Development as Change of System Dynamics:
Stability, Instability, and Emergence.
G. Schöner
Institut für Neuroinformatik
Ruhr-Universität Bochum
Germany
October 13, 2007
Schöner, G.: Development as Change of System Dynamics: Stability, Instabilty, and
Emergence. In: Toward a New Grand Theory of Development? Connectionism and
Dynamic Systems Theory Re-Considered, J.P. Spencer, M. Thomas, & J. McClelland
(Eds.), Oxford University Press (2007, in press)
1
Introduction
The conference out of which this book grew was dedicated to Esther Thelen. Three
themes that permeated Esther Thelen’s work are central also to Dynamical Systems
thinking (Thelen, Smith, 1994; Smith, Thelen, 1993; Smith, Thelen, 2003). First is the
conviction that cognition cannot be separated from its sensory and motor embodiment
in a rich and changing environment. Second is the idea that cognition and other complex forms of behavior may emerge in suitable behavioral and environmental contexts.
The soft-assembly of such skills may be multi-causally linked to diverse processes, none
of which need be the single cause for the emergence of the skill. Third, all behavior
and development unfold in time. The behavioral history in a task context and the
developmental history matter for when and how a skill emerges. Development occurs
therefore through individual developmental trajectories.
These themes structure Dynamical Systems thinking at different levels. Esther
Thelen liked to emphasize how Dynamical Systems thinking can provide metaphors
that help ask new questions, generate new experimental paradigms and new measures
and lead to new kinds of explanations. In this chapter I will review Dynamical Systems
Theory (DST) as a set of concepts that formalize such metaphors and thus turn these
into a scientific theory of considerable rigor. While Dynamical Systems Theory is
sometimes also viewed as a collection of certain kinds of models, this chapter will not
focus on any specific model nor review modelling work in general.
In fact, DST is not the idea to model things using differential equations or dynamic
iterative maps. Instead, DST is a much more specific set of concepts, of which I
shall emphasize five in this chapter: (1) Behavioral patterns resist change, that is,
they are stable. This may be mathematically characterized by considering behavioral
patterns as attractor states of a dynamical system. (2) Behavioral change is brought
about by a loss of stability. (3) Representations possess stability properties as well
and can be understood as the attractor states of dynamic fields, that is, of continuous
distributions of neural activation. (4) Cognitive processes emerge from instabilities of
dynamic fields. (5) Learning consists of changes in the behavioral or field dynamics,
that shift the behavioral and environmental context in which instabilities occur.
What I shall do below is walk the reader through each of these conceptual steps
to explain and illustrate the key ideas, evoke exemplary model systems in which these
ideas have been brought to fruitition, and finally link the ideas to other approaches
and problems.
2
Dynamical Systems Theory (DST)
The central nervous system is tightly interconnected. As a result, any given state of
the central nervous system is exposed to a range of influences some of which may be
2
pushing to change the neural state. The very flexibility of the central nervous system, the interconnectedness and multi-functionality of many of its components make
such perturbations the norm rather than the exception. Only neural states that resist
such perturbations will persist long enough to be observable, to influence down-stream
processes, 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 world
through a continuous link to sensory information. Organisms are embodied nervous
systems in that their perceptual, motor, and cognitive processes are intermittently
coupled to sensory information from the sensory and motor systems. Given complex
environments and a complex body with many more degrees of freedom than recruited
for any particular task, such couplings are sources of perturbation. Only stable functional states persist in the face of such perturbations.
Mathematically, stability is the constitutive property of attractors. Illustrated in
Figure 1, an attractor is an invariant (unchanging in time) solution of a dynamical
system, toward which solutions converge if they start nearby. If perturbations push
the 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 pushing
in opposite directions converge and balance. A stabilization mechanism is implied
whenever the rate of change, dx/dt, of a dynamical state variable, x, depends on the
current state in the way illustrated in Fig 1. One may think of stability as an abstract
generalization 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 positive
velocities by decelerating the ball. If the ball moves in the opposite direction, its
velocity is formally negative, and friction changes that velocity to zero. Thus, the state
of zero velocity is a stable state of this system. The figure and the analogy illustrate the
concept of a fixed point attractor, that is, a stable state that is itself a constant solution
of the dynamical system. Stability may be defined for more complex solutions as well,
such as oscillatory solutions (limit cycle attractors), oscillatory solutions with multiple
frequencies (quasi-periodic attractors), or solutions with very complex time structure
(strange attractors). In this review I shall limit myself to fixed point attractors, which
can go a long way toward accounting for states of the nervous system.
Neurons and neural networks are naturally described as dynamical systems (Wilson, 1999) and provide through their intrinsic dynamics the mechanisms that stabilize
attractors (see Hock, Schöner, Giese, 2003, for a discussion). In that sense, stability
comes for free from the neuronal dynamics prevalent in the central nervous system,
although the sensory-motor periphery (through muscle viscosity and elasticity, for instance) may also contribute. Once a stabilization mechanism is in place, there is no
need for other computational mechanisms to determine the output of a dynamic neural
3
dx/dt=f(x)
x
attractor
Figure 1: A differential equation model of a dynamical system is defined by how the rate
of 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 the
arrows: A negative rate of change for values larger than the attractor state leads to
a decrease in time toward the attractor, a positive rate of change for values smaller
than the attractor leads to increase in time toward the attractor. The time-invariant
attractor thus structures the temporal evolution of the state of the dynamical system
in its vicinity.
