Attractor neural networks and Attractor neural networks and concept formation in psychological concept formation in psychological

spacesspaces::mind from brainmind from brain??

Attractor neural networks and Attractor neural networks and concept formation in psychological concept formation in psychological

spacesspaces::mind from brainmind from brain??

Włodzisław Duch Włodzisław Duch Department of Informatics, Department of Informatics,

Nicholas Copernicus University, Toruń, Poland. Nicholas Copernicus University, Toruń, Poland.

www.phys.uni.torun.pl/~duch www.phys.uni.torun.pl/~duch

Bioinspired Computational Models of Learning Bioinspired Computational Models of Learning and Memory, Lejondal Castle, Sept. 2002and Memory, Lejondal Castle, Sept. 2002

Plan:Plan:Plan:Plan:• Intro: gap between neuroscience and psychology.

•From molecules to brain, forgetting the mind•Psychological spaces•Platonic mind model - static version•Some applications•Dynamic extensions•Related ideas•Conclusions

Cognitive Cognitive ScienceScienceCognitive Cognitive ScienceScienceThe 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?

Very few general laws in psychology (mostly psychophysical).Psycho-logy lost the soul (psyche)?

Cognitive science: mixture (syntopy) of cognitive psychology, neurosciences, AI, linguistics, philosophy of mind, psychophysics, anthropology ...

No central model of mind in cognitive science.

Philosophical problems in foundations of cognitive sciences.

Mind the GapMind the GapMind the GapMind the GapGap between neuroscience and psychology: cognitive science is 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, a potentially conscious subset of brain processes.

How to approximate the dynamics of the brain to get satisfactory (geometric) picture of the mind?

From molecules ...From molecules ...From molecules ...From molecules ...

10-10 m, molecular level: ion channels, synapses, membraneproperties, neurochemistry, biophysics, psychopharmacology, mind from molecular perspective (Ira Black)?

10-6 m, single neurons: biophysics, computational neuroscience (CS), compartmental models, spikes, LTP, LTD, neurochemistry & neurophysiology.

10-3 m, neural assemblies: cortical columns, multielectrode & large electrode recordings, microcircuits, neurodynamics,neuroscience, CS.

10-4 m, small networks: neurodynamics, recurrence, spiking neurons,

synchronization, neural code (liquid?), memory effects, multielectrode recordings, neurophysiology, CS.

… … to behavior.to behavior. … … to behavior.to behavior.10-2 m, mesoscopic networks: self-organization, sensory and motor

maps, population coding, continuous activity models, mean field theories, brain imaging, EEG, MEG, fMRI.

10-1 m, transcortical networks, large brain structures: simplified models of cortex, limbic structures, subcortical nuclei, integration of functions, concept formation, sensorimotor integration, neuropsychology, computational psychiatry ...

And then a miracle happens …

1 m, CNS, brain level: intentional behavior, psychology, thinking, reasoning, language, problem solving, symbolic processing, goal oriented knowledge-based systems, AI.

Where is the inner perspective?

Usually: transcortical networks => finite state automata => behavior

Alternative: Platonic model => mental events.

Static Platonic model: motivationStatic Platonic model: motivationStatic Platonic model: motivationStatic Platonic model: motivation

Plato believed in reality of mind, ideal forms recognized by intellect.

A useful metaphore: perceived mind content is like a shadow of ideal, real world of objects projected on the wall of a cave.

(drawing: Marc Cohen)

Real mind objects: shadows of neurodynamics?

Physics and psychologyPhysics and psychologyPhysics and psychologyPhysics and psychologyR. Shepard (BBS, 2001):

psychological laws should be formulated in appropriate psychological abstract spaces.

Physics - macroscopic properties results from microscopic interactions.

Description of movement - invariant in appropriate spaces:

• Euclidean 3D => Galileo transformations;

• (3+1) pseudo-Euclidean space => Lorentz x-t transformations;

• Riemannian curved space => laws invariant in accelerating frames.

