Dynamical Systems Theory

(Teoria Sistemelor Dinamice)

• Netwon (Galilei), Poincare, Landau (‘44)

• Ecological approach (Gibson '66, '79)

• Ecological psychologists (Turvey et al. '81)

• Turvey Kluger Kelso ('80)-Motor coordination

• Thelen & Smith (’90s) for cognition

• Embodied cognition (Gibson, Agre and Chapman, Hutchins)

• Situated action (Gibson → Barwise and Perry '81, '83 Pfeifer and Scheier, Glenberg, Brooks)

• Extended mind (Clark '01, '08)

van Gelder & Port (1995)

• Dynamical and computational approaches to cognition are fundamentally different

• Dynamical approach = Kuhnian revolution

• Brain (inner, encapsulated) vs. Brain + + body + environment

• Discrete static Rs vs. Mutually + simultaneously influencing changes between brain, body and environment

• Geometrical Rs → To conceptualize how system change!

• A plot of states traversed by a system through time = System’s trajectory through state space

• Trajectory – Continuous (real time) or discrete (sequence of points)

• A dimension = A variable of a system A point = A state Ex - Solar system: Position +

Momentum of planets - Mathematical laws relate changes over time → A mathematical dynamical model

• Dynamic systems theory (DST) - Physics• Dynamical system: Set of state variables

+ dynamical law (governs how values of state variables change with time)

• Set of all possible values of state variables = Phase space of system (state space)

• All possible trajectories = Phase portrait• Parameters → Dimensions of space• The sequence of states represents

trajectory of system

1. State space of a system = Space defined by set of all possible states system could ever be in.

2. A trajectory (path) = Set of positions in state space through which system might pass successively. Behavior is described by trajectories through state space.

3. An attractor = Point of state space - system will tend when in surrounding region

4. A repeller = Point of state space away from which system will tend when in surrounding region

5. The topology of a state space = Layout of attractors and repellors in state space

6. A control parameter = Parameter whose continuous quantitative change leads to a noncontinuous, qualitative change in topology of a state space

7. Systems - modeled with linear differential equations = Linear systems

- with nonlinear differential equatio-s = Nonlinear systems8. Linear systems are decomposable = Modeled as collections of

separable components. Nonlinear systems = nondecomposable9. Nondecomposable, nonlinear systems - characterized - collective

variables and/or order parameters, variables/parameters of system that summarize behavior of system’s components (Chemero ’09)

• Goal: Changes over time (and change in rate of change over time) of a system (Clark '01)

• DST → Understanding cognition

• Cognitive systems = Dynamical systems

• “Cognitive agents are dynamical systems and can be scientifically understood as such.” (van Gelder '99)

• Change vs. state

Geometry vs. structure (van Gelder '98)

• Behavior of system (changes over time): Sequence of points = Phase space (Numerical space - differential equations)

• Geometric images → Trajectory of evolution

• Collective variables (relations between variables)

• Control parameters = Factors that affect evolution (Ex: Solar system)

• Rates of change: Differential equations(van Gelder + Port '95)

• DST: Cognition - “in motion” • No distinction between mind-body

Mind-body-environment:• Dynamical-coupled systems • Interact continuously, exchanging

information + influencing each other• Processes - in real continuous time

Quantities (scientific explanation) vs. qualities (Newell & Simon “law of qualitative structure”, van Gelder '98)

“What makes a system dynamical, in relevant sense? … dynamical systems are quantitative. … they are systems in which distance matters.

Distances between states of system/ times that are relevant to behavior of system” → Rate of change (t) (Van Gelder '98)

• DST: Time – involved• Geometric view of how structures in

state space generate/ constrain behavior + emergence of spatio-temporal patterns

→ Kinds of temporal behavior - translated in geometric objects of varying topologies

• Dynamics = Geometry of behavior (Abraham & Shaw '83)

The computational governor vs. the Watt centrifugal governor

Computational governor - Algorithm:

(1)Operating internal Rs and symbols,

(2)Computational operations over Rs

(3)Discrete, sequential and cyclic operations

(4)“Homuncular in construction”, Homuncularity = Decomposition of system in components, each - a subtask + communicating with others (van Gelder '95)

Centrifugal governor (G)

Constant speed for flywheel of steam engine:• Vertical spindle to flywheel - Rotate at a speed

proportionate to speed of flywheel• 2 arms metal balls - free to rise + fall • Centrifugal force-in proportion to speed of G• Mechanical linkage: Angle of arms - change

opening of valve → Controlling amount of steam driving flywheel

• If flywheel - turning too fast, arms - rise → Valve partly close: Reduce amount of steam available to turn flywheel = Slowing it down

• If flywheel - too slowly, arms - drop → Valve – open: More steam = Increase speed of flywheel

Centrifugal governor (G):

Nonrepresentational + noncomputational

• Relationship betw. 2 quantities (arm angle and engine speed) = Coupled

• Continuously reciprocal causation through mathematical dynamics

Clark ('97)

• Such mechanisms = “Control systems” – noncomputational, non-R-l

• No Rs or discrete operations • Explanation = Only dynamic analysis• Relationship arm angle-engine speed: no

computational explanation• These 2 quantities - continuously

influence each other = “Coupling”• Relation brain-body-environ. =

= Continuous reciprocal causation

DST- 2 directions for R: (1) Radical embodied cognition =No Rs/computation

“Maturana and Varela 80; Skarda and Freeman 87; Brooks 1991; Beer and Gallagher 92; Varela, Thompson, + Rosch 91; Thelen + Smith 94; Beer 95; van Gelder 95; van Gelder + Port 95; Kelso 95; Wheeler 96; Keijzer 98 +