4
dx/dt=f(x)
x
Figure 2: When a dynamical system changes (from the form shown as a dashed line
to the form shown as a solid line), the stability of the new attractor state leads automatically 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 perturbations
also help the system track any changes to the attractor state incurred as inputs to the
network change (Figure 2). What such changing inputs do is move the attractor state
to a new location in state space. The old attractor state is then no longer a constant
solution (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, the
attractor is called set-point and deviations from the set-point are called errors. The
control system is designed to reduce such errors to zero. A conceptual step beyond
this analogy is made, however, when multiple attractors co-exist. The simplest such
case, bistability, is illustrated in Figure 3. Which of the two attractor states is realized
depends on the prior history of the system. As long as the state of the system lies within
the basin of attraction of attractor 1, the system tracks that state as the dynamics
change. Note that it makes no longer sense to talk about the deviation from either
attractor 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 over
another attractor is a simple form of decision making and the dynamics stabilize such
5
dx/dt=f(x)
repellor
x
attractor 1
attractor 2
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 sufficiently
far so that it crosses over into the alternative basin of attraction. This induces a
stochastic switch of state.
Such stochastic switches out of an attractor become more likely if a change of the
dynamics reduces the stability of the attractor. Figure 4 illustrates what happens
when 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 then
both disappear. No zero crossing of the rate of change remains in the vicinity of the
former attractor and repellor. If the system was originally in or near attractor 1, it
tracked this attractor as it moved toward the repellor. But when attractor and repellor
collide and disappear, the system must switch to attractor 2 (if it did not escape
with the help of a stochastic perturbation earlier). Mathematically, such a change
in the number and stability of attractors and/or repellors is a bifurcation. In simple
dynamical systems such as the one shown in Figure 4, bifurcations can only occur by
collision 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 rubber
band. 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 function
of the state). The only way to eliminate a zero-crossing of the function is thus to make
6
two or more zero-crossings collide. The case shown in Figure 4 is the simplest such
bifurcation in which the rubber band lifts off from the x-axis just as it becomes tangent
to 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 are
most 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 tracking of a continuously changing state. The change is preceded by tell-tale signatures
of instability such as increasing variability and increasing time needed to recover from
perturbations. These can be exploited to detect instabilities and thus to distinguish
qualitative from quantitative change (review, Schöner, Kelso, 1988; van der Maas,
Molenaar, 1992). In non-linear dynamical systems, instabilities arise naturally in response to even relatively unspecific changes of the dynamics. In the illustration of Figure 4, for instance, the changes to the dynamics are not specifically localized around
the attractor 1 that is losing stability, but rather amount essentially to an increasing
bias toward larger values of the state variable “x”.
This illustrates that attractors are not fixed entities. When they disappear, they are
not stored somewhere, or simply “deactivated”. Attractors may emerge out of nowhere
when the conditions (the dynamics) are right. This can be visualized by looking at the
scenario 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 branching
off 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 kind
of variables “x” would describe behaviors, patterns, decisions? Many readers may be
familiar with now classical examples of DST in interlimb coordination, reviewed for
instance in Scott Kelso’s (1995) book. In this work, the relative phase between two
rhythmically moving limbs has been shown to be sufficient to characterize patterns
of movement coordination. Each of the two common patterns of coordination can be
characterized 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 shared
descending input to the two limbs as well as excitatory coupling. In the “anti-phase”
pattern of coordination (relative phase equal to 180 degrees), homologous muscles
alternate. 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 periodic
limb 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
(Schöner, Kelso, 1988).
Relative phase may be the only obvious example of a single variable that clearly
7
dx/dt=f(x)
x
attractor 1
repellor
attractor 2
Figure 4: Changes of a bistable dynamical system (from dashed via dotted to solid
line) lead to an instability, in which the attractor 1 collides with the repellor, leaving
attractor 2 behind. This bifurcation thus leads to a change from bistable to monostable
dynamics.
describes a pattern (of relative timing) and that is independent of other details of how
the 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 oscillator
models in the literature of coordination of rhythmic movement are formulated in terms
of variables that describe the spatial position and velocity of the moving effector,
although these variables are not identical to the associated physical quantities. When
the limb is mechanically perturbed, for instance, the physical position and velocity is
changed, but the oscillator driving the movement is not necessarily affected (Kay et al.,
1987). Conversely, the oscillator variables are not directly related to neural activations
either (but see Grossberg, Pribe, Cohen, 1997, for an account at the level of neuronal
oscillators).
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 two
variables 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 briefly
sketch, even if a full explanation goes beyond what can be achieved in a survey chapter.
8
The ensemble of neural processes and their coupling to the sensory and motor
surfaces form a very high-dimensional dynamical system (Wilson, 1999). Stable states
in such a high-dimensional dynamics are the persistent macroscopic states which are
observable at the behavioral level. Stability means that in all directions of the highdimensional 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 stability
fail in multiple directions at the same time. The direction along which the restoring
forces 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 is
slower than in any other direction of the high-dimensional state space. This is because
the restoring forces are weaker in this direction leading to slower movement toward the
attractor state along this direction than along any other direction. Perpendicular to the
Center-Manifold, in contrast, restoring forces are strong and the system quickly moves
from wherever it started out to some point on the Center-Manifold. Thus, the longterm evolution of the system is essentially dictated by movement within the CenterManifold. This intuition is formalized in the Center-Manifold-Theorem, which says
that knowing how the system evolves along the Center-Manifold uniquely determines
how the system evolves in the original high-dimensional space. Thus, to capture the
macroscopic states of the high-dimensional dynamics and their long-term evolution, it
is sufficient to model the dynamics along those dimensions along which stability breaks
down.
