From continuous dynamics to symb ols

Herb ert Jaeger

GMD St Augustin

herbertjaegergmdde

February



to app ear in the pro ceedings of the rst Joint Conference on Complex

in Psychology Dynamics Autonomous Agents Gstaad Switzerland

March

Abstract

This article deals with mathematical mo dels of discrete identiable symb olic

events in neural and cognitive dynamics These dynamical symbols are the sup

p osed correlates of identiable motor action patterns from phoneme utterances

to restaurant visits In the rst main part of the article mo dels of dynamical

symb ols oered by dynamical systems theory are reviewed attractors bifurca

tions spatial segregation and b oundary formation and several others In the

second main part transient attractors TAs are oered as yet another mathe

matical mo del of dynamical symb ols TAs share with ordinary attractors a basic

prop erty namely lo cal phase space contraction However a TA can disapp ear

almost as so on as it is created which could not very rigorously b e interpreted

as a bifurcation induced by quickly changing control parameters Such fast

bifurcation sequences standardly o ccur in neural and cognitive dynamics

Intro duction

This pap er is ab out symb ols viewed as identiable events in neural dynamical

systems

The pap er is not ab out symb ols in general That would b e imp ossible The

empirical phenomenology of symb ols is to o rich and the term symb ol is used

with to o manyintentions to allow a comprehensive treatment Compare eg

the multiple roles of symb ols i as mathematical ob jects amenable to a set

theoretic reconstruction ii as signs or signals whichinducephysical or mental

reactions in humans in and certain scho ols in linguistic semantics

iii as aesthetical ob jects in graphical arts iv as physical identiable

states in computer circuitry which can b e manipulated algorithmically

The latter view on symb ols has had a constitutive inuence on articial

intelligence and cognitive science In one of its strong versions it has b ecome

known as the physical symb ol systems hyp othesis This hyp othesis has

b een fervently criticised by some philosophers and psychologists who found that

exp eriential asp ects of a symb ols meaning had b een lost The ensuing debate

of the symb ol grounding problem has grown into an entangled mesh of

claims and counterarguments

As a side line there amed a debate on whether connectionist mo dels can

sustain symb olic reasoning The attacks of classical symb olicists

tickled connectionists so sorely that within a short p erio d they came up with

dozens of connectionist mo dels for variable binding the buildup of representa

tional hierarchies and other symb olic mechanisms which had b een claimed

inaccessible to connectionist mo deling

Many of these connectionist architectures relied on dynamical phenomena in

recurrentnetworks These developments help ed an increasing number

of researchers in articial intelligence and cognitive science to op en up for ideas

