development as change of system dynamics

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Development as Change of System Dynamics:

Stability, Instability, and Emergence.G. Schoner

Institut fur NeuroinformatikRuhr-Universitat Bochum

Germany

October 13, 2007

Schoner, G.: Development as Change of System Dynamics: Stability, Instabilty, and

Emergence. In: Toward a New Grand Theory of Development? Connectionism andDynamic Systems Theory Re-Considered, J.P. Spencer, M. Thomas, & J. McClelland(Eds.), Oxford University Press (2007, in press)



pushing to change the neural state. The very flexibility of the central nervous sys-tem, the interconnectedness and multi-functionality of many of its components makesuch perturbations the norm rather than the exception. Only neural states that resistsuch perturbations will persist long enough to be observable, to influence down-streamprocesses, and to induce behavioral and long-term effects. Stability, the capacity to

resist perturbations, is thus a key property of the functional states of nervous systems.Stability is needed not only to protect functions against distractive internal couplings,but also to enable neural states to maintain sustained coupling to the external worldthrough a continuous link to sensory information. Organisms are embodied nervoussystems in that their perceptual, motor, and cognitive processes are intermittentlycoupled to sensory information from the sensory and motor systems. Given complexenvironments and a complex body with many more degrees of freedom than recruitedfor any particular task, such couplings are sources of perturbation. Only stable func-tional states persist in the face of such perturbations.

Mathematically, stability is the constitutive property of attractors. Illustrated inFigure 1, an attractor is an invariant (unchanging in time) solution of a dynamical

system, toward which solutions converge if they start nearby. If perturbations pushthe state away from an attractor, the dynamical system restores the attractor state.Attractors emerge from the dynamics of a system as points at which the forces pushingin opposite directions converge and balance. A stabilization mechanism is impliedwhenever the rate of change, dx/dt, of a dynamical state variable, x, depends on thecurrent state in the way illustrated in Fig 1. One may think of stability as an abstractgeneralization of the physical concept of friction. Imagine a ball moving through water.The velocity of the ball along a line is the state variable, x. Friction reduces positivevelocities by decelerating the ball. If the ball moves in the opposite direction, itsvelocity is formally negative, and friction changes that velocity to zero. Thus, the stateof zero velocity is a stable state of this system. The figure and the analogy illustrate theconcept of a fixed point attractor, that is, a stable state that is itself a constant solutionof the dynamical system. Stability may be defined for more complex solutions as well,such as oscillatory solutions (limit cycle attractors), oscillatory solutions with multiplefrequencies (quasi-periodic attractors), or solutions with very complex time structure(strange attractors). In this review I shall limit myself to fixed point attractors, whichcan go a long way toward accounting for states of the nervous system.

Neurons and neural networks are naturally described as dynamical systems (Wil-son, 1999) and provide through their intrinsic dynamics the mechanisms that stabilizeattractors (see Hock, Schoner, Giese, 2003, for a discussion). In that sense, stabilitycomes for free from the neuronal dynamics prevalent in the central nervous system,

although the sensory-motor periphery (through muscle viscosity and elasticity, for in-stance) may also contribute. Once a stabilization mechanism is in place, there is noneed for other computational mechanisms to determine the output of a dynamic neural

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x

dx/dt=f(x)

attractor

Figure 1: A differential equation model of a dynamical system is defined by how the rateof change, dx/dt, of the state variable, x, depends on the current state: dx/dt = f (x).The present state thus determines the future evolution of the system. In the presence

of an attractor the system evolves by converging to the attractor as indicated by thearrows: A negative rate of change for values larger than the attractor state leads toa decrease in time toward the attractor, a positive rate of change for values smallerthan the attractor leads to increase in time toward the attractor. The time-invariantattractor thus structures the temporal evolution of the state of the dynamical systemin its vicinity.

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x

dx/dt=f(x)

Figure 2: When a dynamical system changes (from the form shown as a dashed lineto the form shown as a solid line), the stability of the new attractor state leads auto-matically to an updating of the state through relaxation to the new attractor, that is,through change of state until the rate of change reaches again zero.

network. In fact, the very forces that restore an attractor state following perturbationsalso help the system track any changes to the attractor state incurred as inputs to thenetwork change (Figure 2). What such changing inputs do is move the attractor stateto a new location in state space. The old attractor state is then no longer a constantsolution (zero rate of change). It is instead associated with non-zero rates of change

that drive the system toward the new attractor state.The idea of a system tracking a stable state that moves as input varies is shared

with the conceptual framework of cybernetics or control theory. In cybernetics, theattractor is called set-point and deviations from the set-point are called errors. Thecontrol system is designed to reduce such errors to zero. A conceptual step beyondthis analogy is made, however, when multiple attractors co-exist. The simplest suchcase, bistability, is illustrated in Figure 3. Which of the two attractor states is realizeddepends on the prior history of the system. As long as the state of the system lies withinthe basin of attraction of attractor 1, the system tracks that state as the dynamicschange. Note that it makes no longer sense to talk about the deviation from eitherattractor as an “error” (relative to which of the two attractors?) and thus the function

of the dynamical system is not to reduce an error to zero. Instead, selecting one overanother attractor is a simple form of decision making and the dynamics stabilize such

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x

dx/dt=f(x)

attractor 1 attractor 2

repellor

Figure 3: A non-linear dynamical system with two attractors separated by a repellor,a time-invariant solution from which the system diverges.

decisions. On rare occasions stochastic perturbations may shift the system sufficientlyfar so that it crosses over into the alternative basin of attraction. This induces astochastic switch of state.

Such stochastic switches out of an attractor become more likely if a change of thedynamics reduces the stability of the attractor. Figure 4 illustrates what happenswhen changes of the dynamical system reach a critical point at which an attractor(number 1 on the left) loses stability. In this particular instance, the attractor on the

left and the repellor in the middle move toward each other, until they collide and thenboth disappear. No zero crossing of the rate of change remains in the vicinity of theformer attractor and repellor. If the system was originally in or near attractor 1, ittracked this attractor as it moved toward the repellor. But when attractor and repellorcollide and disappear, the system must switch to attractor 2 (if it did not escapewith the help of a stochastic perturbation earlier). Mathematically, such a changein the number and stability of attractors and/or repellors is a bifurcation. In simpledynamical systems such as the one shown in Figure 4, bifurcations can only occur bycollision of attractors and repellors with each other. To see this, visualize the graph of the functional dependence of the rate of change of the current state as a flexible rubberband. This graph and the associated function can be deformed by parametric changes

of the dynamics, but not cut (because the rate of change must be a continuous functionof the state). The only way to eliminate a zero-crossing of the function is thus to make

