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Humans move with ease over rough terrain (and so do many other species), identify and orient toward perceived objects in cluttered environments, or maintain objects and the intention to reach for them in memory while sensory information about the objects is no longer directly available and other tasks must ﬁrst be accomplished. How do nervous systems generate persistent patterns of behavior in the face of very complex sensory environments, and given the continuous scale of possible behaviors within which any particular action or representation must be stabilized? Dynamical systems approaches address the fundamental conﬂict between stability and ﬂexibility of nervous function. Stability is the property of a behavioral or neural state to persist in the face of systematic variation or random perturbation. All observable behavioral or neural states possess stability to some extent, as loss of stability leads to change of state. The generation of behavior or of neural patterns of activity is ﬂexible in the sense that a variety of behavioral or neural patterns may be assembled from a multitude of sensory, motor, and neural systems in response to the sensory situation, the inner state of the neural or behavioral system, and its recent history. Flexibility may therefore involve pattern change and hence require the release of behavioral or neural states from stability. Dynamical systems approaches provide methods with which stable states can be identiﬁed and characterized as ‘attractors.’ The coexistence of multiple attractors and changes in the number and nature of attractors, as sensory information, environmental conditions, or task requirements vary, are important in dynamic systems approaches because they demarcate different elementary behaviors or functional modules.
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The functioning of technical systems (for example, maintaining an airplane on course) must be stable against random perturbations (for example, from turbulence). Stability may be brought about by the physical properties of the system (for example, by the viscous properties of air), by control systems designed speciﬁcally to measure deviations from the desired state and change actuator parameters to minimize such deviations, or by a human operator acting as a control system. The theoretical framework of control theory has been, and is being, used to understand simple aspects of nervous function, an approach known as biological cybernetics. To analyze the capacity of the houseﬂy to orient toward a moving visual stimulus, for instance, Reichardt and Poggio (1976) identiﬁed the control dynamics linking the detected motion stimulus to the input to the ﬂy’s ﬂight motor. Such cybernetic analysis is aimed at identifying relevant sensory signals, the motor system, and the control circuitry transforming sensory signals into motor parameters. This approach is successful as long as the number of components is small, their function is independent of each other, and a single behavioral pattern is generated.
Any behavior involves many sensory subsystems, many motor degrees of freedom, and takes place in complex environments. Certainly true of daily life human activities, this also holds for laboratory experiments, even when they seem to limit the number of relevant components. The simple behavioral task, for instance, of pressing a key in response to a visual stimulus, actually involves many sensory subsystems (including proprioception, haptics, reafference, and vision) that contribute to the estimation of the ﬁnger’s current position, with additional subsystems (maintaining reference frames for eye, head, and body position, different visual features including depth and motion) contributing to the processing of the visual stimulus. Those subsystems go unnoticed because the information they provide is coherent. This is, in part, a property of the environment (in which, for instance, haptic and visual cues about ﬁnger position co-vary), and, in part, due to the experimental task setting which generates a unique relationship between sensory stimulation and motor action.
A unique match of sensory onto motor conﬁgurations is the exception more than the rule, however. Even Reichardt and Poggio’s (1976) houseﬂies were more ﬂexible than that: When two separate patches of the visual array moved, the ﬂies steered toward an averaged direction as long as the two patches lay in similar directions, but locked on to only one of the two patches when these were at larger angular separation. Frequent random switches occurred between epochs in which the ﬂies tracked one or the other visual target selectively. More generally, not all possible couplings between sensory and motor degrees of freedom can be effective all the time. The selective activation of particular couplings between sensory and motor components, and the release from stable coupling of others, is a necessary prerequisite for the generation of ﬂexible behavior. Initiating a motor act, for instance, requires releasing a ﬁxation or postural system from stability.
3. The Dynamical Systems Approach
Broadly speaking, dynamical systems approaches (Port and van Gelder 1995) generalize cybernetic ideas so as to encompass ﬂexibility. The speciﬁc variant reviewed in the rest of this research paper, a dynamical systems approach emphasizing attractor dynamics and their instabilities, grew out of analogies between patter formation in nonequilibrium systems (Haken 1983) and patterns of coordination in biological movement (Schoner and Kelso 1988), but has since been extended to address many aspects of action, cognition, and perception (e.g., Thelen and Smith 1994, Giese 1999). A link to dynamical systems ideas in neural network theory (Amari 1977, Grossberg 1980) has been established through the concept of dynamical ﬁelds (Schoner et al. 1997).