from bio cyb ernetics neuroscience and articial neural network research It now

b ecomes apparent that neural dynamics can b e quite directly related to high

level prop erties of cognitive pro cesses A muchcited example for the insights

aorded by a neural dynamics for cognitivelevel pro cesses are chaotic neural

attractors in classication of sensoric stimuli and concept representation

Awealth of other neuro dynamical phenomena relevant for cognition is do c

umented eg in the handb o oks edited by Gazzaniga and Arbib in the

annual Computation and Neural Systems pro ceedings or in the Behavioral

and Brain Sciences journal

A related recent trend in cognitive science and psychology is to view cog

nitive systems as dynamical systems without necessarily dealing with the un

derlying brain pro cesses I need not say more ab out this

to the participants of the Gstaad workshop In this article I will frequently

use the term neuralcognitive dynamics when referring to matters relevant

on b oth levels of description

All of these debates and strands of research form the background for the

present article I will investigate the topic of symb ols as discrete identiable

phenomena in neuralcognitive dynamics I will pursue this investigation from

a purely dynamical systems p oint of view ignoring most of the deep er episte

mological questions In particular I will not touch the question of a symbols

meaning

The article has to main sections In section I motivate why it is natural

to assume that in neuralcognitive pro cesses there emerge discrete identiable

phenomena which I will call dynamical symb ols I will then review several

candidate mechanisms oered by dynamical systems theory which mathemati

cally describ e the nature and the of dynamical symbols attractors

bifurcations spatial segregation and b oundary formation and others

In the second section I describ e a kind of discrete identiable phenomenon

in nonautonomous dynamical systems which can amply b e termed transient

attractors TA TAs share one crucial prop erty with ordinary attractors

namely lo cal phase space contraction However a TA can disapp ear almost

as so on as it is entered which could not very rigorously b e interpreted as a

bifurcation induced by quickly changing control parameters Such fast bifur

cation sequences o ccur standardly in neural and cognitive dynamics

Dynamical symb ols

In this section I shall rst clarify the notion of dynamical symb ols Then I

shall review some of the mechanisms oered or not yet oered by dynamical

systems theory for mo deling dynamical symb ols

Humans b ehave and their b ehavior can b e observed by other humans Very

generally sp eaking the b ehavior exhibited byahuman is a continuous pro cess

in manyvariables Sometimes in this stream of b ehavior there app ear phe

nomena which i can b e singled out by observers and ii which can b e more

or less reliably classied as an instance of a particular kind of event Examples

of such discrete identiable events are

Having a meal in a restaurant

Blinking ones eyes

Saying sun

Pro ducing the sound s

These events dier from each other in many ways They have dierent

temp oral extensions Some of them are subevents of others Some are more

variable than others there are many dierentways of how the restaurant visit

script can unfold while an eye blinking is stereotyp ed Some pro ceed

in silence others are accompanied by oral utterances and still others are oral

utterances And so on

Despite this diversity all of these events can be isolated and classied by

human observers Isolatability and classiability is certainly a matter of degree

a drunken p ersons utterances can b e slurred to the p oint of b ecoming unin

telligible For the present purp oses however the fringe fuzzyness of b ehavioral

event categorization is irrelevant All we shall make use of is the fact that a

human observer often can isolate and classify and therefore name a b ehavioral

event without much doubt

A crucial observation is that the isolatability and classiability of those

events is to some degree nonarbitrary In the complex pro cesses of blinking

or vo calizing s there is something intrinsic which leads observers to isolate

just these events and which leads dierent observers to the same kind of iso

lation and classication judgements It would b e in some way unnatural to

isolate from the observed facial dynamics of another p erson an event which

starts when an eyeblink is p er cent nished and extends ms after the

eyeblink

Thus there must b e something in the highdimensional tra jectory of a hu

mans stream of b ehavior which enables observers to isolate and classify par

ticular p erio ds and particular subsets of b ehavioral variables due to intrinsic

features of the pro cess which are expressed in those p erio ds Lo osely sp eaking

wemust exp ect some kind of avored lumps to exist in the pro cess lumps

there must b e since there are some entities which can b e isolated and avoured

these lumps must b e since they can b e classied

Since much in this article hinges on the notions of intrinsic isolatabilityand

intrinsic classiability I will try to explain these notions a bit more An event

in some complex ongoing pro cess is intrinsical ly isolatable if the event itself

yields information ab out when it o ccurs ab out its onset and ab out its end

This information must be not at least not completely relative to arbitrary

conventions made by the observer Dierent observers who do not know of

each other must nd it likewise natural to isolate roughly the same event from

the pro cess background An example for intrinsic isolatability would be a

steep rising ank in some variable which indicates the onset of something A

nonexample would b e the mere crossing of a threshold value of some variable

since this way of indicating an onset would dep end on the entirely conventional

xing of a numerical value

In a rst approximation intrinsic classiability means that eacheventcarries

with itself enough qualitative information to enable the observer to classify

it within a typically huge classicatory Somehow each event must

display enough features to allow its classication Again these features

must not be merely conventional A nonexample for intrinsic classiability

would b e to use the rst ve binary digits of a numerical measurementasve

features this b eing an arbitrary way of classication A p ositive example would

b e to use geometrical features from the shap e of a chaotic attractor they are

in some sense prop er prop erties of the attractor event

After this attempt at getting twointrinsically vague concepts clearer let us

return to the main line of argument

We know that the highdimensional overt b ehavior of a human is accompa

nied by neuralcognitive pro cesses in that humans brainmind The internal

pro cess must be in some sense at least as rich as the externally visible

motor b ehavior since the motor b ehavior is in some sense controlled by neu

ralcognitive pro cesses However from a dynamical systems p ersp ective the

internal dynamics dier tremendously from the external b ehavior and a direct

identication or even comparison of internal with external dynamics seems out

of the question However it seems reasonable to exp ect that for every or most

of the intrinsically identiable and classiable events in the externally observ

able motor b ehavior there exists an accompanying event in the neural dynamics

which is also intrinsically identiable and classiable given suitable observation

techniques for neuralcognitive dynamics In other words we exp ect avored

lumps in the neuralcognitive dynamics to o

These latter avored lumps I shall call dynamical symbols In a nutshell thus

dynamical symb ols are any kind of intrinscally isolatable intrinscally classiable

events in neuralcognitive dynamics which correlate with likewise isolatable and

classiable events in overt motor b ehavior

This is of course a very narrow framing of a symb ol concept Still narrow

as it is this sp ecic outlo ok on symb ols leads to interesting questions concerning