Psychology - behavior, categorization, results from neurodynamics.

Neural networks: microscopic description, too difficult to use.

Find psychological spaces that result from neural dynamics and allow to formulate general behavioral laws.

P-spacesP-spacesP-spacesP-spacesPsychological 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 (PCP), geometry of psychological spaces as alternative to logic.

A complete theory of cognition, action, learning and intention.

PCP network, society, journal, software …

P-space definitionP-space definitionP-space definitionP-space definitionP-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 should have (Shepard 1957-2001): • minimal dimensionality• distances that monotonically decrease with increasing similarity

This is done with multi-dimensional non-metric scaling, reproducing similarity relations in low-dimensional spaces.

Laws of generalizationLaws of generalizationLaws of generalizationLaws of generalizationShepard (1987), Universal law of generalization. Tenenbaum, Griffith (2001), Bayesian framework unifying set-theoretic approach (introduced by Tversky 1977) with Shepard ideas.

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.

Minds work in low D!Minds work in low D!Minds work in low D!Minds work in low D!Mind uses only those features that are useful to act/decide. The structure of the world is internalized in the brain.

3 examples of elegant low-D mental principles in vision:

• In a 3-D vector space, in which each variation in natural illumination is cancelled by application of its inverse from the three-dimensional linear group of terrestrial transformations of the invariant solar source, color constancy is achieved.

• Positions and motions of objects represented as points and connecting geodesic paths in the 6-D manifold (3-D Euclidean group and 3-D symmetry group of each object) conserve their shapes in the geometrically fullest and simplest way.

• Kinds of objects support optimal generalization/categorization when represented as connected regions with shapes determined by Bayesian revision of maximum-entropy priors.

Object recognitionObject recognitionObject recognitionObject recognitionObject recognition theory, S. Edelman (1997) Second-order similarity in low-dimensional (<300) space is sufficient. Population of columns as weak classifiers working in chorus - stacking.

Static Platonic modelStatic Platonic modelStatic Platonic modelStatic Platonic modelNewton introduced space-time, arena for physical events.Mind events need psychological spaces.

Goal: integrate neural and behavioral information in one model, create model of mental processes at intermediate level between psychology and neuroscience.

Static version: short-term response properties of the brain, behavioral (sensomotoric) or memory-based (cognitive).

Applications: object recognition, psychophysics, category formation in low-D psychological spaces, case-based reasoning.

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?How to make static model?How to make static model?How to make static model?From neural responses to stimulus spaces.

Bayesian analysis of multielectrode responses (Földiak). P(ri|s), i=1..N computed from multi-electrode measurements The posterior probability P(s|r) = P(stimulus | response) Bayes law:

1

1 2

' 1

( ) || | , ..

( ') | '

N

ii

N N

is i

P s P r sP s r P s r r r

P s P r s

Population analysis: visual object represented as population of column activities.

Same for words and abstract objects (evidence from brain imaging).

Semantic memorySemantic memorySemantic memorySemantic memoryAutoassociative 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 (Rij rij)2

from internal neural activations;

from original 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-spacesFrom neurodynamics to P-spacesFrom neurodynamics to P-spacesFrom neurodynamics to P-spaces

Modeling input/output relations with some internal parameters.

Walter Freeman: model of olfaction in rabbits, 5 types of odors, 5 types of behavior, very complex model in between.Simplified models: H. Liljeström.

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.

More neurodynamicsMore neurodynamicsMore neurodynamicsMore neurodynamicsAmit group, 1997-2001, simplified spiking neuron models of column activity during learning.

Formation of new attractors =>formation of mind objects.

PDF: p(activity of columns| given presented features)

Stage 1: single columns respond to some feature. Stage 2: several columns respond to different features. Stage 3: correlated activity of many columns appears.

Category learning.Category learning.Category learning.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 dimensions:

shape (square/triangle), color (black/white), size (large/small). 4 objects in each of the two categories presented during learning.