In the example of Figure 4, for instance, the dimension, x, would correspond to
the direction in a much higher-dimensional state space along which the stabilization
mechanism breaks down. The true dynamical system may have many more dimensions
(e.g., activation levels of many neurons involved in stabilizing attractor 1). But when
the bifurcation occurs, the system is still sitting in the attractor state along all the
other dimensions except the one, shown in the Figure, along which the instability
occurs. The switch to the new attractor arises from movement long the unstable
direction. The other dimensions do not add anything qualitative to the dynamics and
can, therefore, be left out of the description.
The Center-Manifold Theorem implies a huge reduction in the number of dimensions that need to be measured and modelled to understand macroscopic states and
their change. Although the theorem is mathematically true only exactly at a bifurcation, 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 systems
are almost always near an instability and can thus be described by low-dimensional
dynamical systems most of the time.
9
x3
center
manifold
x2
x1
attractor about
to become unstable
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 typically
one direction in the high-dimensional space along which the restoring forces begin to
fade (shorter arrows) while in other directions the stabilization mechanism still works
(longer arrows). That first direction spans the Center-Manifold.
10
Does this mean that the low-dimensional dynamical systems are purely descriptive
while a fully mechanistic account must take place in the original, high-dimensional
space? The answer depends on what is meant by “fully mechanistic account”. If that
means, literally, an account that captures the state of all neural processes, then, by definition, only extensive high-dimensional computational modelling will be satisfactory.
If this means, however, that an account is sufficient to actually generate a behavior in
a real system, then capturing the macroscopic, low-dimensional dynamics qualifies. A
proof of sufficiency in that sense has been provided, for instance, by generating simple
robotic behaviors such as target acquisition, target selection, obstacle avoidance, and so
on from low-dimensional attractor dynamics with appropriate bifurcations interfaced
with very simple sensory and motor systems (see Bicho, Schöner, 1997, for an example
and Schöner, Dose, Engels, 1995, for a review). Such robotic implementations also
prove that DST models are embodied and situated in the sense that no new concepts
are needed when dynamical systems models are acted out with real bodies moving in
real environments based on real sensors. Such “acting out” in the real world is an
interesting challenge to theoretical accounts of cognition. Accounts rooted in information processing have traditionally relied on relatively high-level interfaces with the
sensory and motor processes necessary to act in the real world (e.g., world models and
configuration space planning). These high-level interfaces have typically been difficult
to 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 abstract
models may result, models that cut through the high-dimensional space describing the
neural systems supporting behavior in ways that depend on the task, on the state studied, and on the particular parametric and input conditions under which an instability
is observed. On the other hand, over the last few years a closer alliance of Dynamical
Systems models with neurophysiological principles has contributed much to reducing
the gap between the low-dimensional dynamical descriptions and the neural networks
that implement them. This will be a theme in the next section.
Given the abstract nature of DST accounts, why is DST often perceived to be
primarily about motor behavior? Many of the exemplary model systems that influenced
the development of Dynamical Systems ideas did come from the motor domain (as
reviewed in Kelso, 1995; but see Hock, Kelso, Schöner, 1993, for early work using
DST in perception). On the other hand, much work has since shown that the ideas
are not intrinsically tied to motor behavior. Van der Maas and Molenaar (1992),
for instance, applied these concepts to characterize a wide variety of developmental
changes including cognitive development. Similarly, van Geert (1998), has used the
abstract setting of DST to think quite generally about continuity vs. discontinuity
in 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
11
increased substantially when moving from simple motor skills to cognitive skills. This
is due to a conceptual problem, that must be confronted when stepping beyond the
domain 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 excessive
abstraction. For instance, the position of my arm, the values of the joint angles in my
arm, their level of neuronal activation, the frequency or phase of my arm’s rhythmical
movement are all perfectly good candidates for dynamical state variables that are not
particularly abstract. They have well-defined values that evolve continuously in time.
When we move beyond pure motor control we encounter problems with these variables, however. What value, for instance, does the phase of my arm’s rhythmical
movement have before I start the movement or after I stopped moving? Which value
do the movement parameters “amplitude” or “direction” have before I have selected
the target of my movement? Obviously, the selection and initiation of motor acts, but
also the generation of perceptual patterns and the commission to memory of a perceptual or motor state require a different kind of variable than those used to describe
motor control. These variables must capture the more abstract state of affairs in which
variables appear to have well-defined values some of the time but not at all times. More
fundamentally, we must understand how state variables may change continuously even
during 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 high
levels of activation impact on down-stream systems, including ultimately on the motor
system, while low levels of activation do not. This captures the fundamental sigmoidal
nonlinearity of neural function: Only activated neurons transmit to their projection
targets, while insufficiently activated neurons do not. There are multiple neuronal
mechanism through which activation may be realized neuronally (e.g., through the
intra-cellular electrical potential in neurons, through the firing rate of neurons, through
the firing patterns of neural populations, or even through the amount of synchrony
between the spike trains of multiple neurons, see, e.g., Dayan and Abbott, 2001). But
the concept of activation does its work for DST independently of the details of its
neural 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,
12
there is little doubt that a wide range of perceptual, cognitive and motor processes can
be characterized by such dimensions (Shepard, 2001). Below I will argue that neuronal
representations in the higher nervous system provide guidance in identifying relevant
dimensions. Note that activation, u, is now taking on the role of the continuous state
variable 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). Localized 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 structures and behavior; the location of the peaks indicate the current values along the
field dimension that are handed on to the down-stream structures. By contrast, flat
patterns of low-level activation (middle) represent the absence of specific information
about 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 bringing
about an effect (bottom).
