID and the last two optimization steps in DPAD, these two prioritized techniques are still able to learn the overall neural dynamics accurately if state dimension is high enough ( d ). Overall, the performance of linear DPAD and PSID 6 are similar for the special case of linear modeling.

Extended Data Fig. 2 DPAD successfully identifies the origin of nonlinearity and learns it in numerical simulations.

DPAD can perform hypothesis testing regarding the origin of nonlinearity by considering both behavior decoding (vertical axis) and neural self-prediction (horizontal axis). a , True value for nonlinear neural input parameter K in an example random model with nonlinearity only in K and the nonlinear value that DPAD learned for this parameter when only K in the learned model was set to be nonlinear. The true and learned mappings match and almost exactly overlap. b , Behavior decoding and neural self-prediction accuracy achieved by DPAD models with different locations of nonlinearities. These accuracies are for data generated from 20 random models that only had nonlinearity in the neural input parameter K . Performance measures for each random model are normalized by their ideal values that were achieved by the true model itself. Pluses and whiskers are defined as in Fig. 3 ( N  = 20 random models). c , d , Same as a , b for data simulated from models that only have nonlinearity in the recursion parameter A ′. e - f , Same as a , b for data simulated from models that only have nonlinearity in the neural readout parameter C y . g , h , Same as a , b for data simulated from models that only have nonlinearity in the behavior readout parameter C z . In each case ( b , d , f , h ), the nonlinearity option that reaches closest to the upper-rightmost corner of the plot, that is, has both the best behavior decoding and the best neural self-prediction, is chosen as the model that specifies the origin of nonlinearity. Regardless of the true location of nonlinearity ( b , d , f , h ), always the correct location (for example, K in b ) achieves the best performance overall compared with all other locations of nonlinearities. These results provide evidence that by fitting and comparing DPAD models with different nonlinearities, we can correctly find the origin of nonlinearity in simulated data.

Extended Data Fig. 3 Across spiking and LFP neural modalities, DPAD is on the best performance frontier for neural-behavioral prediction unlike LSTMs, which are fitted to explain neural data or behavioral data.

a , The 3D reach task. b , Cross-validated neural self-prediction accuracy achieved by each method versus the corresponding behavior decoding accuracy on the vertical axis. Latent state dimension for each method in each session and fold is chosen (among powers of 2 up to 128) as the smallest that reaches peak neural self-prediction in training data or reaches peak decoding in training data, whichever is larger ( Methods ). Pluses and whiskers are defined as in Fig. 3 ( N  = 35 session-folds). Note that DPAD considers an LSTM as a special case ( Methods ). Nevertheless, results are also shown for LSTM networks fitted to decode behavior from neural activity (that is, RNN decoders in Extended Data Table 1 ) or to predict the next time step of neural activity (self-prediction). Also, note that LSTM for behavior decoding (denoted by H) and DPAD when only using the first two optimization steps (denoted by G) dedicate all their latent states to behavior prediction, whereas other methods dedicate some or all latent states to neural self-prediction. Compared with all methods including these LSTM networks, DPAD always reaches the best performance frontier for predicting the neural-behavioral data whereas LSTM does not; this is partly due to the four-step optimization algorithm in DPAD that allows for overall neural-behavioral description rather than one or the other, and that prioritizes the learning of the behaviorally relevant neural dynamics. c , Same as b for raw LFP activity ( N  = 35 session-folds). d , Same as b for LFP band power activity ( N  = 35 session-folds). e - h , Same as a - d for the second dataset, with saccadic eye movements ( N  = 35 session-folds). i , j , Same as a and b for the third dataset, with sequential cursor reaches controlled via a 2D manipulandum ( N  = 15 session-folds). k - n , Same as a - d for the fourth dataset, with random grid virtual reality cursor reaches controlled via fingertip position ( N  = 35 session-folds). Results and conclusions are consistent across all datasets.

Extended Data Fig. 4 DPAD can also be used for multi-step-ahead forecasting of behavior.

a , The 3D reach task. b , Cross-validated behavior decoding accuracy for various numbers of steps into the future. For m -step-ahead prediction, behavior at time step k is predicted using neural activity up to time step k − m . All models are taken from Fig. 3 , without any retraining or finetuning, with m -step-ahead forecasting done by repeatedly ( m −1 times) passing the neural predictions of the model as its neural observation in the next time step ( Methods ). Solid lines and shaded areas are defined as in Fig. 5b ( N  = 35 session-folds). Across the number of steps ahead, the statistical significance of a one-sided pairwise comparison between nonlinear DPAD vs nonlinear NDM is shown with the orange top horizontal line with p-value indicated by asterisks next to the line as defined in Fig. 2b (N = 35 session-folds). Similar pairwise comparison between nonlinear DPAD vs linear dynamical system (LDS) modeling is shown with the purple top horizontal line. c - d , Same as a - b for the second dataset, with saccadic eye movements ( N  = session-folds). e - f , Same as a - b for the third dataset, with sequential cursor reaches controlled via a 2D manipulandum ( N  = 15 session-folds). g - h , Same as a - b for the fourth dataset, with random grid virtual reality cursor reaches controlled via fingertip position ( N  = 35 session-folds).

