Robert F. Port, Indiana University

December 5, 2000 draft

-draft entry for *`Encyclopedia of Cognitive Science * MacMillan Reference Ltd,

London. Amy Lockyer,
Assoc. Ed.

Charge from the editor.

Article Title: *DYNAMICAL SYSTEMS
HYPOTHESIS IN COGNITIVE SCIENCE*

Maximum text length in words: 2000

Level of article: Focussed (suitable for
undergraduates/postgraduates)

Suggested first level headings:

*1. The computational framework and the
dynamical systems framework*

*2. Strengths of the dynamical systems
approach*

*3. The role of time in cognitive models*

*4. The role of discrete and continuous
representations in cognitive models*

*5. Dynamical systems models of
sensory-motor capabilities*

*6. Dynamical systems models of decision
making and high-level cognition*

*7. Situated cognition and the coupling of
systems*

*8. Connectionism and the dynamical systems
hypothesis*

*9. Summary*

**Outline of Draft
(current size excluding references = 4,099 words)**

A. Overview
(proposed 1)

B. Mathematical
Context (proposed 1, 3)

C. Perceptual
and Motor Models (proposed 5)

D. High-level
Cognition (proposed 6)

E. Relation
to Situated Cognition and Connectionism (proposed 7, 8)

F. Contrasting
the Dynamical Hypothesis with Traditional Approaches (proposed 1, 3)

G. Strengths
and Weaknesses of the Dynamical Models (proposed 2)

H. Discrete
vs. Continuous Representations (proposed 4)

The dynamical hypothesis in cognition refers to various research
paradigms applying the mathematics of dynamical systems to understanding
cognitive function. The approach is allied with and inspired by research in
neural science over the past fifty years for which dynamical equations have
been found to provide excellent models for the behavior of single neurons
(e.g., Hodgkins and Huxley, 1952). It also derives inspiration from work on
gross motor activity by the limbs (e.g., Bernstein, 1967, Feldman, 1966). In the early 1950s, Ashby made the startling
proposal that all of cognition might be accounted for with dynamical system models
(1952), but little work directly followed from his speculative suggestion due
to a lack of appropriate mathematical tools as well as the lack of
computational methods to implement such models in a practical way. More recently, the connectionist movement
(Rumelhart and McClelland, 1986) provided insights and mathematical
implementations of learning, for example, that have helped restore interest in
dynamical modeling.

The dynamical approach to cognition is also closely related to
ideas about the ** embodiment** of mind and the environmental

and the gross motor system
is coupled with both audition and environmental sound. Because of this commonality between the
world, the body and cognition, according to this view, the method of
differential equations is applicable to events at all levels of analysis over a
wide range of time scales. This approach directs explicit attention to change
over time and rates of change over time of system variables.

The mathematical models employed by dynamical systems research
derive from many sources in biology and physics. Only two schemas will be
pointed out here. The first is the
neural network idea, partially inspired by the remarkable equations of Hodgkins
and Huxley (1952) which account for many known phenomena about neurons in terms
of the dynamics of cell membrane. They
proposed a set of differential equations for the flow of sodium and potassium
ions through the axonal membrane during the passage of an action potential down
the axon. These equations, which turn
out to apply with slight modification to nearly all neurons, inspired
extensions to account for whole cells (rather than just a patch of membrane) in
terms of its likelihood to fire given various excitatory and inhibitory
inputs. Interesting circuits of
neuron-like units were also constructed and simulated on computer. The
Hodgkin-Huxley equations also inspired many psychological models, like those of
Grossberg (1976;1980), the neural network models (Rumelhart and McClelland,
1986; Hinton, 1986) and models of neural oscillations (Kopell, 1995).