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.Canonical dynamics.Canonical dynamics.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:

22 2 2

2 2 2

2 2 2

2 2 2

1, , 3

4

3

3

3

V x y z xyz x y z

Vx yz x y z x

xV

y xz x y z yy

Vz xy x y z z

z

Corresponding feature space for relevant dimensions A, B

Inverse based ratesInverse based ratesInverse based ratesInverse based ratesRelative frequencies (base rates) of categories are used for classification:

if on a list of disease and symptoms disease C associated with (PC, I) symptoms is 3 times more common as R, then symptoms PC => C, I => C (base rate effect).

Predictions contrary to the base: inverse base rate effects (Medin, Edelson 1988).

Although PC + I + PR => C (60% answers) PC + PR => R (60% answers)

Why? Psychological explanations are not convincing.

Effects due to the neurodynamics of learning?

I am not aware of any dynamical models of such effects.

IBR explanationIBR explanationIBR explanationIBR explanationPsychological explanation: J. Kruschke, Base Rates in Category Learning (1996).

PR is attended to because it is a distinct symptom, although PC is more common.

Basins of attractors - neurodynamics; PDFs in P-space {C, R, I, PC, PR}.

PR + PC activation leads more frequently to R because the basin of attractor for R is deeper.

Construct neurodynamics, get PDFs. Unfortunately these processes are in 5D.

Prediction: weak effects due to order and timing of presentation (PC, PR) and (PR, PC), due to trapping of the mind state by different attractors.

LearningLearningLearningLearning

Neurodynamics Psychology

I+PC more frequent => stronger synaptic connections, larger and deeper basins of attractors.

Symptoms I, PC are typical for C because they appear more often.

To avoid attractor around I+PC leading to C, deeper, more localized attractor around I+PR is created.

Rare disease R - symptom I is misleading, attention shifted to PR associated with R.

Point of view

ProbingProbingProbingProbing

Neurodynamics Psychology

Point of view

Activation by I leads to C because longer training on I+PC creates larger common basin than I+PR.

I => C, in agreement with base rates, more frequent stimuli I+PC are recalled more often.

Activation by I+PC+PR leads frequently to C, because I+PC puts the system in the middle of the large C basin and even for PR gradients still lead to C.

I+PC+PR => C because all symptoms are present and C is more frequent (base rates again).

Activation by PR+PC leads more frequently to R because the basin of attractor for R is deeper, and the gradient at (PR,PC) leads to R.

PC+PR => R because R is distinct symptom, although PC is more common.

Automatisation of actionsAutomatisation of actionsAutomatisation of actionsAutomatisation of actionsHow does the sensorimotor and cognitive learning takes place?

Initially conscious decisions are needed, at the end it is automatic, subconscious, intuitive, and well-localized in the brain.

Formation of attractors during learning => model of Amit & co.

Reinforcement learning requires observing and evaluating the actions that the brain has planned and is executing.

Relating current performance to memorized episodes of performance requires evaluation + comparison (Gray – subiculum) followed by emotional reactions that provide reinforcement and increase neuromodulation, facilitating rapid learning. Working memory is essential to perform such complex task.Errors are painfully conscious and are remembered. Conscious experiences: transferring data from WM to motor cortex and memory in the ERTAS loop? Is this the main role of consciousness?

PsychophysicsPsychophysicsPsychophysicsPsychophysicsStatic Platonic model is useful for immediate, memory-based behavior. Local maxima of PDF are due to the potential activations of the long-term memory, “mind landscape” slowly changing with time.

Working memory, content of mind - currently active objects.

Psychophysical phenomena, like masking: the circle exposed for 30 ms is seen, but not if the ring follows.

Dennett (1991): Stalinist and Orwellian scenarios: preventing conscious experience or erasing the history?

How to describe such phenomena?

