Functional architectures and structured flows on manifolds: a dynamical framework for motor behavior.

Psychological Review 2014, Vol. 121, No. 3, 302–336

© 2014 American Psychological Association 0033-295X/14/$12.00 http://dx.doi.org/10.1037/a0037014

Functional Architectures and Structured Flows on Manifolds: A Dynamical Framework for Motor Behavior Raoul Huys

Dionysios Perdikis

Aix-Marseille Université; Institut National de la Santé et de la Recherche Médicale, Marseille, France; and Centre National de la Recherche Scientifique, France

Aix-Marseille Université; Institut National de la Santé et de la Recherche Médicale, Marseille, France; and Max Planck Institute for Human Development, Berlin, Germany

This document is copyrighted by the American Psychological Association or one of its allied publishers. This article is intended solely for the personal use of the individual user and is not to be disseminated broadly.

Viktor K. Jirsa Aix-Marseille Université; Institut National de la Santé et de la Recherche Médicale, Marseille, France; and Centre National de la Recherche Scientifique, France We outline a dynamical framework for sequential sensorimotor behavior based on the sequential composition of basic behavioral units. Basic units are conceptualized as temporarily existing lowdimensional dynamical objects, or structured flows, emerging from a high-dimensional system, referred to as structured flows on manifolds. Theorems from dynamical system theory allow for the unambiguous classification of behaviors as represented by structured flows, and thus provide a means to define and identify basic units. The ensemble of structured flows available to an individual defines his or her dynamical repertoire. We briefly review experimental evidence that has identified a few basic elements likely to contribute to each individual’s repertoire. Complex behavior requires the involvement of a (typically high-dimensional) dynamics operating at a time scale slower than that of the elements in the dynamical repertoire. At any given time, in the competition between units of the repertoire, the slow dynamics temporarily favor the dominance of one element over others in a sequential fashion, binding together the units and generating complex behavior. The time scale separation between the elements of the repertoire and the slow dynamics define a time scale hierarchy, and their ensemble defines a functional architecture. We illustrate the approach with a functional architecture for handwriting as proof of concept and discuss the implications of the framework for motor control. Keywords: motor control, dynamical systems, sequential behavior, behavioral repertoire Supplemental materials: http://dx.doi.org/10.1037/a0037014.supp

numerous processes and constraints (internal and external to the actor) operating on a multitude of spatial and temporal scales. Nevertheless, behaviors typically manifest themselves as lowdimensional patterns. For instance, whole-body motion patterns typically involve a few characteristic dynamic patterns (or eigenmodes) only; several examples include walking (Lamoth, Daffertshofer, Huys, & Beek, 2009; Troje, 2002) and running (Lamoth et al., 2009), executing a tennis (Huys, Smeeton, Hodges, Beek, & Williams, 2008) or handball shot (Bourne, Bennett, Hayes, & Williams, 2011), whole-body reaches (Fautrelle, Berret, Chiovetto, Pozzo, & Bonnetblanc, 2010), and piano playing (Furuya, Flanders, & Soechting, 2011). That is, the multitude of degrees of freedom at various scales associated with motor behavior is reduced to a smaller number operating at the behavioral level. Third, although different realizations of a given action are never exact copies of each other (Bernstein, 1967), they do share certain common features, allowing us to effortlessly distinguish one action, actor, and/or action outcome from another (Huys, Smeeton, et al., 2008; Johansson, 1973, 1976; Runeson & Frykholm, 1981; Smeeton & Huys, 2011; Troje, 2002). This suggests that some of the variance associated with perceptual–motor acts sets one type apart from others; that is, it points to the presence of class-defining invariance. Classes constitute the basis for a behavioral repertoire.

Sensorimotor behavior comprises at least three related key aspects that have been discussed in the scientific literature at length. First, it is stable—successful task performance is often preserved despite perturbations, yet flexible—and goals can be successfully achieved in numerous ways. Second, it invariantly arises from

Raoul Huys, Institut de Neurosciences des Systèmes, Aix-Marseille Université; Institut National de la Santé et de la Recherche Médicale UMR S 1106, Marseille, France; and Centre National de la Recherche Scientifique, France; Dionysios Perdikis, Institut de Neurosciences des Systèmes, Aix-Marseille Université; Institut National de la Santé et de la Recherche Médicale UMR S 1106, Marseille, France; and Max Planck Institute for Human Development, Berlin, Germany; Viktor K. Jirsa, Institut de Neurosciences des Systèmes, Aix-Marseille Université; Institut National de la Santé et de la Recherche Médicale UMR S 1106, Marseille, France; and Centre National de la Recherche Scientifique, France. The research reported herein was supported by the Brain Network Recovery Group through the James S. McDonnell Foundation, the FP7ICT BrainScales, and the AMIDEX project Coord-Age. Correspondence concerning this article should be addressed to Raoul Huys, Institute de Neurosciences des Systèmes, Faculté de la Médicine, Aix-Marseille Université, 27 Boulevard Jean Moulin, 13005 Marseille, France. E-mail: [email protected] 302


A DYNAMICAL FRAMEWORK FOR MOTOR BEHAVIOR

The idea of a behavioral repertoire has been exploited in various domains, often driven by the motivation to tie the behavioral components together into sequential behavior, as per example in bird song production (Yu & Margoliash, 1996) and its perception (Kiebel, Daunizeau, & Friston, 2009), human speech production (Goldstein, Pouplier, Chen, Saltzman, & Byrd, 2007; Liberman & Whalen, 2000) and perception (Kiebel et al., 2009), typing (Viviani & Terzuolo, 1980), applications in robotics (Shevchenko, Windridge, & Kittler, 2009), and so on. We present a general framework for (perceptual–)motor behavior, referred to as structured flows on manifolds (SFM), in the tradition and spirit of coordination dynamics (cf. Kelso, 1995) and synergetics (cf. Haken, 1983, 1996), which incorporates the aspects alluded to above. In brief, synergetics, Haken’s theory of self-organized pattern formation in open systems, provided the theoretical foundation for coordination dynamics, which conceptualizes (perceptual–motor) behavior in terms of self-organized pattern formation sculpted by functionally meaningful information (intentional change, learning, etc.; cf. Kelso, 1995). The corresponding macroscopic patterns, often referred to as modes in physics, can be viewed as compact, abstract descriptions capturing the temporally existing functional organization of the action system’s degrees of freedom implicated in an act (i.e., coordinative structures or synergies; see, e.g., Beek, Peper, & Stegeman, 1995; Kelso & Tuller, 1984; Kugler, Kelso, & Turvey, 1980; Turvey, 1990). Accordingly, due to the nonlinear interactions between the numerous “fast” elements on the microscopic level, one or a few “slow” macroscopic patterns (or order parameters) emerge. In turn, the macroscopic patterns exhibit an organizing influence on the microscopic elements, “enslaving” their dynamics. Consequently, the macroscopic dynamics fully describe the system, such that in order to investigate the latter, it suffices to study the order parameter dynamics, since all other degrees of freedom are enslaved. The synergetics formalism, however, is technically valid only in the vicinity of instabilities, that is, regions where patterns appear and disappear via so-called bifurcations. Nevertheless, concepts of the synergetic framework, in particular the focus on order parameters, have often been applied at systems far from such instabilities. The theory of SFM, however, applies more generally also far away from instabilities, but is mathematically more constrained to a normal form. In short, SFM posits that a functional, low-dimensional dynamics constrained onto a manifold emerges from a high-dimensional (internally coupled) system for the duration of a given process. The time scale associated with the functional dynamics is much slower than that of its generating subsystems. The low-dimensional dynamics can be unambiguously classified according to its associated phase space topology, which allows for the classification of the corresponding behavior as well as for tremendous intraclass variation. We view qualitatively distinct structured flows, to which we refer as functional modes, as representing self-organized elementary patterns of behavior (building blocks). In that regard, coordination dynamics has focused primarily on rhythmic behaviors, and the coordination between rhythmic phenomena, which are conceived of as limit cycle and fixed-point attractors (and/or repellors), respectively. SFM extends this tradition by allowing for a rich repertoire of dynamical forms, and by its explicit focus on (phase) flows rather than (attractor) structure. The functional modes can be combined into functional architectures to perform sequential behavior. Typically, perceptual–motor acts are investigated “in isolation.” Through the construction of functional architectures, we propose an explanation of how such acts can be embedded (or

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nested) into a larger behavior. Below, we outline SFM in detail and provide experimental and computational findings consistent with the approach. Next, we present the notion of functional architectures using distinct low-dimensional structured flows as elementary behavioral units and illustrate the approach in the context of handwriting. Finally, we discuss several implications of our perspective for perceptual–motor control. First, however, we briefly review the main perspectives on perceptual–motor control and serial behavior that have been prominent over the last few decades.

Background Cognitive Psychology The cognitive approach treats the human mind as an information-processing device, in which algorithmic computations operate on central representations. In this view, motor control is exerted centrally, establishing a fundamental divide between the controller (the mind) and the controlled (the muscular apparatus). For instance, Schmidt’s schema theory (R. A. Schmidt, 1975; R. A. Schmidt & Lee, 2005) posits that generalized motor programs (GMPs) containing a set of algorithms allow for the control of its corresponding class of action, thereby assuring action-classinvariant features (the order of events, relative timing, and relative force). Context-specific constraints are satisfied by the tuning of adjustable parameters (time duration, force, and effectors used) each time a GMP is used (see also R. A. Schmidt, 2003). The schemata are thought of as abstract memory or rule representations, and they relate GMP parameterization, action outcome, perception (sensory consequences), and learning. While schema theory has been among the most successful motor control theories since its introduction, it has faced strong criticisms, in particular against the algorithmic nature of information processing and its disconnectedness from physical and biological reality (cf. Edelman, 1987; Kugler et al., 1980; Meijer & Roth, 1988).

Computational Approach The computational approach holds that the nervous system is a computational device, and can thus be viewed as a modern successor of the information processing approach. This approach comes in various flavors, with a dominant focus on internal models (Hwang & Shadmehr, 2005; Wolpert, 1997; Wolpert & Ghahramani, 2000), optimization (Guigon, Baraduc, & Desmurget, 2007b, 2008; Todorov, 2004; Todorov & Jordan, 2002), Bayesian decision theory (Chater, Tenenbaum, & Yuille, 2006; Kersten & Yuille, 2003; Körding & Wolpert, 2004; Wolpert, 2007; Yuille & Kersten, 2006) or a combination thereof. One of its main concepts, the internal model (cf. Craik, 1943; Kawato, 1999), comes in two kinds. Forward models predict sensory consequences using efference copies of motor commands.1 If these predictions deviate from the actual percepts, the motor commands are adjusted. Inverse models compute the (feed-forward) motor commands to accom1 As pointed out by one reviewer, the notions “corollary discharge” and “efference copy” are often taken to be equivalent but may be conceptually distinct: the input and output codes are the same in corollary discharge but different in efference copy (where the copy of commands [efference] is registered against input [afference]; cf. Gallistel, 1980; Kelso, 1992).


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plish a desired trajectory, which introduces the (computational) problem of how the redundant degrees of freedom pertaining to the motor system can be harnessed to achieve a desired behavioral solution (the inverse problem). This problem can be solved by optimizing a cost function, minimizing for example energy (Anderson & Pandy, 2001; Hatze & Buys, 1977), jerk (Flash & Hogan, 1985; Hogan, 1984), joint torque (Nakano et al., 1999; Uno, Kawato, & Suzuki, 1989), muscle force (Pandy, Garner, & Anderson, 1995; Pedotti, Krishnan, & Stark, 1978), impedance (Burdet, Osu, Franklin, Milner, & Kawato, 2001), and endpoint variance (Harris & Wolpert, 1998). Evidence in favor of as well as against the minimization of specific proposed cost has been reported repeatedly, and rather than propose an ubiquitous cost function, a more realistic performance criterion may consist of a mixture of costs terms (Todorov, 2004). Although the approach is quite popular these days, it has not been without criticism. For one, an implicit assumption underlying the approach is what Ostry and Feldman (2003) referred to as the force control hypothesis, that is, the notion that the nervous system controls movement through the specification of forces (Guigon, Baraduc, & Desmurget, 2007a; Todorov, 2000, 2005; Wolpert, 1997). The authors argue that this idea, already questioned by Bernstein (1967), results in predictions that are at odds with empirical data when the corresponding models integrate neuromuscular realism.2 Furthermore, new insights into the motor cortex somatotopic map do not support the notion that its organization is dominantly muscle based (as longtime thought) but in terms of functional behaviors (Aflalo & Graziano, 2006; Graziano & Aflalo, 2007). Another criticism concerns one of the key assumptions, optimality, which is often motivated by making reference to evolution. Evolution, however, is driven by random factors (mutation), and the mechanism of selection operates at the level of populations, not individuals. That is, optimality is not an evolutionary driving force— biology seems to work on a whatever-fits basis rather than seeking optimal solutions, and in fact, evolution often settles on suboptimal (Dawkins, 1982; Mayr, 2002; Varela, Thompson, & Rosch, 1992) or “habitual” solutions (de Rugy, Loeb, & Carroll, 2012). Nonoptimal strategies may also apply to perception and motor control (cf. Withagen & Chemero, 2009). These arguments cannot refute the possibility that optimality defined relative to a given individual at a given moment is the driving principle governing underlying motor control. They do show, however, that optimality is a proposition rather than a biological fact (as it is often presented).