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two or more zero-crossings collide. The case shown in Figure 4 is the simplest suchbifurcation in which the rubber band lifts off from the x-axis just as it becomes tangentto that axis. This so-called tangent bifurcation is one of four elementary bifurcations(the others being the transcritical, the pitchfork, and the Hopf-bifurcation), which aremost likely to be observed in real systems, because they make the smallest number of

demands on the parameter values of the dynamical system (Perko, 1991).Instabilities may thus lead to qualitative change, as contrasted to the mere track-ing of a continuously changing state. The change is preceded by tell-tale signaturesof instability such as increasing variability and increasing time needed to recover fromperturbations. These can be exploited to detect instabilities and thus to distinguishqualitative from quantitative change (review, Schoner, Kelso, 1988; van der Maas,Molenaar, 1992). In non-linear dynamical systems, instabilities arise naturally in re-sponse to even relatively unspecific changes of the dynamics. In the illustration of Fig-ure 4, for instance, the changes to the dynamics are not specifically localized aroundthe attractor 1 that is losing stability, but rather amount essentially to an increasingbias toward larger values of the state variable “x”.

This illustrates that attractors are not fixed entities. When they disappear, they arenot stored somewhere, or simply “deactivated”. Attractors may emerge out of nowherewhen the conditions (the dynamics) are right. This can be visualized by looking at thescenario of Figure 4 in the reverse order: As the bias to larger values of “x” is reduced,a new attractor may be formed spontaneously, coming “out of nowhere” and branchingoff an associated repellor that forms a new boundary between two basins of attraction.

So far, I have talked about the “state” of a system in the abstract. What kindof variables “x” would describe behaviors, patterns, decisions? Many readers may befamiliar with now classical examples of DST in interlimb coordination, reviewed forinstance in Scott Kelso’s (1995) book. In this work, the relative phase between tworhythmically moving limbs has been shown to be sufficient to characterize patternsof movement coordination. Each of the two common patterns of coordination can becharacterized through a specific, constant value of the relative phase. In the “in-phase”pattern of coordination (relative phase equal to zero), homologous muscles co-contract.This is typically the most stable pattern of coordination, neuronally based on shareddescending input to the two limbs as well as excitatory coupling. In the “anti-phase”pattern of coordination (relative phase equal to 180 degrees), homologous musclesalternate. This pattern is less stable. In the laboratory one may push the “anti-phase”pattern through an instability by, for instance, increasing the frequency of the periodiclimb motion. At a critical frequency, the stability of “anti-phase” coordination is lost,leading to increased fluctuations of relative phase, increased time needed to recover from

perturbations and, ultimately, to a switch to the “in-phase” pattern of coordination(Schoner, Kelso, 1988).

Relative phase may be the only obvious example of a single variable that clearly

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x

dx/dt=f(x)

attractor 1 attractor 2repellor

Figure 4: Changes of a bistable dynamical system (from dashed via dotted to solidline) lead to an instability, in which the attractor 1 collides with the repellor, leavingattractor 2 behind. This bifurcation thus leads to a change from bistable to monostabledynamics.

describes a pattern (of relative timing) and that is independent of other details of howthe pattern is generated (e.g., the movement amplitude or the exact trajectory shape).Other examples from the literature are less obvious. For instance, the many oscillatormodels in the literature of coordination of rhythmic movement are formulated in termsof variables that describe the spatial position and velocity of the moving effector,

although these variables are not identical to the associated physical quantities. Whenthe limb is mechanically perturbed, for instance, the physical position and velocity ischanged, but the oscillator driving the movement is not necessarily affected (Kay et al.,1987). Conversely, the oscillator variables are not directly related to neural activationseither (but see Grossberg, Pribe, Cohen, 1997, for an account at the level of neuronaloscillators).

That Dynamical Systems models may require only a small number of variables(i.e., may be low-dimensional) is a central assumption of DST. What is it based on?Why should it be possible to describe by a simple differential equation of one or twovariables what happens when a nervous system generates behavior, when millions of neurons engage sensory and motor processes and couples them by feedback through

the outer world? There is a mathematical answer to this question, which I will brieflysketch, even if a full explanation goes beyond what can be achieved in a survey chapter.

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The ensemble of neural processes and their coupling to the sensory and motorsurfaces form a very high-dimensional dynamical system (Wilson, 1999). Stable statesin such a high-dimensional dynamics are the persistent macroscopic states which areobservable at the behavioral level. Stability means that in all directions of the high-dimensional space, restoring forces secure the state against perturbations. An attractor

becomes unstable when the restoring forces in one particular direction begin to fail(Figure 5). Only under exceptional circumstances caused by symmetries would stabilityfail in multiple directions at the same time. The direction along which the restoringforces become weak defines the low-dimensional Center-Manifold (see, e.g., Perko, 1991,for a textbook treatment).

The temporal evolution of the state of the system along the Center-Manifold isslower than in any other direction of the high-dimensional state space. This is becausethe restoring forces are weaker in this direction leading to slower movement toward theattractor state along this direction than along any other direction. Perpendicular to theCenter-Manifold, in contrast, restoring forces are strong and the system quickly movesfrom wherever it started out to some point on the Center-Manifold. Thus, the long-

term evolution of the system is essentially dictated by movement within the Center-Manifold. This intuition is formalized in the Center-Manifold-Theorem, which saysthat knowing how the system evolves along the Center-Manifold uniquely determineshow the system evolves in the original high-dimensional space. Thus, to capture themacroscopic states of the high-dimensional dynamics and their long-term evolution, itis sufficient to model the dynamics along those dimensions along which stability breaksdown.

In the example of Figure 4, for instance, the dimension, x, would correspond tothe direction in a much higher-dimensional state space along which the stabilizationmechanism breaks down. The true dynamical system may have many more dimensions(e.g., activation levels of many neurons involved in stabilizing attractor 1). But whenthe bifurcation occurs, the system is still sitting in the attractor state along all theother dimensions except the one, shown in the Figure, along which the instabilityoccurs. The switch to the new attractor arises from movement long the unstabledirection. The other dimensions do not add anything qualitative to the dynamics andcan, therefore, be left out of the description.

The Center-Manifold Theorem implies a huge reduction in the number of dimen-sions that need to be measured and modelled to understand macroscopic states andtheir change. Although the theorem is mathematically true only exactly at a bifur-cation, in practice the low-dimensional description provides a fair representation of the fuller dynamics whenever the system is near an instability, even if not exactly at

the instability (Haken, 1983). DST is based on the assumption that nervous systemsare almost always near an instability and can thus be described by low-dimensionaldynamical systems most of the time.

