Dynamic Systems

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 ExampleAge A P15.0 80 615.5 70 716.0 61 816.5 53 917.0 46 1017.5 40 1118.0 36 1218.5 34 1319.0 32 1419.5 31 1520.0 30 1720.5 31 2021.0 32 2421.5 34 3022.0 37 3822.5 41 4823.0 46 6023.5 53 7524.0 57 8524.5 59 8925.0 60 90

0

10

2030

40

50

60

7080

90

100

15 16 17 18 19 20 21 22 23 24 25

AGE (in Years)

ARBITRARY SCALE

A

P

Time series of Assertiveness (A) andPlanning 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 ExampleScatter Plot of A vs P for Maturity SystemTrajectory interpolated onto the scatter plot

0102030405060708090

100

0 10 20 30 40 50 60 70 80 90 100

PLANNING ABILITY

ASSERTIVENESS

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.

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QuickTime™ and aTIFF (Uncompressed) decompressorare needed to see this picture.

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

From Beltrami

relaxed

contractedElectrochemical

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 & CatastrophesLow tension Weak beat

Normal beat

High tensionCardiac 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.