Dynamic Systems

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

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

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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.

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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?

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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.

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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.

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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.

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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.

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

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

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

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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.

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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.

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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.

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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.

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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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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

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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.

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Maturity Example
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Bifurcations Catastrophes
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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.

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Bifurcations Catastrophes
relaxed
contracted
Electrochemical
From Beltrami
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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.

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

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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.

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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.

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Chaos

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