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x1

centermanifold

x3

attractor aboutto become unstable

x2

Figure 5: When an attractor of a high-dimensional dynamical system (of which 3

dimensions, x1, x2, and x3 are sketched here), becomes unstable, there is typicallyone direction in the high-dimensional space along which the restoring forces begin tofade (shorter arrows) while in other directions the stabilization mechanism still works(longer arrows). That first direction spans the Center-Manifold.

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Does this mean that the low-dimensional dynamical systems are purely descriptivewhile a fully mechanistic account must take place in the original, high-dimensionalspace? The answer depends on what is meant by “fully mechanistic account”. If thatmeans, literally, an account that captures the state of all neural processes, then, by def-inition, only extensive high-dimensional computational modelling will be satisfactory.

If this means, however, that an account is sufficient to actually generate a behavior ina real system, then capturing the macroscopic, low-dimensional dynamics qualifies. Aproof of sufficiency in that sense has been provided, for instance, by generating simplerobotic behaviors such as target acquisition, target selection, obstacle avoidance, and soon from low-dimensional attractor dynamics with appropriate bifurcations interfacedwith very simple sensory and motor systems (see Bicho, Schoner, 1997, for an exampleand Schoner, Dose, Engels, 1995, for a review). Such robotic implementations alsoprove that DST models are embodied and situated in the sense that no new conceptsare needed when dynamical systems models are acted out with real bodies moving inreal environments based on real sensors. Such “acting out” in the real world is aninteresting challenge to theoretical accounts of cognition. Accounts rooted in infor-

mation processing have traditionally relied on relatively high-level interfaces with thesensory and motor processes necessary to act in the real world (e.g., world models andconfiguration space planning). These high-level interfaces have typically been difficultto put into practice in real implementations (Brooks, 1991).

A related question is how abstract the low-dimensional dynamical descriptions of behavior end up being. The Center-Manifold argument suggests that fairly abstractmodels may result, models that cut through the high-dimensional space describing theneural systems supporting behavior in ways that depend on the task, on the state stud-ied, and on the particular parametric and input conditions under which an instabilityis observed. On the other hand, over the last few years a closer alliance of DynamicalSystems models with neurophysiological principles has contributed much to reducingthe gap between the low-dimensional dynamical descriptions and the neural networksthat implement them. This will be a theme in the next section.

Given the abstract nature of DST accounts, why is DST often perceived to beprimarily about motor behavior? Many of the exemplary model systems that influencedthe development of Dynamical Systems ideas did come from the motor domain (asreviewed in Kelso, 1995; but see Hock, Kelso, Schoner, 1993, for early work usingDST in perception). On the other hand, much work has since shown that the ideasare not intrinsically tied to motor behavior. Van der Maas and Molenaar (1992),for instance, applied these concepts to characterize a wide variety of developmentalchanges including cognitive development. Similarly, van Geert (1998), has used the

abstract setting of DST to think quite generally about continuity vs. discontinuityin development, using test scores in a broad variety of tasks to map the changes of dynamics over development. In many of these cases, however, the level of abstraction

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increased substantially when moving from simple motor skills to cognitive skills. Thisis due to a conceptual problem, that must be confronted when stepping beyond thedomain of motor behavior and that I shall address now.

3 Dynamic Field Theory (DFT)

It is easy to talk about the dynamical state of a motor system without excessiveabstraction. For instance, the position of my arm, the values of the joint angles in myarm, their level of neuronal activation, the frequency or phase of my arm’s rhythmicalmovement are all perfectly good candidates for dynamical state variables that are notparticularly abstract. They have well-defined values that evolve continuously in time.

When we move beyond pure motor control we encounter problems with these vari-ables, however. What value, for instance, does the phase of my arm’s rhythmicalmovement have before I start the movement or after I stopped moving? Which valuedo the movement parameters “amplitude” or “direction” have before I have selected

the target of my movement? Obviously, the selection and initiation of motor acts, butalso the generation of perceptual patterns and the commission to memory of a per-ceptual or motor state require a different kind of variable than those used to describemotor control. These variables must capture the more abstract state of affairs in whichvariables appear to have well-defined values some of the time but not at all times. Morefundamentally, we must understand how state variables may change continuously evenduring such seemingly discrete acts as the initiation or termination of a movement.

The classical concept of activation can do this work for us. As a neural concept,activation is invoked in much of cognitive psychology and in all connectionist models.Activation may be mapped onto observable behavioral states by postulating that highlevels of activation impact on down-stream systems, including ultimately on the motor

system, while low levels of activation do not. This captures the fundamental sigmoidalnonlinearity of neural function: Only activated neurons transmit to their projectiontargets, while insufficiently activated neurons do not. There are multiple neuronalmechanism through which activation may be realized neuronally (e.g., through theintra-cellular electrical potential in neurons, through the firing rate of neurons, throughthe firing patterns of neural populations, or even through the amount of synchronybetween the spike trains of multiple neurons, see, e.g., Dayan and Abbott, 2001). Butthe concept of activation does its work for DST independently of the details of itsneural implementation.

The second concept that we will need is that of an activation field, that is, of a set of continuously many activation variables, u(x), defined over a continuous dimension, x,that spans a range of behaviors, percepts, plans, and so on. While the identification of psychologically meaningful dimensions is one of the core problems of cognitive science,

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there is little doubt that a wide range of perceptual, cognitive and motor processes canbe characterized by such dimensions (Shepard, 2001). Below I will argue that neuronalrepresentations in the higher nervous system provide guidance in identifying relevantdimensions. Note that activation, u, is now taking on the role of the continuous statevariable that was played by x in the previous section. The dimension, x, now is a

continuously valued index of multiple such state variables. In a moment I will show,that this shift of notation keeps the concepts of DST and DFT aligned.Different states of affairs can be represented by activation fields (Figure 6). Lo-

calized peaks of activation (top) indicate two things: the presence of large values of activation means that the activation field is capable of influencing down-stream struc-tures and behavior; the location of the peaks indicate the current values along thefield dimension that are handed on to the down-stream structures. By contrast, flatpatterns of low-level activation (middle) represent the absence of specific informationabout the dimension. Graded patterns of activation may represent varying amounts of information, probabilities of a response or an input, or how close the field is to bringingabout an effect (bottom).

