Mind from brain:
psychological spaces and neuroscience.
Włodzisław Duch
Department of Computer Methods,
Nicholas Copernicus University, Toruń, Poland.
http://www.phys.uni.torun.pl/kmk
Plan:
• Intro: gap between neuroscience and psychology.
•
•
•
•
•
•
From molecules to mind
Platonic mind model - static version
Some applications
Related ideas
Dynamic extensions
Conclusions
Cognitive Science
Cognitive science: mixture (syntopy) of cognitive psychology,
neurosciences, AI, linguistics, philosophy of mind, psychophysics,
anthropology ...
No central model of mind in cognitive science.
Very few general laws in psychology (mostly psychophysical).
Psycho-logy lost the soul ?
Philosophical problems in foundations of cognitive sciences: mindbody problem, qualia, symbol grounding, Searle critique, the
binding problem, Fodor's critique of connectionist approach ...
The Central Paradox of Cognition: how can the structure and
meaning, expressed in symbols and ideas at the mental level, result
from numerical processing at the brain level?
Mind the Gap
Gap between neuroscience and psychology:
cognitive science - at best incoherent mixture of various branches.
Is a satisfactory understanding of the mind possible ?
Roger Shepard, Toward a universal law of generalization for
psychological science (Science, Sept. 1987)
“What is required is not more data or more refined data but a
different conception of the problem.”
• Mind is what the brain does.
How to approximate the dynamics of the brain to get satisfactory
picture of the mind ?
From molecules ...
to mind.
Static Platonic model: motivation
Plato believed in reality of mind, ideal forms recognized by intellect.
Perceived mind content is like a shadow of ideal, real world of objects
projected on the wall of a cave.
Real mind objects: shadows of neurodynamics.
R. Shepard (BBS, 2001): psychological laws should be formulated
in appropriate psychological abstract spaces.
Physics - macroscopic properties results from microinteractions.
Description of movement - invariant in appropriate spaces:
• Galileo transformations in Euclidean 3D;
• Lorentz transformations in (3+1) pseudo-Euclidean;
• Riemannian curved space, laws invariant in accelerating frames.
Psychology - categorization, behavior, results from neurodynamics.
Neural networks: microscopic description, too difficult to use.
Find psychological spaces resulting from neural dynamics, allowing for
general behavioral laws.
P-spaces
Psychological spaces:
K. Lewin, The conceptual representation and the measurement of
psychological forces (1938), cognitive dynamic movement in
phenomenological space.
George Kelly (1955), personal construct psychology, geometry of
psychological spaces as alternative to logic.
A complete theory of cognition, action, learning and intention.
P-space: region in which we may place and
classify elements of our experience,
constructed and evolving, „a space without
distance”, divided by dichotomies.
P-spaces (Shepard 1957-2001):
• minimal dimensionality
• distances that monotonically decrease
with increasing similarity
(multi-dimensional non-metric scaling).
Some evidence
Universal law of generalization, Shepard (1987)
Tenenbaum, Griffith (2001), Bayesian framework unifying set-theoretic
approach (Tversky 1977) with Shepard.
Generalization gradients tend to fall off approximately exponentially with
distance in an appropriately scaled psychological space.
Distance - from MDS maps of perceived similarity of stimuli.
G(D) = probability of response learned to stimulus for D=0, for many
visual/auditory tasks, falls exponentially with the distance.
More evidence
Object recognition theory, S. Edelman (1997)
Second-order similarity in low-dimensional (<300) space is sufficient.
Population of columns as weak classfiers working in chorus - stacking.
Static Platonic model
Newton: introduced space-time, arena for physical events.
Mind events: need psychological spaces.
Goal: integrate neural and behavioral information in one model, connect
psychology and neuroscience, create mind model at intermediate level.
Static version: short-term response properties of the brain, behavioral
(sensomotoric) or memory-based (cognitive).
Applications: object recognition, category formation in low-dimensional
psychological spaces, models of mind.
Approach:
• simplify neural dynamics, find invariants (attractors), characterize
them in psychological spaces;
• use behavioral data, represent them in psychological space.
How to make static model?
From neural responses to stimulus spaces.
Bayesian analysis of multielectrode responses (Foldiak).
P(ri|s), i=1..N computed from multi-electrode measurements
The posterior probability P(s|r) = P(stimulus | response)
Bayes law:
N
P  s | r   P  s | r1 , r2 ..rN  
P( s ) P  ri | s 
i 1
N
 P(s ') P  r | s '
i
s'
i 1
Population analysis: visual object represented as
population of column activities.