Equilibrium Point Models In contrast to the approaches discussed above, equilibrium point (EP) control models are motivated by the biomechanics of the neuromuscular system. In particular, the viscoelastic (spring-like) properties of muscles guarantee that a joint spanned by agonists and antagonists will be at equilibrium given specific length-force relations of the muscles. By changing these relations, the nervous system can alter the EP and thus generate movements. Multiple EP models have been proposed, but two classes of models have received particular attention: In the ␣ model, EPs are set through the muscles’ rest length and stiffness, which are determined by ␣-motoneuron activity (McIntyre & Bizzi, 1993); in the ␭ model, EPs are set through the threshold (␭) of the stretch reflex of the

ensemble of muscles involved (Feldman, 1986; Feldman & Levin, 2009). Novel formulations involve hybrid EP models, combining ␣ and ␭ model characteristics (McIntyre & Bizzi, 1993), intermittent control (Kistemaker, Van Soest, & Bobbert, 2006), and movement synergies (Balasubramaniam & Feldman, 2004; Latash, 2010). The common denominator, and one of the attractive features of EP approaches, is that they evade the “inverse problem.” EP models have been criticized, however, based on a mismatch between experimentally extracted stiffness and thereon predicted EP trajectories and observed trajectories (Gomi & Kawato, 1996, 1997). These negative results, however, may have been flawed by simplifications of the musculoskeletal model (the negligence of higher order terms) used to estimate stiffness (Kistemaker, Van Soest, & Bobbert, 2007).

Free Energy and Active Inference A particularly influential proposal that aims for generality across motor behavior, perception, and learning is the so-called free-energy principle (cf. Friston, 2011; Friston, Daunizeau, Kilner, & Kiebel, 2010; Friston, Mattout, & Kilner, 2011), essentially a mathematical formulation of how adaptive, self-organized systems resist the natural (thermodynamical) tendency to disorder. Free energy is an information theoretic quantity that sets a bound on “surprise,” or the negative log probability of an agent’s sensory input, given a generative (i.e., internal) model of how the sensory data were generated. In order to resist disorder (i.e., maintain homeostasis), organisms have to minimize the long-term average of the surprise, understood as the deviation from “the expected.” This minimization can be achieved by changing predictions about perceptual causes or by changing the sensory sampling (action). Motor control, then, is geared toward satisfying prior expectations about the (proprioceptive) sensations that the action-to-beperformed will cause. This Bayesian “active inference” differs from optimal control formulations, as it views motor control as an inference problem (Friston, 2011). Accordingly, action is enacted by reflex mechanisms (as in EP control) in response to top-down predictions of the proprioceptive consequences of anticipated (or desired) behavior. This eschews the need for separate inverse and forward models and emphasizes the role of dynamics responsible for the proprioceptive predictions. Because this (purely Bayesian) approach is formulated in terms of inference, there is only one objective function (the marginal likelihood or evidence for a model or beliefs about behavior) such that desired behavior or control signals are replaced by prior expectations about movement trajectories. Consistent with this view is the recently presented neuro2 The issue is that posture-stabilizing neuromuscular (reflex) mechanisms generate muscle force in order to resist perturbations; the question thus arises why intentional movements are not subjected to the corresponding restoring forces (the von Holst paradox; von Holst & Mittelstaedt, 1950/1973). Experimental evidence shows that agonist and antagonist muscle activity is absent prior to and following movement (to another position), indicating that the postural-stabilizing mechanism is somehow reset. In force-control models incorporating muscle properties and reflexes (i.e., neuromuscular realism), the postural-stabilizing forces are counteracted via muscle activity at the new posture, which is inconsistent with experimental findings. Model formulations in which electromyography is specified via a reflex and central component run counter to observations when a limb is unexpectedly unloaded. In the absence of neuromuscular realism, these problems do not arise.


anatomical and physiological evidence suggesting that the motor cortex is organized so as to support the generation of (motor) predictions rather than commands (Adams, Shipp, & Friston, 2013). Because prior expectations can in principle be expressed as SFM, active inference may provide a plausible framework for linking high-level abstract movement dynamics—in terms of SFM—to movement execution through peripheral mechanisms.


Ecological Psychology The ecological approach to perception and action was given form by Gibson (1966, 1986). Central to the approach is the organism– environment system, claimed as the proper unit of analysis in the study of behavior. In stark contrast to the dominant view in the 1960s to 1980s, Gibson opposed the idea that information reaching the senses is impoverished (or ambiguous) and that therefore the formation of meaningful percepts requires intervention from the nervous system. (Ironically, these days, the ambiguity of the environmental information is key to Bayesian approaches; cf. Körding & Wolpert, 2004; Wolpert, 2007; Yuille & Kersten, 2006) In contrast, he proposed that perceptual information resides in the ambient energy arrays and flows that directly specify environmental properties. Meaningful structured informational patterns are furthermore created as an animal moves. Perception and action are thus reciprocally coupled: Action is information based, and information is constantly generated as the animal moves. In the further development of these ideas, the notion that environment, information, and perception (all) relate one-to-one to each other became a cornerstone of the neo-Gibsonian program (R. E. Shaw, Turvey, & Mace, 1982; Turvey & Shaw, 1999; Turvey, Shaw, Reed, & Mace, 1981). As such, the ecological program has been mostly concerned with identifying specifying informational variables and their discovery in learning (Jacobs & Michaels, 2007; Jacobs, Silva, & Calvo, 2009; Lee, 1976; B. K. Shaw, McGowan, & Turvey, 1991; Turvey, Burton, Amazeen, Butwill, & Carello, 1998; Warren, 2007; Zaal & Michaels, 2003; see also Withagen & Chemero, 2009). Consequently, the approach’s concern with the corporal aspect of action has been rather limited. A noticeable exception, in that regard, is Warren’s (2006) “behavioral dynamics,” which aims to integrate concepts from ecological psychology and dynamical systems theory. Warren treats an agent and the environment within which the agent acts as two dynamical systems that are coupled via mechanics and information. Accordingly, changes in the environment are a function of the environmental state and forces (due to the actor or otherwise) external thereto. Informational laws at the ecological scale (optics, haptics, etc.) map the environment–actor system properties onto informational variables through which the actor, in turn, is coupled to the environment. In other words, perception–action comprises an environment and an agent that are both defined as dynamical systems. The agent affects (is coupled to) the environment via an effector function; in turn, the environment guides (is coupled to) the actor through an information function. Control laws define the functional adjustments of an agent’s intrinsic dynamics through information. The dynamics defined across the environment–actor system constitute the behavioral dynamics. As will become clear below, SFM provides a general perspective to frame the behavioral dynamics.

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The Dynamical Approach The dynamical approach to (perceptual–motor) behavior is a phenomenological perspective to behavioral patterns that is founded on concepts of physical theories of self-organized pattern formation (Haken, 1983, 1996; Nicolis & Prigogine, 1977) and dynamical systems theory (Perko, 2006; Strogatz, 1994). While it is mostly known for its rhythmic coordination paradigm, insights into the nature of perceptual–motor behavior as dynamical systems have been gained through various lines of research. For instance, investigation of the structure of the variability (or “noise”) in behavior has revealed the presence of so-called 1/f noise in behaviors such as walking, tapping, visual search, reaction time, and decision tasks (Cutting, DeLong, & Nothelfer, 2010; Delignières, Torre, & Lemoine, 2008; Diniz et al., 2011; Gilden, 2001; Kello, Beltz, Holden, & Van Orden, 2007; Torre & Wagenmakers, 2009; Van Orden, Holden, & Turvey, 2003; for a skeptical note as to the relevance and ubiquity of 1/f noise in psychology, see Farrell, Wagenmakers, & Ratcliff, 2006; Wagenmakers, Farrell, & Ratcliff, 2005). Importantly, 1/f fluctuations have no single characteristic time scale, but rather point to interactions between subsystems operating on multiple time scales. An issue of debate is whether these fluctuations are intrinsic to the system under study as a whole and are no further localizable or whether 1/f sources can (in principle) be pinned down in some of the subsystems constituting the whole. Another approach to performance variability aims to identify its task-relevant and task-irrelevant components. Upon defining controlled variables as those that are stable against (phasic) perturbations, Schöner (1995) conjectured that not all variables are controlled and that many joint configurations may support task success. The variance pertaining to task performance can be partitioned into two orthogonal components, of which one, the uncontrolled manifold, does not affect task achievement. The uncontrolled manifold approach has been applied to a multitude of task settings such as sit-to-stand (Scholz & Schöner, 1999), reaching for objects (Krüger, Borbély, Eggert, & Straube, 2012; Mattos, Latash, Park, Kuhl, & Scholz, 2011) and reaching with objects (van der Steen & Bongers, 2011), multiple fingers force production (Park, Singh, Zatsiorsky, & Latash, 2012; Park, Zatsiorsky, & Latash, 2010), etc. (for an overview, see Latash, 2012). The insight from these studies is that the abundant degrees of freedom are a bliss for the motor system—it allows for flexibility, rather than the redundancy being a curse (Latash, 2012). Important as these insights may be, interpretations of abundance as a bliss rather than a curse are not mutually exclusive, as they are defined relative to different issues (flexibility versus controllability). A different approach that has been developed over the last decade is referred to as dynamic field theory (Erlhagen & Schöner, 2002). Dynamic fields are dynamical systems with a continuous spatial extent modeled via Amari-type equations (originally developed to capture spatiotemporal patterns of activity in extended neural systems; Amari, 1977). In the context of movement, the so-called activation field encodes a movement-relevant metric (e.g., direction or amplitude). For instance, a one-dimensional field may specify the degree of activation for all possible parameter values. The dynamics of the field is governed by the space and time-continuous state variables, a constant specifying the “resting level,” external input (e.g., a stimulus in a reaction time task), and an interaction kernel that guarantees local excitation and global inhibition. In the absence of input, the system typically has two stable solutions, corresponding to an off (low


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activation) and on (high activation) state. This bistability is at the heart of dynamic field theory. In particular, it allows for the emergence of local excitation following input that persists even after the input has vanished. For multiple inputs various local excitations may appear and interact. Task input, which represents task context and is defined in terms of the same parameter as the activation field, shapes the field prior to specific input. The resulting field interactions predict interference phenomena in behavior. Dynamic fields have been applied to movement preparation (Erlhagen & Schöner, 2002), saccadic eye movements (Wilimzig, Schnieder, & Schöner, 2006), visual working memory (Johnson, Spencer, & Schöner, 2009), and visual–spatial cognition (Johnson, Spencer, & Schöner, 2008). Dynamic field theory thus portrays specification as a dynamic process in a continuous task-parameter space. It remains, to our best knowledge, silent, however, on the trajectory formation itself. The dynamical approach is most widely known, however, for its work on rhythmic behaviors and their coordination as formulized in the by now paradigmatic Haken–Kelso–Bunz (HKB) model (Haken, Kelso, & Bunz, 1985). The investigation of rhythmic movements in terms of limit cycles has instigated several important insights (cf. Beek, 1989; Beek, Rikkert, & van Wieringen, 1996; Beek, Schmidt, Morris, Sim, & Turvey, 1995; Kay, Kelso, Saltzman, & Schöner, 1987; Kay, Saltzman, & Kelso, 1991). For instance, the (task-specific) identification of oscillator ingredients reveals which variables determine energy injection and when this occurs (“energy bookkeeping”; cf. Beek et al., 1996; Post, de Groot, Daffertshofer, & Beek, 2007). The theoretical investigation of limit cycles resulted in a catalogue of oscillator ingredients relevant for motor control and their impact on movement frequency and accuracy constraints (Beek & Beek, 1988). The impact of some of these ingredients was confirmed in ecologically relevant task settings involving the speed–accuracy trade-off (Bootsma, Fernandez, & Mottet, 2004; Mottet & Bootsma, 1999; van Mourik, Daffertshofer, & Beek, 2008) and juggling (Beek, 1989). While these (and other) studies have confirmed that rhythmic movements, as a first approximation, obey two-dimensional limit cycle properties, small deviations from theoretically derived expectations have also been found, however. For instance, following a perturbation, the return to the attractor (the phase response) is not fully consistent with a two-dimensional limit cycle (Kay et al., 1991), which may indicate that the two-dimensional approximation is incorrect (Daffertshofer, 1998; Kay, 1988), potentially calling for a twolayered model involving a neural and an effector component (Beek, Peper, & Daffertshofer, 2002; Jirsa, Fuchs, & Kelso, 1998, 1999; Peper, Ridderikhoff, Daffertshofer, & Beek, 2004). Also, twodimensional oscillator models are incapable of generating the wellknown lag-one correlation (Daffertshofer, 1998) typically observed in finger tapping (Wing & Kristofferson, 1973a, 1973b), which may suggest that such intervals are not generated by an autonomous limit cycle (Delignières, Lemoine, & Torre, 2004). Regardless, the two-dimensional approximation accounts for most of the experimentally observed phenomena, which justifies their (continued) use as building blocks in the HKB model. In that regard, Kelso (1981, 1984) observed that when people start oscillating their fingers in an antiphase manner (i.e., with a 180° phase difference) and gradually increase movement frequency at a certain moment, an abrupt transition occurs toward the in-phase (i.e., a 0° phase difference) pattern (see also Yamanishi, Kawato, & Suzuki, 1979, 1980). Prior to the transition, the variability of the phase difference between the fingers increased (critical fluctuations), which is a signature of