Conceptually, localized peaks are the units of representations in DFT. The locationof a peak represents the value along the dimension that this peak specifies. The di-mensional axis thus encodes the metrics of the representation, what is being prepared,perceived, or memorized and how different the various possible values along the di-mension are. During the preparation of a goal-directed hand movement, for instance,a peak of activation localized along the dimension “movement direction” is a move-ment plan, that both specifies the direction in which the movement target lies and thereadiness to initiate the movement (Erlhagen, Schoner, 2002). If the target is shiftedduring the preparation of the movement, the peak may shift continuously along thefield dimension and thus provide an update of the movement plan to changed sensoryinformation. In this situation of a continuously moving peak we are back to the simplerpicture of DST, in which the value of the state variable, x, now represented by the peaklocation, changes continuously. In this simpler description, the peak location, x, is thedynamical variable.

This form of representation is essentially the space code principle of neurophysiol-ogy (e.g., Dayan and Abbott, 2001), according to which the location of a neuron inthe neural network determines what the neuron encodes, while its level of activationrepresents how certain or important or imminent the information is that the neurontransmits. Feature maps are possible neuronal realizations of such activation fields,consisting of ensembles of neurons that code for the feature dimensions. Such mapsconserve topology, so that neighboring neurons represent neighboring feature values.

The activation field is the approximation of such a network by a continuum of neurons, justified because neural tuning curves overlap strongly.

The spatial arrangement of neurons along the cortical surface does not actually

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dimension

activation

field

specified value

dimension

activation

field

no value specified

dimension

activation

field

metric contents

information, probability, certainty

Figure 6: An activation field defined over a continuous dimension, x, may represent

through a localized peak of activation both the presence of information and an estimateof the specified value along the dimension (top), while flat, low-level distributions of activation indicate the absence of information about the dimension (middle). Gradedactivation patterns may represent the amount of certainty about different values alongthe metric dimension (bottom).

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matter. The function of a neuronal network is solely determined by its connectivity,after all. The spatial arrangement is merely a matter of organizing the axons and den-dritic trees in efficient ways (critical, no doubt, for growth processes). The principle of organization in feature maps that makes neuronal fields an appropriate description of their function are the broad, overlapping turning functions of neurons as well as the

patterns of neuronal interaction that I will discuss below. Overlapping tuning curvesimply that whenever a particular value of a feature dimension is specified, an entirepopulation of neurons is active. Based on this insight, distributions of populationactivation may be constructed from the neural activation levels of many neurons, irre-spective of where these neurons are located in a cortical or subcortical area (Erlhagenet al., 1999). Figure 7 shows an example. About 100 neurons in motor cortex wererecorded while a monkey performed center-out hand movements toward visual tar-gets. These neurons were tuned to movement direction and their tuning curves wereweighted with the current firing rate of each neuron to define its contribution to thedistribution of population activation (the data are from Bastian, Schoner, and Riehle,2003, the representation is generated based on the optimal linear estimator method as

described in Erlhagen et al., 1999). That distribution is essentially an estimate of thedynamic activation field that represents the planned movement direction and evolvesover time reflecting the process of specification of a directed movement. In the shownexample, a graded pattern of activation first arises when a preparatory signal indicatestwo possible movement targets. Low-level activation is centered over the two associ-ated movement directions. One second later, a response signal indicates to the animal,which of the two targets must be selected. This leads to the generation of a peak of activation centered over the selected movement direction. The movement is initiatedwhen activation reaches a critical level (Bastian, Schoner, Riehle, 2003, show how theactivation level predicts response times).

How may activation fields be endowed with stability and attractors? Naturally,the fields themselves are assumed to form dynamical systems, consistent with thephysiology and physics of the corresponding neural networks. The temporal evolutionof these Dynamic Fields is thus generated by forces that determine the rate of changeof activation at each field site. Two factors contribute to the field dynamics, inputs andinteractions. Inputs are contributions to the field dynamics that do not depend on thecurrent state of the field. The role of inputs may therefore be understood separately forevery field site. The rate of change of activation, du(x)/dt, at a particular field site, x,and its dependence on input are illustrated in Figure 8. The fundamental stabilizationmechanism originating with the biophysics of neurons is modelled by a monotonicallydecreasing dependence of the rate of change on the current level of activation. This

leads to an attractor state at the level of activation at which this function intersectsthe activation axis (where the rate of change of activation is zero). In the absence of inputs, the attractor is at the resting level of the activation variable (assumed negative

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preparatory

signal

m o v e m e n t d i r e c t i o n

pre-cued

movement

directions

response

signal

PS250

500750

RS

4

5

6

1

2

3

0

0.5

1

a c t i v a t i o n

movement

direction

specified by

response

signal

t i m e [ m s ]

Figure 7: A distribution of population activation over the dimension “movement direc-

tion” evolves in time under the influence of a preparatory signal, which specifies twopossible upcoming movement directions, and a response signal, which selects one of the two. The distribution is estimated from the tuning curves of about 100 neurons inmotor cortex and their firing rate in 10 ms time intervals.

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activation field

dimension

sensory

surfaceobject 1 object 2

input

Figure 9: Input to a dynamic activation field may derive from a sensory surface. Themap from the surface to the field may involve feature extraction or complex transfor-mation, although it is represented by a homogenous one-to-one mapping here. Sensorycells with broad tuning generate inputs to a range of field sites as illustrated here for asample of only five sensor cells. In this example, the input arises from two objects onthe sensory surface. The dynamic field selects object number 2 through interaction.

These may lead to distributed activation rather than localized patterns of input areare not directly compatible with DFT.

Interactions are all contributions to the field dynamics that depend on the current

activation level at any location of the field. Connectionists refer to networks with in-teractions as “recurrent” networks. Dynamic Field Theory is based on the assumptionthat there is a universal principle of interaction — local excitation/global inhibition(Figure 10). First, neighboring field sites which represent similar values of the dimen-sion are assumed to interact excitatorily, that is, activation at either site generatespositive rates of change at the other site. This form of interaction stabilizes peaksof activation against decay and thus contributes to the stability of peaks. Second,field sites at any distance are assumed to interact inhibitorily, that is, activation ateither site generates negative rates of change at the other site. This form of interactionprevents peaks from broadening through lateral diffusion and thus also contributes tothe stability of peaks. This pattern of interaction is ubiquitous in cortex and many

subcortical structures. Only field sites with a sufficient level of activation contributeto interaction, a principle of neural function described by sigmoidal transmission func-

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dimension, x

local excitation: stabilizespeaks against decay

global inhibition: stabilizespeaks against diffusion

input

activation field u(x)

σ(u)

u

Figure 10: Right panel: Locally excitatory and globally inhibitory interaction enabledynamic fields (solid line) to generate localized peaks of activation (left), centered onlocations of maximal input (dashed line), as well as to suppress activation at competinglocations (right). Left panel: Interaction passes through a sigmoidal function, σ(u),so that only activated sites contribute (where u(x) > 0 is sufficiently large and henceσ(u(x)) is larger than zero). The activated site on the left, for instance, stronglysuppresses activation at the site on the right (see difference between input and field).The much less activated field site on the right has only very little inhibitory influencebecause its activation level is low.