the mathematical mo deling of neuralcognitive information pro cessing

I will now review briey some of the known mathematical candidates for

dynamical symbols comment on their shortcomings and on the way clarify

further what I mean by isolatability and classiability or lumpiness and a

vor

One of the most widely used ways to extract discrete events from continuous

dynamics is via partition cel ls The basic recip e is to dene some volume cells

c in phase space lab el them with symb ols E and when the system

i iI i iI

tra jectory passes through cell c say that the event E has o ccured

j j

This way of transforming a continuous tra jectory into a symbol sequence

is constitutive for ergo dic theory and chaotic symb ol dynamics eg

It also is a common strategy in the interpretation of recurrent neural

networks eg or the theory of qualitative resoning in classical AI eg

However dening discrete events via the tra jectorys passing through a par

tition cell yields no mo del for dynamical symb ols since these events are neither

intrinsically isolatable nor intrinsically classiable

The delimiting co ordinates of a particular volume cell stem from arbitrary

converntions They are extrinsic to the pro cess

Likewise volume cells p er se are not avored If we only know that the

tra jectory passes through c now and through c next then wehave no infor

 

mation whatsover to tell us what kind of eventwehave b een witnessing Mere

hittingavolumecell events are not intrinsically classiable

Often of course the observer will have some extra clues telling him to delimit

volume cells in a particular way and these clues may come from particular

dynamical phenomena that are exhibited when the tra jectory passes through

these cells Then the events might be intrinsically isolatable and classiable

alb eit only due to the involvment of some extra cluegiving phenomena

Another quite common approach to picking discrete events from continuous

dynamics is to use point attractors The general scheme is to rep ort an event

whenever the system has relaxed into a stable equilibrium

This approach is p opular with articial recurrent neural networks used for

classication eg or constraint satisfaction problems eg Recently

even logical inferences have b een rigorously reinterpreted as xedp oint relax

ation of neural networks Altogether it seems quite natural to equate

discrete cognitive units symb ols concepts with p oint attractors and related

cognitive pro cesses constraint satisfaction classication inferences with re

laxation dynamics

Point attractors are intrinsically isolatable stable equilibria are system prop

erties not observational conventions

One obvious shortcoming of p oint attractor mo dels is that a true p oint at

tractor like any attractor terminal ly captures the system tra jectory If one

wants to describ e neural dynamics which exhibit a sequence of p oint attractor

events one has to intro duce extra mechanisms to kick the tra jectory out of

attractors Such extra mechanisms are eg noise p opular in Hopeld net

works or inputinduced bifurcations I shall treat the issue of bifurcations and

attractors extensively b elow

A less obvious deciency of p oint attractor mo dels is that p oint attractors

per se are hardly intrinsically classiable Like partition cells they have al

most no avor By this I mean that there are no obvious nonconventional

features bywhichpoint attractors might gain individuality Theoreticallyone

mightcharacterize dierent p oint attractors by the magnitude of their Lyapunov

exp onents by the size of their basin of attraction and other such measures But

this rep ertoire of distinguishing features seems to b e quite small to o small in

any case to account for the enormous variability of dynamical symb ols

Currently the most prominent candidate for dynamical symb ols is attractors

with complex p erio dic or semip erio dic orbits and chaotic attractors Such com

plex attractors have b een detected and induced b oth in articial and biological

neural systems

Complex attractors are intrinsically isolatable like any kind of attractor

Their great charm lies in the fact that they are also intrinsically classiable Two

chaotic attractors typically lo ok quite dierent even to dierent observers

who do not knowof each other Freemans et al graphical representations of

chaotic attractor states in the olfactory bulb and the way they geometrically

change due to sensory input are deeply inspiring

Thus are complex in particular chaotic attractors go o d candidates for

dynamical symb ols

I am sceptical ab out the ultimate value of chaos for the practical mo deling of

neuralcognitive phenomena basically b ecause identifying a highdimensional

chaotic attractor in an empirical time series typically requires more data than

can be gathered while the attractor is extant for more detailed criticism cf

for an enlightening case study cf I am afraid that in live brains

under reallife conditions chaotic attractors cannot b e monitored long andor

precisely enough to certify their existence

Iwould stickoutmy neckeven further and question that chaotic attractors

are the right mathematical metaphor at all for what wewould like to observe

What I nd dubious is the idea of a highdimensional complex attractor in the

rst place The work of Freeman Babloyantz and others has op ened our eyes

for the extreme richness subtlety and exibility of assumedly chaotic activity

in recurrent neural systems Babloyantz and her colleagues in particular have

put emphasis on the hyp othesis that it is the negrained dynamical variants

of chaotic attractors which hold promise as mo dels for conceptual memory ie

as mo dels for certain dynamical symb ols Now having negrained subtle

highdimensional chaotic attractors also means only marginal stability which

has b enets for swift and exible reactions as has b een p ointed out by the cited

