Dynamical Systems Approach

(Teoria Sistemelor Dinamice)

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

• Ecological approach (Gibson 66, 79)

• Ecological psychologists (Turvey et al. 81)

• Turvey Kluger Kelso (80s)-Motor coordinatio

• 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 (95)

• Dynamical and computational approaches to cognition are fundamentally different

• Dynamical approach = Kuhnian revolution

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

• Discrete static Rs vs. Mutually + simultaneously influencing changes

• 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: Height-weight; 2 neurons; 4 or 60 neurons = High dimensional state space

• Dynamic systems theory (DST) - Physics

• Dynamical system: Set of state variables + dynamical law (governs how values of state variables change with time)

• The 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

Dynamical Systems Terminology1. The state space of a system = space defined by set of all possible states

system could ever be in.2. A trajectory or 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 Systems - modeled with nonlinear differential equatio-s = nonlinear systems8. Only 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, p. 36)

• Goal: Changes over time (and change in rate of change over time) of a system (Clark 2001)

• 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 described by differential equations)

• Geometric images → Trajectory of evolution• Collective variables (relations bet. variables)• Control parameters = Factors affect evolut.• Ex: Solar system - Position + Momentum of

planets - Mathematical laws relate changes over time → A math-ical dynamical model

• Rates of change: Differential equations(van Gelder 1995, + Port 1995)

• 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 1998)

• DST: Time – involved

• Geometric view of how structures in state space generate/ constrain behavior + emergence of spatiotemporal patterns

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

• Dynamics = Geometry of behavior (Abraham & Shaw 1983; Smale 1980 in Crutchfield, 95)

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 (Gelder 95)

Centrifugal governor (G):

• Norepresentational + noncomputational

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

• Continuously reciprocal causation through mathematical dynamics

• Clark (p. 126)

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

• 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

We might also add 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 has argued several times (97, 01, 08; Clark and Toribio 94 (Miner & Goodale ’95, ventral vs. dorsal); Clark and Grush 1999) 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: Pattern + 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)

• DST rejects Rs, introduces time

• Bodily actions (T&S 98, child’s walking)

• Movement of fingers (HKB 87, Kelso 95)

→ Extrapolate from sensoriomotor processes to cognition processes!

• No decision making/contrafactual reason

• 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, p. 39)

• Against REC - Clark and Toribio (94): certain tasks cannot be accomplished without Rs

• “Hungry Rs problems” (decision making, counterfactual reasoning) - 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 97a)

• Compromise: Milner and Goodale (95), Norman (02)

• TDS - Change:

a) Interactions betw. (ensembles) neurons

b) Constitutive relations betw. Rs

→ No prediction but explanation

• Dynamics among Rs

(Fisher and Bidell 98; van Geert 94)

• Radical dynamicists: Cognition = Result of evolution of perception + sensoriomotor control systems

• Dynamical models - “having” R-s: Attractors, trajectories, bifurcations, and parameter settings

→ DS store knowledge + Rules defined over numerical states

(van Gelder & Port 95)

• DST manages 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”

• If cognition = particular structure in space and time, mission - discover how “a stable state of brain in context of body + environ”. (van Gelder and Port 95)

Distinction on-line/off-line processes

• “Off-line cognition = Decision making + contrafactual reasoning

• Subject thinks about Rs in their absence” → Not rejecting computation of brain that presuposses Rs (Clark)

Van Gelder’s in BBS (98)

• “Open Peer Commentary”: Many commentaries - DST can explain only perception + sensoriomotor control systems, not cognitive processes

• Van Gelder & Port: Everything in motion→ No static discrete Rs → “Everything is simultaneously affecting everything else.”

Cognitive processes

• Conceptualize in geometric terms

• Unfolds over time = How total states system passes through spatial location

• Unfold in real time their behaviors - by continuities and discretenesses

• Structures - not present from first moment, but emerge over time - operate over many times scales and events at different times scales

(van Gelder & Port 95)

Skarda & Freeman’s model of olfactory bulb

• Freeman’s network (85) (Bechtel, p. 259) • 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)

• Neural system does not exhibit behavior that can be modeled with point attractors, except (anesthesia or death)

• Instead, 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

Excitatory + inhibitory neurons (different cell types) = Separate components:

• Second-order nonlinear diff-tial equations

• Coupled via excitatory/inhibitory connec-s

→ Interactive network

• Conditioned rabbits respons to odors

• EEG recordings:

- Exhalation = Pattern of disorderly

- Inhalation = More orderly

• 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)

• Freeman’s model - Logistic equation (figure 8.2, p. 242) = Chaotic dynamics in a region with values of A beyond 3.6.

• Within this region there existed values of A for which dynamics again became periodic

→ Moving from chaotic to temporarily stable (and back to chaotic ones) through small changes in parameter values

• Ability could be extremely useful for a nervous system (Bechtel 02)

Haken-Kelso-Bunz model (fingers’ movements)

• 2 basic patterns (in phase-antiphase)

• Increase oscillation frequency in time:

1) People: in antiphase motion → in-phase (at a certain frequency of movement ‘‘critical region’’)

2) Subjects: in-phase = NO in phase motion

2 stable patterns of low frequencies,

1 pattern = Stable, frequen. beyond critical point

↔ 2 stable attractors at low frequencies bifurcation at a critical point → 1 stable attractor at high frequencies (Kelso in Walmsley 2008)

“coordination - not as masterminded by a digital computer … but as an emergent property of a nonlinear dynamical system self-organizing around instabilities” (van Gelder 98)

Fischer & Bidell (98), van Geert (93)• Continuity + discreteness • Dynamical combinations of R-s → Dynamical structuralism: Variations within

stability + Structure in motion[Ecological, dynamic, interactive, situated,

embodied approaches]

Melanie Mitchell (98)

• Theory of cognition: both computational and dynamical notions

• How functional information-processing structures emerge in complex dynamical system

• DST - Do not explain information-processing content of states over which change is occurring because either tasks with no complex information processing or high-level information-related primitives pp. a priori

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 • Complexity + Structure (van Gelder 98)

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

• Clark 97, 01

• Markman & Dietrich 00, 02

• Wheeler 96, 05

• Fisher & Bidell 98

• van Geert 94

• “no decomposition into distinct functional modules + no aspect of agent’s state need be interpretable as a R. (Beer 95, p. 144)