<p>Given your background and interest in modeling, I would highly recommend <a href="http://rads.stackoverflow.com/amzn/click/0521674107">The Cambridge Handbook of Computational Psychology</a>. The book provides an overview for several of the prominent modeling paradigms in cogsci, including dynamical systems, as well as many concrete examples--albeit most using other computational paradigms.</p>

<p>Dynamical systems, to my knowledge, are most used in the field of motor control. Though to be honest, dyn systems is not 'my' paradigm, and most of my knowledge comes from the aforementioned book.</p>

<p>In particular, the chapter "<a href="http://fog.its.uiowa.edu/~icdls/dft-school/documents/DFT_Publications/cambridge_chapter_schoner.pdf">Dynamical Systems Approaches to Cognition</a>" by 
G. Schoner gives this example that has always stuck in my mind:</p>

<blockquote>
  <p>A highly illustrative example comes from the orientation behaviors of
  the common house ﬂy (Reichardt &amp; Poggio, 1976; Poggio &amp; Reichardt,
  1976). Flies orient toward moving objects, which they chase as part of
  their mating behavior. Detailed analysis revealed that the circuitry
  underlying this behavior forms a simple controller: a motion detection
  system fed by luminance changes on the ﬂy’s facet eye drives the ﬂight
  motor, generating an amount of torque that is a function of where on
  the sensory surface motion was detected. If the speck of motion is
  detected on the right, a torque to the right is generated. If the
  speck is detected on the left, a torque to the left is generated. The
  level of torque passes through zero when the speck is right ahead. The
  torque changes the ﬂight direction of the ﬂy, which in turn changes
  the location on the facet eye at which the moving stimulus is
  detected. Given the aerodynamics of ﬂies, the torque and its on-line
  updating generate an orientation behavior, in which the insect orients
  its ﬂight into the direction in which a moving stimulus is detected.</p>
</blockquote>

CC BY-SA 3.0
2012-02-06T00:18:44.730Z
2012-02-06T00:18:44.730Z
Jeff
<p>Great to hear you're interested in applying your quantitative background to cognitive science; the field can definitely use more individuals like you!</p>

<p>I'm not sure to what degree these models might be meet your definition of "dynamical systems", but here are some off the top of my head:</p>

<ul>
<li>stochastic models of human response time and accuracy in 2-alternative speeded choice experiments. <a href="http://www.newcl.org/publications/DonkinCrossFitsRevision2.pdf">Here's a recent paper</a> that does well to summarize two dominant models in this field.</li>
<li><a href="http://www.indiana.edu/~clcl/Papers/Holowords_BRM.pdf">Holographic models of semantic memory</a> (includes dynamic encoding and retrieval processes)</li>
<li>Jeff Hawkins' (yes, that Jeff Hawkins) very cool Hierarchical Temporal Memory Model (<a href="http://www.numenta.com/htm-overview/education.php">lots of info here</a>)</li>
</ul>

CC BY-SA 3.0
2012-02-06T02:39:16.563Z
2012-02-06T02:39:16.563Z
Mike Lawrence
<p>From my social psychology perspective, there has been some computational modelling work on things like attitudinal influence dynamics. See:
<a href="http://cognitrn.psych.indiana.edu/rgoldsto/complex/nowak90.pdf">Nowak, A., Szamrej, J., &amp; Latané, B. (1990). From private attitude to public opinion: a dynamic theory of social impact. <em>Psychological Review, 97</em>, 362-476.</a></p>

<p>Robin Vallacher and Andrzej Nowak crop up a lot in this field, and they're a joy to read. Perhaps this is less up your alley, since social psychologists are less interested in formulating differential equations and more interested in just applying the <em>ideas</em> of complexity science to their research methods, but it's certainly a step forward from what's been going on in the field so far. If anyone's interested in those broad concepts, try reading:
<a href="http://psr.sagepub.com/content/8/2/183.full.pdf+html">Nowak, A. (2004). Dynamical Minimalism: Why Less is More in Psychology. <em>Personality and Social Psychology Review, 8</em>, 183-192.</a></p>