researchers Marginal stability means for an attractor that it is easily disrupted

by noise and that it takes long for the tra jectory to settle even in the absence of

noise Biological brain subsystems are noisy they are driven with strong signals

from sensors and other subsystems and they are highly adaptive and learning

ie a brain subsystem do es not stay itself very long All of these conditions

render a brain subsystem a hostile environment for marginally stable subtly

complex attractors I doubt that in a live brain and under reallife working

conditions a chaotic attractor ever really has a chance to stabilize

Thus I fear that chaotic attractors are more a myth than a realityinlive

situated brains However it seems undisputable that the investigations of Free

man Babloyantz and others have touched on something fundamental and that

this fundamental thing is somehow connected with chaos From this p ersp ec

tive a promising route wouldbetoinvestigate neural activity under the auspices

of chaos but without relying on attractors This implies that chaos has to b e

dened in a novel way which works for randomly driven dynamics Such a

denition is in fact available

Partition cells and attractors are probably the most common but by no

means the only candidates for dynamical symb ols that dynamical system theory

can oer The candidate that I am going to describ e next will turn out to

be inherently classiable but not inherently isolatable This combination of

prop erties is remarkable since attractor mo dels of dynamical symbols may easily

make one b elieve that isolability can b e considered an implicit consequence of

classiability Since this topic touches basic asp ects of the dynamical systems

outlo ok on neuralcognitive systems I will explain this p oint in some more

detail

One basic mechanism for explaining how a system tra jectory can get caught

in a sequence of dierent attractors is bifurcations The tra jectory is released

by one attractor due to the fact that the attractor itself vanishes and is caught

bythe next b ecause that attractor newly comes into existence Two complex

attractors which are separated in time from eachotherby bifurcations of the

entire system will typically have dierent top ological features This implies that

they cannot b e smo othly morphed into each other The bifurcation that o c

curs b etween them marks a singularity in the reshaping of the systems phase

p ortrait Thus in this case inherent classiability granted by the attractors

top olgical features implies inherent isolatability since top ologcial features can

app ear only in catastrophes which are markers for isolation of the newly

app earing attractor

For a long time I b elieved that these observations reected a deep er general

law namely that a qualitative change of a systems dynamics cannot occur

smo othly or expressed more casually that dierent complex pro cesses cannot



b e morphed into each other Misguided by the fundamental phenomenon of

bifurcations I b elieved that a dynamical systems qualitativetyp e of dynamics

can change into another qualitativetyp e another phase p ortrait only through

some kind of catastrophic transition To me this seemed an extremely valuable

insight b ecause it seemed to p oint to a fundamental necessity in continuous

nature to pro duce discontinuities From here the road seemed paved toward

understanding how dynamical symb ols arise in neuralcognitive dynamics The

hop e was that a suciently rich dynamics would by necessity show some sort

of discrete klicking and ratcheting

The candidate for dynamical symb ols which I am going to describ e now

demonstrates that this insight was false Qualitatively dierent stochastic

pro cesses can b e smo othly morphed into each other This means that wehave

inherent classiability without inherent isolatability

Discretetime discretevalue sto chastic systems are convenient mathemati

cal to ols for mo deling the dynamics of cognitiveand neural pro cesses There

are many variants of such systems Markov mo dels hidden Markov mo dels

sto chastic automata of various kinds sto chastic cellular automata to name but

a few Ihave added to this multitude myself byintro ducing dynamical symbol

systems and observable op erator mo dels a generalization of hidden



In computer graphics the smo oth transformation of a picture into another is sometimes

called morphing

Markov mo dels with nicer mathematical prop erties Systems of this kind make

for coarsegrained transparent and often computationally ecient mo dels of

continuous systems among them recurrent neural networks In

the programming of mobile rob ots they are widely used as learnable memory

mo dules for representing temp oral exp eriences in particular in navigation

A simple example of such a system is given in g The gure shows a

twostate sto chastic transition graph which generates sto chastic sequences of

as and bs as follows Atany time t where t the system is either

in state a or in state b If it is in state a then it jumps to state b at time t

with probability p If it is in state b it jumps to state a with probability

A sequence of states pro duced that way is a system tra jectory

1 p a b

1-p

Figure A simple sto chastic transition graph

Now interpret p as a control parameter If we set p we observe a

sequence aaaaa which consists entirely of as with a p ossible leading b as an

inital transient The other extreme would b e to set p whichwould result

in an alternating sequence abababa Intermediate settings of the control

parameter would yield all sorts of mixtures

If a dynamical systems theorist would b e oered for analysis just the two time

series aaa and ababa hewould probably susp ect to have b een presented

with a coarse version derived by partitioning a phase space into two cells a

and b of a dynamical system that has undergone a p erio d doubling bifurcation

between the two observed time series

This presumable sp ontaneous reaction of a system theorist demonstrates

that in some intuitive sense the sequences aaa and ababa are quali

tatively dierent This view is however very much nourished by the inter

pretation that a familiar p erio d doubling bifurcation has given rise to the two