In this general framework, each
cell (or group of cells) in a network of interconnected cells (or cell groups)
is hypothesized to follow an equation like:

Equation 1:* dA/dt* = g*A(t*) + d[*aE(t) – bI(t) + cS(t*)] *+
bias*

indicating that the change in activation (i.e., the likelihood of
firing) at time *t*, *dA/dt,* depends on the decay, g, of
the previous value of *A* plus a term representing inputs from other cells
that are either excitatory (that is, tending to increase the likelihood of
firing), *E(t),* or inhibitory
(tending to decrease the likelihood of firing), *I(t).* For some units there may be an external
physical stimulus, *S(t*). The
nonlinearity, d(*x*), encourages all-or-none
firing behavior and the bias term adjusts the value of the firing
threshold. An equation of this general
form can describe any neuron. Over the
past 50 years, networks of units like these have demonstrated a wide variety of
behaviors, including many specific patterns of activity that animal nervous
systems exhibit.

A second concrete schema for the dynamical approach to cognition
is simply the classical equation for a simple oscillator like a pendulum.
Indeed, it is obvious that arms and feet have many of the properties of
pendula. Students of motor control
have discovered that pendular motion is a reasonable architype for many limb
gestures. A nearly identical system (lacking the complication of arc-shaped
motion) is the equation for a mass-spring system. In this form:

Equation 2:* md(d ^{2}x)dt^{2} + d(dx/dt) + k(x-x_{0}) = 0*

it specifies simple harmonic motion in terms of the mass, *m*, times the acceleration,*
(d ^{2}x)dt^{2}*, the damping,

In the most general terms, **a dynamical system** may be
defined as **a set of quantitative variables that change simultaneously and
interdependently over quantitative time in accordance with some set of
equations** (van Gelder and Port, 1995).
From this perspective, Newton's equations of motion for physical bodies
were the earliest dynamical models. Mathematical developments over the past 30
years have revolutionized the field.
Whereas up until the 1950s, the analysis of dynamical models was
restricted to linear systems (where the dynamic rule has linearly proportional
effects on system variables such as in Equation 2 above) and only when they
contain no more than a couple of variables, now, through the use of
computational simulations (using discrete approximations, of course) and
computer graphics to facilitate geometric interpretations of these systems,
practical methods for studying nonlinear models with many variables are now
possible.

Although models of this general class have proven useful for many
problems, it remains to be seen whether such models will be viable in the long
run to account for the full range of cognitive behavior.

Dynamical models seem particularly appropriate to account for
recognition and motor control, since research leading to temporal information
about the process of perception has been gathered for many years (in contrast
to `general thinking' or reasoning) and because motor control is clearly a task
that requires refined temporal control.

One well-known example of a dynamical model for general perception
is the *ART model* (*adaptive resonance theory*) of Grossberg (1995).
This neural network is defined by a series of differential equations, similar
to the network equation shown above, describing how the activation of a node is
increased or decreased by stimulus inputs, excitation and inhibition from other
nodes and intrinsic decay. This depends on weights (represented above as
matrices for *a, b* and *c* in Equation 1) which are modified by
successful perceptual experiences (to simulate learning from experience). The
model can discover the low-level features that are most useful for
differentiating frequent patterns in its stimulus environment (using *unsupervised
learning*), identify specific high-level patterns (even from noisy or
incomplete inputs) and reassign resources whenever a new significant pattern
appears in its environment without forgetting earlier patterns.

To recognize an object such as a letter from visual input, the
signal from a spatial retina-like system excites low-level features in a set of
nodes called collectively *F1.* The pattern of activated features in *F1*
feeds excitation through weighted connections to all the nodes of *F2*,
the set of nodes that will do the identification. These nodes compete with each other by inhibiting each other in
proportion to their activation. Thus, the best matching unit in *F2* will
quickly win and suppress all its neighbors. But this is only the first step in
the process. At this point, the winning *F2* node feeds activation back to
those *F1* nodes that it predicts should be active (as determined by the
weighted connections). If not enough of the *F1* nodes turn out to be
active (that is, if the pattern does not match well enough), then the system
rejects this identification by shutting down the *F2* node that had won
(for some time interval). Then the activation of *F2* by *F1* begins
again and a different *F2* node will win. If on feedback this one matches
sufficiently well, then a ``resonance loop’’ is established where *F1* and
*F2* reinforce each other. Only at this point is successful (and,
according to Grossberg, conscious) identification achieved. This perceptual model is dynamic because it
depends on differential equations that gradually increase and decrease the
activation of nodes in a network at various rates. Grossberg's group has shown
that variants of this model can account in a very natural way for many
phenomena of visual perception, including those involving backward masking,
reaction time and so on.