tions and source of the fundamental non-linearity of neural dynamics (e.g., Grossberg,1973).When interaction is strong it may dominate over inputs, so that the attractor states

of the dynamic field are no longer dictated by input. In this regime, dynamic fieldsare not described by input-output relationships. Instead, strong interactions supportdecision making. This may be demonstrated even for the simplest response of a dy-namic field, the detection of a single localized object on the sensory surface (Figure 11).For weak input strength, the field simply reproduces the input pattern as its stableactivation pattern. This is the input driven regime, in which most neural networksare operated in connectionist modelling. When input becomes stronger, this state be-comes unstable because local excitatory interaction begins to amplify the stimulated

field site. The field relaxes to the other stable state available, in which the activatedpeak is self-stabilized by the combination of local excitation and global inhibition (seeAmari, 1977, for seminal mathematical analysis).

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xinput input

activation

fieldactivation

field

just below instability just above instability

x

u(x) u(x)

Figure 11: In response to a single localized input (dashed line) a dynamic field generatesa input-defined pattern of activation while the input is below the detection instability

(left). When the input drives activation beyond a critical level, the field goes throughan instability, in which the input-defined pattern becomes unstable and the field relaxesto an interaction-dominated peak (right). This peak is stabilized by local excitatoryinteraction, which pulls the activation peak higher than specified by input, and globalinhibition, which strongly suppresses all locations outside the peak.

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monostable bistable

resting level

u(x)

x

x

u(x)

u(x)

x

0

Figure 12: Self-sustained activation peaks (top right) and flat activation patterns atresting level (bottom right) coexist as attractors (bistable) in an appropriate parameterregime. When activation is globally lowered (e.g., by lowering the resting level), thesustained activation mode becomes unstable and a mono-stable regime results (left).

Under appropriate conditions, a self-stabilized peak of this kind may persist as anattractor of the field dynamics in the absence of input. It may then serve as a formof sustained activation, supporting working memory (Thelen, et al., 2001; Schutter,Spencer, Schoner, 2003; Spencer, Schoner, 2003; for neurocomputational models see,e.g, Durstewitz, Seamans, Sejnowski, 2000; Wang, 1999). Such sustained peaks may

coexist with input-defined patterns of activation, forming a bistable dynamical system(Figure 12) with two accessible attractor states. By lowering the resting level, theself-sustained solution may be made unstable, returning the system to a mono-stableinput-driven state and “resetting” working memory. The detection instability enablesthe “setting” of memory by making the input-driven state unstable.

The capability of dynamic fields to make decisions by selecting one of multiplesources of input emerges similarly from an instability (Figure 13). When such sourcesare metrically close, a peak positioned over an averaged location is monostable. Whensuch sources are far from each other in the metric of the dimension, x, the field selectsone of the sources and suppresses activation at the other locations. An account forthe transition from averaging to selection in visually-guided saccades based on thisinstability has successfully described behavioral and neural features of saccade initia-tion in considerable detail (Kopecz, Schoner, 1995; Trappenberg et al., 2001; Wilimzig,

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fusion

selection

input

input input

u(x)

x

u(x)

x

u(x)

x

Figure 13: Two closely spaced peaks in the input (dashed line) may generate a monos-table fused activation peak (solid) positioned over an averaged location (top). This isdue to local excitatory interaction. When the two input peaks are increasingly moreseparate (bottom), this fused solution becomes unstable and the fused attractor bifur-cates into two attractors. In each, one input peak is selected while activation at theother is suppressed.

Schneider, Schoner, 2006). The basic mechanism for how dynamic fields make selec-tion decisions is the same as that in competitive activation networks (e.g,. Usher,McClelland, 2001).

Why is it important that representations are endowed with stability? And whatrole do instabilities play? The computational metaphor is fundamentally time-less,conceiving of cognition as the computation of responses to inputs. The inputs aregiven at some point in time, the response is generated after a latency which reflectsthe amount of computation involved. An embodied and situated cognitive system, bycontrast, acts under the continuous influence of inputs. New processes are started ona background of ongoing processes. To provide for any kind of behavioral coherence,cognitive processes must be stabilized against the continuous onslaught of new sensoryinformation, of sensory feedback from ongoing action, and of internal interactions fromparallel, possibly competing processes. The localized activation peaks of DFT areinstances of stable states that support representation and resist change through the self-

stabilization induced by interaction. Resistance to change induces the reverse problemof achieving flexibility, being capable to release one process from stability to give way

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to a new process. In DFT, stable peaks of activation must be destabilized to allow forthe generation of new peaks. Instabilities play this crucial role. Through instabilities,moreover, cognitive properties emerge such as sustained activation, the capacity toselect among inputs, to fuse inputs, to couple to time-varying inputs, or to suppresssuch coupling.

Stabilities and instabilities thus form the foundation for embodied cognition be-cause they enable the emergence of cognitive states while retaining close ties to thesensory and motor surfaces, which are situated in rich, structured, and time-varyingenvironments. Within DFT, the mechanisms for creating stability and instability areneurally plausible. Finally, the Dynamical Systems account both in the classical formof DST and in the expanded form of DFT is open to an understanding of learning anddevelopment.

4 Learning and development

In DST learning means changing the dynamics of the system. Although easily statedand seemingly obvious, this insight has far-reaching consequences. Work on how peoplelearn new patterns of interlimb coordination may serve to illustrate the ideas (Schoner,1989; Zanone, Schoner, Kelso, 1992; Zanone, Kelso, 1992).

Participants practiced a new pattern of bimanual coordination, a 90 degrees phaserelationship over 5 sessions in so many days. They were provided with knowledge of results after each trial. Before and after each session, they performed other phaserelationships sampling the range from in-phase (0 degrees) to phase alternation (180degrees) in 10 steps. These required relative phases were invoked by presenting twometronomes with the requested phase relationship. Figure 14 highlights the mainresults. During learning, both the constant and the variable error decrease, consistent

with increasing stability of the practiced pattern. Before learning, performance of other patterns of relative timing is systematically biased toward phase alternation, oneof the intrinsically stable patterns of coordination. After learning, performance hasnot only improved at the practiced 90 degrees pattern, but also for all other patterns.There is now a systematic bias toward 90 degrees, reflecting the changed dynamics of coordination which have acquired a new force attracting to 90 degrees relative phase.