Same for words and abstract objects (evidence
from brain imaging)
Semantic memory
Autoassociative network, developing internal
representations (McClleland-Naughton-O’Reilly,
1995).
After training distance relations between different
categories are displayed in a dendrogram, showing
natural similarities/ clusters.
MDS mappings: min S (Rij-rij)2
from internal neural activations;
from orginal data in the P-space - hypercube, dimensions
for predicates, ex. robin(x)  {0, 1};
from psychological experiments, similarity matrices;
show similar configurations.
From neurodynamics to P-spaces.
Modeling input/output relations with some internal parameters.
Freeman: model of olfaction in rabbits, 5 types of odors, 5 types of
behavior, very complex model in between.
Attractors of dynamics in high-dimensional space => via fuzzy symbolic
dynamics allow to define probability densities (PDF) in feature spaces.
Mind objects - created from fuzzy prototypes/exemplars.
Case-based reasoning: static model.
Geometric properties.
Geometric representation of mental events should be understandable.
Problem of all Euclidean models: similarities are non-metric.
Re-entry connections between columns are not symmetric.
Asymmetric MDS requires change of perspective for each object.
Solution: Finsler geometry (ex: time as distance)
A curve X(t) parameterized by t, distance between t1=A, t2=B depends on
the positions X(t+dt) and derivative dX(t)/dt.
B
s  A, B   min  L  X (t ), dX (t ) / dt  dt
A
where L(.) is the metric function (Lagrangian in physics).
Distance = „action” , fundamental laws of physics have such form.
To get nonsymetric distance s(A,B), potential may be introduced, for
example proportional to probability density.
More neurodynamics.
Amit group, 1997-2001,
simplified spiking neuron
models of column activity
during learning.
Stage 1: single columns
respond to some feature.
Stage 2: several columns
respond to different features.
Stage 3: correlated activity of
many columns appears.
Formation of new attractors =>
formation of mind objects.
PDF: p(activity of columns,
given presented features)
Category learning.
Large field, many models.
Classical experiments: Shepard, Hovland and Jenkins (1961), replicated by
Nosofsky et al. (1994)
Problems of increasing complexity; results determined by logical rules.
3 binary-valued dim:
shape (square/triangle), color (black/white), size (large/small).
4 stimuli in each of the two categories presented.
Type I - categorization using one dimension only.
Type II - two dim. are relevant (XOR problem).
Types III, IV, and V - intermediate complexity between Type II - VI. All 3
dimensions relevant, "single dimension plus exception" type.
Type VI - most complex, 3 dimensions relevant, logic = enumerate stimuli in
each of the categories.
Difficulty (number of errors made): Type I < II < III ~ IV ~ V < VI
Canonical dynamics.
What happens in the brain during category learning?
Complex neurodynamics <=> simplest, canonical dynamics.
For all logical functions one may write corresponding equations.
For XOR (type II problems) equations are:
1 2
2
2 2
V  x, y, z   3 xyz   x  y  z 
4
V
x -3 yz -  x 2  y 2  z 2  x
x
V
y -3 xz -  x 2  y 2  z 2  y
y
V
z -3 xy -  x 2  y 2  z 2  z
z
Corresponding feature space for relevant
dimensions A, B
Inverse based rates.
Relative frequencies (base rates) of categories are used
for classification: if C is 3 times as coomn as R, and C is
associated with (PC, I) symptoms then PC => C, I => C.
Predictions contrary to the base: inverse base rate effects
(Medin, Edelson 1988).
Although PC + I + PR => C (60%)
PC + PR => R (60%)
Basins of attractors - neurodynamics;
PDFs in P-space {C, R, I, PC, PR}.
Psychological interpretation (Kruschke 1996):
PR is attended to because it is a distinct
symptom, although PC is more common.
PR + PC activation leads more frequently to R
because the basin of attractor for R is deeper.
Feature Space Mapping.
FSM (Duch 1994) - neurofuzzy system for modeling PDFs using separable
transfer (membership) functions.
Categorization (classification), extraction of logical rules, decision support.
Set up (fuzzy) facts explicitly as dense regions in the feature space;
Initialize by clusterization - creates rough PDF landscape.
Train by tuning adaptive parameters P;
novelty criteria allow for creation of new nodes as required.
Self-organization of G(X;P) = prototypes of objects in the feature space.
g p ( X; P )   g p ,i  xi ; Pi p 
N
p
i 1
F ( X; P)  W p g p  X; P p 
n
p 1
Recognition: find local maximum of
the F(X;P) function.