bifurcations or so-called phase transitions. Reducing movement frequency when in the in-phase pattern did not lead to a transition to the antiphase pattern. These phenomena were modeled by Haken et al. (1985). Given its primacy in coordination dynamics, we discuss the HKB model in more detail. The model is composed of two nonlinearly coupled limit cycle oscillators: u¨ 1 ⫹ 共Au21 ⫹ Bu˙ 21 ⫺ ␥兲u˙ 1 ⫹ ␻2u1 ⫽ (u˙ 1 ⫺ u˙ 2)共␣ ⫹ ␤(u1 ⫺ u2)2兲

u¨ 2 ⫹ 共Au22 ⫹ Bu˙ 22 ⫺ ␥兲u˙ 2 ⫹ ␻2u2 ⫽ (u˙ 2 ⫺ u˙ 1)共␣ ⫹ ␤(u2 ⫺ u1)2兲 , (1) where the dot notation signifies the time derivative. The left-hand side of the equations represents the so-called hybrid oscillators, describing the limbs’ movements, consisting of Rayleigh (⬃u˙u˙2) and van der Pol terms (⬃u˙u2; cf. Haken et al., 1985; Kay et al., 1987, 1991); the right-hand side represents the (difference) coupling. This formulation allows for the mathematical derivation of the relative phase dynamics, ˙

␾ ⫽ ⫺a sin (␾) ⫺ 2b sin (2␾),

(2)

where a and b are frequency-dependent parameters derived from the oscillators’ dynamics (Equation 1). The popularity of the rhythmic coordination paradigm owes much to the visualization of dynamics via the potential function V(␾) ⫽ ⫺a cos (␾) ⫺ b cos (2␾).

(3)

In this picture, the dynamics of the relative phase ␾ between the oscillating limbs can be seen as the overdamped motion of a ball in the potential landscape where minima represent stable states and the maxima unstable states (see Figure 1).3 Many of the features predicted by the HKB model (i.e., a bifurcation from bistability to monostability with increasing frequency, critical slowing down and fluctuations, hysteresis, etc.) have been repeatedly reported in rhythmic coordination scenarios such as in interlimb, sensorimotor (Byblow, Chua, & Goodman, 1995; Kelso, Delcolle, & Schöner, 1990), and interpersonal coordination (Issartel, Marin, & Cadopi, 2007; R. C. Schmidt & Richardson, 2008), learning (Kelso & Zanone, 2002; Kostrubiec, Zanone, Fuchs, & Kelso, 2012; Smethurst & Carson, 2001; Zanone & Kelso, 1992, 1997), and the impact of attention (Monno, Temprado, Zanone, & Laurent, 2002; Temprado, 2004; for a concise overview, see Kelso, 1995). The rhythmic paradigm has further instigated the model-driven investigation of physiological mechanisms (Carson & Kelso, 2004; G. N. Lewis, Byblow, & Carson, 2001; Roberts, Stinear, Lewis, & Byblow, 2008) and the relation between behavioral and brain dynamics (Boonstra, Van Wijk, Praamstra, & Daffertshofer, 2009; Daffertshofer, Peper, & Beek, 2005; Jirsa et al., 1999; Kelso, Dumas, & Tognoli, 2013; Kelso et al., 1998)—a still continuing research program spanning over 2 decades. The success of the rhythmic paradigm is not equaled in the non-rhythmic domain, however. Discrete movements and the co3 Critical fluctuations and pattern switching can only occur in the presence of noise. In order to explicitly incorporate the importance of noise, Schöner, Haken, and Kelso (1986) formulated a stochastic extension of the HKB model. For a discussion on stochastic dynamics, see Daffertshofer (2011).


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Figure 1. The Haken–Kelso–Bunz potential. The vertical axes represent the potential V(␾); the horizontal axis represent the relative phase ␾. The local (⫾␲) and global (0) minima represent the system’s stable states. The ratio b/a decreases from the left to the right panel. Accordingly, the local minima (left two panels) disappear and only the global minimum remains.

Sequential Behavior

Whalen, 2000). Cursive handwriting has a principal frequency component of about 5 Hz (Teulings, Thomassen, & Maarse, 1989), but also a slower periodicity of about 1 Hz (corresponding to three to four characters; Thomassen & Meulenbroek, 1998). The implication of multiple processes operating on distinct characteristic time scales raises questions about the processes’ functional roles and interactions among them and how the time scales relate to the functional differentiation of the distinct processes. By and large, three lines of thinking on serial behavior have been developed. Response chaining posits that the execution of any one unit (response) generates feedback, which triggers the following unit (Bain, 1868; James, 1890). This stimulus–response reflex chaining is supposedly independent of sensory feedback. More modern versions of associative response chaining theories, recurrent neural networks models, are based on reciprocal interactions among all subprocesses of a sequence (Cleeremans & McLelland, 1991; Dominey & Arbib, 1992; Elman, 1995, 2004; Jordan, 1989). The interactions form a representation of a sequence’s history in a high-dimensional space through learning. Each sequence’s unit (network output) is fed back to the network and integrated with its current state to produce the next response. Consequently, the slow time scale of a sequence emerges from the interactions of its constituent fast time scale processes in a fundamentally serial manner. The sequence as a whole is not represented a priori: The information determining the sequence is generated as the sequence unfolds. The absence of postulating a priori higher level representations renders such approaches parsimonious. At the same time, however, in the presence of perturbations or noise, such architectures may not guarantee behavioral integrity on the sequence level. Lashley (1951), who put the problem of serial order (back) on the agenda of experimental psychology, rejected response chaining in favor of a hierarchical organization of serial behavior. In such

Serial behavior inherently contains multiple processes operating at different time scales (for a classic study, see Sternberg, Monsell, Knoll, & Wright, 1978; for a review of modeling approaches, see Rhodes, Bullock, Verwey, Averbeck, & Page, 2004). For instance, in birdsong notes and syllables are associated with distinct time scales (Yu & Margoliash, 1996). A similar time scale separation supposedly underlies the perception of birdsong (Friston & Kiebel, 2009) as well as speech (Kiebel et al., 2009). In speech, the slower time structure is thought to emerge from the concatenation of “gestures” (articulatory sound-generating “primitives” underlying linguistic communication; Goldstein et al., 2007; Liberman &

4 One problematic but often overlooked issue is that the concepts that have been developed in the rhythmic coordination paradigm do not trivially transfer (if at all) to other (experimental– behavioral) paradigms. Take the notion of a stable state, corresponding to the 0° and 180° relative phase patterns in rhythmic coordination, for instance. Discrete phenomena are transient by definition; the notion of a stable state is not very meaningful—it applies to moments prior to and following the transients only. In the coordination of discrete and rhythmic phenomena, it does not apply at all. In other words, the expansion of the behavioral domain beyond rhythmicity will benefit from the development of novel concepts. One proposal thereto is that of divergence and convergence of (phase space) trajectories (Jirsa & Kelso, 2005; see also Calvin & Jirsa, 2011).

ordination between them have only rarely been studied experimentally (Fink, Kelso, & Jirsa, 2009; Kelso, Southard, & Goodman, 1979a, 1979b), and the first theoretical formulation incorporating rhythmic as well as discrete movements (Schöner, 1990) has found little empirical follow-up (but see Zaal, Bootsma, & van Wieringen, 1999). Similarly, investigations into the coordination between rhythmic and discrete movements have been far and few in between (but see Calvin, Huys, & Jirsa, 2010; de Rugy & Sternad, 2003; Sternad, 2008; Sternad, de Rugy, Pataky, & Dean, 2002; Wei, Wertman, & Sternad, 2003). Moreover, in several cases modeling deviated from the HKB formulation, such that an encompassing framework was not reached. Overall, this state of affairs has obviously limited the scope of the dynamical approach. Even when incorporating discrete and rhythmic trajectory formation within a single (two-dimensional) framework (Jirsa & Kelso, 2005; Schöner, 1990), the number of distinct functional forms accounted for remains limited. While the combination of rhythmic and discrete movements leads to a considerable behavioral repertoire, it seems unlikely that the corresponding two-dimensional attractors fully cover our behavioral spectrum, definitively so if expansion into the cognitive domain is strived for.4 In order to overcome this limitation Saltzman and Kelso (1987) proposed that qualitatively distinct dynamical topologies, defined in an abstract task space, underlie the performance of correspondingly distinct tasks. Our approach, as outlined below, follows the essence of this task-dynamics idea. However, in contrast to the task-dynamics framework that was illustrated in the context of a few (to some extent) arbitrarily chosen tasks, SFM aims for a general formalization for a multitude of (task-specific) topologies resulting from the interplay between a system’s microscopic and macroscopic dynamics in the spirit of synergetics.


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conceptions, associated higher level representations take the form of tree-like branching structures (Book, 1908; Miller, Galanter, & Pribram, 1960). Units of behavior of increasing complexity (from the bottom to the top) form the levels of the hierarchy such that the higher levels deal with longer term movement planning and the lower ones fill in the movements’ details. For example, Verwey and colleagues (Verwey, 2001; Verwey, Abrahamse, & de Kleine, 2010) proposed a dual (motor and cognitive) processor model for sequential typewriting. Whereas the prime role of the cognitive processor shifts (with practice) from executing to initiating sequences, the gradual development of motor chunks allows the motor processor to execute them. Similarly, it has been postulated that handwriting involves parallel functions that are related to processing across multiple levels of a hierarchy: the linguistic, semantic or word, graphemic, allographic or letter, and stroke level (van Galen, 1991; van Galen, Meulenbroek, & Hylkema, 1986). Thus, hierarchical control implies a direct connection between the characteristic time scale of a process at a specific level of the hierarchy and its functional role. A third class of models, competitive queuing models (Bullock & Rhodes, 2003; Houghton, 1990; Page & Norris, 1998), is based on parallel representations of (learned) sequences and on competitive interactions among their items. The information specifying a sequence is contained in initial activations of the participating items. These activations establish a so-called primacy gradient (i.e., a parallel representation of the sequence; Page & Norris, 1998). At each stage of the sequence the component processes compete for their activation (via mutually inhibitory couplings), following an order of priority defined by the primacy gradient. The “winning” item is executed and subsequently inhibited; the competition continues among the remaining available items. In the case of an emergent gradient, the initial primacy gradient is modulated (through feedback) by the ongoing evolution of the sequence (Houghton, 1990; Ward, 1994). The competition outcome may also be affected by noise (Page & Norris, 1998). In a more recent advancement, the (time-continuous) N-STREAMS neural network model combines parallel competitive queuing processes with serial (chain-like) and context-specific ones, thus reconciliating mechanisms previously treated as mutually exclusive. Consequently, state-of-the-art competitive queuing models involve functionally different processes that are characterized by their associated time scales: an emergent slow unfolding of a sequence of activations of the component subprocess (controlled by the initial primacy gradient and/or emergent gradient), a faster execution of the subprocess, and a still faster time scale competition prior to the activation of each item of the sequence. The competition involved in competitive queuing, referred to as “winner take all” (WTA) competition, it involves a single welldefined winner at each competition round and has a long history in the modeling of biological systems, and dynamical systems more generally (Grossberg, 1980; Haken, 2004; Smale, 1976). WTA competition among different patterns is the main principle of functioning of the synergetic computer (Haken, 2004), a computational paradigm inspired by synergetics (Haken, 1983) and applied to pattern recognition such as switches in the visual perception of ambivalent patterns (Ditzinger, 2011; Ditzinger & Haken, 1989, 1990). There, two alternative patterns (“percepts”) compete to dominate (win) the competition temporarily, to subsequently be inhibited due to feedback, and give way to the activation of the