Conceptually, localized peaks are the units of representations in DFT. The location
of a peak represents the value along the dimension that this peak specifies. The dimensional axis thus encodes the metrics of the representation, what is being prepared,
perceived, or memorized and how different the various possible values along the dimension are. During the preparation of a goal-directed hand movement, for instance,
a peak of activation localized along the dimension “movement direction” is a movement plan, that both specifies the direction in which the movement target lies and the
readiness to initiate the movement (Erlhagen, Schöner, 2002). If the target is shifted
during the preparation of the movement, the peak may shift continuously along the
field dimension and thus provide an update of the movement plan to changed sensory
information. In this situation of a continuously moving peak we are back to the simpler
picture of DST, in which the value of the state variable, x, now represented by the peak
location, changes continuously. In this simpler description, the peak location, x, is the
dynamical variable.
This form of representation is essentially the space code principle of neurophysiology (e.g., Dayan and Abbott, 2001), according to which the location of a neuron in
the neural network determines what the neuron encodes, while its level of activation
represents how certain or important or imminent the information is that the neuron
transmits. Feature maps are possible neuronal realizations of such activation fields,
consisting of ensembles of neurons that code for the feature dimensions. Such maps
conserve 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
13
activation
field
dimension
specified value
activation
field
dimension
no value specified
information, probability, certainty
activation
field
dimension
metric contents
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 estimate
of the specified value along the dimension (top), while flat, low-level distributions of
activation indicate the absence of information about the dimension (middle). Graded
activation patterns may represent the amount of certainty about different values along
the metric dimension (bottom).
14
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 dendritic 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 curves
imply that whenever a particular value of a feature dimension is specified, an entire
population of neurons is active. Based on this insight, distributions of population
activation may be constructed from the neural activation levels of many neurons, irrespective of where these neurons are located in a cortical or subcortical area (Erlhagen
et al., 1999). Figure 7 shows an example. About 100 neurons in motor cortex were
recorded while a monkey performed center-out hand movements toward visual targets. These neurons were tuned to movement direction and their tuning curves were
weighted with the current firing rate of each neuron to define its contribution to the
distribution of population activation (the data are from Bastian, Schöner, 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 the
dynamic activation field that represents the planned movement direction and evolves
over time reflecting the process of specification of a directed movement. In the shown
example, a graded pattern of activation first arises when a preparatory signal indicates
two possible movement targets. Low-level activation is centered over the two associated 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 initiated
when activation reaches a critical level (Bastian, Schöner, Riehle, 2003, show how the
activation 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 the
physiology and physics of the corresponding neural networks. The temporal evolution
of these Dynamic Fields is thus generated by forces that determine the rate of change
of activation at each field site. Two factors contribute to the field dynamics, inputs and
interactions. Inputs are contributions to the field dynamics that do not depend on the
current state of the field. The role of inputs may therefore be understood separately for
every 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 stabilization
mechanism originating with the biophysics of neurons is modelled by a monotonically
decreasing 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 intersects
the 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
15
activation
1
0.5
0
3
pre-cued
movement
directions
2
time
1
movement
direction
specified by
response
signal
6
5
mo 4
v
dire emen
ctio t
n
PS
250
500
750
preparatory
signal
[ms]
RS
response
signal
Figure 7: A distribution of population activation over the dimension “movement direction” evolves in time under the influence of a preparatory signal, which specifies two
possible 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 in
motor cortex and their firing rate in 10 ms time intervals.
16
S(x)
du(x)/dt =
-u(x) + h + S(x)
u(x)
attractor
without
input u=h
attractor
with input
u=h+S(x)
Figure 8: The generic dynamics of an activation variable, u(x), is illustrated in the
absence of interaction. The fundamental stabilization mechanism (“−u”) generates
an attractor state at the (negative) resting level h in the absence on input (dashed
line). Input, S(x), shifts the rate of change upwards, generating a new attractor at
u = h + S(x) (solid line).
by convention although the units are arbitrary). Excitatory inputs, S(x), shift the rate
of change upwards by a constant amount. This shifts the attractor toward positive
levels of activation.
Different field sites, x, receive different inputs, S(x). For sensory representations,
the way inputs vary along the field dimension, x, defines the meaning of that dimension. The typical idea is that inputs derive from the sensory surfaces, so that the
input function varies continuously as a function of x and neighboring sites receive
overlapping inputs from any item on the sensory surface (Figure 9). For instance, if
x represents retinal location, then inputs coming from the retina are determined by
the receptive field of every field site. If x represents orientation, then the neuronal
wiring that extracts the orientation of edges generates a link between the retina and
the associated locations in a direction field (Jancke, 2000). For motor fields, the meaning of the dimension is established by how the field projects onto the motor surface,
that is, onto the space of all possible effector configurations. Connectionist networks
sometimes make the same assumption of a continuous one-to-one mapping between
sensory surfaces and networks (e.g., when modelling cortical feature maps). In other
instances, however, connectionist networks use much more complex input functions.
17
activation field
dimension
input
sensory
object 1 object 2 surface
Figure 9: Input to a dynamic activation field may derive from a sensory surface. The
map from the surface to the field may involve feature extraction or complex transformation, although it is represented by a homogenous one-to-one mapping here. Sensory
cells with broad tuning generate inputs to a range of field sites as illustrated here for a
sample of only five sensor cells. In this example, the input arises from two objects on
the sensory surface. The dynamic field selects object number 2 through interaction.