Learning a motor skill thus amounts to generating new forces which stabilize thenew patterns. To see what such learning processes look like in neural language weexamine the simplest learning processes in DFT. Patterns of activation may be stabi-lized simply by leaving a memory trace of ongoing activity (Schoner, Kopecz, Erlhagen,1997; Erlhagen, Schoner, 2002; Thelen et al., 2001). Such a memory trace in effect pre-activates and thus preshapes the activation field when new stimuli or new task demandsarise. The memory trace thus biases neural representations toward previously activated

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learning time

relative phase

180 deg

variability of relative phase

required

relative phase

constant error:

performed - required relative phase

90 deg

after learning

before lear ning

90 deg 180 deg

learning time

Figure 14: Schematic summary of the results of the Zanone-Schoner-Kelso (1992) ex-periment on the learning of a bimanual coordination pattern of 90 degrees relativephase. Over learning, the mean relative phase (top) approached correct performance

at 90 degrees from an initial bias toward phase alternation (180 degrees), while the vari-ability of relative phase (middle) decreased. The bottom graph compares the constanterror of relative phase at a range of required relative phases before (dashed) and afterlearning (solid). Note the bias toward phase alternation before learning (constant erroris positive for required relative phases below 180 degrees) and bias to 90 degrees afterlearning (constant error is positive below, negative above 90 degrees). Constant erroris reduced after learning at all patterns, not only at the practiced 90 degree pattern.

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patterns. The concept of a memory trace has a long history in psychology, dating backat least to William James (1899, his chapter IV on habit formation), to Gestalt psy-chology and also plays a role in modern accounts of human memory (Baddeley, 1997).Memory traces may be generated through a variety of neuronal mechanisms. In par-ticular, Hebbian strengthening of those inputs that have been successful in inducing an

activation peak would be a simple neural mechanism for the functionality of a memorytrace. The combined mechanisms of a memory trace and its role of preshaping the fieldaround previously activated locations provide an elementary form of learning, that is,experience dependent change of the dynamics of an activation field. Because a memorytrace mechanism is formally a dynamical system as well, we sometimes refer to thislearning mechanism as the preshape dynamics of a dynamic field (Erlhagen, Schoner,2002; Thelen et al., 2001).

Support for the concept of preshaping and a preshape dynamics comes from ac-counts for a wide range of phenomena, including how the probability of choices influ-ences reaction times (Erlhagen, Schoner, 2002), how motor habits are formed and theninfluence motor decisions in the “A not B” paradigm (Thelen et al., 2001), and how

spatial memory is biased toward previously memorized locations (Spencer, Smith, The-len, 2001; Schutte, Spencer, Schoner, 2003). As an illustration consider perseverativereaching in the A not B paradigm (review in Wellman, Cross, Bartsch, 1986). In thistask setting, infants are shown two reachable locations “A” and “B”, typically two wellswithin a small box. While the infant is watching, a toy is hidden either in the “A” well(on an initial set of trials called the “A”-trials) or in the “B” well (on the subsequenttest trials called the “B” trials). After a short delay of a few seconds, the box is pushedwithin reach of the infant, who will typically then will make a motor decision of reach-ing either toward the “A” or the “B” location. Young infants below 11 to 12 months of age make the “A not B” error of reaching toward the “A” location on “B” trials, thusperseverating in the action they have performed on the preceding “A” trials. Smith,Thelen, Titzer, and McLin (1999) showed that perseverative reaching is observed in atoyless version of the paradigm, in which the experimenter merely attracts the atten-tion of the infants to either the “A” or the “B” location rather than actually hiding atoy there. A DFT account of the rich phenomenology of this paradigm was provided byThelen and colleagues (2001). An activation field representing the planned movementreceives input from the attention getting stimulation, but also from the perceptuallymarked “A” and “B” locations as well as from a memory trace built up during the firstfew reaches to the “A” location. On the first “B” trial, the activation induced by theattention getting stimulus at the “B” location competes with the activation induced bythe memory trace at the “A” location. In young infants, the field is largely dominated

by these inputs. For sufficiently long delays, the attention induced activation has de-cayed and the preshape induced by the memory trace promotes reaching to the “A”instead of the “B” location, thus generating the error. In older infants, in contrast, the

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learn to reach to the “A” location. This creates an error when the cued locationswitches, but stabilizes their reaching behavior in an unchanging environment. Doeslearning always have a stabilizing effect? In the domain of motor learning we havealready seen that learning may destabilize patterns that were stable before learning:Learning a new pattern of coordination may reduce the stability of old patterns of

coordination (see Schoner, 1989, and Schoner, Zanone, Kelso, 1992, for how instabil-ities may arise during motor learning). In DFT, a form of learning that destabilizespreviously stable patterns accounts for habituation (Schoner, Thelen, 2006). Habit-uation is the gradual reduction of sensitivity to a repeated form of stimulation andis observed across a wide range of behaviors in many species (Thompson, Spencer,1966). Habituation plays an important role in developmental psychology as a probeof infant perception and cognition (Kaplan, Werner, Rudy, 1990; Spelke, Breinlinger,Macomber, Jacobson, 1992).

To illustrate how habituation is used to this end, I will briefly describe the influentialdrawbridge paradigm (Baillargeon, Spelke, Wasserman, 1985). Infants are presentedwith a visual stimulus that consists of a flap moving repeatedly through a 180 degrees

rotation (as illustrated in the top left of Fig. 16, the infant would observe this stimulusfrom the right). The infants habituate to this stimulus, which manifests itself in areduced amount of time they look at the stimulus. They are then tested with twodifferent stimuli. The “impossible” stimulus (middle left panel of Fig. 16) involves thesame 180 degree flap motion (in that respect a familiar stimulus), although initially ablock is visible that would appear to block the path of the flap (in that respect violatingan expectancy if infants’ visual representations are structured by such “knowledge”about the world). In the “possible” stimulus (bottom left panel of Fig. 16) the flapmotion is stopped at an angle of 112 degrees (in that respect a novel stimulus). Thisis consistent with the initially visible block remaining in place and preventing furthermotion of the flap (in that respect not violating an expectancy). A novel stimulus ispredicted to elicit more renewed looking on the test trials than a familiar stimulus.When instead the familiar, but “impossible” stimulus arouses more looking, then thisobservation is used to infer that infants are, in fact, building expectations about theperceived scene based on “knowledge” they have about the world (such as that objectscannot penetrate each other).

