Transcript Dynamical system approach

Dynamical Systems
Approach
(Teoria Sistemelor Dinamice)
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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 Terminology
1. 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 systems
8. Only linear systems are decomposable = modeled as collections of
separable components. Nonlinear systems = nondecomposable
9. 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 headbeliefs, 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 odorspecific 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)