These may lead to distributed activation rather than localized patterns of input are
are 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 interactions as “recurrent” networks. Dynamic Field Theory is based on the assumption
that there is a universal principle of interaction — local excitation/global inhibition
(Figure 10). First, neighboring field sites which represent similar values of the dimension are assumed to interact excitatorily, that is, activation at either site generates
positive rates of change at the other site. This form of interaction stabilizes peaks
of 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 at
either site generates negative rates of change at the other site. This form of interaction
prevents peaks from broadening through lateral diffusion and thus also contributes to
the stability of peaks. This pattern of interaction is ubiquitous in cortex and many
subcortical structures. Only field sites with a sufficient level of activation contribute
to interaction, a principle of neural function described by sigmoidal transmission func-
18
activation field u(x)
local excitation: stabilizes
peaks against decay
global inhibition: stabilizes
peaks against diffusion
u
input
σ(u)
dimension, x
Figure 10: Right panel: Locally excitatory and globally inhibitory interaction enable
dynamic fields (solid line) to generate localized peaks of activation (left), centered on
locations of maximal input (dashed line), as well as to suppress activation at competing
locations (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 hence
σ(u(x)) is larger than zero). The activated site on the left, for instance, strongly
suppresses 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 influence
because 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 fields
are not described by input-output relationships. Instead, strong interactions support
decision making. This may be demonstrated even for the simplest response of a dynamic 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 stable
activation pattern. This is the input driven regime, in which most neural networks
are operated in connectionist modelling. When input becomes stronger, this state becomes unstable because local excitatory interaction begins to amplify the stimulated
field site. The field relaxes to the other stable state available, in which the activated
peak is self-stabilized by the combination of local excitation and global inhibition (see
Amari, 1977, for seminal mathematical analysis).
19
just below instability
u(x)
input
x
just above instability
u(x)
input
x
activation
field
activation
field
Figure 11: In response to a single localized input (dashed line) a dynamic field generates
a 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 through
an instability, in which the input-defined pattern becomes unstable and the field relaxes
to an interaction-dominated peak (right). This peak is stabilized by local excitatory
interaction, which pulls the activation peak higher than specified by input, and global
inhibition, which strongly suppresses all locations outside the peak.
20
bistable
monostable
u(x)
x
u(x)
x
u(x)
x
0
resting level
Figure 12: Self-sustained activation peaks (top right) and flat activation patterns at
resting level (bottom right) coexist as attractors (bistable) in an appropriate parameter
regime. When activation is globally lowered (e.g., by lowering the resting level), the
sustained activation mode becomes unstable and a mono-stable regime results (left).
Under appropriate conditions, a self-stabilized peak of this kind may persist as an
attractor of the field dynamics in the absence of input. It may then serve as a form
of sustained activation, supporting working memory (Thelen, et al., 2001; Schutter,
Spencer, Schöner, 2003; Spencer, Schöner, 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, the
self-sustained solution may be made unstable, returning the system to a mono-stable
input-driven state and “resetting” working memory. The detection instability enables
the “setting” of memory by making the input-driven state unstable.
The capability of dynamic fields to make decisions by selecting one of multiple
sources of input emerges similarly from an instability (Figure 13). When such sources
are metrically close, a peak positioned over an averaged location is monostable. When
such sources are far from each other in the metric of the dimension, x, the field selects
one of the sources and suppresses activation at the other locations. An account for
the transition from averaging to selection in visually-guided saccades based on this
instability has successfully described behavioral and neural features of saccade initiation in considerable detail (Kopecz, Schöner, 1995; Trappenberg et al., 2001; Wilimzig,
21
fusion
u(x)
input
selection
u(x)
x
u(x)
input
input
x
x
Figure 13: Two closely spaced peaks in the input (dashed line) may generate a monostable fused activation peak (solid) positioned over an averaged location (top). This is
due to local excitatory interaction. When the two input peaks are increasingly more
separate (bottom), this fused solution becomes unstable and the fused attractor bifurcates into two attractors. In each, one input peak is selected while activation at the
other is suppressed.
Schneider, Schöner, 2006). The basic mechanism for how dynamic fields make selection 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 what
role do instabilities play? The computational metaphor is fundamentally time-less,
conceiving of cognition as the computation of responses to inputs. The inputs are
given at some point in time, the response is generated after a latency which reflects
the amount of computation involved. An embodied and situated cognitive system, by
contrast, acts under the continuous influence of inputs. New processes are started on
a background of ongoing processes. To provide for any kind of behavioral coherence,
cognitive processes must be stabilized against the continuous onslaught of new sensory
information, of sensory feedback from ongoing action, and of internal interactions from
parallel, possibly competing processes. The localized activation peaks of DFT are
instances of stable states that support representation and resist change through the selfstabilization induced by interaction. Resistance to change induces the reverse problem
of achieving flexibility, being capable to release one process from stability to give way
22
to a new process. In DFT, stable peaks of activation must be destabilized to allow for
the generation of new peaks. Instabilities play this crucial role. Through instabilities,
moreover, cognitive properties emerge such as sustained activation, the capacity to
select among inputs, to fuse inputs, to couple to time-varying inputs, or to suppress
such coupling.
Stabilities and instabilities thus form the foundation for embodied cognition because they enable the emergence of cognitive states while retaining close ties to the
sensory and motor surfaces, which are situated in rich, structured, and time-varying
environments. Within DFT, the mechanisms for creating stability and instability are
neurally plausible. Finally, the Dynamical Systems account both in the classical form
of DST and in the expanded form of DFT is open to an understanding of learning and
development.
4
Learning and development
In DST learning means changing the dynamics of the system. Although easily stated
and seemingly obvious, this insight has far-reaching consequences. Work on how people
learn new patterns of interlimb coordination may serve to illustrate the ideas (Schöner,
1989; Zanone, Schöner, Kelso, 1992; Zanone, Kelso, 1992).