Masking in P-spaceMasking in P-spaceMasking in P-spaceMasking in P-spaceP-space: basic feature of objects. Mind state, object seen: initially blank screen, object O1 attractor. O1 is an active object, mind state has momentum and inertia.

A series of masking stimuli following each other – no time to settle in an attractor, no conscious experience at all?

External stimulus (circle) 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. Solid state physics: use effective mass, forget microscopic interactions.

Geometric propertiesGeometric propertiesGeometric propertiesGeometric propertiesGeometric 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.

, min ( ), ( ) /B

A

s A B L X t dX t dt dt

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.

Feature Space MappingFeature Space MappingFeature Space MappingFeature Space MappingPlatonic Model: inspiration for FSM (Duch 1994) - neurofuzzy system for modeling PDFs using separable transfer (fuzzy membership) functions. 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.

Recognition: find local maximum of the F(X;P) function.

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Np p

p p i i ii

np

p pp

g g x P

F W g

X P

X P X P

Intuitive thinkingIntuitive thinkingIntuitive thinkingIntuitive thinking

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.

Question in qualitative physics: if R2 increases, R1 and Vt are constant, what will happen with current and V1, V2 ?

Intuitive reasoningIntuitive reasoningIntuitive reasoningIntuitive reasoning5 laws are simultaneously fulfilled, all have the same representation:

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 consistent combinations out of 2187 (5%) exist.

5

1 2 1 21

( , , , , , , ) ( , , )t i i i ii

F V R I V V R R F A B C

Intuitive reasoning, no manipulation of symbols; heuristics: select variable giving unique answer. Soft constraints or semi-quantitative => small |FSM(X)| values.

Platonic mind model.Platonic mind model.Platonic mind model.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 attention mechanisms.Feelings: gradients in the global space.

Language of thought.Language of thought.Language of thought.Language of thought.Precise language, replacing folk psychology, reducible to neurodynamics. Mind state dynamics modeled by gradient dynamics in mind space, „sticking” to PDF maxima, for example:

where g(x) controls the „sticking” and (t) is a noise + external forces term.

Mind state has inertia and momentum; transition probabilities between mind objects should be fitted to transition prob. between corresponding attractors of neurodynamics (QM-like formalism). Primary mind objects - from sensory data.Secondary mind objects - abstract categories.

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inp

S

S X

S t M S t g M S t t

Some connectionsSome connectionsSome connectionsSome connectionsGeometric/dynamical ideas related to mind may be found in many fields:

Neuroscience: D. Marr (1970) “probabilistic landscape”. C.H. Anderson, D.C. van Essen (1994): Superior Colliculus PDF mapsS. Edelman: “neural spaces”, object recognition, global representation space approximates the Cartesian product of spaces that code object fragments, representation of similarities is sufficient.

Psychology: K. Levin, psychological forces.G. Kelly, Personal Construct Psychology.R. Shephard, universal invariant laws.

Folk psychology: to put in mind, to have in mind, to keep in mind (mindmap), to make up one's mind, be of one mind ... (space).

More connections More connections More connections More connections AI: problem spaces - reasoning, problem solving, SOAR, ACT-R, little work on continuous mappings (MacLennan) instead of symbols.

Engineering: system identification, internal models inferred from input/output observations – this may be done without any parametric assumptions if a number of identical neural modules are used!

Philosophy: P. Gärdenfors, conceptual spacesR.F. Port, T.van Gelder, ed. Mind as motion (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, Psych. Rev. (1997) Semantic for 60 k words corpus requires about 300 dim.

ConclusionsConclusionsConclusionsConclusionsPlatonic model - a unified paradigm for cognitive science? Complex neurodynamics replaced by simpler dynamics in P-spaces.Searching for low-dimensional representation of mental events.Relations between different levels of modeling are important, eg. recurrent neural network => psychological spaces.Useful technical/psychological applications of the static Platonic model.

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? Many choices.

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 ?And in the end ?

A lot of work to do ...A lot of work to do ...