In recent years, dynamical models have also been applied to more
high-level cognitive phenomena. First, Grossberg and colleagues have elaborated
the *ART* model with mechanisms like `*masking fields'* that permit the model to be extended to
tasks like word recognition from auditory input arriving over time. Several
time-sensitive phenomena of speech perception can be successfully modeled this
way (Grossberg, 1986). Second, models
of human decision making for many years have applied `*expected utility
theory,'* where a choice is based on evaluation of relative advantages and
disadvantages of each choice at some particular point in time. But Townsend and Busemeyer (1995) have been
developing their *decision field theory* that not only accounts for the
likelihood of each eventual choice, but also accounts for many time-varying
aspects of decision making, such as approach-avoidance effects, vacillations,
and the fact that some decisions need more time than others.

Finally, it’s important to note that phenomena that at first
glance seem to depend on high-level cognitive skills may turn out to reflect
much more low-level properties of cognition.
One of the most startling results of this kind is in the *``A-not-B
problem''*. This is a traditional puzzle in early cognitive development first
discovered by Piaget (1954) and interpreted by him as showing that very young
children (roughly 9-12 months) do not have the concept of `*object permanence'*, that is, that
children have inadequate representations of objects, thinking falsely that objects
intrinsically belong to the specific place where they were first found. Here is
the supporting experiment procedure in canonical form: With the child watching,
the experimenter places an interesting object under a cover on the table (at
position A) in front of the child and lets the child reach over to lift the
cover and grab the toy. This is done this several times. Now the object is put
under a different cover in front of the child (at position B). Astonishingly
children at this age will often reach back to lift the cover they had reached
to earlier rather than the correct one – to A not to B! Many experiments have
explored this phenomenon over the past forty years and a wide variety of
cognitive accounts have been proposed in addition to Piaget’s notion of inadequate `object permanence’. Recently,
Thelen, Schöner, Schrier and Smith have demonstrated that the hypothesized
cognitive representation of the object has nothing to do with this behavior.
Instead, their dynamical model for choosing the direction of reach shows that
the children have generated a strong bias (or habituation) toward reaching in
the A direction due to the repeated earlier reaches. Watching the experimenter put the object in a new location is not
sufficient to overcome the bias to repeat the same gesture as before. The
authors show that the bias can be overcome, however, by such simple revisions
as (a) inserting a time delay between the first reaches and the final one
(giving time for the first bias to decay), or (b) only letting the child reach
once to the first location rather than multiple times (so the directional bias
to A is less strong) or (c) using an object in the B location that is new and
more interesting to the child than the earlier one (thus boosting the
directional bias toward the new position relative to the old). Their dynamical model predicts sensitivity
to just the same variables. The lesson is that sometimes what seems at first to
be a property of abstract, high-level, static representations may turn out to
result from rather low-level time-sensitive effects – most of which are
naturally modeled in terms of dynamical equations.

Although the issue of ** situated cognition** can be
interpreted in many ways that have little to do with dynamical systems, the
dynamical systems approach is highly compatible with this concern. The primary
reason is that, from this perspective, the world, the body and the cognitive
functions of the brain can all be analyzed using the same conceptual tools.
This is important because it greatly simplifies our understanding of the
relationship between these systems, and is readily interpreted as an instance
of the biological adaptation of the body and the brain to the environment.