Participants practiced a new pattern of bimanual coordination, a 90 degrees phase
relationship 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 phase
relationships sampling the range from in-phase (0 degrees) to phase alternation (180
degrees) in 10 steps. These required relative phases were invoked by presenting two
metronomes with the requested phase relationship. Figure 14 highlights the main
results. 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, one
of the intrinsically stable patterns of coordination. After learning, performance has
not 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 the
new patterns. To see what such learning processes look like in neural language we
examine the simplest learning processes in DFT. Patterns of activation may be stabilized simply by leaving a memory trace of ongoing activity (Schöner, Kopecz, Erlhagen,
1997; Erlhagen, Schöner, 2002; Thelen et al., 2001). Such a memory trace in effect preactivates and thus preshapes the activation field when new stimuli or new task demands
arise. The memory trace thus biases neural representations toward previously activated
23
relative phase
180 deg
90 deg
learning time
variability of relative phase
learning time
constant error:
performed - required relative phase
before learning
after learning
90 deg
180 deg
required
relative phase
Figure 14: Schematic summary of the results of the Zanone-Schöner-Kelso (1992) experiment on the learning of a bimanual coordination pattern of 90 degrees relative
phase. Over learning, the mean relative phase (top) approached correct performance
at 90 degrees from an initial bias toward phase alternation (180 degrees), while the variability of relative phase (middle) decreased. The bottom graph compares the constant
error of relative phase at a range of required relative phases before (dashed) and after
learning (solid). Note the bias toward phase alternation before learning (constant error
is positive for required relative phases below 180 degrees) and bias to 90 degrees after
learning (constant error is positive below, negative above 90 degrees). Constant error
is reduced after learning at all patterns, not only at the practiced 90 degree pattern.
24
patterns. The concept of a memory trace has a long history in psychology, dating back
at least to William James (1899, his chapter IV on habit formation), to Gestalt psychology 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 particular, 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 memory
trace. The combined mechanisms of a memory trace and its role of preshaping the field
around previously activated locations provide an elementary form of learning, that is,
experience dependent change of the dynamics of an activation field. Because a memory
trace mechanism is formally a dynamical system as well, we sometimes refer to this
learning mechanism as the preshape dynamics of a dynamic field (Erlhagen, Schöner,
2002; Thelen et al., 2001).
Support for the concept of preshaping and a preshape dynamics comes from accounts for a wide range of phenomena, including how the probability of choices influences reaction times (Erlhagen, Schöner, 2002), how motor habits are formed and then
influence motor decisions in the “A not B” paradigm (Thelen et al., 2001), and how
spatial memory is biased toward previously memorized locations (Spencer, Smith, Thelen, 2001; Schutte, Spencer, Schöner, 2003). As an illustration consider perseverative
reaching in the A not B paradigm (review in Wellman, Cross, Bartsch, 1986). In this
task setting, infants are shown two reachable locations “A” and “B”, typically two wells
within 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 subsequent
test trials called the “B” trials). After a short delay of a few seconds, the box is pushed
within reach of the infant, who will typically then will make a motor decision of reaching 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, thus
perseverating in the action they have performed on the preceding “A” trials. Smith,
Thelen, Titzer, and McLin (1999) showed that perseverative reaching is observed in a
toyless version of the paradigm, in which the experimenter merely attracts the attention of the infants to either the “A” or the “B” location rather than actually hiding a
toy there. A DFT account of the rich phenomenology of this paradigm was provided by
Thelen and colleagues (2001). An activation field representing the planned movement
receives input from the attention getting stimulation, but also from the perceptually
marked “A” and “B” locations as well as from a memory trace built up during the first
few reaches to the “A” location. On the first “B” trial, the activation induced by the
attention getting stimulus at the “B” location competes with the activation induced by
the 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 decayed 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
25
Figure 15: Dynamic fields representing the reaching direction in the A not B task are
shown here (top) together with the associated preshape fields (bottom). Along the time
axis, the time structure of the task is reflected by a sequence of 6 A trials (stimulation
at A) and 2 B trials (stimulation at B). On each trial a peak is generated either at A
or at B, and leaves a matching memory trace, which in turn preshapes the field on the
next trial. This generates a tendency for perseveration both in the conventional sense,
perseveration to A after a number of reaches to A on A trials (left) as well as in a new
sense, correctly reaching to B after a number of reaches to B on A trials (right).
field is dominated by interaction which stabilizes the initial decision to reach to the
“B” location against the competing influence of the memory trace.
Recent work by Evelina Dineva (2005; Schöner, Dineva, 2006; Dineva, Schöner,
Thelen, in preparation) has elaborated how the behavioral history in the task predicts
performance. Figure 15 illustrates the evolution of a dynamic field representing reaching directions and its preshape over six “A” and two “B” trials. Reaches to “A” leave
a memory trace, which preshapes the field at “A” and thus promotes further reaches
to “A”, inducing the “A not B” error in the “B” trials. Spontaneous errors, that is,
reaches to “B” on “A” trials, in contrast, leave a memory trace at “B”, which promotes
further reaches to “B” and supports correct reaching to “B” on “B” trials, reducing
the “A not B” error. Later chapters in this book examine the “A not B” paradigm in
more detail.