other pattern. Feedback is mediated by so-called attentional parameters, which also regulate the time each pattern remains active. Such dynamics can be scaled up to N patterns (items), each of which is associated with an attentional parameter, thus generating oscillatory sequences by means of an autonomous dynamical system of dimensionality 2 ⫻ N. The transitions between the items of a sequence are generally much faster than the time an item is active (dominating), thus introducing a time scale separation. Of late, an alternative to WTA competition, the so-called winnerless competition, has been advocated by Rabinovich and colleagues (Rabinovich, Huerta, Varona, & Afraimovich, 2008; Seliger, Tsimring, & Rabinovich, 2003). Accordingly, behavioral or cognitive sequences are driven by a heteroclinic dynamics, which is characterized by sequential transitions from one unstable equilibrium point (a saddle) to another. Each equilibrium point corresponds to a particular (cognitive, emotional, or behavioral) mode. Each saddle attracts the flow in the phase space from all directions except one—the repelling manifold (a one-dimensional subset of the phase space). The unstable direction is connected to an attracting branch of the next saddle node in the sequence. Thus, robust, transient sequential dynamics can be generated that are characterized by asymptotic stability in a neighborhood of the phase space resembling a hypertube (see Krupa & Melbourne, 2008, Figure 1.3). As the dynamics is slower, close to the equilibrium points, the transitions are typically fast and short-lived relative to the time spent in their neighborhood (time scale separation). The competition is referred to as winnerless because the system settles only temporarily at the neighborhood of a sequence’s node. Heteroclinic sequences can be connected in various ways (Postlethwaite & Dawes, 2005) and evolve on different time scales so that a hierarchical multiscale dynamics can emerge. Such dynamics has been designed for the recognition of sequences (Kiebel, von Kriegstein, Daunizeau, & Friston, 2009) that combined heteroclinic sequences’ dynamical models with the freeenergy minimization principle (cf. Friston & Stephan, 2007). In a similar vein, Friston et al. (2011) used a heteroclinic sequence to “drive” an EP like dynamics to generate cursive handwriting. In both types of competition (i.e., WTA and winnerless) the competitive interactions among items are at the basis of the resulting emergent sequence as well as the time scale separations. Heteroclinic cycles require N dimensions only for an N–item sequence; this parsimony, however, comes at the expense of a relative loss of flexibility, which is guaranteed by the (additional) attentional parameters in the WTA competition. In addition to these classes of (modeling) approaches, Saltzman and colleagues (Saltzman & Munhall, 1992; Saltzman, Nam, Krivokapic, & Goldstein, 2008) have developed a framework to model speech production. Accordingly, the spatiotemporal dynamics of speech emerges from two interacting but functionally distinct levels. The coordination among articulators (e.g., lips, jaw, tongue) are viewed as the basic functional elements of speech production (gestures). The corresponding dynamics, typically fixed-point attractors, contains tract (or goal) variables that are transformed into model articulator variables. The patterns of (relative) timing among the articulatory gestures implicated in a given utterance are defined at the intergestural level. Each distinct gesture is associated with a so-called activation variable that determines the gesture’s contribution to the vocal tract movement at a given instance in time. In an early version of the model (Saltzman



& Munhall, 1989), the intergestural timing, which defines a gestural score for a given utterance, was implemented “by hand.” More recently, the intergestural timing follows from an ensemble of “planning oscillators” that are pairwise bidirectionally coupled (and can be represented in a coupling graph; Saltzman et al., 2008; see also Saltzman & Byrd, 2000). The coupling among elements gives rise to relative phase patterns that are mapped into corresponding gestural activation patterns that define the gestural score. Although this approach contains two levels and two time scales (relating to the gestures and their sequencing), the time scales are not explicitly modeled but emerge from the dynamics. Common to most of the latter modeling approaches is the implication that interactions between time scales are essential for the organization of serial behavior— even though this functionality is not always explicitly recognized. The interactions between processes acting on (at least) two time scales (one pertaining to the sequence, one to each item) play a central role characterized by circular causality: The slow sequential dynamics modifies the fast dynamics at the level of the sequence’s items, often resulting in nonstationary processes (top-down influence; Wolpert, Doya, & Kawato, 2003; Wolpert, Ghahramani, & Flanagan, 2001); at the same time, complex behavior emerges as functional units are brought into meaningful relationships (i.e., serial order; bottom-up composition). Circular causality becomes explicit in models including output feedback, such as in the cases of attentional parameters, the emergent primacy gradient or the inclusion of context-specific recurrent neural network dynamics. We discussed the various theoretical frameworks within which the emergence of perceptual motor behavior can be understood. Beyond the historical perspective that our discussion offers, we attempted to provide an integrated point of view of the conceptual range spanned by the various approaches. Our objective to do so has been to demonstrate that none of the existing theoretical frameworks offer a sufficiently wide perspective to understand (perceptual) motor behavior as a whole. Each approach focuses on a particular aspect of behavior, or subset of behaviors, for a given level of organization and ignores the others. For instance, the actual trajectory formation of movement (or generally behavior) is left open in dynamic field theory or heteroclinic cycles. Synergetics and coordination dynamics rest mostly on the center manifold theorem, which is at the heart of the slaving principle, and hence are concerned with behaviors describing general reorganizations (bifurcations). Obviously, when traversing scales of organization, some dynamic organizations naturally fall out of the scope of a level of description, such as the dynamics of learning or the spiking patterns of neuronal interactions (for an account of traversing scales of organization, see Fuchs, Jirsa, & Kelso, 1999; Jirsa et al., 1999; Kelso, Jirsa, & Fuchs, 1999). What we wish to advocate is the need for a formal framework offering a perspective of a behavioral organizational principle for the same level of spatiotemporal organization and perceptual motor behavior as a whole. In that regard, in the next section we outline the theoretical framework of SFM, which will be essential for the development of an integrated perspective of functional architectures discussed in the subsequent section.

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Structured Flows on Manifolds: Basic Concepts In the following we first define the notion of phase flow and related concepts and then place it in the context of SFM. We illustrate the concepts via computational examples and provide proof of concept using motor behavioral data. Although our formalizations are typically deterministic, we hold that the emergence of, and switching between SFM are inherently stochastic processes (cf. Haken et al., 1985). Accordingly, all provided simulations include additive (Gaussian) noise. The formal stochastic extension of SFM, however, is beyond our present scope. The main gist of the SFM framework is (a) that a low-dimensional dynamics collapses onto a subspace (manifold) of the intrinsically high-dimensional system from which it emerges, (b) that this low-dimensional dynamics is appropriately cast in terms of phase flows that represent functional processes, and (c) that different phase flow topologies set qualitatively distinct processes apart from another.

State Spaces: Phase Flows and Topologies In the following we will introduce our notation and present an introduction to the key mathematical concepts in nonlinear dynamics. A single time-dependent state variable, say, q1(t), evolves in a one-dimensional state space (i.e., a line). Similarly, a two-dimensional state space is spanned by two variables, (q1(t), q2(t)), and more generally, an N-dimensional state space is spanned by N variables, (q1(t), q2(t), . . . , qN(t)). These N variables define a state vector, which, as time evolves, traces out a trajectory in state space. In general, not every degree of freedom is used by the dynamic system, and only a lower dimensional subspace with dimension M ⬍⬍ N is occupied by the trajectories. This subspace represents a form of a workspace, in which most of the important dynamics evolves. The various theoretical frameworks are concerned with the constraints to be imposed upon the dynamic system for a desired workspace to emerge. Hence, the lower dimensional subspace referred to as workspace is spanned by the set of variables {ui(t)} with i ⫽ 1, . . . , M, whereas all other degrees of freedom (the set of variables {sj(t)}) are recruited (note that we use {} to denote the complete set of variables). How this recruitment occurs, including the relations of the various sets of variables, will be discussed in more detail in the following sections. Further, it is often useful to consider so-called manifolds to describe parts of trajectories that share common features. Manifolds are P-dimensional subspaces, which are locally Euclidian in nature, in an N-dimensional space (P ⬍ N). For instance, a point is a zero-dimensional manifold, a line and a circle are examples of a one-dimensional manifold, and a plane and a sphere are examples of two-dimensional manifolds, where typically P ⱕ ⌴ ⬍⬍ N. A key concept of SFM is the phase flow described by a vector field in the state space. The phase flow indicates the rate of change of the state variables at every point in the state space. Phase flows and differential equations are two sides of the same coin: They are a graphical and symbolical (mathematical) representation of a given dynamical system, respectively. Phase flows offer a language to describe dynamic processes. They are not limited to particular behaviors and hence offer a powerful set of predictive tools. Dynamical systems theory states that


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every deterministic, time-continuous and autonomous system can be described unambiguously through its flow in state space (also referred to as phase space). Autonomous systems are those without explicit time dependency; the system’s evolution depends on its state variables only. The behavior of nonautonomous systems is further shaped by a time-dependent impact. A general description of dynamical systems in terms of ordinary differential equations is then given by q˙ 1 ⫽ f 1(q1, . . . qN)


É ˙qN ⫽ f N(q1, . . . qN).

(4)

When dealing with experimental data, the workspace and its dimensionality are generally not known beforehand but can be identified via phase space reconstruction techniques (cf. Abarbanel, 1996). In particular, state variables are here assumed to be already in the workspace, since effectively in the experiment these are the accessible variables. In the motor control literature, it is then commonplace to depict the state space of a movement along a single (physical) dimension as the (work)space spanned by the system’s position u and its time derivative, the velocity u˙. In this case, state space and workspace are identical, or at least not easily distinguishable. The flow in state space uniquely describes the direction of the system’s evolution as a function of its current state, (u, u˙). The phase flow is a structural property of the system’s dynamics. A trajectory is an actualization of a system’s evolution in state space governed by the flow. Bearing in mind that each state space location has a unique flow associated with it, it can easily be appreciated that phase flows are structured and that trajectories cannot cross.5 The flow topology defines the system’s behavior. As the type of topological structures that can occur in phase space depends on its dimensionality, the same applies to the possible behaviors. All topologies that are possible in N-dimensional systems are also possible in N ⫹ 1 dimensional systems; the dynamic repertoire thus increases with a system’s dimensionality. In one-dimensional systems (i.e., a flow on a line) only so-called fixed-point dynamics can occur. Fixed points are points in phase space where the system’s rate of change is zero ( u˙ ⫽ 0); if stable (unstable), all trajectories eventually converge toward it (diverge away from it). Unstable fixed points separate two regions of attraction around stable fixed points and hence are called separatrices. Fixed points are also found in twodimensional systems, relative to one-dimensional systems; however, they come in a larger variety as the flow no longer necessarily is directed straight toward (or away from) it but may also approach (or recede) it spiraling (in which case they are called spirals) or approach it along one direction and recede from it in another direction (in which case they are called saddles). In addition, two-dimensional systems allow for so-called limit cycles, which are closed orbits that may be stable or unstable. Here, separatrices are “lines” that divide the state space in regions with locally distinct phase flows. Unstable (stable) fixed points necessarily exist in the space surrounded by stable (unstable) limit cycles. A system on a limit cycle will repetitively traverse the same trajectory in state space and sustain a periodic motion. Two-dimensional phase flows including fixed points, separatrices, and a limit cycle are shown in Figure 2. Fixed points and limit cycles can be associated with discrete and rhythmic be-

havior, respectively. In higher dimensions, all the aforementioned objects exist plus higher dimensional attractors, including deterministic chaos, which allows for irregular behavior in a fully deterministic system. Changing the parameters of a system changes its phase flow: In a bifurcation, the number and/or nature of the solutions changes. We illustrate the phase flow concept along a recently introduced (two-dimensional) class of models, referred to as the excitator (Jirsa & Kelso, 2005), u˙1 ⫽ 共u1 ⫹ u2 ⫺ g1(u1)兲␶

˙u2 ⫽ ⫺ 共u1 ⫺ a ⫹ g2(u1, u2) ⫺ I兲 ⁄ ␶,

(5)

with u˙ 1 and u˙ 2 representing the time derivatives of u1 and u2, respectively, and ␶ and I representing a time constant and an input (external to the system), respectively. Different flows and distinct topologies can be implemented through g1(u1) and g2(u1, u2); under their appropriate choice, this model belongs to the class of excitable systems (Izhikevich, 2007; Murray, 1993; Wilson, 1999). Setting g1(u1) ⫽ u13/3 and g2(u1, u2) ⫽ b·u2 provides the minimal realization that allows for the dynamic repertoire of phase flows discussed in motor control: a single stable fixed point (monostable condition; a ⫽ 1.05, b ⫽ 0), two stable fixed points (bistable condition; a ⫽ 0, b ⫽ 2), and a limit cycle (a ⫽ 0, b ⫽ 0; see also Figure 2). The presence or absence of particular topological structures can be readily illustrated via the system’s nullclines, that is, the curves in phase space where the flow is purely horizontal (i.e., u˙ 2 ⫽ 0) or vertical (i.e., u˙ 1 ⫽ 0). The nullclines in the present realization are defined by ˙u1 ⫽ 0 ⫽ u1 ⫹ u2 ⫺ u31 ⁄ 3 u2 ⫽ 0 ⫽ ⫺u1 ⫹ a ⫺ bu2 ˙

冎再 )

u2 ⫽ ⫺u1 ⫹ u31 ⁄ 3 u2 ⫽ (a ⫺ u1) ⁄ b

(6)

(with I ⫽ 0). A fixed point is found at the intersection of both nullclines. From these equations it can be appreciated that only the u˙ 2 ⫽ 0 nullcline is affected by the parameterization and that its translation (via a) or rotation (via b) determines whether we find a single stable fixed point (monostable condition), two stable fixed points interspersed with an unstable one (referred to as the bistable condition), or a stable limit cycle. An important theorem in dynamical systems theory is the Poincaré–Bendixson theorem (cf. Perko, 2006; Strogatz, 1994); it states that if a trajectory is confined to a closed bounded region that contains no fixed points, then the trajectory must eventually approach a closed orbit. By implication, the only topological structures possible in two-dimensional systems are those mentioned above. Chaos is thus ruled out, and the possible dynamics are limited. Within a dynamic class, however, a large variety of quantitatively distinct behaviors are possible. In the excitator system, this can be achieved by the substitution and/or addition of terms in the functions g1 and g2 as well as by adjusting the parameters a and b (changing the nullclines) without changing the topology. The Hartman–Grobman theorem states that the flow in 5 Experimentally obtained trajectories typically reveal crossing, which may be due to noise and/or indicates that the dimension of the phase space in which the trajectories embedded is too low (see also Kay, 1988).



