Lecture Notes in Biomathematics

Pellionisz, A. & Llinas, R. (1982) Tensor Theory of Brain Function. The Cerebellum as a Space-Time Metric. Chap~r 23. in: Competition and Cooperation in Neural Nets. Proceedinqs of the u.s.-Japan Joint Seminar held at Kyoto, Japan February 15..;..19, 1982. (ed. by S.Arnari and M.A·. ~bib) , Lecture Notes in Biomathematics, #45 (managing ed. S. Levin), Springer-Verlag~ Berlin, Heidelberg, New York, pp.394-417.

Lecture Notes in Biomathematics Managing Editor: S. Levin

45 Competition and Cooperation in Neural Nets Proceedings of the U.S.-Japan Joint Semir.ar held at Kyoto, Japan February 15-19,1982

Edited by S. Amari and MA. Arbib

G~J

~l!J

Springer-Verlag Berlin Heidelberg New York 1982

Editorial Board

W. Bossert H. J. Bremermann J. D. Cowan W. Hirsch S. Karlin J. B. Keller M. Kimura S. Levin (Managing Editor) R. C. Lewontin R. May G. F. Oster A S. Perelson T. Poggio L A Segel Editors

S.Amari University of Tokyo Dept of Mathematical Engineering and Instrumentation Physics Bunkyo-ku, Tokyo 113, Japan MA.Arbib Center for Systems Neuroscience Computer and Information Science, University of Massachusetts Amherst, MA 01003, USA

AMS Subject Classifications (1980): 34 A 34, 34 D 05, 35 B 32, 35.0 20, 58 F 13, 58F14,60G40, 60J70, 68D20, 68G10, 92.06, 92A15,92A27 ISBN 3-540-11574-9 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-11574-9 Springer-Verlag New York Heidelberg Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re·use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsgesellschaft Wort", Munich. (' by Springer·Verlag Berlin Heidelberg 1982 Printed in Germany Printing and binding: Beltz Offsetdruck. Hemsbach/Bergstr. 2141/3140·543210

23

CHAPTER

TENSOR THEORY OF BRAIN FUNCTION. THE CEREBELLUM AS A SPACE-TIME METRIC

Andr~s PELLIONISZ & Rodolfo LLINAS Department of Physiology & Biophysics New York Univeversity Medical Center 550 First Ave, New York, 10016, USA

1. EXPERIMENTAL NEUROSCIENCE AND BRAIN THEORY -TO KNOW AND TO UNDERSTAND THE BRAIN-

1.1.

THE GOAL OF

BRAIN

THEORY

is

to

transform

knowledge

relating

to

the

properties of the central nervou=s system into an undet"standing of brain function• As in other fields of science using theories as heuristic tools of understandinq, not only the quality of a particular theory has to be carefully measure
usefulness

must

also

be

considered.

Usefulness

serves

independent test of the deqree of understandinq provided by "

then

as

an

qi ven abstraction.

An understanding, e.q. of the cerebellum, may be qauqed by the ability of a theory

to

provide

a

"blueprint"

of

a

device

capable

function, in this case motor coordination.

of

accomplishing

the

described

Such an understandinq of brain function

has qreat potential for robotics, an ultimate beneficiary of neuroscience.

In thi9

paper we wish to make the point that the Tensot' Network Theory of Central Nervous System

(a qeometrical abstt"action and qeneralization of

the

approaches) is capable of yielding a usable understandinq.

known

vector-matrix

We offer in support of

this assertion a tensorial network model capable of qeneratinq motor coordination via cerebellar-type neuronal circuits.

1.2. PRINCIPAL METHODS OF BRAIN THEORY: CONCEPTUALIZATION AND ABSTRACTION. achieve

its

qoals,

abstraction,

usinq

brain

theory

model-makinq

relies

and

chiefly

mathematics

on

as

conceptualization

its

basic

tools.

To and Such

enterprise in all fields of science commences by collectinq data that are perceived relevant for the goal of t"eseat"ch. collectively

addressed

as

Havinq qathered a

KNOWLEDGE,

abstraction

is

set of

SYSTEM which relates these facts to a given global function. relations amonq facts,

fragmented facts,

introduced

to

set of points.

