Dynamic Systems Thanks to Derek Harter for having notes on the web. Also see, Port & Van Gelder and Beltrami.

Agenda Dynamic systems Haken, Kelso, & Bunz, 1985 Bit of history for cognition. Dynamic systems vocabulary. Bifurcations & catastrophes. Chaos. Haken, Kelso, & Bunz, 1985

From Symbols to Dynamics Computational view of mind Symbolic atoms. Serial processing. Syntactic manipulation as in logic or language. Worry about syntax, not semantics. Connectionism Distributed representations. Parallel processing. Good generalization. Graceful degradation. Recurrent nets incorporate temporal dynamics.

From Symbols to Dynamics Is flight best understood by Flapping or Dynamics of airfoils, airflow, mass, etc? Is cognition best understood by Symbolic and logical reasoning or Some underlying system of temporal dynamics?

Dynamical Cognitive Hypothesis The cognitive system is not a discrete sequential manipulator of static representational structures; rather, it is a structure of mutually and simultaneously influencing change.

Dynamical Cognitive Hypothesis Cognitive processes do not take place in the arbitrary, discrete time of computer steps; rather, they unfold in the real time of ongoing change in the environment, the body, and the nervous system.

Dynamical Cognitive Hypothesis The dynamical approach at its core is the application of the mathematical tools of dynamics to the study of cognition. Natural cognitive systems are dynamical systems, and are best understood from the perspective of dynamics.

Basic Concepts System - a set of interacting factors (called state variables) whose values change over time. Learning, perception, maturity, sensation, communication, feeding, attitude, motion, etc. State - vector of values, one for each variable of the system at a given moment.

Maturity Example Time series of Assertiveness (A) and Planning Ability (P) as a function of Age

Basic Concepts State Space - all possible states of the system. State Variables - the variables used to define the state space. Trajectory - a curve connecting temporally successive points in a state space.

Maturity Example Scatter Plot of A vs P for Maturity System Trajectory interpolated onto the scatter plot

Basic Concepts Phase Portrait - a state space filled with trajectories of a given model.

Vectorfields Instantaneous Velocity Vector - the instantaneous rate and direction of change in the state of the system at a point in time. Describes the tendency of the system to change when in that state. It says in what direction and how fast the system should change on all variables simultaneously.

Vectorfields Vectorfield - the collection of all of the instantaneous velocity vectors. Technically a Dynamical System is equivalent to this vectorfield. A vectorfield summarizes all the possible changes that can occur in the system.

Vectorfields The trajectories (Phase Portrait) gives the history of change of the system over time. The vectorfield gives the rules for the tendency of change for each state in the system.

Properties of Phase Portraits Fixed(constant, critical, rest) point - a point in the state space with zero instantaneous velocity. Periodic (cyclic, closed) trajectory – a trajectory that closes upon itself.

Properties of Phase Portraits Chaotic (strange) trajectory – trajectories that are neither fixed nor cyclic but which fill up a constrained region of the state space. Does not go to a fixed point or a cycle, but remains constrained in a region of phase space.

Properties of Phase Portraits Attractor – limit sets to which all nearby trajectories tend towards. Fixed attractor, periodic attractor, chaotic attractor Basin – a region of the state space containing all trajectories which tend to a given attractor

Properties of Phase Portraits Separatrix – consists of points and trajectories which are not in any basin (i.e. do not tend toward any attractor). Repellor – Points and periodic trajectories from which trajectories only leave Saddles – limit sets which some trajectories approach and others depart.

Maturity Example

Bifurcations & Catastrophes

Bifurcations & Catastrophes A bifurcation is a major change in the phase portrait when some control parameter is changed past a critical value. A catastrophic bifurcation is when a limit set appears or disappears when the control parameter is changed.

Bifurcations & Catastrophes relaxed contracted Electrochemical From Beltrami

Bifurcations & Catastrophes If the heart muscle is already slightly stretched before beating, a larger beat will result. The stretching is caused by tension which results from increased blood pressure at the moments of stress. More tension, faster rate of pumping. Less tension, weaker pumping.

Bifurcations & Catastrophes

Bifurcations & Catastrophes Low tension Weak beat Normal beat High tension Cardiac arrest

Chaos A chaotic system is roughly defined by sensitivity to initial conditions: infinitesimal differences in the initial conditions of the system result in large differences in behavior. Chaotic systems do not usually go out of control, but stay within bounded operating conditions.

Chaos Chaotic systems, like people, Tend to revisit similar “states”. Are unpredictable, although may be deterministic. Are sensitive to internal and external conditions. Are typically bounded.

Chaos Chaos is often found in the dynamic systems used to model cognition, e.g., neural nets. Chaos has been found in the brain processes. E.g., chaos is integral to a model of the olfactory system, it provides a “ready” state for the system.

Chaos Chaos provides a balance between flexibility and stability, adaptiveness and dependability. Chaos lives on the edge between order and randomness.