CC BY-SA 3.0
2012-06-26T09:17:48.540Z
2012-06-26T09:17:48.540Z
adb
<p>I have a similar background to you, and have found a lot of interesting things in <a href="http://en.wikipedia.org/wiki/Evolutionary_game_theory" rel="nofollow noreferrer">evolutionary game theory</a> (you can follow <a href="http://egtheory.wordpress.com/" rel="nofollow noreferrer">links</a> from my profile for more). But on the specific content of your question: I have come across to uses of dynamic systems on the opposite ends of cognition. Beer's work on modeling minimal cognition, and Busemeyer &amp; Townsend's work on human decision making.</p>
<h3>Minimal cognition</h3>
<p>Beer's work models simple cognitive agents as dynamic recurrent neural networks. He then either hand-builds the agents, or evolves them, and analyzes them from a <a href="http://en.wikipedia.org/wiki/Situated_cognition" rel="nofollow noreferrer">situated cognition</a> perspective by treating them as dynamic systems.</p>
<ul>
<li><p>Beer, R. D. (1995). On the Dynamics of Small Continuous-Time Recurrent Neural Networks. Adaptive Behavior, 3(4), 469-509. doi:10.1177/105971239500300405</p>
</li>
<li><p>Beer, R. D. (2003). The Dynamics of Active Categorical Perception in an Evolved Model Agent. Adaptive Behavior, 11(4), 209-243. doi:10.1177/1059712303114001</p>
</li>
</ul>
<p>Minimal cognition research tends to use dynamic systems much more heavily than other fields I am familiar with. <a href="http://minimalcognition.wordpress.com/about/" rel="nofollow noreferrer">This blog</a> provides a very thorough survey of papers in the field. A short 'manifesto':</p>
<ul>
<li>Harvey, I. (1997). <a href="http://minimalcognition.wordpress.com/2012/06/19/cognition-is-not-computation-evolution-is-not-optimisation/" rel="nofollow noreferrer">Cognition is not Computation: Evolution is not Optimisation</a>. In Artificial Neural Networks – ICANN97, Gerstner, W., Germond, A., Hasler, M, and Nicoud, J-D (eds). Springer-Verlag LNCS 1327: 685-690. [<a href="http://www.sussex.ac.uk/Users/inmanh/CogCompEvOp.pdf" rel="nofollow noreferrer">pdf</a>]</li>
</ul>
<h3>Human decision making</h3>
<p>Busemeyer &amp; Towsend on the other hand try to describe how humans deliberate and make decisions. One of the key aspects they want to capture is not just the outcome of human decision making (with all of its flaws and <a href="https://cogsci.stackexchange.com/q/251/29">irrationality</a>) but also the deliberation time. To do this, they introduced <a href="http://en.wikipedia.org/wiki/Decision_field_theory" rel="nofollow noreferrer">decision field theory</a> which models human decision making as a stochastic dynamic process.</p>
<ul>
<li><p>Busemeyer, J. R., &amp; Townsend, J. T. (1993) Decision Field Theory: A dynamic cognition approach to decision making. Psychological Review, 100, 432-459. [<a href="http://mypage.iu.edu/%7Ejbusemey/psy_rev_1993.pdf" rel="nofollow noreferrer">pdf</a>]</p>
</li>
<li><p>Busemeyer, J. R. &amp; Diederich, A. (2002) Survey of decision field theory. Mathematical Social Sciences, 43, 345-370. [<a href="http://www.cogs.indiana.edu/Publications/techreps2000/242/242.pdf" rel="nofollow noreferrer">pdf</a>]</p>
</li>
</ul>
<h3>Development</h3>
<p>Esther Thelen and Linda Smith wrote a nice book examining human development from a dynamic systems approach:</p>
<p><a href="https://mitpress.mit.edu/books/dynamic-systems-approach-development-cognition-and-action" rel="nofollow noreferrer">https://mitpress.mit.edu/books/dynamic-systems-approach-development-cognition-and-action</a></p>

CC BY-SA 3.0
2012-02-06T02:44:26.540Z
Is Busemeyer's work really anything beyond application of decades-old models of perceptual decision making (diffusion, various accumulator models) to more abstract non-perceptual judgements?
Mike Lawrence
CC BY-SA 3.0
2012-02-06T02:51:40.323Z
@Mike I don't know enough about the decades-old models of perceptual decision making to comment. I don't think the authors viewed their work as just routine application though. They obviously use diffusion equations... but deeming a work as a basic application only on account of that would make almost everything a basic application of statistical mechanics (which might be a true judgement ;)).
Artem Kaznatcheev
CC BY-SA 3.0
2012-02-06T02:37:17.627Z
2015-10-21T17:32:07.937Z
2020-06-17T09:55:31.633Z
Artem Kaznatcheev
<p>I'm a mathematics and physics student very much interested in cognitive science. Recently I've been hearing about "a new approach" in cognitive science via dynamical system theory. </p>

<ul>
<li>What are some good examples of using differential equations to model cognitive phenomena?</li>
<li>What papers would you recommend?</li>
</ul>