3.1 Behavioral Or Neural Variables
Variables that are very closely linked to the peripheral motor or sensory subsystems are often not suited to describe behavioral or neural patterns, which may involve many different such subsystems or even different combinations of such subsystems. The dynamical systems approach employs, instead, variables that characterize directly the behavioral or neural patterns themselves by differentiating the different observed forms of the patterns and quantifying deviations from behavioral or neural states. When, for instance, Reichardt and Poggio’s (1976) houseﬂies switch stochastically between two patterns of locking on to either of two moving targets, their heading direction in an outer reference frame is a useful variable in terms of which both states and deviations from them can be described, even though this variable is not directly available to sensor or motor systems. Other examples of behavioral variables that are useful to characterize multiple behavioral states are the relative phase between different limbs or articulators (Schoner and Kelso 1988), the position and velocity of an endeffector in outer (task) space to characterize the formation of trajectories in goal-directed movements (Saltzman and Kelso 1987), or the directions of reaching movements in outer space to characterize how movement targets are represented (Georgopoulos et al. 1986).
3.2 Behavioral Or Neural Dynamics
Any behavioral or neural system is exposed to multiple sources of potential perturbation. Stability, resistance to change, is thus a necessary prerequisite for a pattern to be reproducible and observable. Even when transient patterns are generated (in response to stimulation, for instance) it is the asymptotic state and its stability properties that determine the form of such transients. This stability requirement is central to the dynamical systems approach, which aims to characterize and analyze behavioral and neural systems in terms of its stable states.
Given behavioral or neural variables, the approach can be formalized in terms of dynamical systems that describe the evolution in time of the system. The dynamical system is characterized by its asymptotically stable states or attractors to which the variables converge in time (Fig. 1). Linear dynamical systems have exactly one state, and linear systems theory provides formal methods for how to identify the dynamical system given such measures as the means, variances, and co-variances of the behavioral or neural variables. For a system to have more than one attractor, the dynamics must be strongly nonlinear (Fig. 2). There is no formal procedure for identifying such systems. One heuristic approach is to characterize the different attractors by local linear models which can by fused into a nonlinear model by taking the limited range of validity of the linear models into account (Schoner et al. 1995). Noise induced switches between different attractors provide additional information about the nonlinear dynamics.
The strongest tool for the identiﬁcation of such nonlinear dynamics is the observation of bifurcations, also called instabilities. At bifurcations, the number or nature of attractors changes (Fig. 3). The detection of bifurcations as sensory information, task constraints, or motor parameters vary helps identify qualitatively different contributions to the behavioral or neural dynamics. States that may coexist multistably or are linked through bifurcations reﬂect separate elementary behaviors or neural patterns. The various quadrupedal gaits, for instance, are different elementary behaviors rather than variants of the same basic behavior, because changes from one gait to another may take the form of bifurcations. Kelso and colleagues studied a laboratory model of such patterns of coordination (Schoner and Kelso 1988) when they asked human participants to move two homologous limbs rhythmically (e.g., ﬁngers, arms, legs). In-phase and anti-phase patterns of coordination were identiﬁed as two elementary behaviors, which coexist bistably at low frequency of the rhythmic movement. Anti-phase coordination (syncopation) becomes unstable in a pitchfork bifurcation (Fig. 3) when movement frequency is increased. Other domains in which this form of analysis can be applied include the integration of multiple sources of sensory information to control action (such as in posture), the establishment of stable coupling between perceived events and action (such as when catching a ball), or the control of timing itself (such as when generating polyrhythms). At the neural level, central pattern generators and their interactions with sensory and motor subsystems can be analyzed this way. This is of particular interest in view of the ubiquitous multifunctionality of such circuits.