sequences in this view intermediate mixtures b etween the two sequences would

not b e p ossible

If bycontrast the sto chastic transition graph is known to b e the generating

system the judgment that aaa and ababa are qualitatively dierent gets

shaky After all the two sequences are just the two extremes on a continuum

of pro cesses with intermediate phenomenologies

A similar morphing can o ccur in continuoustime continuousvalued pro

cesses For instance consider a dynamical system governed by a control pa

rameter where there is a bifurcation between and Let the

system run and while running sto chastically switch between and with

varying average frequency and with varying average relative duration of vs

In the two extreme cases and ie zero switching frequency we

will observe the two clean bifurcativevariants of the system Dep ending on

the settings of average switching frequency and of relative duration however

all kinds of intermediate dynamics will b e observable to o

In order to preclude a p ossible misunderstanding let me emphasize that

in the transition graph example it is not the state symbols a and b whichare

the candidates for dynamical symb ols Rather the candidate for a dynamical

symbol would b e the qualitative sequence pattern altogether This is completely

analogous to the complex attractors discussed previously where the dynamical

symb ol is the overall pattern of the tra jectory

I b elieve that the existence of intrinsically classiable but not isolatable

dynamical entities is not just an academic mathematical p eculiarity The oc

curence of such entities but are they entities if they are not isolatable

has to b e exp ected whenever a dynamical system is driven by sto chastic input

which app ears to b e the standard case for neural and cognitive subsystems

This situation calls for the development of new mathematical concepts

What seems needed here is a dynamical mixing theory which can give us

a clearer picture of what it means for temp oral patterns to mix Is it p ossible

in a sto chastic pro cess to somehow factor out pure dynamical subpro cesses

of which the observed pro cess is a mixture Iamworking toward such a theory

but it is to o early to rep ort results

Besides the relatively prominent candidates p ointed out so far there is an

unfathomable wealth of others lesser known ones only few of whichhaveyet

been explored as mo dels for dynamical symb ols I shall pro ceed in a more

summary fashion

A basic prop erty of biological neural structures is their spatial organization

Somatotopic or top ographic maps ab ound and primary visual cortices app ear to

exhibit b eneath the overall top ographic representation a negrained columnar

pattern where the activation of columns represents sp ecic features in the visual

stimulus One striking feature of the cereb ellum is the organization of its

surface into b eams which have b een interpreted among other options as

adaptive detectors of motor control signal sequences

Findings of this kind suggest localist or more generally spatial mo dels of

dynamical symbols A dynamical symbol would corresp ond to the activation of

some spatially dened collection of neurons It would b e intrinsically isolatable

if the activated neural collectivewould exhibit a nonarbitrary b oundary of

some kind This b oundary could b e anatomically dened through discontinu

ities in neural connectivity as in columns or b eams Boundaries of some sort

can also arise in homogeneous neural substrates by nonlinear comp etitive spa

tiotemp oral neural eld dynamics Reactiondiusion typ e of dynamics

would b e another p ossibility

Abatch of active neural tissue can b e intrinsically classiable for many rea

sons eg by its anatomical structures or by its b eing lo calized in a particular

part of the brain Furthermore its activation dynamics may b e complex enough

to distinguish it from other batches

Another exciting challenge for mo deling dynamical symbols is spatiotemporal

dynamics Unfortunately a satisfactory qualitative mathematical theory of such

dynamics is presently beyond our reach Only a few phenomena we have yet

learnt to discern eg solitons or spiral patterns in reactiondiusion dynamics

A glimpse into the future can be cast through Binghams article on the

p erception of spatiotemp oral patterns byhumans The exp erimental scheme

describ ed by Bingham is to present sub jects with dynamical patterns of a

few white dots on a blackbackground The patterns are derived from lmed

sequences of natural dynamical scenes eg a eld of high grass swaying in the

wind or honey o ozing out of a jug The white dots which are shown mark

selected tips of grass or particles oating with the honey etc Sub jects can

correctly recognize these emp overished stimuli Their p erformance can only b e

explained when one assumes that humans have rich mo dels of spatiotemp oral

patterns This contrasts starkly with what mathematicians can yet reconstruct

The observations rep orted by Bingham do not directly p ertain to dynami

cal symb ols Ihave mentioned them here b ecause his article indicates research

directions for dynamical systems theory which are equally relevant for the qual

itative phenomenology of spatiotemp oral neural dynamics

The last and p ossibly most elusive candidate for dynamical symbols is

species Sp ecies are intrinsically isolatable and classiable entites which arise in