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Figure 2. Two-dimensional phase flows corresponding to a linear system (upper left), a bistable system (upper right), a monostable system (lower left), and a limit cycle (lower right). The arrows represent the flow; the lines represent trajectories. In the upper left panel, the flow converges to a single stable fixed point. In the upper right panel, two stable fixed points exist separated by one unstable fixed point (in the middle). In the lower left panel, a single stable fixed point exists. In the upper right and lower left panels, it can be seen that trajectories starting nearby may diverge, indicating the presence of a separatrix. In the lower right panel, the flow converges onto a stable limit cycle, which has an unstable fixed point enclosed.

the vicinity of a hyperbolic fixed point6 is topologically equivalent to that of its linearization, implying the existence of a continuous invertible mapping (a homeomorphism) between both local phase spaces. The existence of a homeomorphism (and only then) implies that the topologies are equivalent; the systems thus belong to the same dynamical class. While the Hartman–Grobman theorem is valid for fixed points only, the notion of topological equivalence is utilized for other topological structures too. Intuitively, dynamical systems are topological equivalent under “distortions” if the underlying structure remains invariant. For instance, bending and stretching are allowed, but disconnecting closed trajectories is not (cf. Stewart, 1995, chapters 10 and 12, for an intuitive nontechnical introduction). The notion of topological equivalence thus provides for an unambiguous classification of dynamical systems. In sum, the flow structure prescribes a dynamics, which is governed by distinct irreducible topologies and hence defines distinct behavioral classes in a principled way (see also Saltzman & Kelso, 1987).

Synergetics Since synergetics and SFM are closely related, we briefly outline some of the concepts of synergetics first, which will aid in better understanding the concepts of SFM. Synergetics is the theory of self-organized pattern formation in open systems (i.e., those that are in contact with the environment through matter, energy, and/or information fluxes) far from thermoequilibrium that

are composed of numerously weakly interacting microscopic elements (Haken, 1983, 1996). Due to the interaction among the microscopic elements, such systems may become organized in spatially as well as temporally ordered patterns that are typically macroscopic in nature and can be described by a limited number of so-called order parameters (or collective variables). Spontaneous switches between macroscopic patterns (i.e., non-equilibrium phase transitions) are accompanied by loss of the stability of the original pattern. The stability of a system’s state (or phase) implies that every perturbation out of that state causes the system to return to it; when stability is lost, the system does not return but rather switches to another stable state. Close to such points of (macroscopic) instability, the time to return increases tremendously. As a result, the macroscopic state evolves rather slowly in response to perturbation, whereas the underlying microscopic components maintain their individual time scale. Consequently, the time scales of their dynamics differ tremendously (time scale separation). From the perspective of the slowly evolving macroscopic state, the microscopic components change so quickly that they can adapt instantaneously to macroscopic changes. Thus, even though the macroscopic patterns are generated by the subsystems, they order 6 A fixed point of an nth order system is hyperbolic if all eigenvalues ␭ of the linearization have a non-vanishing real part (i.e., ᑬ(␭i) ⫽ 0 for i ⫽ 1 . . . n), which is the case for most fixed points (and those under consideration at present).



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the microscopic components or, put metaphorically, enslave them (Haken, 1983). Ordered states can always be described by a very few variables (at least in the close vicinity of a bifurcation), and consequently the state of the originally high-dimensional system can be summarized by a few variables, or even a single collective variable, the order parameters. The order parameters then span the workspace. The circular relationship between the enslaving order parameters and the enslaved microscopic components, which generate the order parameters, is sometimes referred to as circular causality, which effectively allows for a low-dimensional description of the dynamical properties of the system. The notion of circular causality and the emergence of low-dimensional dynamics in self-organizing systems are at the heart of Hermann Haken’s synergetics. The mathematical formalism of synergetics can be briefly sketched as follows. Let us start with a state vector Q(t) ⫽ (q1(t), q2(t), . . . , qN(t)), where Q 僆 ᑬN. The evolution of the state vector is captured by a nonlinear ordinary differential equation, ˙ Q(t) ⫽ N[Q(t), {␣}] ⫹ F(t),

(7)

where N is a nonlinear vector function that depends on the state variables Q and {␣} is the set of control parameters. F represents the internal and external fluctuations that are present in the system. The state vector and its evolution equations can be extended to incorporate spatial dependence of the state variables, but it suffices here to consider only spatially discrete systems. We also ignore the fluctuation term F in the subsequent discussion. For a detailed treatment, see Haken (1983). Synergetics is primarily concerned with the qualitative changes (state transitions) on the macroscopic level of description. Hence, a first step is to determine the stationary state for a given parameter configuration and its stability. A given state typically corresponds to a (stationary) fixed point or periodic (limit) cycle in state space, and synergetics aims at describing the flow in its vicinity—it is in this sense that the synergetic formalism is local. The dynamics in this vicinity can be studied by perturbing the system around the state, which we illustrate for a fixed-point solution below. Periodic states are treated equivalently after some preprocessing allowing to rewrite the equation as a fixed-point solution. The deviation ε(t) from the steady state Q0 satisfies Q(t) ⫽ Q0 ⫹ ε(t), and the local approximation of the flow in the neighborhood of a state is operationalized through a Taylor expansion, leading to ˙ Q(t) ⫽ N[Q0, {␣}] ⫹ L[Q0, {␣}]ε ⫹ A[Q0, {␣}]ε : ε ⫹ . . . , (8) where we omitted to indicate the time dependence of ε on the right hand side of the equation. By definition, for the steady state N[Q0,{␣}] ⫽ 0. The matrix L is the Jacobian matrix; second-order polynomials are captured by the expression A, where we use the following tensor product convention for its ith component: 共A : ε : ε兲i ⫽ jk Aijkε jεk; higher orders are indicated by the horizontal dots. In the neighborhood of instability, that is, close to the bifurcation, there always exists a separation into (at least) two time scales. To identify these, that is, to capture the nature of the flow in the neighborhood of the steady

兺

state Q0, requires that one spans a linear vector space so as to separate the entire state space into a linear subspace, in which the transient dynamics collapses rapidly toward an attractive subspace (i.e., the workspace); and the workspace itself, in which the dynamics evolves slowly. We decompose hence N ε共t兲 ⫽ i⫽1 ␰i共t兲Vi and LVi⫽␭iVi, Vi†L ⫽ ␭iⴱVi†, where Vi†Vj ⫽ ␦ij. Multiplying Equation 8 from the left with Vk†, we obtain

兺

␰˙k ⫽ ␭k␰k ⫹ V†k A[Q0, {␣}]ε : ε ⫹ . . . ⫽ ␭k␰k ⫹

N

兺 A˜kij␰i␰j ⫹ . . . , i ,j⫽1 (9)

˜ absorbs all the constant parameters from the projection where A kij of the expression A onto the linear vector space. In Equation 9, ␭k plays the role of a rate parameter and therefore encodes the (separation of) time scales shown by the variables. Equation 9 describes the evolution of small perturbations in a linear vector space around a steady state. The equations are now linearly decoupled and allow for a classification of time scales in terms of slow and fast variables. The slow variables are close to the instability point defined by |ᑬ(␭i)| ⬇ 0 for i ⫽ 1, . . . M, where the fast variables are stable and act on a faster time scale, |ᑬ(␭j)| ⬎⬎ |ᑬ(␭i)| ⫽ 0 for i ⫽ 1, . . . M, j ⫽ M ⫹ 1, . . . , N. Here, typically M ⬍⬍ N; in fact, often M is 1 or 2. Consistent with our previously introduced notation, we now rename these two sets of variables as follows:

ⱍ ⱍ ⱍᑬ(␭ )ⱍ ⬎ 1 for j ⫽ M ⫹ 1, . . . , N.

ui(t) ⫽ ␰i(t) where 1 ⬎⬎ ᑬ(␭i) ⱖ 0 for i ⫽ 1, . . . , M ⬍⬍ N s j⫺M(t) ⫽ ␰ j(t) where

j

(10) For some critical values of the control parameters, the real part of one or more eigenvalues become zero or positive, defining the corresponding modes as unstable modes, which are called the order parameters. The stable modes evolve on a faster time scale. Now the dynamics of the entire system can be expressed as u˙ i ⫽ ␭iui ⫹ Qi[ui, s j, {␣}] ˙s j ⫽ ␭ js j ⫹ P j[ui, s j, {␣}].

(11)

This time scale separation ᑬ(␭u) ⬍⬍ ᑬ(␭s) of the stable and unstable modes allows us to apply the slaving principle of synergetics to the system. The slaving principle is based on the center manifold theorem. As a consequence of this theorem, we can set the dynamics of the fast, stable modes to zero, that is, s˙j ⫽ 0, which allows us to perform what is known as adiabatic elimination, as follows: u˙ i ⫽ ␭iui ⫹ Q[ui, G j({ui}), {␣}] 1 s j ⫽ ⫺ P j[ui, s j, {␣}] ⫽ G j({ui}, {␣}) ⫹ higher orders of ui. ␭i (12) Equation 12 presents the final result of the synergetic formalism. All stable modes sj collapse adiabatically onto the center manifold Gj. Hence the dynamics of the entire system is prescribed by the now closed differential equation system in terms of the order


parameters ui and the fact that the stable modes sj may be expressed as a function of {ui}.