Extended Data Fig. 5 Neural self-prediction accuracy of nonlinear DPAD across recording electrodes for low-dimensional behaviorally relevant latent states.

a , The 3D reach task. b , Average neural self-prediction correlation coefficient (CC) achieved by nonlinear DPAD for analyzed smoothed spiking activity is shown for each recording electrode ( N  = 35 session-folds; best nonlinearity for decoding). c , Same as b for modeling of raw LFP activity. d , Same as b for modeling of LFP band power activity. Here, prediction accuracy averaged across all 8 band powers ( Methods ) of a given recording electrode is shown for that electrode. e-h , Same a - d for the second dataset, with saccadic eye movements ( N  = 35 session-folds). For datasets with single-unit activity ( Methods ), spiking self-prediction of each electrode is averaged across the units associated with that electrode. i - j , Same as a , b for the third dataset, with sequential cursor reaches controlled via a 2D manipulandum ( N  = 15 session-folds). White areas are due to electrodes that did not have a neuron associated with them in the data. k - n , Same as a - d for the fourth dataset, with random grid virtual reality cursor reaches controlled via fingertip position ( N  = 35 session-folds). For all results, the latent state dimension was 16, and all these dimensions were learned using the first optimization step (that is, n 1  = 16).

Extended Data Fig. 6 Nonlinear DPAD extracted distinct low dimensional latent states from neural activity for all datasets, which were more behaviorally relevant than those extracted using nonlinear NDM.

a , The 3D reach task. b , The latent state trajectory for 2D states extracted from spiking activity using nonlinear DPAD, averaged across all reach and return epochs across sessions and folds. Here only optimization steps 1-2 of DPAD are used to just extract 2D behaviorally relevant states. c , Same as b for 2D states extracted using nonlinear NDM (special case of using just DPAD optimization steps 3-4). d , Saccadic eye movement task. Trials are averaged depending on the eye movement direction. e , The latent state trajectory for 2D states extracted using DPAD (extracted using optimizations steps 1-2), averaged across all trials of the same movement direction condition across sessions and folds. f , Same as d for 2D states extracted using nonlinear NDM. g-i , Same as d - f for the third dataset, with sequential cursor reaches controlled via a 2D manipulandum. j - l , Same as d - f for the fourth dataset, with random grid virtual reality cursor reaches controlled via fingertip position. Overall, in each dataset, latent states extracted by DPAD were clearly different for different behavior conditions in that dataset ( b , e , h , k ), whereas NDM’s extracted latent states did not as clearly dissociate different conditions ( c , f , i , l ). Of note, in the first dataset, DPAD revealed latent states with rotational dynamics that reversed direction during reach versus return epochs, which is consistent with the behavior roughly reversing direction. In contrast, NDM’s latent states showed rotational dynamics that did not reverse direction, thus were less congruent with behavior. In this first dataset, in our earlier work 6 , we had compared PSID and a subspace-based linear NDM method and, similar to b and c here, had found that only PSID uncovers reverse-directional rotational patterns across reach and return movement conditions. These results thus also complement our prior work 6 by showing that even nonlinear NDM models may not uncover the distinct reverse-directional dynamics in this dataset, thus highlighting the need for dissociative and prioritized learning even in nonlinear modeling, as enabled by DPAD.

Extended Data Fig. 7 Neural self-prediction across latent state dimensions.