evolutionary pro cesses

Cognitive pro cesses have b een describ ed in some detail building on the idea

of concepts sp ecies However in that work only the shortterm in

evolutionary p ersp ective population dynamics is used to mo del cognitive

phenomena treating sp ecies as givens I should also p oint out the inspiring

young research strand of evolutionary linguistics where the very emergence of

language is mo deled with concepts from evolution theory This approach

sheds a bright light on the genesis of phonemes words and grammar and should

not b e missed byanyone interested in the nature of symbols Finally the b est

known attempt to tame evolutionary dynamics for mo deling cognitive dynamical

systems and the evolution thereof is classier systems and genetic algorithms

Sadly the mathematical theory of evolutionary dynamics is still in its in

fancy Eigens and Schusters hyp ercycle mo del notwithstanding This

renown mathematical achievement only captures sp eciation in certain chemi

cal reaction systems Spatial segregation or the emergence of ever more complex

inheritance mechanisms are not addressed The hyp ercycle describ es one mecha

nism but biological evolution is very much a story of the op enended generation

of a pluralityofmechanisms Although the theory and practice of classier

systems and genetic algorithms has b een develop ed further and broader than

that of other evolutionary mo dels they do not oer even a convincing mo del

of sp ecies p This corresp onds with the situation in the biological

theory of evolution where it is not at all clear on which units selection actually

works genes or sp ecies or symbiotic multisp ecies systems

I feel that our lack of understanding of evolutionary pro cesses cannot be

fundamentally remedied since evolution is qualitatively productive ever new

mechanisms emerge and even mechanisms of evolution of mechanisms evolve

There is no master mechanism the knowledge of whichwould giveus

the multitude as a corollary

Brains are the pro duct of evolution and p ossibly the developmentof cog

nitive systems in ontogenesis also b ears some marks of evolutions qualitative

pro ductivity Inasmuch as dynamical symb ols can b e interpreted as sp ecies

or genes or p opulations or any other kind of lumps inhabiting brainscognitive

systems it seems fundamentally imp ossible that wecanachieve a unied math

ematical mo del of them

This was a sweeping pass over some mathematical mo dels for avored

lumps It has led us from simple partition cells to the most elusive pro ducts of

evolutionary dynamics The general message I wantedtoconvey is summarized

in the following p oints

There are manyways how symb olic units can arise in neuralcognitive

dynamics

We should not lo ok for the correct mathematical mo del Nature loves to

play not to ll out forms

The mathematical mo deling and our intuitive understanding of sym

b ols has barely started Dynamical systems theory still has to integrate

sto chastic spatial and evolutionary asp ects Exciting discoveries are wait

ing for mathematicians and braincognition researchers

Transient attractors

In this section I shall motivate and explain transient attractors TAs This

mathematical ob ject generalizes the notion of attractors Unlike classical at

tractors TAs can exist in systems driven bystochastic input and in systems

whose variables dynamically change their relative time scales Thus TAs can

serve as mo dels for dynamical symb ols in some cases where classical attractors

are not dened

I havementioned in section an intrinsic diculty with attractor mo dels

for dynamical symbols Namely an attractor by denition terminally captures

the system tra jectory By contrast dynamical symb ols sequentially arise and

vanish in neuralcognitive dynamics As I have noted in the previous section an

apparentway out of this dilemma is by bifurcations which generate and destroy

attractors The control parameters which induce the bifurcations presumably

are input quantities from sensors or other neuralcognitive subsystems

A problem with bifurcations is that they are welldened only when the

dynamics of the control parameters is at least an order of magnitude slower

than the time scale of the controlled system But this is not typically the case

with neuralcognitive systems Quite to the contrary the dynamics of input

variables which control a subsystem is typically just as fast as the dynamics

of the subsystem With the exception of some slow somatosensory mo dalities

temp erature hunger certain kinds of pain etc the brain is under constant

re of fast sensory input visual auditory kinaesthetic Furthermore dierent

brain subsystems will often tightly interact with each other each one giving a

p ortion of its own dynamics as input to the other

From a mathematical p ersp ective wehave to admit that the notion of bi

furcation and hence of attractors is no longer welldened in such situations

The following formal example of a transient attractor shows what it means

foracontrol parameter to have a dynamics which is as fast as the controlled

system

Consider the system sp ecied in p olar co ordinates by r r

r sin Its phase p ortrait is characterized in the vicinity of the origin byanti

clo ckwise closed lo ops among them a lo op on the unit circle g a

When one follows any two tra jectories the xed p oint tra jectory at the

origin excepted through increasing values of one nds that they come closer r ϕ 1

(a) (b)