Structured Flows on Manifolds In contrast to synergetics, which is valid in the neighborhood of phase transitions, SFM is not local: rather than being (Taylor) expanded around a local point in state space, SFM assumes the existence of a subspace in which an invariant attractive manifold is defined. In this sense SFM represents an expansion around processes rather than states as in synergetics. This assumption imposes constraints onto the mathematical formulation of the system’s equations, effectively splitting the state space spanned by {qi} into the two subspaces spanned by {ui} and {sj}. These ideas have been paraphrased mathematically in more technical terms as follows (Jirsa, Huys, Pillai, Perdikis, & Woodman, 2012; Pillai, 2008; Pillai & Jirsa, 2007; Woodman & Jirsa, 2013): The collapse onto the manifold occurs much faster than the system’s evolution on it; that is, the system’s dynamics has a slow and fast time scale associated with it. The general model equations of an SFM read u˙ i ⫽ ⫺f({ui}, {s j})ui ⫹ ␮g({ui}, {s j}) ˙s j ⫽ ⫺s j ⫹ h({s j}, {ui})

(13)

u 僆⺢M, s 僆⺢N⫺M, M ⬍⬍ N. For 0 ⬍ ␮ ⬍⬍ 1, f(.) constrains the dynamics to a particular subspace and defines the manifold; g(.) defines the flow on the manifold that governs the slow dynamics and is not any further constrained; g(.) may hence represent a wide variety of processes such as excitator (Jirsa & Kelso 2005) and Lotka–Volterra (Lotka, 1920; Murray, 1993) dynamics; s˙ represents the fast dynamics that collapses onto the manifold. The time scale separation is a consequence of the choice of ␮, the smallness parameter, being small. The properties of f(.) under which the manifold is attractive are discussed elsewhere (Pillai, 2008; Woodman & Jirsa, 2013). If f(.) is zero, then Equations 13 almost resemble the synergetic Equations 11 (except a scaling of the nonlinearity with the smallness parameter), in which the manifold is effectively zero-dimensional (i.e., a point). Systems of the SFM form contain a so-called inertial manifold (cf. Constantin, Foias, Nicolaenko, & Teman, 1989; Iannelli, 2009), which, like the center manifold, is used for dimensionality reduction of dynamical systems. In contrast to the center manifold theorem, however, no general theoretical basis for inertial manifolds yet exists. Consequently, systems containing inertial manifolds have to be dealt with individually. We illustrate SFM via simulations of two-dimensional flows on a plane and sphere in (n ⫹ 2)-dimensional and (n ⫹ 3)dimensional space, respectively. In order to obtain the corresponding flow that collapses onto a planar manifold, we use u˙ 1 ⫽ ␮共u1 ⫹ u2 ⫺ u31 ⁄ 3兲␶ u˙ 2 ⫽ ⫺␮(u1 ⫺ a ⫹ bu2 ⫺ I) ⁄ ␶

(14)

˙s1.. n ⫽ ⫺s1.. n (the dynamics corresponding to the flow in Equations 14 and 15 is governed by excitator dynamics [Equation 5]). As can be appreciated in Figure 3, the dynamics of a system of this type is constrained onto the plane (sj ⫽ 0 @ j) following an initial transient toward it. A spherical manifold with similar flow can be obtained via

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u˙ 1 ⫽ u1共⫺u21 ⫺ u22 ⫺ u23 ⫹ 1兲 ⫹ ␮共u1 ⫹ u2 ⫺ u31 ⁄ 3兲␶

u˙ 2 ⫽ u2共⫺u21 ⫺ u22 ⫺ u23 ⫹ 1兲 ⫺ ␮(u1 ⫺ a ⫹ bu2 ⫺ I) ⁄ ␶

u˙ 3 ⫽ u3共⫺u21 ⫺ u22 ⫺ u23 ⫹ 1兲 ˙s1.. n ⫽ ⫺s1.. n.

(15)

Figure 3 (left panels) shows multiple trajectories from different initial conditions that were generated by Equations 14 and 15. As can be seen, the trajectories rapidly collapse onto the manifold after which they evolve according to the structured lowdimensional flow. The rapid collapse and subsequent stationary behavior is further testified to by the principal component analysis (PCA; see Figure 3). These examples visualize the main ideas behind SFM, that is, the collapse of a high-dimensional system onto a functionally relevant subspace (manifold) and the structured dynamics (flow) on that subspace. Moreover, comparison of Equations 14 and 15 demonstrates that the same flow, or order parameter equations, can emerge and determine a system’s collective dynamics even though they are constrained to completely different workspaces (they live on different manifolds). It should be noted that no established principled method for the reconstructing of SFM yet exists: Its identification from experimental data requires the insight of the researcher. Data reduction techniques (such as PCA or independent component analysis) provide an entry point to the reconstruction of an SFM’s manifold from behavioral data, as exemplified in the context of team dynamics in Dodel et al. (2011; or see Pillai, 2008). There the authors reconstructed low-dimensional manifolds (trajectories evolving on them) from three 16-dimensional team-dynamics spaces (pertaining to novices, intermediates, and experts’ performances) using PCA and quantified the reconstruction quality. Whenever the dimensionality and state variables of the low-dimensional functional dynamics (or order parameters) are known or can be safely assumed, however, the corresponding phase flows can be recovered via the Kramers– Moyal expansion (see below). This latter method may also be employed after the data have been projected onto the relevant low-dimensional subspace. Moreover, transition paradigms (Kelso, 1995; Kelso et al., 2013) and perturbations (Kay et al., 1991) may help the experimenter to identify the relevant variables underlying behavioral patterns. In that regard, because order parameter equations are universal, their identification is independent of the manifold (onto which they inscribe a flow).

A Well-Known Example: The HKB Model as an SFM We illustrate the SFM concept (see Figure 4) by applying it to the HKB model (Haken et al., 1985). As a first step, we examine the component dynamics (Equation 1) when uncoupled. Each osciullator can be represented in a phase plane. In conjunction, however, they span a four-dimensional space in which the whole system is represented by a single limit cycle. When plotting the trajectories of both limit cycles in a three-dimensional subspace, it is clear that all phase relations between both oscillators are possible— the flow on the manifold is uniform. Whichever relative phase between the oscillators will be established depends on the initial conditions only. Introducing the coupling between the oscillators structures the flow on the manifold so that only the in-phase (0°) and antiphase


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Figure 3. Structured flows on manifolds on a planar (A) and spherical (B) manifold. The left panels show multiple trajectories from different initial conditions that were generated by Equations 14 (A) and 15 (B). In both cases, a 16-dimensional system with limit cycle dynamics was simulated 40 times with different initial conditions (only some trajectories are shown). Regardless of the manifold’s geometry and initial condition, trajectories rapidly collapse onto the manifold and stay on it with little variation orthogonal to it only, which is due to the manifolds’ attractiveness. Once on the manifold, the trajectories evolve according to the flow. (The flow is not shown due to the difficulty of displaying the relevant information in vector fields of a dimension larger than 2.) The upper middle panel depicts the eigenvalues corresponding to the principal component analysis (PCA) across the 40 simulations as a function of time; the lower middle panel depicts the associated eigenvector corresponding to the first mode. For visualization purposes, only the initial time development is depicted so as to include the initial transient part as well as the following convergence toward a stationary state (evidenced via the oscillatory behavior in the eigenvalues and [first] eigenvector). Notice the effective dimensionality reduction associated with the fast collapse onto the manifold. The right panels show multiple reconstructed trajectories (via PCA; for details of the analysis and reconstruction, see Dodel et al. 2011); the structure of low-dimensional flows on the manifolds are well identified.

relationships (180°) remain stable. The flow now shows two regions, convergence zones, set apart by a separatrix. Within a zone, the flow converges onto a particular subspace corresponding to the in-phase or antiphase relation. The separatrix coincides with the (unstable) 90° relative phase. As the coupling parameters are increased beyond a critical point, the flow restructures; the separatrix disappears and only a single convergence zone remains (corresponding to the 0° pattern). Thus, rather than conceptualizing a coordination as an ensemble of its constituting components, as an SFM it is conceived of as a single dynamics that is characterized by its form and orientation in its “embedding” space. Above, we outlined the theory of SFM, dynamical objects that capture the (temporal) emergence of a structured dynamics in terms of phase flows on low-dynamical manifolds arising from a highdimensional system. We view the low-dimensional structured flow as the structure underlying functional behavior (and processes, more in general). Our framework can be seen as a further development of the dynamical approach to (sensorimotor) behavior: It extends that tradition through its explicit incorporation of the temporal collapse of a

high-dimensional system onto a low-dimensional subspace, the allowance of a functional dynamics of various dimensions (typically low) and forms, the explicit focus on phase flows and topologies (rather than attractor structures— even though these are mutually inclusive), and the classification of functional patterns according to the flow topology. Below, we will first briefly discuss methods that allow for the identification of phase flows and SFM, and point out recent studies explicitly directed at identifying (low-dimensional) phase flow topologies of various dimensions and as a function of different task constraints. These studies have shown the existence (in humans) of a few distinct dynamical classes. Next, we outline a theoretical framework for functional architectures characterized by multiple dynamics operating on various time scales (i.e., hierarchies) that allow for the generation of complex behaviors. Importantly, SFM is one of the key ingredients for functional architectures. As proof of concept, we will illustrate the power of this framework in the context of handwriting (via an engineered toy example). Finally, we will discuss the significance of our framework and findings for (phenomenological) theories of sensorimotor control.



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Figure 4. The Haken–Kelso–Bunz system represented as a three-dimensional system. Upper left two panels: Trajectories representing two individual oscillators. Upper right: The two oscillators are uncoupled. Consequently, all phase relationships are possible. Each plotted trajectory (blue, or dark gray) represents the two uncoupled oscillators at a particular phase relationship. Lower left: The two oscillators are coupled with a strength that allows for the in-phase (0°; blue, or dark gray, trajectory) as well as the antiphase (180°; red, or light gray, trajectory) relationship. The arrows indicate the flow (as usual) and show the 90° phase relation as a separatrix. Lower right: As in the middle panel but now for a coupling strength in which only the in-phase (0°; blue, or dark gray, trajectory) relationship is stable. The flow (arrows) converges onto a one-dimensional line, which represents the coupled oscillator system in the sole stable solution. (The cylinder serves as a visual aid only.) See the online article for the color version of this figure.

Phase Flow Topologies in Empirical Data Attempts to delineate movements into types, mainly in terms of “discrete,” “rhythmic,” or “continuous,” is traditionally attempted via kinematic properties (Buchanan, Park, Ryu, & Shea, 2003; Guiard, 1993; R. A. Schmidt & Lee, 2005; van Mourik & Beek, 2004). Given the diversity of kinematic profiles that can be generated by a given dynamic class, however, class identification via kinematic profiles is prone to failure (van Mourik, 2006). Regardless, movements have regularly been classified as more or less discrete, rhythmic, etc. Such a classification, which seems to make sense in terms of kinematic properties, has the backdrop that its classes are not unique and that any mapping onto generating mechanisms is ambiguous at best. A classification of dynamic phenomena motivated in phase flow topologies does not suffer this shortcoming. Only quite recently, however, was a technique developed that explicitly allows one to recover a system’s underlying phase flow from empirical data. The method is based on the fact that the future state of a stochastic process u(t ⫹ ⌬t) depends (is conditional) on its present state u(t), which can be expressed in terms of a conditional probability (or transition) matrix P(u,t ⫹ ⌬t | u,t). Conditional probability matrices can be computed from (behavioral) data, after which the Kramers–Moyal expansion allows for the extraction of the corresponding system’s deterministic (drift) and stochastic (diffusion) components (Friedrich & Peinke,

1997a, 1997b; for an overview, see Daffertshofer, 2011). Utilizing that technique, Huys, Studenka, Rheame, Zelaznik, and Jirsa (2008) recently examined the dynamical structure underlying repetitive movements. The participants executed flexion– extension finger movements at frequencies from 0.5 Hz to 3.5 Hz (step size 0.5 Hz). In different conditions, they were instructed to move as fast as possible or as smooth as possible, or had no specific instruction (the fast, smooth, and natural conditions, respectively). In addition, Huys, Studenka, et al. investigated the behavioral capacity of the excitator system (Equation 5) in the monostable and limit cycle regime under a wide range of parameters including movement frequency. This analysis revealed that at low frequencies the monostable (or discrete) system produced trajectories similar to those of the human participants under the natural and fast condition at low frequencies. At higher frequencies (⬃1.5 to 2 Hz), however, the “external pacing” (I) interfered with the system’s intrinsic dynamics (causing a period doubling) and the required frequency could no longer be achieved. That is, an externally driven system cannot produce movements at high frequencies while satisfying the required temporal constraints. In the limit cycle regime, however, the system was able to comply with all temporal demands. The vector fields reconstructed from the behavioral data identified a fixed point in the phase flows under the fast condition at low frequencies (and for half of the participants in


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the natural condition). At frequencies above ⬃1.5 to 2 Hz, however, no indications for the existence of fixed points were found regardless the instructions. The participants’ movements were thus governed by limit cycle dynamics (under the two-dimensional assumption; see above). These results (see also Figure 5) thus showed that humans are able to utilize (at least) two control mechanisms and naturally switch from a discrete mechanism to a rhythmic mechanism at a certain frequency. Although Huys, Studenka, et al. (2008) found clear indications of the implication of a fixed point under certain task constraints, they did not explicitly investigate the existence of a separatrix existing close to the fixed point. This issue was investigated by Fink et al. (2009) with a perturbation experiment. Recall, a separatrix can be thought of as a threshold that, in the absence of or before voluntary movement initiation, can be crossed only due to perturbations and/or noise. In the presence of noise, the crossing of the separatrix becomes stochastic. In the presence of noise and perturbations, it is more likely to bring the system across the separatrix (and execute a movement) if the perturbation is in the intended movement direction rather than in the opposite direction. Participants performed a reaction time task in which they had to flex their index finger upon an auditory stimulus. In 25% of the trials, a perturbation was applied in either the flexion or extension

direction (at variable intervals before the tone). In absence of perturbations, false starts were observed in 2% of the trials. In flexion and extension perturbations, false start occurrence was 34% and 9%, respectively. That is, the data were consistent with the prediction that follows from the presence of a separatrix. Of relevance, in this regard, is that the existence and utilization of a system involving a fixed point and separatrix implies that an external impact is required to initiate the movement (a “go” signal). Once issued, an initiated movement can no longer be stopped. This phenomenon has been referred to as the “point of no return” (Bartlett, as cited in R. A. Schmidt & Lee, 2005). Numerous studies indicate, for instance in type writing, that the point of no return is situated before movement onset (Logan, 1982, 1994; Verbruggen & Logan, 2008), which can be accounted for easily when considering neural conduction delays. To investigate the generality of the implication of the distinct classes reported by Huys, Studenka, et al. (2008), and rule out task idiosyncrasies, Huys and colleagues performed two further studies involving markedly different task constraints. One study examined a circle drawing task under the same movement frequency and instruction conditions as in Huys, Studenka, et al. (Huys, Studenka, Zelaznik, & Jirsa, 2010). This experimental paradigm differs from the one discussed above in that two physical dimen-