** Connectionist models** are discrete approximations of dynamical systems and so are the
learning algorithms used with them. But
when computational models are used simply to do identification or make a choice
by settling to a fixed point (as was the case in the early connectionist models
of the mid 1980s), they seem to be focused on tasks that are largely inspired
by issues in symbolic cognitive models. Even if they attempt to solve temporal
problems yet address only time specified as serial order, they would seem to be
minimally dynamical. The touchstone of
a thoroughly dynamical approach is the study of phenomena that occur in
continuous time. The source
`situatedness’, we would say, is situation in real time. Of course, neural
networks are frequently used to study such phenomena, but other dynamical
methods are also available for some
problems that do not employ network simulations. Thus the development of connectionist modeling since the 1980s
has certainly helped to move the field in the direction of dynamical thinking,
but these models are not always good illustrations of the dynamical hypothesis
of cognition.

It should be acknowledged that the most widespread
conceptualization of the mechanism of human cognition is that cognition
resembles computational processes, like deductive reasoning or long division,
by making use of symbolic
representations of objects and events in the world that are manipulated by
cognitive operations (typically serially ordered) which might reorder or
replace symbols, and draw deductions from them. This approach has been called the ** computational approach**
and its best-known articulation is the

It should be noted that the dynamical systems hypothesis for
cognition is in no way incompatible with serially ordered processes on discrete
symbol tokens. Some possible examples
are forms of human cognition like doing arithmetic and generating a sentence.
Still the dynamical systems approach denies that all cognition (or even most)
can be satisfactorily understood in computational terms and insists that, since
the physiological brain, the body and the environment are best accounted for in
dynamical terms. Any explanation of human symbolic processing must include an
account of their implementation in dynamical terms. The dynamical approach points out the inadequacy of simply
assuming that a `symbol processing mechanism' is somehow available to human
cognition, the way a computer happens to be available to a programmer. Instead wherever either discrete-time or
digital functions are found in cognition, the continuous processes which give
rise to the discreteness and digitality demand, sooner or later, a complete,
continuous-time account. Indeed the
dynamical approach tends to deny that cognition can be separated from the
physical body or from the environment.
A fundamental contrast is that the discrete time of computational models
is replaced with continuous time for which rate of change and changes in the
rate of change (first and second derivatives) etc. are meaningful at each instant. Thus it is invariably dependent upon
studying parameters that change over time and it attempts to understand those
changes by modelling them.

Dynamical modeling offers many important strengths relative to
traditional symbol processing or computational models of cognition. First, the
biological plausibility of digital, discrete-time models is always a problem.
How and where might there be in the brain, a device that would behave like a
computer chip, clicking along performing infallible operations on digital
units? The answer often offered in the
past was "We don't really know how the brain works, anyway, so this
hypothesis is as plausible as any other." Such an argument does not seem
as reasonable today as it did 30 or 40 years ago. Certainly neurophysiological
function exhibits many forms of discreteness. But that does not justify simply
assuming whatever kind of units we like and operations.

Second, temporal data can finally, by this means, be incorporated
directly into cognitive models. Phenomena like (a) processing time (observable
in data on reaction time, recognition time, response time, etc.), (b) temporal
structure in motor behaviors (like motor patterns of reaching, speech
production, locomotion, dance, etc.), and (c) temporal structure in stimulation
(e.g., for speech and music perception, interpersonal coordination in, e.g.,
watching a tennis match, etc) can now be linked together if events can be
predicted in time.

The language of dynamical systems provides a conceptual vocabulary
that permits unification of cognitive processes in the brain with physiological
processes in our bodily periphery and with environmental events external to the
organism. Unification of processes across these fuzzy and partly artificial
boundaries makes possible a truly embodied and situated understanding of human
behavior of all types. The discrete
time modeling of traditional approaches was always forced to draw a boundary
separating the discrete time, digital aspects of cognition from continuous time
functions (as, e.g., in Chomsky's distinction of *Competence vs. Performance*).

Third, cognitive development and `runtime processing' can now be
integrated, since learning and perceptuo-motor behavior are governed by similar
processes even if on different time scales.
Symbolic or computational models were forced to treat learning and
development as totally different processes unrelated to motor and perceptual
activity.