B

A

(c) (d)

Figure a The system r r r sin b To b e noise and not to b e

noise whichiswhich in an empirical phase p ortrait c Crossing tra jectories

d Phase space contraction

to each other in the upp er half of the plane ie whereas they

recede from each other in the lower half This can b e interpreted as the eect of

a fast bifurcation induced by a fast control parameter as follows Reinterpret

r r r sin as a onedimensional system with a control parameter If

is xed at a value b etween this system exhibits a p oint attractor at

r For the p oint attractor in r turns into a rep ellor The

values and mark bifurcations

Thus one mightinterpret the system shown in g a as consisting of two

coupled onedimensional subsystems in and r where one of the subsystems

yields a fast control parameter for the other Variations of this control

parameter induce a fast creationanddestruction cycle of an attractor a

transient attractor

Examples like this have b een my original motivation for the intro duction

and naming of transient attractors The idea of fast bifurcations do es however

not lead very far in practical applications b ecause control parameters and the

dynamics thereof are mostly unknown The fast control parameters will quite

often come from external input into the system and have essentially sto chastic

dynamics Even worse typically one will not even b e able to identify the relevant

input parameters But without an idea what the control parameters are an

analysis of bifurcation sequences b ecomes all but imp ossible

Empirical phase p ortraits derived from neuralcognitive systems have still

other prop erties which render the classical notions of control parameters and

bifurcations almost useless I mention only two of these unpleasant prop erties

First empirical phase p ortraits are noisy But it is by and large imp ossible to

separate noise from the actual system dynamics b ecause whatever amplitudes

at whatever frequency we observe it might b e an actual sensemaking system

answer to some input which must not be discarded as noise g b The

second unpleasant prop erty of empirical phase p ortraits is that they feature

crossing tra jectories g There are many reasons for tra jectories to cross

eg noise or pro jections of a system which is dened on an ndimensional

manifold on a ndimensional subspace of the emb edding space or observation

of a highdimensional system in only a few of its variables

Prop erties of these kinds force one to abandon autonomous systems ruled

by dierential equations as mo del systems for neuralcognitive pro cesses in

which there are dynamical symb ols to be found A more general framework

of sto chastic pro cesses seems adequate Therefore the question is what is

the intuitive core of TAs when wehave to abandon the descriptivetoolsof

control parameters and bifurcations

I suggest to use as the dening prop ertyofaTA that it lead to a contraction

of phase space volume This eect is illustrated in g d A TA reveals its

existence by tra jectories which approacheach other in time Another wayto

state the same fact is to say that a TA aords us with go o d local predictability of

the pro cess Referring to g d if at time t weknowby some measurement

n

that the system state is in A then if there is a transient attractor we can predict

that the system will b e in B at time t where the volume B is smaller than

n

A This contraction of phase space volume corresp onds to an information gain

over time and this gain is indicative for the presence of a TA

There are manyways how this basic idea of phase space contraction can b e

made precise in sto chastic pro cesses In I gave a denition which I nd

to o narrow and to o complicated to day I will present a more general and more

transparent denition presently The practical use of such denitions is limited

b ecause of their high level of mathematical abstraction Therefore I will not put

much emphasis on the formal denition and present it with little explanation

for readers who are familiar with the terminology of sto chastic pro cesses For

most readers it will be more relevant to know that a simple and transparent

algorithm for detecting TAs in empirical multivariate time series is sketched in



Now one p ossible denition of TAs Let APX b e a stationary

t tR

n n n

sto chastic pro cess with values in the observation space R B where B is

n

the Borel algebra on R Let A b e the sub algebra of A which is generated



byX ie A is the algebra of the pro cesses past up to t

t t 

Let us return for a moment to the situation of a classical ODE systems phase

p ortrait If wewould want to dene a mutual approaching of two tra jectories

 