Figure 5. Movements from distinct classes. The left, middle, and right columns represent position (gray) and velocity (black) time series, phase flows (reconstructed using five trials) with a trajectory of a single trial overlain, and angle diagrams, respectively. The spaces corresponding to the phase flows and angle diagrams are spanned by movement position and velocity. The angle diagrams display the maximum angle between the vector of each state (position–velocity pair) and that at all its direct neighbors (see Huys, Studenka, et al., 2008, for details). The upper, middle, and lower rows represent finger movements at 0.5 Hz (upper and middle row) and 3.5 Hz (lower row), respectively, under instructions stressing smoothness (upper row) or staccato-like movements (middle row). In the upper row, the phase flow is not well structured, and the trajectory (middle panel) is wiggly. The absence of a clear structured flow suggests that the data are nonautonomous, which, alternatively (but not mutually exclusively), may be due to (physiological) tremor. In contrast, the flow in phase space is well structured in the middle and lower. In addition, in the middle row the vectors locally converge, indicative of a fixed point. The lack of local convergence in the well-structured flow in the lower row indicates the existence of a limit cycle. a.u. ⫽ arbitrary units.



sions are (roughly) equally brought into play in drawing a circle; the corresponding dynamics are therefore four-dimensional (i.e., position and velocity along two orthogonal axes). However, rather than analyze the four-dimensional phase flows, which would require a (too) large amount of data points, the analysis was performed in correspondence to a one-dimensional model. The model portrays the evolving angle between both orthogonal drawing components as a flow on a line and exhibits either fixed-point or oscillator dynamics (associated with distinct timing mechanisms). The basic finding replicated the previous one: Movement frequency (and to some extent, instruction) dictated the qualitative dynamics displayed. The transition between the dynamic mechanisms occurred via a so-called saddle-node-on-an-invariant-circle bifurcation. Interestingly, as predicted by the model (shown by simulations), the behavioral data revealed an increased variability of the movement (drawing) trajectories around the transition point—a key signature of phase transitions. This finding, in other words, places the occurrence of both dynamical mechanisms that are in close correspondence to the so-called event and emergent timing (Robertson et al., 1999; Spencer, Zelaznik, Diedrichsen, & Ivry, 2003; Zelaznik, Spencer, & Ivry, 2002; Zelaznik et al., 2005) in the realm of self-organized pattern formation. The other follow-up study examined the utilization of the distinct mechanisms as a function of movement accuracy in the context of Fitts’s task (Huys, Fernandez, Bootsma, & Jirsa, 2010).7 In contrast to the other two studies, in Fitts’s tasks spatial accuracy is systematically varied. Participants moved a stylus between two targets (whose width was varied across conditions under constant target distance) under instructions stressing speed as well as accuracy. As in the timing task (above), the predominant movement dynamics were confined to one physical direction, rendering their dynamics two-dimensional. Under high accuracy requirement the vector field reconstruction clearly indicated the existence of fixed points located at the targets, while a limit cycle dynamics was present when the accuracy constraints were smaller. The transition between control regimes occurred around an (effective) index of difficulty of ~5. Importantly, while the gradual variation of task difficulty evoked a continuous adjustment of the movement kinematics (Bootsma et al., 2004; Mottet & Bootsma, 1999, 2001), the underlying parameterization of the dynamics evoked an abrupt transition in dynamic mechanisms (see Huys, Fernandez, et al., 2010, data supplement). The possibility that this finding was due to movement frequency rather than task difficulty was ruled out in a second experiment where task difficulty was held constant (accuracy constraints were removed) while movement frequency was structurally varied. That is, next to movement frequency, task difficulty (accuracy) also affects the implementation of motor control mechanisms; (fast) precision movements require the utilization of a fixed dynamics control mechanism, maybe so because the precision with which limit cycle movements can be performed is bounded (at least under the combination of accuracy and speed constraints). In all three studies described above (especially so in the timing and Fitts’s task), evidence was found for trajectories that could not be associated either with a mono- or bistable fixed-point dynamics or with a limit cycle. These trajectories were observed only for slow tracking-like movements and were typified by a relative lack of structure of the flows (see Figure 5). Dynamically, such flows (and movements) can be accounted for via a (linear) fixed point

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that moves (continuously or in small steps) at the time scale of the movement (Huys, Jirsa, Studenka, Rheame, & Zelaznik, 2008; Perdikis, Huys, & Jirsa, 2011a). Accordingly, the dynamical organization constitutes a driven, non-autonomous system. Alternatively, however, the lack of structure in the corresponding flows could also stem from the superposition of physiological tremor onto cyclic movements (Feldman, 1980).

Functional Architectures The notion that basic elementary units serve as constituent building blocks (or primitives) for the composition of complex functional processes and behaviors is widely adhered to in the biological and life sciences. Such functional units preserve some of their properties among different utilizations (invariance), which identify them as units. As building blocks, they can be brought into meaningful relationships resulting in larger functional organizations in space (e.g., multisensory integration) and in time (e.g., via their concatenation). Consequently, the resulting processes exhibit a meaningful hierarchical structure spanning multiple spatial and time scales. The ensemble of dynamics and interactions among them defines a functional architecture. We previously proposed that complex behaviors could be explained via functional architectures constituted by functional modes and their interactions (Perdikis et al., 2011a; Perdikis, Huys, & Jirsa, 2011b). Essentially, we consider SFM as patterns in a multilayer hierarchy; in the first layer SFM emerges and in the second layer pattern formation and competition mechanisms are in place acting upon a repertoire of patterns. In contrast to traditional approaches, the SFM patterns are dynamical objects prescribing the evolution of processes on the behavioral level. The resulting dynamics is determined by the SFM emerging from a competition among the SFM in the dynamic repertoire. Perdikis et al. (2011a, 2011b), building on documented examples of distinct movement classes (see above), investigated whether systematic transitions from one elementary process to another can explain the emergence of non-stationary multitime scale behaviors. The functional architectures they proposed comprise functional modes and additional dynamics operating on distinct time scales (referred to as operational signals). Operational signals can be classified in terms of their characteristic time scale relative to that of the functional modes (Perdikis et al. 2011a). Briefly, they may act as “instantaneous” kicks (i.e., very fast) initiating a movement (or process) as required in functional modes that contain a separatrix (as, for instance, in monostable and bistable regimes). These input signals are task unspecific, though they need to be defined in terms of strength and time of occurrence. Alternatively, the operational signal may act on a time scale similar to that pertaining to the functional mode, which is required when the mode contains a (linear) fixed point. The operational signal then drives the fixed point through phase space (the “driven fixed point” mechanism; see above and below). Empirical evi7 Fitts’s task examines the relation between task difficulty and movement time. Task difficulty knows several formalizations; in the one generally adopted, task difficulty is a function of target width W and target distance D, referred to as the index of difficulty ID ⫽ log2(2D/W). The so-called Fitts’s law states that movement time MT is a linear function of ID, MT ⫽ a ⫹ b ID.


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dence in favor of these scenarios was summarized above. The operational signal may also be constant, that is, effectively absent, in which case the functional mode is autonomous. Lastly, and of most relevance for our present purposes, the operational signal may act on a time scale slower than that pertaining to the functional modes. In this case, it temporarily “selects” functional modes and binds them together into a functionally meaningful sequence. These slow dynamics can in principle be of different type (see below).


Mathematical Implementation Functional modes modeled as SFM constitute the basic components of functional architectures. As stated above, we consider an SFM as the macroscopic functional dynamics that emerges from interactions in a high-dimensional system (an agent) under environmental constraints and perturbations. After adiabatically eliminating the fast variables sj of Equation 13 (i.e., by solving sj ⫽ h({ui},{sj}) for sj), the dynamics of a functional mode is described as ␶u˙ i ⫽ ⫺g({ui}, {s j({ui})})ui ⫹ ␮f({ui}, {s j({ui})}) ⫽⫺g({ui})ui ⫹ ␮f({ui}) ⫽F({ui})

(16)

u 僆 RM, M ⬍⬍ N. That is, we consider a functional mode F({ui}) as a transiently emerging M-dimensional dynamics in {ui} space originating from the higher N-dimensional ({ui},{sj}) space. To reiterate, these modes constitute functional units that incorporate desired properties such as low-dimensionality, class-defining invariance combined with within-class variation, executive stability (i.e., performance maintenance in the presence of perturbations), functionally meaningful dynamical structure (referred to as dynamical constituency by Petitot, 1995), and compositionality (i.e., they can be embedded into a larger functional organization). Most generally, we describe a functional architecture through its flow F(.) in phase space potentially subjected to additional operations (see Perdikis et al. 2011a, for details): ␶u˙ i ⫽ F共{ui}, ␴(t)兲 ,

(17)

where {ui} are the system’s state variables and ␴(t) is the operational signal.8 F({ui}) is identified as a specific functional mode in the absence of ␴(t). The dynamical repertoire is the set of modes available to an agent; it represents the ensemble of elementary functions that can be used in a relatively invariant manner across different instances of the agent’s behavior. The key feature of functional architectures for sequential behavior is the presence of a slow process ␰(t) (slow relative to the time scale pertaining to the functional modes) that temporarily determines (or selects) which mode within the repertoire is functionally expressed (in the spirit of physics of pattern formation; cf. Haken, 1983, 2004). More specifically, at each moment in time, the expressed phase flow is given as a linear combination of all functional modes available in an agent’s dynamical repertoire, ␶u˙ i ⫽ F({ui}, t) ⫽

兺j ⱍ␰j(t)ⱍFj({ui}, t) ⫹ ␦i(t),

(18)

where {ui} are the state variables, Fj(.) is the jth mode, and ␰j acts

as a weighting coefficient for the jth mode (constrained to be positive). While the ␰s operates on a (relatively) slow time scale, the transitions between modes involving contraction onto the respective manifold are typically fast (see Figure 6). A particular mode Fj is expressed as its corresponding competition parameter ␰j goes to 1 and all others ({␰k} for k ⫽ j) vanish. The exact moment of pattern switching, however, is of course further affected by stochastic impacts on the competition parameters. The operational signals {␦i} are optionally involved, leaving the flow Fj(.) unaffected because they act instantaneously, just like a functionally meaningful perturbation (e.g., by moving the system beyond a separatrix and thus initiating a significant change in the trajectory’s evolution, as mentioned above). The functional architecture defined across a dynamic repertoire {F(.)} and competition dynamics {␰j} is illustrated in Figure 7. The interactions among functional modes can take various forms, such as the WTA competition (Grossberg, 1980; Haken, 2004; Smale, 1976), the winnerless competition (Afraimovich, Young, Muezzinoglu, & Rabinovich, 2011; Rabinovich et al., 2008; Rabinovich, Muezzinoglu, Strigo, & Bystritsky, 2010; for an application in the present context, see Woodman et al., 2011), or dynamic fields (Sandamirskaya & Schöner, 2010). Here we illustrate the interactions according to the WTA competition in view of its long history and consistency with many observations in the life sciences. Accordingly, the modes’ activations do not overlap in time. The {␰j} dynamics is given by

共

␶c␰˙ j ⫽ L j ⫺ C j

兺k ␰2k兲␰j

L j ⱖ 0, C j ⬎ 0,

(19)

where ␶c is a time constant ensuring that the competition evolves fast. The competition evolves among modes with Lj ⬎ 0, and its outcome is determined by parameters {Cj} and {Lj}: the “winning” ␰j is the one with 共L⁄C兲 j ⫽ max兵共L⁄C兲k其. k

Equation 19 is a particular representation of WTA dynamics; others exist, such as systems of the Lotka–Volterra type (Lotka, 1920; Murray, 1993). Effectively the details of the WTA representation do not play any role for our case here. We briefly illustrate the interaction between SFM and competition in an example that can be interpreted as dynamic pattern recognition (in the spirit of Haken’s synergetic computer). We suppose a functional architecture “containing” a set of functional modes Fj with (two-dimensional in this case) state variables {u1,2} (i.e., a dynamic repertoire), and the existence of a pattern matching mechanism that becomes active whenever the architecture is subjected to a pattern F. The pattern matching (a projection operator between the vector field reconstructed from the input time series and the vector fields of each of the repertoire’s patterns) determines the competition by modifying the Cj parameters (all Lj ⫽ 1 in this example, but the Ls can optionally be modified as well), thus simultaneously driving the pattern formation process formalized in Equation 18. This continuous process is illustrated in the video in the supplemental materials. There, it can be seen that soon 8 Under the assumption that the control demands associated with using functional modes, once established in one’s dynamical repertoire, are negligible relative to the task-varying utilization of operational signals, it can be shown that different degrees of control are required for distinct functional architectures (Perdikis et al., 2011a).