Therefore,

buildinq an understandinq implies the construction in our minds

approach

rept"esentation

may

be

the

addressed as UNDERSTANDING, is similar to the establishing

of the qeometry that may exist among a

qeometrical

explore

Revealinq a system of

of

implemented

the by

relations

in

collaborative

the

external

efforts

amonq

in our view,

of

world.

an

internal

The

scientists.

above For

395

instance,

in our particular case, we found that our interest and expertise were

ccxnplementary and could be integrated into formulating a theory by combining the functional view, by R.L., indicating that CNS function is to be understood in the manner of geometry, with the implementation of such view via tensor analysis, by A.P., and by continuing a dialogue for close to.a decade so far. In this paper we

summarize

our view that

the

fundamental

concept of brain

function is the establishment 'of related geometries (that the brain is a gecxnetric object. in the KllONIAN sense, PELLIONISZ

&

LLINAS, 1979b) •

We stz:ess that, as of

today, the best abstract language for describing geometrical properties in a formal manner appears to be tensor analysis (PELLIONISZ & LLINAS, 1979a, 1980, 1982).

For

an·overview of broader aspects of this approach see, e.g. MELNECHUK, 1979.

1.3. THE VARIETY OF CONCEPTS OF ORGANIZATION OF CNS FUNCTION.

Firstly,

the

most significant initial step in the process of abstraction is the introduction of AN APPROPRIATE FUNDAMENTAL CONCEPT. datal

becomes

thereby

possible.

Model-making (a conceptual representation of The

next

step

of

abstraction

is

then

the

rigorously self-consistent formalization of the concept that we may call a theory. Secondly, important.

finding

the

-most

SUITABLE

METHOD

OF

ABSTRACTION

is

equally

Indeed, no matter how sound and powerful the mathematical apparatus may

be, the results of the abstraction can only be as good as the quality of the basic concept it relies on.

Metaphorically speaking,

-c.f. ARBIB, 1972-, the need for

having both a proper concept and a suitable method of abstraction is comparable to the need, when fishing, of both knowing where to fish and also of net to entrap them securely.

hav1n~

a proper

In these terms, formulating a brain theory without a

suitable mathematical system of abstraction (e.g. as in ECCLES, 1981) is comparable to fishing in the proper spot but with bare hands.

On the other hand, having the

finest and strongest net a·nd casting it into empty waters later-

i~

-as pointed out

McCULLOCH & PITTS, 1943) is, again, futile.

:Indeed,

THE

PRIMARY

TASK

OF

CONCEPTUALIZATION

IS

FUNDAMENTAL ORGANIZING PRINCIPLES OF BRAIN FUNCTION. significant 1906).

(e.g.

views

relating

to

brain was

the

THE

SEARCH

FOR

ADEQUATE

Historically one of the most

concept

of

REFLEXES

( SHERRINGTON,

The next major change was to consider brain function as embodying LOGICAL

CALCULUS (McCULLOCH

&

PITTS, 1943).

approach missed its mark,

However, it is quite evi.,ent today that this

as the control paradigm used by the brain in say,

coordinated goal-oriented sensorimotor action,

is not

~OOLEAN.

A different,

a

but

scxnewhat primitive concept of motor actions is that of assuming the existence of a "LOOKUP-TABLE" (RAYBERT, 1977, 1978), i.e. to suppose that a motor action relies on the selection of a

~Dvement-sequence

from a stored set of solutions.

Yet another

concept, borrowed from engineering, is the one of LINEAR SYSTEM THEORY

(e.g.

for

the representation of eye movements, c.f. ROBINSON, 1968) which provides a highly

396

practical

framework

for

sensorimotor systems.

an

abstract

mathematical

interpretation

of

special

In highly nonlinear visuomotor systems (such as the visual

control of a fly's navigation) NONLINEAR SYSTEM THEORY has provided algorithms for the movement-control (POGGIO &· REICHARDT, 1981).

However, these authors do admit

that

relies

"• •• the visual

well ••• ", e.g.

control of flight

Taylor-se~ies

certainly

expansion (ibid).

on

other

algorithms

as

Recently BIZZI (1981) proposed the

view that regards the control paradigm used in the visuomotor system primarily as a ~or

COMPUTATIONAL problem.

a review of various other organizing concepts proposed

for brain function, see SZENTAGOTKAI

1.4.

&

ARBIB (1975),

THE GRADUALLY INCREASING LEVEL OF ABSTRACTION

IN NEUROSCIENCE.

At the

level of simple description of the morphology and physiology of neuronal networks, abstraction (relating the detailed knowledge to the ultimate function) was avoided altogether. geometry,

Such a traditional view of the brain was directed at the structural aiming

at

establishing

the

system

of

different types of c'ells in neuronal networks. circuitry drawings by RAMON. Y CAJAL ( 1911). intuitively clear,

it has

serious

spatial

relations

among

the

This approach is best known from While such a descriptive method is

limitations

as

it

does

not

explain

how

the

network provides the global function, nor offers a way to handle its complexity in a quantitative manner.