CC BY-SA 3.0
2012-02-06T00:02:46.683Z
Welcome to the site. Those were both great and welcome questions; I hope you don't mind; but I was hoping you could ask your second question : "For a person with mathematical background what would be a good way to dive into the field of cognitive science?" as a separate question; the site works best when there is one topic per question (unless they are highly related, which in this case, I don't think they are).
Jeromy Anglim
CC BY-SA 3.0
2012-02-08T14:46:03.530Z
You might also want to check out a recent "Great Debate" issue of *Topics in Cognitive Science* about the [Complex Systems Approach to Cognitive Science](http://onlinelibrary.wiley.com/doi/10.1111/tops.2012.4.issue-1/issuetoc)
Dan M.
CC BY-SA 3.0
2017-10-09T07:34:06.907Z
Nobody did mention so far Scholarpedia's editor-in-chief Eugene M. Izhikevich's monography Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting. As user @Memming did mention, "this book is mostly focused on single neuron level dynamics, and hence has little to do with cognition". I mention it here anyway, because nevertheless it *has* to do with cognition.
Hans-Peter Stricker
CC BY-SA 4.0
2023-05-04T14:49:43.260Z
you can have look at this recent model: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC9945965/
Dadep
<reference-request><cognitive-modeling><mathematical-psychology>
What are good examples of applying dynamical systems in cognitive science?
Tomi
# What are good examples of applying dynamical systems in cognitive science?

(Asked by: username_0 on 2012-02-05)

I'm a mathematics and physics student very much interested in cognitive science. Recently I've been hearing about "a new approach" in cognitive science via dynamical system theory.

* What are some good examples of using differential equations to model cognitive phenomena?
* What papers would you recommend?

**Tags:** reference-request, cognitive-modeling, mathematical-psychology

> **Comments:**
> **username_1**: Welcome to the site. Those were both great and welcome questions; I hope you don't mind; but I was hoping you could ask your second question : "For a person with mathematical background what would be a good way to dive into the field of cognitive science?" as a separate question; the site works best when there is one topic per question (unless they are highly related, which in this case, I don't think they are).  
> **username_2**: You might also want to check out a recent "Great Debate" issue of *Topics in Cognitive Science* about the [Complex Systems Approach to Cognitive Science](http://onlinelibrary.wiley.com/doi/10.1111/tops.2012.4.issue-1/issuetoc)  
> **username_3**: Nobody did mention so far Scholarpedia's editor-in-chief Eugene M. Izhikevich's monography Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting. As user @Memming did mention, "this book is mostly focused on single neuron level dynamics, and hence has little to do with cognition". I mention it here anyway, because nevertheless it *has* to do with cognition.  
> **username_4**: you can have look at this recent model: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC9945965/  

---

## 4 answers

### Answer by username_5 on 2012-02-06. Score: 12

Given your background and interest in modeling, I would highly recommend [The Cambridge Handbook of Computational Psychology](http://rads.stackoverflow.com/amzn/click/0521674107). The book provides an overview for several of the prominent modeling paradigms in cogsci, including dynamical systems, as well as many concrete examples--albeit most using other computational paradigms.

Dynamical systems, to my knowledge, are most used in the field of motor control. Though to be honest, dyn systems is not 'my' paradigm, and most of my knowledge comes from the aforementioned book.

In particular, the chapter "[Dynamical Systems Approaches to Cognition](http://fog.its.uiowa.edu/~icdls/dft-school/documents/DFT_Publications/cambridge_chapter_schoner.pdf)" by
G. Schoner gives this example that has always stuck in my mind:

> A highly illustrative example comes from the orientation behaviors of
> the common house ﬂy (Reichardt & Poggio, 1976; Poggio & Reichardt,
> 1976). Flies orient toward moving objects, which they chase as part of
> their mating behavior. Detailed analysis revealed that the circuitry
> underlying this behavior forms a simple controller: a motion detection
> system fed by luminance changes on the ﬂy’s facet eye drives the ﬂight
> motor, generating an amount of torque that is a function of where on
> the sensory surface motion was detected. If the speck of motion is
> detected on the right, a torque to the right is generated. If the
> speck is detected on the left, a torque to the left is generated. The
> level of torque passes through zero when the speck is right ahead. The
> torque changes the ﬂight direction of the ﬂy, which in turn changes
> the location on the facet eye at which the moving stimulus is
> detected. Given the aerodynamics of ﬂies, the torque and its on-line
> updating generate an orientation behavior, in which the insect orients
> its ﬂight into the direction in which a moving stimulus is detected.


---

### Answer by username_8 on 2012-02-06. Score: 9

I have a similar background to you, and have found a lot of interesting things in [evolutionary game theory](http://en.wikipedia.org/wiki/Evolutionary_game_theory) (you can follow [links](http://egtheory.wordpress.com/) from my profile for more). But on the specific content of your question: I have come across to uses of dynamic systems on the opposite ends of cognition. Beer's work on modeling minimal cognition, and Busemeyer & Townsend's work on human decision making.

