Dynamic Systems Thanks to Derek Harter for having

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

Agenda • Dynamic systems – 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.