A more detached way to relate the structure to function is

the use of computer models (e.g. PELLIONISZ, 1970, PELLIONISZ et al., latter based on quantitative histological data (LLINAS, 1971).

1977), the

While such mo,iels

make it possible to address quantitative properties of large networks, they still do not explain their function. A somewhat

more

abstract handling

of

the, functioning

of

neuronal

relies on the notion of "patterns" of activities over a field of neurons.

networks It is of

historical interest to note that the idea of patterns was first used in a poetic manner by SHERRINGTON Later,

such

patterns

(1906) were

who envisioned the

handled

either

by

brain

as

hand-drawn

an

"enchanted

sketches

(e.g.

loom". ECCLES,

1981), or by mathematical abstraction (e.g. BEURLE, 1962, KATCHALSKY et al., 1974) or by computer-simulation graphics (e.g. PELLIONISZ, 1970). studies AMARI

of

the

ARBIB

&

intrinsic ( 1977)

dynamical

developed

features

the

of

concept of

For exact mathematical

excitation-inhibition competition

and

patterns,

cooperation

in

neuronal nets. The concept of "patterns" of activities raises the deep question of whether nervous

function

is

to be

deterministic or rather, consensus stochastic

on

the

methods

understood

as a

deterministic may

still

at

an

abstract

stochastic process.

be

nature applied

of

brain

both

for

level

as

a

fundamentally

While there appears to be a function,

it

investigation

is

clear

(e.g.

that

HARTH

&

TZANAKOU, 1974) and even for functional analysis of neuronal systems, given that

397.

the level of description is close to the QUANTA inherent in the described phenomena (c.f.

HOLDEN,

1976).

An

important

step

towards

a

more

abstract

mathematical

analysis of local neuronal mechanisms in networks is represented by the worlt of & WILLSHAW, 1976, and LEGENDY, 1978, who already used some aspects of the

MAL~BURG

next most important step of abstraction, the vector and matrix approach.

2. THE VECTOR-MATRIX APPROACH IN BRAIN THEORY

While

the

vector-matrix

approach

opens

the

way for

tremendous

mathematical brain theory, technically it appears almost trivial. and

M output

neurons,

the

morphological

system

mathematically characterized by an array of N x activity of the quantities

(a

frequencies

interconnections

M quantities

input can also be formally described

vector,

of

of

the

where

N

or

the

components

M neurons).

can

be,

However,

as

advances

for

vector-matrix approach has also proved to be treacherous.

can

(a matrix).

by an ordered

we

in

Having N input

example,

The

set the

elaborate

be

of N firing

below,

the

Indeed, it generated an

enormous impetus for the mathematically-minded to build up the apparatus of the abstraction, but the biological relevance of the application was not always beyond question.

While the formalism utilized by many workers was usually mathematically

appropriate this MEANS OF ANALYSIS occasionally became a GOAL in itself, with its advancement biological

the

vector-matrix

problems.

representation

However,

before

not

pointing

always out

the

remaining

relevant

deficiencies

that

to we

perceive in the applications on record, a brief overview of the works in question seems appropriate. Even (PITTS

though

vectorial

& McCULLOCH,

notation

1947

and

WIENER,

applicability was not pointed out. time

the

overwhelming

(McCULLOCH computers

&

and

PITTS,

view

was

1943).

brains

was

in brain

theory

1948),

was

the

introduced

profound

quite

reason

early

for

its

This is not altogether surprising since at that that

neuronal

Accordingly,

hardly

nets

the

conceptual

distinguishable,

overshadowed brain theory for more than a decade.

perform

and

logical

operations

difference

between

computer

science

thus

Not until von NEUMANN

( 1957)

stated the important theoretical difference between the two information processing systems did these two .efforts gradually grow distinct. transition to the more (1967).

recent,

An interesting paper in the

non-BOOLEAN approaches

is

that

of

von

FOERSTER

His work was technically a vector-matrix approach, but the conceptology

was mixed: the networks were in part assumed to perform operations of mathematical logic, but at other times the network was Modern

vector-matrix

PERCEPTRON (ROSENBLATT,

approaches

to

characterize~

as a computational machine.

brain theory

can

1962, MINSKY & PAPERT, 1969).

be

dated

back

to

the

Percept ron is basically a

computational geometric theory that was applied, unfortunately, to conjectural

398

Therefore,

circuits rather than to real neuronal systems. gradually moved away from neuroscience

this

line of

into the realm of artificial

research

intelligence

which aims at developing brain-like machines independent of the modes of operation of

the

brain.

