dynamical systems theory

*An area of mathematics used to describe the behavior of complex systems by employing differential and difference equations. Recently this approach has been advanced by some as the best way to describe human cognition. See also symbolicism, connectionism*

**Details:**

Proponents of the dynamical systems theory approach to cognition believe that systems of differential or difference equations are the most appropriate tool for modeling human behavior. These equations are interpreted to represent an agent's cognitive trajectory through a high dimensional state space. In other words, cognition is explained as a multidimensional space of all possible thoughts and behaviors that is traversed by a path of thinking followed by an agent under certain environmental and internal pressures, all of which is captured by sets of differential equations. The terminology of dynamical systems theory is also adapted. Thus, cognition is spoken of in terms of state spaces; point, cyclic and chaotic attractors; trajectories; and deterministic chaos.

Dynamicists, including van Gelder, Port, Thelen and Smith, believe that they have a mandate to prove that this dynamicist conception of cognition is the correct one to the exclusion of symbolicism and connectionism. Van Gelder has formulated this as the Dynamicist Hypothesis (van Gelder, 1995, p. 4):

Natural cognitive systems are certain kinds of dynamical systems, and are best understood from the perspective of dynamics.

Those 'certain kinds' of systems are identified as (van Gelder and Port, 1995, p. 5):

state-determined systems whose behavior is governed by differential equations... Dynamical systems in this strict sense always have variables that are evolving continuously and simultaneously and which at any point in time are mutually determining each other's evolution.

In sum, van Gelder and Port assign the following criteria to dynamicist explanations of cognition. Dynamicist descriptions must be:

- deterministic
- generally complex (i.e. nonlinear)
- described with respect to the independent variable of time
- of low dimensionality
- intimately linked (i.e. coupled)

There are also various claims made about the status of computation and representation in such systems. Often, it is claimed that dynamical systems are noncomputational and nonrepresentational.

Criticisms of dynamic systems theory are numerous. Some focus on the rejection of representation and computation. These either claim that the systems are indeed representational and computational or that dynamic systems theory will not be able to explain how cognitive systems are able to be both representational and computational. As well, the similarity between behaviorism and the dynamicist approach is often used discredit dynamicism because of the numerous and debilitating difficulties faced by early behaviorists.

Another important criticism stems from the relation between dynamic systems theory and connectionist networks. Though many dynamicists feel that connectionism should be replaced by dynamicism, it is not clear why. Connectionist networks exhibit many of the behaviors dynamicist note as being central to cognition and incompatible with classical, or symbolicist AI. Though many connectionist networks do not live up to the expectations of dynamicists, there are a number which do, while not discarding the notions of computation and representation. Though a strong normative claim may be made concerning the importance of dynamical systems theory to connectionist models, it is far from clear that dynamicist cognitive approach will, or should,

*replace*the connectionist approach.Chris Eliasmith