### Minimal cognition

Beer's work models simple cognitive agents as dynamic recurrent neural networks. He then either hand-builds the agents, or evolves them, and analyzes them from a [situated cognition](http://en.wikipedia.org/wiki/Situated_cognition) perspective by treating them as dynamic systems.

* Beer, R. D. (1995). On the Dynamics of Small Continuous-Time Recurrent Neural Networks. Adaptive Behavior, 3(4), 469-509. doi:10.1177/105971239500300405
* Beer, R. D. (2003). The Dynamics of Active Categorical Perception in an Evolved Model Agent. Adaptive Behavior, 11(4), 209-243. doi:10.1177/1059712303114001

Minimal cognition research tends to use dynamic systems much more heavily than other fields I am familiar with. [This blog](http://minimalcognition.wordpress.com/about/) provides a very thorough survey of papers in the field. A short 'manifesto':

* Harvey, I. (1997). [Cognition is not Computation: Evolution is not Optimisation](http://minimalcognition.wordpress.com/2012/06/19/cognition-is-not-computation-evolution-is-not-optimisation/). In Artificial Neural Networks – ICANN97, Gerstner, W., Germond, A., Hasler, M, and Nicoud, J-D (eds). Springer-Verlag LNCS 1327: 685-690. [[pdf](http://www.sussex.ac.uk/Users/inmanh/CogCompEvOp.pdf)]

### Human decision making

Busemeyer & Towsend on the other hand try to describe how humans deliberate and make decisions. One of the key aspects they want to capture is not just the outcome of human decision making (with all of its flaws and [irrationality](https://cogsci.stackexchange.com/q/251/29)) but also the deliberation time. To do this, they introduced [decision field theory](http://en.wikipedia.org/wiki/Decision_field_theory) which models human decision making as a stochastic dynamic process.

* Busemeyer, J. R., & Townsend, J. T. (1993) Decision Field Theory: A dynamic cognition approach to decision making. Psychological Review, 100, 432-459. [[pdf](http://mypage.iu.edu/%7Ejbusemey/psy_rev_1993.pdf)]
* Busemeyer, J. R. & Diederich, A. (2002) Survey of decision field theory. Mathematical Social Sciences, 43, 345-370. [[pdf](http://www.cogs.indiana.edu/Publications/techreps2000/242/242.pdf)]

### Development

Esther Thelen and Linda Smith wrote a nice book examining human development from a dynamic systems approach:

<https://mitpress.mit.edu/books/dynamic-systems-approach-development-cognition-and-action>

> **Comments:**
> **username_6**: Is Busemeyer's work really anything beyond application of decades-old models of perceptual decision making (diffusion, various accumulator models) to more abstract non-perceptual judgements?  
> **username_8**: @Mike I don't know enough about the decades-old models of perceptual decision making to comment. I don't think the authors viewed their work as just routine application though. They obviously use diffusion equations... but deeming a work as a basic application only on account of that would make almost everything a basic application of statistical mechanics (which might be a true judgement ;)).  

---

### Answer by username_6 on 2012-02-06. Score: 6

Great to hear you're interested in applying your quantitative background to cognitive science; the field can definitely use more individuals like you!

I'm not sure to what degree these models might be meet your definition of "dynamical systems", but here are some off the top of my head:

* stochastic models of human response time and accuracy in 2-alternative speeded choice experiments. [Here's a recent paper](http://www.newcl.org/publications/DonkinCrossFitsRevision2.pdf) that does well to summarize two dominant models in this field.
* [Holographic models of semantic memory](http://www.indiana.edu/~clcl/Papers/Holowords_BRM.pdf) (includes dynamic encoding and retrieval processes)
* username_5 Hawkins' (yes, that username_5 Hawkins) very cool Hierarchical Temporal Memory Model ([lots of info here](http://www.numenta.com/htm-overview/education.php))


---

### Answer by username_7 on 2012-06-26. Score: 5

From my social psychology perspective, there has been some computational modelling work on things like attitudinal influence dynamics. See:
[Nowak, A., Szamrej, J., & Latané, B. (1990). From private attitude to public opinion: a dynamic theory of social impact. *Psychological Review, 97*, 362-476.](http://cognitrn.psych.indiana.edu/rgoldsto/complex/nowak90.pdf)

Robin Vallacher and Andrzej Nowak crop up a lot in this field, and they're a joy to read. Perhaps this is less up your alley, since social psychologists are less interested in formulating differential equations and more interested in just applying the *ideas* of complexity science to their research methods, but it's certainly a step forward from what's been going on in the field so far. If anyone's interested in those broad concepts, try reading:
[Nowak, A. (2004). Dynamical Minimalism: Why Less is More in Psychology. *Personality and Social Psychology Review, 8*, 183-192.](http://psr.sagepub.com/content/8/2/183.full.pdf+html)


---