ANDERSON

(see,

More

directly

e.g.

1981).

independent line of

wo~k

related His

t.o

early

real

work

by COOPER (1974).

brains

was

(ANDERSON

the

et

approach

al.,

1972)

using

extensive

vector-matrix

led

to

by an

In the field of associative memory, one

of the highest brain functions, pioneering work was done by KOHONEN. while

taken

calculations,

reveals

limitation of the vector-matrix approach quite explicitly:

a

His approach,

common

conceptual

"considerations of the

network should be understood as a SYSTEM-THEORETICAL APPROACH only; no assumptions will be made at this stage about the actual data represented by the patterns of activity" (KOHONEN et al., 1981). An altogether different approach was represented by papers that focused on the physical geometrical properties of the circuitry (e.g. SCHWARTZ, 1977, COWAN, 1981).

MALSBURG & WILLSHAW,

1976,

While the central concept in these papers appears to

be geometry, the relation of the physical and functional geometries of the networks has not been established by a

conceptually homogeneous treatment merging the two

together.

2.1. one

THE VECTOR-MATRIX APPROACH IN CEREBELLAR RESEARCH.

of

us

(A.P.,

mentioned

in

SZENTAGOTHAI,

1968)

An early notion from

related

to

a

vector-matrix

approach to compare cerebellar circuitries with the STEINBUCH-matrix extensive studies

mathematical as

well,

discriminators GREENE

(GROSSBERG,

regarding

spaces".

analyses

having

motor

by

GROSSBERG

featured 1970).

Purkinje

have

relevance

cells

Conceptually quite

generators

as

devices

as

to

( 1961 l.

the

cer

the

geometrical

properties

of

the

space

particular space involved is well known.

can

only

be

ignored

THE CONCEPT OF VECTOR,

because

HOWEVER,

the

IS NOT

LIMITED TO THE PRIMITIVE .CASE OF CARTESIAN,

THREE-DIMENSIONAL ORTHOGONAL VECTORS

WITH EUCLIDEAN GEOMETRY IN THE VECTORSPACE.

THEREFORE, WHEN DEALING WITH "BRAIN

VECTORS", NONE OF THE ABOVE SIMPLIFYING GENEROSITIES CAN BE PERMITTED.

2.4.

THE ABSTRACTION AND GENERALIZATION OF THE VECTOR-MATRIX TREATMENT INTO A

TENSORIAL APPROACH. neurons

is

only

Fundamentally the vectorial description of the activity of N

permissible

if

the

general

definition

of

the

vector,

as

a

mathematical point of an abstract space, is used: " ••• it is essential _to adopt the fundamental

convention

of

using

the

term

point

of

an

abstract

N-dimensional

manifold (N being any positive integer) to denote a set of N values assigned to any N variables

This

is

an

extension

obvious

of

the

use

of

the

term in the one-to-one correspondence which can be established between pairs or triplets of co-ordinates and the points of a plane or space ••• " (LEVI-CIVITA, 1926). Thus,

a

pivotal point of our tensor-argument

is

that

in the

CNS

the

same

event-point (an invariant) that can be externally represented, e.g. by a Cartesian vector

of

M(x,y,z,t)

mathematical

vector

may

internally

F(f 1 ,f 2 , ••• ,fn)'

be

represented

another

ordered

the firing frequencies of a set of N neurons.

in set

the of

CNS

by

numbers,

another such

as

The tensor concept hinges on the

fact that the two vectorial descriptions (M and F) are equally appropriate; i.e., neither is, a priori, preeminent.

When using such vectors (as F) in a generalized

sense, it is clear that one has both a mathematical instrument, an ordered set of quantities,

(i.e. the mathematical vector)

and a

physical entity (the latter, by

its nature, being invariant to the existence of any set of quantities that may be arbitrarily assigned to it). vector is implemented by and

the

mathematical

mean~

vector

relation exists between a assigned to it,

The co-ordination of an invariant with a mathematical of a reference-frame, through which the invariant become

related

entities.