DFT provides an account for this pattern of looking that makes no use of “knowl-edge” but relies instead on the metrics of the perceptual events induced by this sequenceof stimuli. The three stimuli are embedded into a perceptual dimension, for instance,the spatial location (in depth) at which both flap motion and block are perceived (rightcolumn of Fig. 16). In this embedding it is obvious that the “impossible” test stimulus

overlaps more with the habituation stimulus than the “possible” test stimulus.These stimuli provide input to an activation field that controls looking behavior

(Figure 17). Levels of activation larger than a looking threshold lead to looking at the

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inputs

inputs

perceptualdimension

inputs

test impossible (familiar)

test possible (novel)

habituation stimulus

u1u2

block

block

drawbridgedrawbridge

drawbridge

drawbridge

block

Figure 16: Left column: A sketch of the 3 stimuli presented to the observing infant(which looks at these from the right). Top: The moving flap alone during habitu-ation. Middle: The same flap motion and a block on its movement path, which isactually quickly removed through a trap-down as the flap occludes it (“impossible”test stimulus). Bottom: Reduced flap motion and the block (“possible” test stimulus).

Right column: These stimuli can be embedded in a perceptual dimension that spansthe visual depth at which movement and block are seen. The moving flap providesbroad input (solid line) across this spatial dimension, but centered at a larger meanvisual depth for the 180 degree motion (top two panels) than for the stopped motion(bottom). The block provides added input (dashed line) at a larger depth (bottom twopanels). The two marked locations along this perceptual dimension (vertical bars) arerepresented by two activation variables, u1 and u2, in the dynamic model of Schonerand Thelen (2006).

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stimulus, levels below that threshold lead to looks away from the stimulus. The acti-vation field drives a second, inhibitory field, which projects back as inhibition onto theactivation field. During habituation, this accumulated inhibition in the inhibitory fieldgradually reduces the amount of activation a given stimulus induces. This destabilizesthe positive peak of activation induced by the stimulus and promotes looking away

from the stimulus. The coupled dynamics of activation and inhibition thus account forthe typical pattern of looking during the habituation phase (bottom right of Fig. 17).When a new stimulus is presented during the test phase, the amount of dishabitua-

tion depends on how strongly the test and the habituation stimulus overlap (Figure 18).A strongly overlapping stimulus (such as the familiar or “impossible” stimulus in thedrawbridge paradigm) activates field sites that have previously been activated andare thus initially at a higher level of activation, leading to more looking (top panelof Fig. 18). A less overlapping stimulus (such as the novel of “possible” stimulus inthe drawbridge paradigm) activates field sites that were previously at rest, leading toinitially lower rates of looking (middle panel of Fig. 18). The pattern of looking alsodepends on the temporal structure of visual experience. Only the looking times on

the first few test trials are determined by the level of activation at the beginning of the test phase. After these first few presentations of test stimuli, differences in initialactivation have been washed out (bottom right in Fig. 18). At that point, differences inthe level of inhibition determine differences in activation and the model predicts largerlooking times for novel stimuli. In the experimental literature, this time dependenceof looking on test shows up as an interaction between the novelty or familiarity of astimulus and the order of presentation. The preference for the “impossible” stimulusarises only when that stimulus are presented first during test, not when is is presentedsecond (Baillargeon, 1987).

What about development? Central to the dynamical systems approach to devel-opment is the postulate that development depends on experience and is therefore inlarge part a learning process (see Thelen, Smith, 1994, for detailed argument). Simi-lar to what I just described for the learning of motor skills, what changes during thedevelopmental process is the neuronal dynamics supporting a particular function. Inthe DFT model for perseverative reaching, for instance, a change of the field dynamicsfrom largely input-driven to largely interaction-driven accounts for how perseverativereaching subsides with increasing age. This account explains a range of age-relatedchanges, including the tolerance of increasing delays and the resistance to distractorstimuli (Thelen et al., 2001). John Spencer and colleagues have generalized this ac-count to the processes supporting spatial representations in what they call the “spatialprecision hypothesis” (Schutte, Spencer, Schoner, 2003; Spencer, this volume). They

have shown that a shift from weak and spatially diffuse interaction to stronger andspatially focussed interaction in dynamic fields supporting spatial memory and actionplanning accounts for the time dependence and variance of performance in tasks re-

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100 200

2 4 6 8 10

4

8

habituation trials

inhibition

looking time [sec]

time [sec]

stimulus

activation field

lookingthreshold

stimulation

perceptual dimension

inhibition field

activation

Figure 17: Left column: A dynamic field model of habituation consists of two layers, anactivation field (middle) that receives sensory input (top) and represents the propen-sity to look at the stimulus, and an inhibition field (bottom) which is driven by theactivation field which it in turn inhibits. The fields span a relevant perceptual dimen-

sion (here the visual depth of a stimulus in the drawbridge paradigm as schematicallyindicated at the bottom). The stimulus marked by a vertical line (the 180 degree draw-bridge motion) is presented periodically during the habituation phase . Right column:The stimulus (square trace), activation field (zig-zag trace) and inhibition field (thinincreasing trace) at that marked location of the field are shown as a function of timein the top panel. The bottom panel shows the looking time across trials predicted bythis model. For each stimulus presentation, looking time is computed as the amountof time that activation is above the looking threshold (of zero).

30



100 200

-1

0

1

2 4 6 8 20

5

10

lookingtime

-1

0

1u1

u2

v1

v2

1

habituation test

familiar

novel

inhibition

field

time [sec]

100 200 time [sec]

12

activation

field

Figure 18: Left column: Two locations in the field are marked by vertical lines.The right location 1 corresponds to the “impossible” test stimulus of the drawbridgeparadigm with 180 degrees motion. The left location 2 corresponds to the “possible”test stimulus with reduced motion of the flap. Right column: Activation and inhibition

at these two locations are shown as functions of time in the two upper panels. Thetraces are identifiable as in Fig. 17. The bottom panel shows the looking time as a func-tion of stimulus presentations. During the habituation phase, activation and inhibitionbuild for u1 (top), leading to decrease of looking (bottom) to criterion (horizontal line:half of the average looking time during the first 3 habituation trials). During test, theblock provides a boost of input to both variables, leading to renewed looking (dishabit-uation). There is more dishabituation to the familiar than to the novel stimulus whenthe familiar stimulus is presented first as shown here.

31



quiring spatial working memory. This account has implications for a range of otherbehaviors involving spatial representations such as spatial long-term memory, spatialdiscrimination, and the capacity to maintain object-centered reference frames (Simmer-ing, Spencer, Schutte, 2007). In all of these behaviors, the spatial precision hypothesispredicts correlated changes during development.