Kugler, Kelso, + Turvey 1980; Turvey et al. 81; Kugler + Turvey 1987; Harvey, Husbands, + Cliff 94; Husbands, Harvey, + Cliff 95; Reed 96; Chemero 00, 08; Lloyd 00; Keijzer 01; Thompson + Varela 01; Beer 03; Noe and Thompson 04; Gallagher 05; Rockwell 05; Hutto 05, 07; Thompson 07; Chemero + Silberstein 08; Gallagher + Zahavi 08” (Chemero 09)

(2) Moderate = Replace vehicle of Rs or R in a weaker sense (Bechtel '98, '02; Clark '97a,b; Wheeler & Clark 97; Wheeler ’05)

• Clark ('97, '01, '08; Clark and Toribio '94 (Miner & Goodale ’95, ventral vs. dorsal); Clark and Grush '99) that anti-R-ism of radical embodied cognitive science is misplaced. (Chemero, ’09, p. 32)

• Radicals: “R”, “computation”, “symbols”, and “structures” - Useless in explanation cognition (van Gelder, Thelen & Smith, Skarda, etc.)

• “Explanation in terms of structure in the head-beliefs, rules, concepts, and schemata - not acceptable. … Our theory - new concepts … coupling … attractors, momentum, state spaces, intrinsic dynamics, forces. These concepts - not reductible to old”

• “We are not building Rs at all! Mind is activity in time… the real time of real physical causes.” (Thelen and Smith ‘94)

- Notions: Patterns + self-organization + coupling + circular causation (Clark ‘97b; Kelso ‘95; Varela et al. ‘91)

- Patterns - emerge from interactions between organism and environment

- Organism-environment = Single coupled system (composed of two subsystems)

- Its evolution through differential equations (Clark)

• Bodily actions (child walking – T&S) • Movement of fingers (HKB '87, Kelso)

→ Extrapolate from sensoriomotor processes to cognition processes!

(Implicit-explicit → Hybrid models?) No decision making/contrafactuals • Replace static, discrete Rs with

attractors = Continuous movement• At conceptual level attractors seem

static and discrete

• Globus '92, '95; Kelso '95: Reject Rs + computations

• Globus: Replaces computation with constraints between elements-levels

• “[R]ather than computes, our brain dwells (at least for short times) in metastable states”. (Kelso '95) (See Freeman '87)

• Radical embodied cognition: Explores “minimally cognitive behavior” = Categorical perception, locomotion, etc. (Chemero '09)

• Against radicals - Clark and Toribio ('94): certain tasks cannot be accomplished without Rs → “Hungry Rs problems” (decision making, counterfactuals) → Decoupling between R-l system and environment = Off-line cognition (not on-line)

• “Cognitive system has to create a certain kind of item, pattern or inner process that stands for a certain state of affairs, in short, a R.” (Clark)

• TDS - Change:

a) Interactions between (ensembles) neurons

b) Constitutive relations between Rs

→ No prediction, but explanation

• Dynamics among Rs

(Fisher and Bidell '98; van Geert '94)

• Radicals: Cognition = Result of evolution of perception + sensoriomotor control systems [see Barsalou]

• Dynamical models - “having” R-s: Attractors, trajectories, bifurcations, and parameter settings → DS store knowledge + Rules defined over numerical states

(van Gelder & Port '95)

• DST - discrete state transitions

(a) Using discrete states (catastrophe model → Bifurcation)

(b) Discreteness: “How a continuous system can undergo changes that look discrete from a distance”

Skarda & Freeman’s model of olfactory bulb

• Freeman’s network ('85) (Bechtel) • Rabbit - Pattern neurons - Smelling A,

then B then again A• Pattern of activity A1 ≠ A2 (even similar) →

No Rs ('88, '90) • “Nothing intrinsically R-l about dynamic

process until observer intrudes. It is experimenter who infers what observed activity patterns represents to in a subject, in order to explain his results to himself.” (Werner '88 in Freeman & Skarda '90)

• Nervous system = Dynamical system, constantly in motion

• Chaos - System continuously changes state; trajectory appears random but determined by equations

• Chaotic systems: Sensitivity to initial conditions = Small differences in initial values → Dissimilar trajectories

• Late exhalation: no input + behaves chaotically

• Inhalation: Chaos → Basin of one limit cycle attractors (Each attractor is a previously learned response to a particular odor)

• System - recognized an odor when lands in appropriate attractor

• Recognition response is not static!

• Odor recognition = Olfactory system alternates between relatively free-ranging chaotic behavior (exhalation) and odor-specific cyclic behavior (inhalation)

Objections • Computers are Dynamical Systems• Dynamical Systems are Computers• Dynamical Systems are Computable• “Description, not Explanation”(Dynamical models = Descriptions of data,

not explain why data takes form it does. Wrong Level - DST operates at micro, lower levels)

• Not focus on specifically cognitive aspects

• Both alternatives (computationalism & DST) = Necessary for explaining cognition

• Clark '97, '01

• Markman & Dietrich '00, '02

• Wheeler '96, '05

• Fisher & Bidell '98

• van Geert '94