Given

that

an

abstract

single invariant and any mathematical vector that is

it follows that if several different vectors are assigned to an

invariant (e.g. by using different frames of reference) then all such vectors will also be related to one another. generalized concept of

vectors

Thus,

concept expressing their relation. A MATHEMATICAL BELONGING TO

DEVICE

THAT

a

further

that

is

abstraction:

implied the

in the

generalized

THE GENERAL CONCEPTUAL DEFINITION OF A TENSOR:

EXPRESSES

INVARIANT ENTITIES.

the abstraction

invokes

The

THE

RELATIONS

relation of

AMONG

the

MATHEMATICAL

invariant

itself

VECTORS to

the

403

mathematical vector is formally expressed by a tensor of rank one (the generalized concept of a "vector"), while the relation of two mathematical vectors, belonging to the· same invariant, is expressed by a tensor of rank two (a generalization of the concept of "matrix"). same· phys1cal entity, another.

Given that both mathematical vectors are assigned to the

i t follows

that M and l' are tensorially

This consideration provides a. basis for

a

related

to

one

tensorial treatment of

the

internal representation of external invariants by the brain.

on

The

e~act

the

properties

mathematical qualities of any particular type of tensor depend both of

the

physical

invariants

in

question

as · well

as

on

the

properties of the mathematical vectors attributed to them by particular kinds of reference frames.

inva~iants

For example, if one deals with

such as a location in

the physical space and assigns three-dimensional orthogonal frames of reference to them, then the relations among the resulting mathematical vectors will be expressed by so-called Cartesian tensors

always,

by

definition,

(of the second rank l.

three-by-three matrices,

according to both covariant and contravariant rules reference (c.f. TEMPLE, 1960). a

Such Cartesian tensors are

with

components upon

that

rotating

the

transform frame

of

However, if one assigns to the same invariant both

covariant N-dimensional- and a

contravariant H-dimensional mathematical vector

(by using two different frames of reference, having N and H axes respectively, and using a different method of assigning the components in each),

then the relation

between these two mathematical vectors will be expressed by a tensor which has N x H components, transforming differently than those of the previous tensor, since the latter is obviously non-Cartesian.

The lesson from this example is,

that for an

application in which fundamental features of the applied vectors are unknown, one has to be very careful not to define tensors overly strictly, just as vectors must not be limited to Cartesian vectors

if one deals with a

RIEMANNIAN space.

For

further reading of tensors in general unrestricted coordinate systems consult, e.g. SYNGE & SCHILD, 1949 or WREDE, 1972. In defining a vector as mathematical point, a vector itself becomes the least interesting general

entity:

definition

an

ordered

clearly

set

a

trivial

task

to

quantities.

distinguishes

invariant to which it was assigned. not

of

identify

the

set

More of

significantly,

numbers

from

the

such

a

physical

However, in the case of "brain vectocs" it is the

particular

particular vector in the CNS is assigned.

physical

invariant

to

which

a

The subsequent task may be even more

difficult: to determine what frame of reference is implied in such a co-ordination of an invariant with a mathematical point. the

way

of

assigning

contravariant procedure. external reflex,

invariants ( c.f.

is

the

components

A further task is to establish whether to

the

invariant

is

a

covariant-

or

These tasks of relating sensorimotor "brain vectors" to probably

the

PELLIONISZ & LLINAS,

easiest 1980 a).

in

the

so-called

In such a

vestibulo-ocular

primitive system it is

404

evident that the brain-vectors at both ends of the reflex arc are directly related to

an

obvious

iden~ical

p,yslcal

invariant,

the

head-displacement

eye-displacement, respectively.

and

the

(optimally)

Another extreme, when the exact system

of relations of the internal brain vectors and external physical entities is rat,er . unclear, may be represented· by a linquistic sensorimotor transformation performed by the CNS: e. q. when an interpreter, whose acoustic sensory input vector is " Japanese word, transforms this to a

correspondinq vocal motor· output vector,

an

Enqlish word., where both vectors are assiqned to the same invariant. General particular

vectors ways

invariant. ABSTRACT

necessitate

anti means

by

even

which

more

a

profound

mathematical

considerations

vector

is

than

assiqned

the

to

the

These are THE QUESTIONS CONCERNING THE PROPERTIES OF THE N-DIHENSIONAL MATHEMATICAL

SPACE,

(that

which

we

call

a

".hyperspace"

il"'

order

to

distinquish it from the three-dimensional physical space) to which the particular mathematical

point

belonqs.

Thus,

the

basic

question

concerns

the

system

of

relations amonq the points of the abstract space: and indirectly, it raises the question nf the type of qeometry of the CNS hyperspace.

This internal qeometry is

related to another qeometry over the invariants themselves "net" of Euclidean physical space).

qeometry that can be use