That development means change of the neuronal dynamics gives a new and concretemeaning to the idea of emergence. If it is the dynamics that develop, then developmen-tal processes must be analyzed by investigating the causes of behavioral patterns, not

just the patterns themselves. Thus, at a particular stage of a developmental processthe system may have a certain propensity to generate patterns of sustained activa-tion supporting working memory, but whether or not such patterns are actually stablemay depend on the perceptual context, the short-term behavioral history, or the totalamount of stimulation (Schoner, Dineva, 2006).

The DFT account for habituation similarly postulates that what changes duringdevelopment is the neuronal dynamics of activation and inhibition (Schoner, Thelen,2006). The activation fields of older infants are assumed to have larger resting levels, so

that less stimulation is required for them to reach the point at which inhibition startsto built. This accounts both for the faster rates of habituation in older infants as wellas their increased preference for novel stimuli under equivalent stimulus conditions.Recent work by Perone, Spencer and Schoner (2007) showed that this account can belinked to a generalization of the spatial precision hypothesis (space now referring tothe space of visual features).

The postulate that development amounts to change of dynamics has been made inslightly different form also by van der Maas and Molenaar (1992). These authors havelooked at more complex cognitive tasks (like Piaget’s task of judging the amount of water in differently shaped containers) that are assessed by overall performance scores.The level of performance at any particular point during development is interpreted asreflective of an attractor state of a dynamical system. It is not entirely clear whatthe status of that dynamical system is. Its attractors do not describe the real-timebehavior (which is a complex sequence of actions in many of the cognitive skills towhich this model refers). These attractors characterize something closer to an overallstate of competence. Time and stability have an uncertain status as well but may referto something like how reproducible successful displays of a competence are. The mainpoint of the account is, however, that graded and continuous changes of the parametersof such attractor dynamics may lead to categorical and temporally discrete changes of the number and nature of the attractors of such a system. Using catastrophe theory(with noise added, so that the account is closer to the theory of nonequilibrium phase

transitions of Haken and colleagues, 1983), van der Maas and Molenaar (1992) describevarious signatures of such qualitative changes of a competence dynamics. These includethe critical fluctuations and critical slowing down for which direct evidence has been

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provided in the context of movement coordination (Schoner, Kelso, 1988; Schoner,Haken, Kelso, 1986). Here, however, fluctuations and resistance to perturbations arelooked at in a different way. They occur in the manifestations of a general competence,so that performance in a cognitive task may vary from one test to another, for instance.

But how about the developmental processes itself through which the neuronal dy-

namics change? To date, the DFT approach has not addressed these processes at alevel of specificity and detail that would comparable to that achieved in capturing thebehavioral dynamics. This is clearly a task for the future. Knowing what it is thatchanges during development is, of course, a critical prerequisite to providing a processaccount for such change.

In contrast, Paul van Geert (1991; 1998) and colleagues have used dynamical sys-tems thinking primarily at the time scale of development. Their postulate is that theprocess of development can be characterized as some sort of dynamical system, evolvingon the slower time scale over which development unfolds. Specifically, they propose tothink of development as a growth process. Qualitatively different patterns of growthcan be modelled such as continuous growth, characteristic variation of growth rate

from slow to fast to slow (S-shaped), and oscillatory growth rates. Moreover, cou-pling among various growth processes may induce complex patterns of growth, such asstage-like alternations between slow and fast rates of change.

The concrete mathematical models that are used to illustrate these ideas encounterthe same kind of problem discussed above for the account of van der Maas and Mole-naar (1992). The state variable that reflects a particular stage in development is notdirectly linked to real-time behavior. It represents again something like a general levelof competence, assessed through such measures as the sizes of vocabularies or the fre-quency of use of certain competences. In fact, one version of the model (van Geert,1998) explicitly introduces a large ensemble of possible competences, which are givenweights that may reflect how likely it is to display the competences. These weightsare the dynamical variables that evolve over developmental time. There is clearly agap between these abstract, disembodied variables and the concrete neuronal processthat generate the behavior that is used to assess competences. Moreover, the com-petences are preformed and only need to be activated by increasing their respectiveweight value. Both limitations make clear that this account does not yet link the devel-opmental process to the stream of behavioral experience. Implicitly, the model seemsto assume that there is an intermediate level at which the developing nervous systemtakes stock of its current set of skills, and updates its probability of use dependent onthis form of assessment. How the skills are implemented in the first place, where theycome from, how they are employed to deal with incoming sensory information, and

what sort of sensory information drives the learning process, such questions remainopen. The strength of the model consists, instead, of a demonstration that qualitativechanges may emerge from developmental processes that are governed by fairly simple

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dynamical laws.Consistent with this kind of use of models, Paul van Geert and colleagues emphasize

that dynamical systems thinking may be powerful at the level of metaphor (see vanGeert, Steenbeek, 2005, where the relationship between the “Bloomington” and the“Groningen” approaches is discussed in some detail). Dynamical Systems as metaphor

has been an important source of new ideas, new methods of analysis, new questions, inboth flavors of the approach. An example is the emphasis on variability as a measure of underlying stabilization mechanisms, important on the time scale of behavior (Thelen,Smith, 1994), on the somewhat indeterminate time scale of expression of competence(van der Maas, Molenaar, 1992), and on the developmental time scale (van Geert,1998).

5 Conclusions

In this overview, I have shown how four concepts, attractors, their instabilities, dynamic

activation fields, and the simple learning mechanism of a memory trace together mayenable an embodied theory of cognition. This is a theory that takes seriously thecontinuous link of cognitive processes to the sensory and motor surfaces, and that takesinto account that cognition emerges when systems are situated in richly structuredenvironments. The theory is open to an understanding of learning, and is based onneuronal principles.

Connectionist thinking shares many of the same ambitions and core assumptions,in particular, the commitment to using neuronal language to provide accounts forfunction and learning. In some instances, connectionist models have used the exactsame mathematics as dynamical systems models (e.g., Usher and McClelland, 2001)and many connectionist networks are formally dynamical systems. Connectionism is

really about change, about how under the influence of a universe of stimuli, statisticalregularities may be extracted by learning systems, which may reflect these properties inthe structure of their representations (Elman et al., 1997). Signatures of that processinclude accounts for learning curves, so that connectionist models speak to the evolutionof systems on the time scale of development. On the other hand, some connectionistmodels also speak to the flow of behavior on its own time scale, the generation of responses to inputs provided in a given task setting. Connectionism thus straddles thetwo time scales that the two flavors of dynamical systems ideas have studied, for themost part, separately.

The detailed comparison of Dynamical Systems thinking and connectionism is notthe goal of this overview (see Smith and Samuelson, 2003, and much of this book).Maybe I can say just this much: The problem of bringing Dynamical Systems think-ing to bear on the process of development itself may be the most constructive point

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development as change of system dynamics

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