a , The 3D reach task. b , Cross-validated neural self-prediction accuracy (CC) achieved by variations of nonlinear and linear DPAD/NDM, for different latent state dimensions. Solid lines and shaded areas are defined as in Fig. 5b ( N  = 35 session-folds). Across latent state dimensions, the statistical significance of a one-sided pairwise comparison between nonlinear DPAD/NDM (with best nonlinearity for self-prediction) vs linear DPAD/NDM is shown with a horizontal green/orange line with p-value indicated by asterisks next to the line as defined in Fig. 2b ( N  = 35 session-folds). c , d , Same as a , b for the second dataset, with saccadic eye movements ( N  = 35 session-folds). e , f , Same as a , b for the third dataset, with sequential cursor reaches controlled via a 2D manipulandum ( N  = 15 session-folds). g , h Same as a , b for the fourth dataset, with random grid virtual reality cursor reaches controlled via fingertip position ( N  = 35 session-folds). For all DPAD variations, the first 16 latent state dimensions are learned using the first two optimization steps and the remaining dimensions are learned using the last two optimization steps (that is, n 1  = 16). As expected, at low state dimensions, DPAD’s latent states achieve higher behavior decoding (Fig. 5 ) but lower neural self-prediction than NDM because DPAD prioritizes the behaviorally relevant neural dynamics in these dimensions. However, by increasing the state dimension and utilizing optimization steps 3-4, DPAD can reach similar neural self-prediction to NDM while doing better in terms of behavior decoding (Fig. 3 ). Also, for low dimensional latent states, nonlinear DPAD/NDM consistently result in significantly more accurate neural self-prediction than linear DPAD/NDM. For high enough state dimensions, linear DPAD/NDM eventually reach similar neural self-prediction accuracy to nonlinear DPAD/NDM. Given that NDM solely aims to optimize neural self-prediction (irrespective of the relevance of neural dynamics to behavior), the latter result suggests that the overall neural dynamics can be approximated with linear dynamical models but only with high-dimensional latent states. Note that in contrast to neural self-prediction, behavior decoding of nonlinear DPAD is higher than linear DPAD even at high state dimensions (Fig. 3 ).

Extended Data Fig. 8 DPAD accurately learns the mapping from neural activity to behavior dynamics in all datasets even if behavioral samples are intermittently available in the training data.

Nonlinear DPAD can perform accurately and better than linear DPAD even when as little as 20% of training behavior samples are kept. a , The 3D reach task. b , Examples are shown from one of the joints in the original behavior time series (light gray) and intermittently subsampled versions of it (cyan) where a subset of the time samples of the behavior time series are randomly chosen to be kept for use in training. In each subsampling, all dimensions of the behavior data are sampled together at the same time steps; this means that at any given time step, either all behavior dimensions are kept or all are dropped to emulate the realistic case with intermittent measurements. c , Cross-validated behavior decoding accuracy (CC) achieved by linear DPAD and by nonlinear DPAD with nonlinearity in the behavior readout parameter C z . For this nonlinear DPAD, we show the CC when trained with different percentage of behavior samples kept (that is, we emulate different rates of intermittent sampling). The state dimension in each session and fold is chosen (among powers of 2 up to 128) as the smallest that reaches peak decoding in training data. Bars, whiskers, dots, and asterisks are defined as in Fig. 2b ( N  = 35 session-folds). d , e , Same as a , c for the second dataset, with saccadic eye movements ( N  = 35 session-folds). f , g , Same as a , c for the third dataset, with sequential cursor reaches controlled via a 2D manipulandum ( N  = 15 session-folds). h , i , Same as a , c for the fourth dataset, with random grid virtual reality cursor reaches controlled via fingertip position ( N  = 35 session-folds). For all DPAD variations, the first 16 latent state dimensions are learned using the first two optimization steps and the remaining dimensions are learned using the last two optimization steps (that is, n 1  = 16).

Extended Data Fig. 9 Simulations suggest that DPAD may be applicable with sparse sampling of behavior, for example with behavior being a self-reported mood survey value collected once per day.

a , We simulated the application of decoding self-reported mood variations from neural signals 40 , 41 . Neural data is simulated based on linear models fitted to intracranial neural data recorded from epilepsy subjects. Each recorded region in each subject is simulated as a linear state-space model with a 3-dimensional latent state, with the same parameters as those fitted to neural recordings from that region. Simulated latent states from a subset of regions were linearly combined to generate a simulated mood signal (that is, biomarker). As the simulated models were linear, we used the linear versions of DPAD and NDM (NDM used the subspace identification method that we found does similarly to numerical optimization for linear models in Extended Data Fig. 1 ). We generated the equivalent of 3 weeks of intracranial recordings, which is on the order the time-duration of the real intracranial recordings. We then subsampled the simulated mood signal (behavior) to emulate intermittent behavioral measures such as mood surveys. b , Behavior decoding results in unseen simulated test data, across N  = 87 simulated models, for different sampling rates of behavior in the training data. Box edges show the 25 th and 75 th percentiles, solid horizontal lines show the median, whiskers show the range of data, and dots show all data points ( N  = 87 simulated models). Asterisks are defined as in Fig. 2b . DPAD consistently outperformed NDM regardless of how sparse behavior measures were, even when these measures were available just once per day ( P  < 0.0005, one-sided signed-rank, N  = 87).

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Supplementary Figs. 1–9 and Notes 1–4.

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Sani, O.G., Pesaran, B. & Shanechi, M.M. Dissociative and prioritized modeling of behaviorally relevant neural dynamics using recurrent neural networks. Nat Neurosci (2024). https://doi.org/10.1038/s41593-024-01731-2

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Received : 22 April 2023

Accepted : 17 July 2024

Published : 06 September 2024

DOI : https://doi.org/10.1038/s41593-024-01731-2

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