T T we would lo ok at the p oints x and x through which they pass at time

t and then consider their future development after they have passed there



Very general sto chastic analogues of the p oints x and x and of the past

  

of T T b efore those tra jectories passed through x and x are sets A A from



A Intuitively A and A are informations that can b e gained ab out the system



state by some observations that were made in the past up to t



ftpable from httpwwwgmddePeopleHerb ertJaegerPublications The algorithm

is currently b eing implemented for a diploma thesis



A likewise general sto chastic analog of the future of T T after t can b e



A A

sp ecied through the conditioned probability measures P and P whichare

t t



A A 

dened by P B P X B j A and P P X B j A where t

t t

t t



n A A

and B B The families P and P could amply b e called fuzzy

t t

t t



tra jectories through fuzzy p oints A and A

Now what do es it mean for two such fuzzy tra jectories to approach each

other One would like to dene some kind of distance b etween the conditioned



A A

probabilities P and P and then to note when this distance shrinks in time

t t



Consider the distance measure for two probability measures P P on

RR

n n  

R B dened by P P kx y kP dxP dy This is actually not a

distance measure in the mathematical sense b ecause P P is not zero for most

probability measures P it is zero only if P is a p oint measure However for our

purp oses it is the right measure In the sp ecial case of classical tra jectories ie



A A

where P and P are p oint measures yields the ordinary metric distance

t t



A A

In the case of fuzzy tra jectories P and P not b eing p oint measures can

t t

intuitively b e interpreted as a kind of mutual information one fuzzy tra jectory

aords ab out the other at time t



Now in order to dene whether the future developments of A and A ap



A A

proach each other we can consider the derivativeof P P int ie

t t



A A

dP P dt If this numb er is negative wehave found an approaching

t t

of future developments

Having these to ols ready transient attractors can b e dened as follows

First dene a handy class of admissible observations of the past ie a

manageable subset C A Probably the simplest choice would be to use



n

C fX x j x R gie the p oint observations of the pro cess in t A

 

n

trickier but still simple variantwould b e C fX x X y j x y R g

  

ie the informations ab out the system attainable from a p oint observations at

the present plus a p oint observation one time step in the past



 A A

Next consider the function ta C C R A A dP P dt

t t

Then dene as a transient attractor every maximal connected region in C C

in which ta

Of course this denition presupp oses that C has b een selected in a way

which allows to dene a top ology on C C in order to make the notion of

connectedness come to b ear

The variants C and C are roughly analog to describing a physical system

 

only through its p ositions vs through p ositions plus velo cities Another anlogue

would b e to describ e a system by a rstorder vs a secondorder Markov pro cess

The second variant allows to disentangle transient attractors that cross each

other in phase space as in g c

This denition is admittedly complex not worked out in detail and to some

degree arbitrary However something or other in this fashion has to b e xed if

we wish to rigorously work out the intuitive idea of fast generationdestruction of

attractors in sto chastic dynamics I feel a bit embarassed ab out the current state

of aair but I cannot oer anything b etter however the practical algorithm

mentioned ab oveismuch easier to understand than this abstract denition

Discussion

This article pursued two goals First I wished to illustrate and emphasize the

phenomenological diversityof dynamical symb ols Complex attractors are an

inspiring and imp ortant class of mo dels but a host of other yet unimaginable

dynamical phenomena awaits us Second I to ok a tentative little step in that

unchartered terrain by oering the concept of transient attractors

In former workIhave assumed dynamical symb ols as givens and develop ed

a mathematical theory which describ es how dynamical symb ols can interact

build up complex resonances and develop into hierarchies in a selforganizing

fashion This mathematical approach dynamical symbol systems ba

sically describ es the temp oral evolution of directed graphs where the edges

are identied with dynamical symb ols and where selforganization into res

onances comes ab out as selfreinforcing of cyclic subgraphs In that work

however I assumed only some abstract prop erties of dynamical symbols and I

was vague ab out the concrete mathematical nature of dynamical symb ols them

selves all I did in this direction was to allude to chaotic attractor states In

the present article conversely I fo cussed on the mathematical nature of single

dynamical symb ols

I required dynamical symb ols to p ossess intrinsic isolatability and classi

ability It seems unlikely that a precise denition of intrinsic isolatabilityand

classiability can b e given I rather b elieve that neural dynamics propp ed by

billions of years of freewheeling evolution intrinsically dees clean denitions

mathematical rigour hardly b eing a tness criterion in natural selection As

a consequence I do not think that the notion of dynamical symb ols can be

mathematically dened It is more a horizon line for op enended quest than

an axiom from which to start Whatever dynamical phenomena we will learn

to see on that way they will enrich our understanding of howwalking reason

ing sp eaking reminding unfold the dynamical everyday stu which prop els us

through our lives

Acknowledgments The ideas presented in this article grew over a long

time during which I b enetted very much from discussions with friends and

colleagues among them Andreas Birk Rene ten Bo ekhorst Thomas Christaller

Tim van Gelder Dieter Jaeger Christoph Lischka Rafael Nunez Frank Pase

mann Rolf Pfeifer Eva Ruhnau Christian Scheier Luc Steels and Jun Tani

I wish to thank them all The work was sustained by a p ostdo ctoral grant

donated by GMD Sankt Augustin

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