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 Systems in Psychology Dynamics Synergetics 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 semiotics 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 emergence 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 system 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