3.3 Dynamic Fields
What would be a dynamical systems description of the behavior of generating, in response to a coded stimulus, a goal-directed movement to one of three previously learned, but unmarked, targets? The range of possible movements may be characterized in terms of an appropriate behavioral variable, e.g., the direction in which the target lies from the initial end-effector position. But how to represent the three possible values of this variable? Dynamical variables have a unique value at all times, and that value must change continuously over time. The missing element is some method for representing variable and graded amounts of information. The classical concept of activation, when combined with the notion of a behavioral variable, may supply this missing element. For each potential value of the behavioral variables, an activation variable represents the amount of information about this value: Absence of information is indicated by low levels of activation, and presence of information by large levels of activation. This leads to a ﬁeld of activation variables deﬁned over the space spanned by the behavioral variables. A single peak of activation represents information about a unique value of the behavioral variables (that over which the peak is localized: Fig. 4(a)). Multiple possible values are represented by multiple peaks, and graded distributions of activation represent graded information about the values of the behavioral variables. In neurophysiological terms this resembles the notion of ‘space code,’ in particular, when topography leads to regular neighborhood relationships within a neural network.
Ideas about attractor dynamics and bifurcations can be ‘lifted’ to this level of description by postulating that such activation ﬁelds evolve toward attractor solutions of a dynamical system, which stabilizes localized peaks of activation (Schoner et al. 1997). Interpreted as neural networks, such dynamical system receive input from sensors or other representations that specify locations at which peaks may be localized. In addition, strong internal interactions within the dynamic ﬁeld stabilize individual localized peaks even when input is distributed across the ﬁeld. Such strongly interactive ﬁelds are thus not primarily input-driven. Dynamical neural networks of this kind have been studied as models of cortical function (Amari 1997, Grossberg 1980).
As a simple example consider saccadic eye movements generated toward visual targets. When two potential targets are presented, two types of responses are possible. For narrowly spaced targets, an eye movement toward an average position between the two targets is generated. Beyond a critical angular separation of the two targets, a bimodal response pattern is obtained in which on some trials the saccade goes to one target, and on other trials, to the other target (Ottes et al. 1984). A dynamic ﬁeld representing the direction of the planned saccade receives input from the target locations. When this input consists of two narrowly spaced peaks, a single peak of activation localized over an averaged direction is generated (Fig. 4(a)). When the two input peaks are more separate, mutual inhibition makes it impossible for peaks to become activated at both input sites. The ﬁeld dynamic is bi-stable and generates a single peak, positioned either at one or at the other stimulated location. Which of the two targets is selected depends on the pattern of prior activation, on ﬂuctuations, or on small differences in stimulation between the two sites.
Through the notion of dynamic ﬁelds, the dynamical systems approach can be extended from the domain of sensori-motor patterns toward problems in perception (Giese 1999) and representation (Thelen et al. 2001). A dynamic ﬁeld may, for instance, change through a bifurcation from an input-driven mode, in which activation remains stable only in the presence of appropriate input, to a short-term memory mode, in which input-induced activation remains stable once input has been removed. The dynamic ﬁeld can be linked to the notion of population coding in neurophysiology (Georgopoulos et al. 1986) by interpreting distributions of population activation as ﬁelds of activation over a relevant behavioral variable, to which neurons are tuned (Erlhagen et al. 1999).
If behavioral or neural patterns emerge as stable states from underlying dynamical systems, as postulated in dynamical systems approaches, then the characterization of this dynamics is sufficient for an understanding of the behavioral or neural system. No additional need for computation or speciﬁcation arises.
With origins in the domains of motor behavior and sensori-motor coupling, the more speciﬁc dynamical system approach, emphasizing attractors and their bifurcations, is now being probed as a wider framework for the understanding of nervous function also at the level of perceptual and cognitive representations. Multistability and bifurcations demarcate qualitatively different modules or elementary behaviors, which are the units of analysis of the dynamical systems approach. Because these units reﬂect both the structure of the neural or behavioral system and that of its environment, the dynamical systems approach may play a natural role in the current research program aimed at understanding how cognition is embodied by systems whose structure reﬂects functional adaptation to an environment and a speciﬁc developmental and evolutionary history.
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