Finally, trumping the reasons given above, there is the important
fact that *dynamical models include discrete-time, digital models as a
special case* whereas the other way around is not possible. (The sampling of
continuous events permits discrete simulation of continuous functions, but the
simulation itself remains discrete and digital at all times and only models a
continuous function up to its *Nyquist frequency*, that is, up to half the
sampling rate. See Port, Cummins and McAuley, 1995). Thus, any actual digital
computer is also a dynamical system with real voltage values in continuous time
that are subjected to discretization by an independent clock. Of course,
computer scientists prefer not to look at them as continuous valued dynamical
systems (because it is much simpler to treat them as digital machines) but
computer engineers certainly do. Hardware engineers have learned to constrain
the dynamics so it is governed with great reliability by powerful attractors
for each binary cell that assure that each bit is able to settle into one of
its two possible states before the next clock tick comes round when each cell
is subject to a `poke’ with a voltage.

These strengths of dynamical modelling are of great importance to
our understanding of human and animal cognition. As for weaknesses of dynamical
modelling, there certainly are several, at least. First, the mathematics of dynamical models is quite a bit more
inscrutable and less developed than the mathematics of digital systems. It is
clearly much more difficult, for the time being, to construct actual models
except for carefully constrained simple cases.

Second, during some cognitive phenomena (such when a student is
performing long division, or a mathematician is doing a proof or planning the
design of an algorithm, and possibly to some degree the human's processes of
sentence generation and interpretation) humans appear to rely on serially
ordered operations on digital symbols.
Although, as noted, dynamical models are in principle capable of
exhibiting digital behavior, how a neurally plausible model could do this
remains beyond our grasp for the time being.
For understanding such phenomena, it seems that computational models
are, at the very least, simpler and more direct, even if they are inherently
inherently inadequate.

One of the great strengths of the classical computational approach
to cognition is the seeming clarity of the traditional notion of a ** cognitive
representation**. If cognition is conceived as functioning rather like a
program in Lisp, then the representations resemble Lisp

What representations might play a role here? -- a knife, perhaps,
but what about moving my body and arm to just the right place at just the right
velocity? How could discrete, wordlike
representations be employed to yield successful slicing of bread? And if this is, instead, to be handled by a
nonrepresentational system (perhaps a dynamical one), then how could we
determine the boundary between these two distinct and seemingly incompatible
types of systems?

The development of ** connectionist models** in the
1980s, employing networks of interconnected nodes, provided the first
alternative to the view of representations as discrete context-invariant
manipulable tokens. In connectionist models, the result of a process of
identification (of, say, a letter of the alphabet or a human face) is only a temporary
pattern of activations across a particular set of nodes, not something
resembling a context-free, self-contained object. The possibility of representation in this more flexible form led
to the notion of

The development of dynamical models of perception and motor tasks
has led to further extension of the notion of representation to include
time-varying trajectories, limit cycles, coupled limit cycles and attractors
toward which the system state may tend but which may never be achieved. From the dynamical viewpoint, static
representations play a far more limited role in cognition. Indeed, a few researchers in this tradition
deny that static representations are ever needed (Brooks, 1997).

Ashby, R.
(1952) *Design for a Brain*. (Chapman-Hall, London)

Brooks, Rodney (1997) Intelligence
without Representation. In J. Haugeland
(ed.) *Mind Design II*
(MITP, Cambridge, MA), pp. 395-420.

Chomsky, Noam (1961) On the notion
`rule of grammar.’ *Proceedings of
the 12 ^{th} Symposium in Applied Mathematics*, 6-24.

Chomsky, Noam (1965) *Aspects of
the Theory of Syntax*. (MITP, Cambridge, MA)

Fel’dman, A. G. (1966) Functional
tuning of the nervous system with control of movement or maintenance of a
steady posture---III. Mechanographic analysis of the execution by man of the
simplest motor tasks. *Biophysics 11*, 766-775.

Grossberg, Stephen (1995) Neural
dynamics of motion perception, recognition, learning and spatial cognition. In
Port and van Gelder (eds) *Mind as Motion: Explorations in the Dynamics of
Cognition *(MITP, Cambridge, MA), 449-490.