The DFT account for perseverative reaching postulates that over the course of the
few reaches performed during the experiment, infants acquire a motor habit, that is,
26
learn to reach to the “A” location. This creates an error when the cued location
switches, but stabilizes their reaching behavior in an unchanging environment. Does
learning always have a stabilizing effect? In the domain of motor learning we have
already 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 Schöner, 1989, and Schöner, Zanone, Kelso, 1992, for how instabilities may arise during motor learning). In DFT, a form of learning that destabilizes
previously stable patterns accounts for habituation (Schöner, Thelen, 2006). Habituation is the gradual reduction of sensitivity to a repeated form of stimulation and
is observed across a wide range of behaviors in many species (Thompson, Spencer,
1966). Habituation plays an important role in developmental psychology as a probe
of 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 influential
drawbridge paradigm (Baillargeon, Spelke, Wasserman, 1985). Infants are presented
with 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 stimulus
from the right). The infants habituate to this stimulus, which manifests itself in a
reduced amount of time they look at the stimulus. They are then tested with two
different stimuli. The “impossible” stimulus (middle left panel of Fig. 16) involves the
same 180 degree flap motion (in that respect a familiar stimulus), although initially a
block is visible that would appear to block the path of the flap (in that respect violating
an expectancy if infants’ visual representations are structured by such “knowledge”
about the world). In the “possible” stimulus (bottom left panel of Fig. 16) the flap
motion is stopped at an angle of 112 degrees (in that respect a novel stimulus). This
is consistent with the initially visible block remaining in place and preventing further
motion of the flap (in that respect not violating an expectancy). A novel stimulus is
predicted to elicit more renewed looking on the test trials than a familiar stimulus.
When instead the familiar, but “impossible” stimulus arouses more looking, then this
observation is used to infer that infants are, in fact, building expectations about the
perceived scene based on “knowledge” they have about the world (such as that objects
cannot penetrate each other).
DFT provides an account for this pattern of looking that makes no use of “knowledge” but relies instead on the metrics of the perceptual events induced by this sequence
of 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 (right
column 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
27
inputs
habituation stimulus
drawbridge
drawbridge
test impossible (familiar)
block
inputs
perceptual
dimension
block
drawbridge
test possible (novel)
inputs
drawbridge
block
u1 u2
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 habituation. Middle: The same flap motion and a block on its movement path, which is
actually 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 spans
the visual depth at which movement and block are seen. The moving flap provides
broad input (solid line) across this spatial dimension, but centered at a larger mean
visual 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 two
panels). The two marked locations along this perceptual dimension (vertical bars) are
represented by two activation variables, u1 and u2 , in the dynamic model of Schöner
and Thelen (2006).
28
stimulus, levels below that threshold lead to looks away from the stimulus. The activation field drives a second, inhibitory field, which projects back as inhibition onto the
activation field. During habituation, this accumulated inhibition in the inhibitory field
gradually reduces the amount of activation a given stimulus induces. This destabilizes
the positive peak of activation induced by the stimulus and promotes looking away
from the stimulus. The coupled dynamics of activation and inhibition thus account for
the 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 dishabituation 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 the
drawbridge paradigm) activates field sites that have previously been activated and
are thus initially at a higher level of activation, leading to more looking (top panel
of Fig. 18). A less overlapping stimulus (such as the novel of “possible” stimulus in
the drawbridge paradigm) activates field sites that were previously at rest, leading to
initially lower rates of looking (middle panel of Fig. 18). The pattern of looking also
depends 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 initial
activation have been washed out (bottom right in Fig. 18). At that point, differences in
the level of inhibition determine differences in activation and the model predicts larger
looking times for novel stimuli. In the experimental literature, this time dependence
of looking on test shows up as an interaction between the novelty or familiarity of a
stimulus and the order of presentation. The preference for the “impossible” stimulus
arises only when that stimulus are presented first during test, not when is is presented
second (Baillargeon, 1987).
What about development? Central to the dynamical systems approach to development is the postulate that development depends on experience and is therefore in
large part a learning process (see Thelen, Smith, 1994, for detailed argument). Similar to what I just described for the learning of motor skills, what changes during the
developmental process is the neuronal dynamics supporting a particular function. In
the DFT model for perseverative reaching, for instance, a change of the field dynamics
from largely input-driven to largely interaction-driven accounts for how perseverative
reaching subsides with increasing age. This account explains a range of age-related
changes, including the tolerance of increasing delays and the resistance to distractor
stimuli (Thelen et al., 2001). John Spencer and colleagues have generalized this account to the processes supporting spatial representations in what they call the “spatial
precision hypothesis” (Schutte, Spencer, Schöner, 2003; Spencer, this volume). They
have shown that a shift from weak and spatially diffuse interaction to stronger and
spatially focussed interaction in dynamic fields supporting spatial memory and action
planning accounts for the time dependence and variance of performance in tasks re29
stimulation
stimulus
perceptual dimension
activation
activation field
inhibition
looking
threshold
200
100
time [sec]
looking time [sec]
8
inhibition field
4
2
4
6
8
10
habituation trials
Figure 17: Left column: A dynamic field model of habituation consists of two layers, an
activation field (middle) that receives sensory input (top) and represents the propensity to look at the stimulus, and an inhibition field (bottom) which is driven by the
activation field which it in turn inhibits. The fields span a relevant perceptual dimension (here the visual depth of a stimulus in the drawbridge paradigm as schematically
indicated at the bottom). The stimulus marked by a vertical line (the 180 degree drawbridge motion) is presented periodically during the habituation phase . Right column:
The stimulus (square trace), activation field (zig-zag trace) and inhibition field (thin
increasing trace) at that marked location of the field are shown as a function of time
in the top panel. The bottom panel shows the looking time across trials predicted by
this model. For each stimulus presentation, looking time is computed as the amount
of time that activation is above the looking threshold (of zero).
30
2
activation
field
inhibition
field
1
1
u1
0
-1 v1
100
200
time [sec]
100
200
time [sec]
1
0
-1
u2
v2
looking
time
10
habituation
test
familiar
5
0
2
4
6
8
1
2
novel
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 drawbridge
paradigm 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. The
traces are identifiable as in Fig. 17. The bottom panel shows the looking time as a function of stimulus presentations. During the habituation phase, activation and inhibition
build 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, the
block provides a boost of input to both variables, leading to renewed looking (dishabituation). There is more dishabituation to the familiar than to the novel stimulus when
the familiar stimulus is presented first as shown here.