Dynamic approach.
Static model - responsible for immediate, memory-based behavior.
Local maxima of PDF - potential activations of the long-term memory.
Working memory, content of mind - currently active objects.
Masking: the circle exposed for 30
ms is seen, but not if ring follows.
Mind state - in attractor, near O1, active object, it has momentum and
inertia. External stimulus pushes the mind state towards O2.
A masking stimulus O3 close to O2 blocks activation of O2; no conscious
recall of the small disk is noted; priming lowers inertia.
Platonic mind model.
Feature detectors/effectors: topographic maps.
Objects in long-term memory (parietal, temporal, frontal): local P-spaces.
Mind space (working memory, prefrontal, parietal): construction of mind
space features/objects using attentional mechnisms.
Language of thought.
Precise language, replacing folk psychology,
reducible to neurodynamics.
Mind state dynamics - gradient dynamics in mind
space, „sticking” to PDF maxima, for example:
S (0)  X inp ;


S (t )    S M ( S ; t ) 1  g  M  S ; t    h (t )
where g(x) controls the „sticking” and h(t) is a
noise + external forces term.
Mind state has inertia and momentum;
transition prob. between mind objects should be
fitted to transition prob. between corresponding
attractors of neurodynamics (QM fromalism).
Primary mind objects - from sensory data.
Secondary mind objects - abstract categories.
Intuitive thinking
Question in qualitative physics:
if R2 increases, R1 and Vt are constant,
what happens with current and V1, V2 ?
Geometric representation of facts:
+ increasing, 0 constant, - decreasing.
Ohm’s law V=I×R; Kirhoff’s V=V1+V2.
True (I-,V-,R0), (I+,V+,R0), false (I+,V-,R0).
5 laws: 3 Ohm’s & 2 Kirhoff’s.
All laws A=B+C, A=B×C , A-1=B-1+C-1,
have identical geometric interpretation!
13 True, 14 False facts; simple P-space,
complex neurodynamics.
Intuitive reasoning
5 laws are simultanously fulfield, all have the same representation:
5
F (Vt , R, I ,V1 ,V2 , R1 , R2 )   Fi ( Ai , Bi , Ci )
i 1
Question: If R2=+, R1=0 and V =0, what can be said about I, V1, V2 ?
Find missing value giving F(V=0, R, I,V1, V2, R1=0, R2=+) >0
Suppose that variable X = +, is it possible?
Not, if F(V=0, R, I,V1, V2, R1=0, R2=+) =0, i.e. one law is not fulfilled.
If nothing is known 111 out of 2187 combinations are consistent (5%).
Intuitive reasoning, no manipulation
of symbols; heuristics: select
variable giving unique answer.
Soft constraints or semi-quantitative
=>
small |FSM(X)| values.
Some connections.
Geometric/dynamical ideas related to mind may be found in many fields:
Philosophy: „Mind as motion”, ed. R.F. Port, T. van Gelder (MIT Press 1995)
Linguistics: G. Fauconnier, Mental Spaces (Cambridge U.P. 1994).
Mental spaces and non-classical feature spaces.
J. Elman, Language as a dynamical system (San Diego, 1997).
Stream of thoughts, sentence as a trajectory in P-space.
Psycholinguistics: T. Landauer, S. Dumais, Latent Semantic Analysis Theory,
Psych. Rev. (1997) Semantic for 60 k words corpus requires about 300 dim.
Neuroscience: Anderson, van Essen (1994): Superior Colliculus maps as PDFs
AI: problem spaces - reasoning, problem solving, SOAR, ACT-R
Folk psychology: to put in mind, to have in mind, to keep in mind, to make up
one's mind, be of one mind ... (space).
Conclusions
Platonic model - a unified paradigm for cognitive science?
Complex neurodynmics replaced by simpler dynamics in P-spaces.
Low-dimensional representation of mind events.
Relations between different levels of modeling are important, eg. recurrent
neural network => psychological spaces.
Useful technical/psychological applications of the static model (FSM).
Open questions:
High-dimensional P-spaces with Finsler geometry needed for visualization
of the mind events - will the model be understandable?
Mathematical characterization of mind space?
Challenge: neurodynamical model => P-spaces for monkey categorization.
Large-scale simulations of models of mind are missing but ... hierarchical
approach: networks of networks in simulated environment, is coming.
At the end of the road: physics-like theory of events in mental spaces ?
And in the end ?
A lot of work to do ...