Grossberg, Stephen (1986) The adaptive self-organization of serial
order in behavior: Speech, language and motor control. N E. Schwab and H. Nusbaum (eds.) *Pattern
Recognition by Humans and Machines: Speech Perception.* Academic Press,
Orlando, FL).

Haken, H., Kelso, J. A. S., & Bunz, H.
(1985). A theoretical model of phase transitions in human hand movements. *Biological Cybernetics*, *51*, 347-356.

Haugeland, John. (1985). *Artificial Intelligence: The Very Idea*. Cambridge, MA: Bradford
Books, MITP.

Kopell, Nancy (1995) Chains of coupled oscillators. In M. Arbib
(ed) *Handbook of Brain Theory and Neural Networks* (MITP; Cambridge MA),
pp. 178-183.

Newell, Allen, & Herbert Simon. (1975) Computer
science and empirical inquiry. *Communications of the ACM*, pp. 113-126.

Piaget, Jean (1954) *The
Construction of Reality in the Child *(Basic Books)

Port, Robert F., Fred Cummins, and J. Devin
McAuley. Naive time, temporal patterns and human audition. In Robert F. Port and Timothy van Gelder,
editors, *Mind as Motion: Explorations in
the Dynamics of Cognition.* MIT P, Cambridge, MA, 1995.

Port, Robert & Timothy van Gelder,
editors. *Mind as Motion: Explorations in
the Dynamics of Cognition. *Bradford Books/MITP, 1995.

Thelen, Esther, G. Schöner, C.
Schrier and L. B. Smith (2000) The dynamics of embodiment: A field theory of
infant perseverative reaching. *Behavioral
and Brain Sciences* **(in press).**

Townsend, James and Jerome
Busemeyer (1995) Dynamic representation of decision making. In Robert F. Port and Timothy van Gelder,
editors, *Mind as Motion: Explorations in
the Dynamics of Cognition.* MITP, Cambridge, MA, 1995.

van Gelder,
Timothy, & Robert Port. It's about time: Overview of the dynamical approach
to cognition. In Robert Port and Timothy van Gelder, editors, *Mind as
motion: Explorations in the dynamics of cognition*, pages 1-43. Bradford
Books/MITP, 1995.

Abraham, Ralph H and Christopher D. Shaw ( 1982) *Dynamics: The Geometry
of Behavior*. Parts 1-4. (Ariel Press; Santa Cruz, CA).

Clark, Andy (1997) *Being There:
Putting the Brain, Body and World Together Again* (MITP, Cambridge, MA).

Haugeland, John.
(1985). *Artificial Intelligence: The Very
Idea*. Cambridge, MA: Bradford Books, MIT Press.

Kelso. J. A. Scott (1995)*
Dynamic Patterns: The Self-organization of Brain and Behavior* (MIT Press;
Cambridge, MA).

Port, Robert and Tim van Gelder
(1995) *Mind as Motion: Explorations in the Dynamics of Cognition.* (MIT Press, Cambridge, MA.).

Thelen, Esther and Linda Smith
(1994) *A Dynamical Systems Approach
to the Development of Cognition and Action* (MITPress; Cambridge MA).

**Articles**

Thelen, E., G. Schöner, C. Schrier
and L. B. Smith (2000) The dynamics of embodiment: A field theory of infant
perseverative reaching. *Behavioral
and Brain Sciences* (in press).

Port,
Robert F., Fred Cummins, and J. Devin McAuley. Naive time, temporal patterns
and human audition. In Robert F. Port
and Timothy van Gelder, editors, *Mind as
Motion: Explorations in the Dynamics of Cognition.* MIT Press, Cambridge,
MA, 1995, pp. 339-371.

van Gelder, Tim,
& Robert Port. It's about time: Overview of the dynamical approach to
cognition. In Robert Port and Timothy van Gelder, editors, *Mind as Motion:
Explorations in the dynamics of cognition*, pages 1-43. Bradford Books/MIT
Press, 1995.