31
quiring spatial working memory. This account has implications for a range of other
behaviors involving spatial representations such as spatial long-term memory, spatial
discrimination, and the capacity to maintain object-centered reference frames (Simmering, Spencer, Schutte, 2007). In all of these behaviors, the spatial precision hypothesis
predicts correlated changes during development.
That development means change of the neuronal dynamics gives a new and concrete
meaning to the idea of emergence. If it is the dynamics that develop, then developmental processes must be analyzed by investigating the causes of behavioral patterns, not
just the patterns themselves. Thus, at a particular stage of a developmental process
the system may have a certain propensity to generate patterns of sustained activation supporting working memory, but whether or not such patterns are actually stable
may depend on the perceptual context, the short-term behavioral history, or the total
amount of stimulation (Schöner, Dineva, 2006).
The DFT account for habituation similarly postulates that what changes during
development is the neuronal dynamics of activation and inhibition (Schöner, 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 starts
to built. This accounts both for the faster rates of habituation in older infants as well
as their increased preference for novel stimuli under equivalent stimulus conditions.
Recent work by Perone, Spencer and Schöner (2007) showed that this account can be
linked to a generalization of the spatial precision hypothesis (space now referring to
the space of visual features).
The postulate that development amounts to change of dynamics has been made in
slightly different form also by van der Maas and Molenaar (1992). These authors have
looked 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 as
reflective of an attractor state of a dynamical system. It is not entirely clear what
the status of that dynamical system is. Its attractors do not describe the real-time
behavior (which is a complex sequence of actions in many of the cognitive skills to
which this model refers). These attractors characterize something closer to an overall
state of competence. Time and stability have an uncertain status as well but may refer
to something like how reproducible successful displays of a competence are. The main
point of the account is, however, that graded and continuous changes of the parameters
of 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) describe
various signatures of such qualitative changes of a competence dynamics. These include
the critical fluctuations and critical slowing down for which direct evidence has been
32
provided in the context of movement coordination (Schöner, Kelso, 1988; Schöner,
Haken, Kelso, 1986). Here, however, fluctuations and resistance to perturbations are
looked 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 dynamics change? To date, the DFT approach has not addressed these processes at a
level of specificity and detail that would comparable to that achieved in capturing the
behavioral dynamics. This is clearly a task for the future. Knowing what it is that
changes during development is, of course, a critical prerequisite to providing a process
account for such change.
In contrast, Paul van Geert (1991; 1998) and colleagues have used dynamical systems thinking primarily at the time scale of development. Their postulate is that the
process of development can be characterized as some sort of dynamical system, evolving
on the slower time scale over which development unfolds. Specifically, they propose to
think of development as a growth process. Qualitatively different patterns of growth
can be modelled such as continuous growth, characteristic variation of growth rate
from slow to fast to slow (S-shaped), and oscillatory growth rates. Moreover, coupling among various growth processes may induce complex patterns of growth, such as
stage-like alternations between slow and fast rates of change.
The concrete mathematical models that are used to illustrate these ideas encounter
the same kind of problem discussed above for the account of van der Maas and Molenaar (1992). The state variable that reflects a particular stage in development is not
directly linked to real-time behavior. It represents again something like a general level
of competence, assessed through such measures as the sizes of vocabularies or the frequency 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 given
weights that may reflect how likely it is to display the competences. These weights
are the dynamical variables that evolve over developmental time. There is clearly a
gap between these abstract, disembodied variables and the concrete neuronal process
that generate the behavior that is used to assess competences. Moreover, the competences are preformed and only need to be activated by increasing their respective
weight value. Both limitations make clear that this account does not yet link the developmental process to the stream of behavioral experience. Implicitly, the model seems
to assume that there is an intermediate level at which the developing nervous system
takes stock of its current set of skills, and updates its probability of use dependent on
this form of assessment. How the skills are implemented in the first place, where they
come from, how they are employed to deal with incoming sensory information, and
what sort of sensory information drives the learning process, such questions remain
open. The strength of the model consists, instead, of a demonstration that qualitative
changes may emerge from developmental processes that are governed by fairly simple
33
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 van
Geert, 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, in
both 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 may
enable an embodied theory of cognition. This is a theory that takes seriously the
continuous link of cognitive processes to the sensory and motor surfaces, and that takes
into account that cognition emerges when systems are situated in richly structured
environments. The theory is open to an understanding of learning, and is based on
neuronal principles.
Connectionist thinking shares many of the same ambitions and core assumptions,
in particular, the commitment to using neuronal language to provide accounts for
function and learning. In some instances, connectionist models have used the exact
same 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, statistical
regularities may be extracted by learning systems, which may reflect these properties in
the structure of their representations (Elman et al., 1997). Signatures of that process
include accounts for learning curves, so that connectionist models speak to the evolution
of systems on the time scale of development. On the other hand, some connectionist
models 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 the
two time scales that the two flavors of dynamical systems ideas have studied, for the
most part, separately.
The detailed comparison of Dynamical Systems thinking and connectionism is not
the 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 thinking to bear on the process of development itself may be the most constructive point
34
of convergence. While connectionist networks have shown how the statistics of stimulation drive the development of representations, the models to date have not taken
into account how infants and children control their own stimulation through their own
behavior, how individually different developmental trajectories may emerge from that
fact, and how the space-time constraints of behavior determine what can be learned
and when. DST provides the framework within which online linkage to the environment
can be modelled and demonstrates how cognitive functions emerge from instabilities.
By combining the structured account for real-time behavior of DST with the neural
substrate for learning provided by connectionism, maybe we will be able to understand
not only what develops, but how individuals through their own behavior generate the
experiences on which development is based.
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