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Figure 6. Multiple time scale dynamics. The slow operation signal ␰j comes to temporarily dominate the ␰ dynamics through a fast transient to the jth node (A). Simultaneously, a cylindrical manifold emerges onto which the functional dynamics {ui} collapse and prescribe a spiral flow (B). The jth node’s stability is sustained for the duration of the process after which it destabilizes. Correspondingly, the manifold vanishes and the trajectory parts through a (again) fast transient. The data points’ density is inversely related to the speed of the corresponding processes. See the online article for the color version of this figure. From “Time Scale Hierarchies in the Functional Organization of Complex Behaviors,” by D. Perdikis, R. Huys, and V. K. Jirsa, 2011, PLoS Computational Biology, 7(9), Figure 1. Copyright 2011 by the Authors.

after the architecture is subjected to the to-be-recognized pattern, most of the available patterns (in the dynamic repertoire) start “losing” the competition. The remaining two patterns continue to compete until one patterns wins the competition and a pattern is formed in the output of the architecture, as an interpretation or recognition of the input pattern.

Proof of Concept: Handwriting We have developed a handwriting architecture as proof-ofconcept (Perdikis et al., 2011b), which we outline below. In

order to model serial order, we designed a suitable dynamics guaranteeing the sequential activation of the appropriate functional modes with the correct timing. The chosen implementation was inspired by emergent competitive queuing models of serial behavior mentioned above: At each stage of the sequence, the functional modes compete (via the {␰j}), one of them wins, dominates the output dynamics for the duration of its activation, and is subsequently inhibited, after which the competition continues among the remaining available modes. Equation 19 implements the competition among modes by means of mutual

Figure 7. Functional architecture’s competition dynamics and phase flow transition. (Left) Schematic representation of the functional architecture as a two-level network of interacting processes at two distinct time scales: the top level with the slow winner-take-all competition dynamics ({␰j}, reddish, or light gray) and the bottom level with the fast functional mode dynamics ({ui}, greenish, or dark gray). Note that there are interactions both within and between layers. (Right) Temporal evolution of a phase flow transition of the functional architecture’s emerging dynamics: on top, the time series of a transition from a ␰j process (i.e., ␰j is activated) to a ␰j ⫹ 1 process (␰j ⫹ 1 is activated), and at the bottom, five snapshots of the gradual phase flow change during this (fast) transition, from a limit cycle phase flow to a bistable one. See the online article for the color version of this figure.


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inhibition. The order of activation depends on the {Cj} parameters (the primacy gradient); Lj ⬎ 0 is the condition for mode j to take part at a specific competition round. Consequently, parallel representations of the sequence (encoded in the arrays of {Cj} parameters) are combined with the serial WTA competition. The timely inhibition of an active mode is achieved through a “bottom-up” coupling (feedback) from the output {u } to the slow operational signal: ␰˙ ⫽ f j 共L 共兵u 其兲,C 共 j


i

␰

j

i

j

兵ui其兲兲 ⫽ f ␰j 共兵ui其兲. Thus, the transitions of {Lj} and {Cj}, due to feedback from the output, affect the {␰j} competition, which in turn selects a different functional mode to dominate the architecture’s dynamics at each stage of the sequence. In fact, the {␰j} competition is effectively influenced by all the architecture’s relevant variables and their interactions, and the sequential dynamics becomes an emergent property of the architecture. Moreover, through the various feedback loops, the architecture is inherently informational, even though the nature of the feedback is functionally defined rather than in terms of specific informational variables (optics, haptics, etc.). The dynamics across the three time scales ␶␦ ⬍⬍␶f ⬍⬍ ␶␰ (i.e., those pertaining to the instantaneous ␦ kicks, the functional modes, and the ␰ competition) is coupled, rendering the entire architecture dynamically autonomous. ␦˙ ⫽ f␦ (␦, {ui}) feedback for the appropriate timing of the fast ␦ kicks was also included. It should be noted that the transitions of {Lj} and {Cj}, and thus the transitions in the ␰ competition, are fast. The transition times notwithstanding, during the execution of a particular functional mode the corresponding variables and parameters remain relatively constant. Indeed, the resulting trajectory of the {␰j} dynamics, which passes sequentially through the neighborhood of each mode of the sequence (where it dwells during the mode’s activation; see Figure 6), exhibits a time structure that is defined across the entire sequence. As such, the effective time scale that determines the serial dynamics (referred to as ␶␰ above) is slow (␶␰ ⬎⬎ ␶f). The architecture’s behavior is illustrated in Figure 8, which presents a simulation of the word flow generated repeatedly thrice. The distinct time scales of the interacting processes can be easily observed: The main time scale of the output dynamics pertains to a movement cycle even though a longer (slower) time structure (at the word scale) is present also. This slow time scale dominates the {␰j} dynamics. In contrast, ␦y,z is much faster than the output. Additional simulations were performed whose analysis provided two important insights. First, the functional architecture’s output appeared easily scalable (words could be written smaller or larger) without affecting its structure. Second, identifying transitions between the functional modes that were implicated in generating the writing is possible only via phase flow related measures; the more traditional measures (based on output variability) appeared “contaminated” by the instantaneous ␦ kicks (for details, see Perdikis et al., 2011b).

Discussion Functional Modes: Elementary Units of Behavior We proposed that sequential behavior is constructed by appropriately binding functional modes one after the other, which rests

on the assumption that behavioral sequences can be broken up into smaller units that the action system has at its disposal. This assumption has been popular in the biological and life sciences for a long time, as it is associated with a “division of labor.” The findings reported above (and others) strongly point toward the existence of behavioral classes that are defined in unambiguous terms (i.e., via the notion of topological equivalence; see also Saltzman & Kelso, 1987). Accordingly, each functional mode corresponds to an invariant structure (the phase space topology), whose details (the quantitative aspects of the flow) can be adjusted to the task constraints at hand. For instance, modifications of precision constraints (as in a Fitts’s task) evoke quantitative adjustments to the phase flows (within a given dynamical regime). A further partitioning in subclasses may well be possible in most, if not all, cases. For instance, as indicated above, the detailed study of rhythmic movement has led to the identification of (sub)classes of limit cycles (Beek & Beek, 1988; Beek et al., 1996; Kay, 1988; Kay et al., 1987, 1991; Kelso, 1995; Mottet & Bootsma, 1999). Their specific properties render each subclass (i.e., limit cycle types) appropriate for some task constraints, but not others. For instance, nonlinear Duffing (Beek & Beek, 1988; Beek et al., 1995; Mottet & Bootsma, 1999) and so-called ␲-mix stiffness (Beek & Beek, 1988; Beek et al., 1995) terms are particularly suited to deal with precision constraints. Particular oscillator ingredients rarely, if ever, show up in isolation, however. That is, while identifying how given oscillator properties deal with particular movement constraints is informative relative to the organization of the corresponding movement, in practice so-defined subclasses will lack uniqueness. While pertinent to the present approach, and that of the GMP, the notion of behavioral classes (coordinative structures or synergies) is to our best knowledge absent (at least not explicitly defined) in both the computational (including Bayesian) approaches and EP models. If referred to, authors from those domains typically refer to the motor primitive concept in terms of neural circuits in the spinal cord that are organized in terms of distinct modules (Bizzi, Cheung, D’Avella, Saltiel, & Tresch, 2008; Bizzi & Mussa-Ivaldi, 1998; Mussa-Ivaldi, Giszter, & Bizzi, 1994). This notion is based on observations in spinalized frogs, which showed that stimulation of a particular spinal cord circuit evokes reproducible contractions in muscle groups that induce module-specific force fields (motor primitives). Simultaneous activation of multiple modules leads to the vectorial superposition of the corresponding force fields and may so generate a large variety of motor behaviors. Motor primitives are thus viewed as hardwired entities laid down in the neural (spinal) circuitry. The importance of these circuits notwithstanding, it is debatable whether they can account for motor control: First, there exists considerable doubt about force-based motor control (see above); second, the modules arguably function as “executors” rather than as “controllers.” By analogy, the corresponding circuits define a “spinal keyboard” serving but not controlling typing. The notion of behavioral units in more abstract terms is at the heart of the GMP concept, with which our scheme shares an architectural resemblance, large ontological differences notwithstanding, as well as the synergy (as dissipative structure) concept. GMPs, however, are devoid of form. In contrast, SFM are mathematical objects of a particular morphology whose nature defines class. This notion was forecasted by Bernstein (1967), who



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Figure 8. Functional architecture “writing” the word “flow.” Word formation and the operational signals and parameters involved. “Flow” is generated thrice after a short transient (black solid line). Four functional modes were used, one for each character (associated with solid blue, green, magenta, and cyan lines, respectively), plus two auxiliary ones at the sequence’s beginning and end (dotted dark and light brown lines, respectively). From top to bottom: three repetitions of the word formation in the handwriting workspace (the plane x-y); the output trajectory in the three-dimensional functional phase space (x-y-z space); x, y, and z time series; the time series of the slow winner-take-all (WTA) competition coefficients {|␰j|} where {␰j} of the nonparticipating modes always have a value close to zero (red line); the instantaneous ␦y,z “kicks” (light and dark red, respectively); and the WTA competition parameters Cj and Lj, which vary on the slow time scale of the whole word, even if they also contain fast, abrupt transitions (for details, see Perdikis et al., 2011). Adapted from “Time Scale Hierarchies in the Functional Organization of Complex Behaviors,” by D. Perdikis, R. Huys, and V. K. Jirsa, 2011, PLoS Computational Biology, 7(9), Figure 4. Copyright 2011 by the Authors.

thought of movements as morphological objects (see also Higgins, 1985). Using theorems and tools from dynamical system theory, we identified distinct flow topologies as a function of task constraints. In this set of experiments, only the end-effector dynamics of relatively simple movement (in terms of number of joints involved) were studied, as they promised to be illustrative for identifying distinct classes. Regardless, the same principles apply to, say, whole-body movements. Both walking and running, for instance, are whole-body actions and contain four (principal) dynamics comprising two frequencies. The locomotions are distinct in terms of the relative phasing between the dynamical compo-

nents (Lamoth et al., 2009); that is, their four-dimensional shapes are different (but topology of the dynamics is not). Indeed, through our definition of what establishes a functional mode through phase-flow invariance during an action, it accordingly makes little sense to break down these units in terms of, say, swing and stand phase (in the case of walking). This is not to negate that such smaller elements have particular defining characteristics, but rather, they are integral parts of a functional mode that is controlled as a whole. An obvious, but far from trivial, question concerns the number and origin of the functional modes. As for the latter, one can only


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speculate, and we deem it likely that their origin reflects the nervous system’s intrinsic dynamics (Deco & Corbetta, 2011; Deco, Jirsa, & McIntosh, 2011; see also below) substantively shaped through development and learning. As for their number, we reported evidence in support of the existence of three classes (limit cycles, fixed point with separatrix, and the driven fixed-point system), and suggestions for a fourth (two fixed points separated by a separatrix) exist in the literature (Jirsa & Kelso, 2005) and agree with point-to-point movements (Schöner, 1990). Other (potential) examples may be found in research on the “one-target advantage” (OTA), that is, the phenomenon that the movement to a target is faster when presented in isolation than when it is immediately followed by a reversal movement to a second target (if the second movement does not return to the starting position; cf. Adam et al., 2000; Lavrysen, Helsen, Elliott, & Adam, 2002). The OTA appears when both targets were presented simultaneously but not when the second target is presented at the first movement’s onset or peak velocity. Also, if the second target was present at trial presentation but disappeared at movement onset or peak velocity, the second movement could no longer be prevented (Khan, Tremblay, Cheng, Luis, & Mourton, 2008). These features suggest that the entire two-movements structure was encoded as a single unit (functional mode), a phenomenon traditionally interpreted in terms of “chunking” (Adam et al., 1995). The OTA vanishes for small targets (Adam, van der Bruggen, & Bekkering, 1993), which, with an eye on the utilization of distinct motor classes as a function of precision constraints (Huys, Fernandez, et al., 2010; see also above), may be due to the fact that movements to small targets require the implication of a fixed point. Further example of the (presumable) use of more complex movement patterns controlled as an entity can be found in (skilled) typing (Logan, 1982) and music performance (Godøy, Jensenius, & Nymoen, 2010).

Informational Sculpting of SFM We conceptualized trajectory formation in terms of structured phase flows, and demarcated systems in terms of invariant flow on the time scale of the movement versus flow changes on that time scale. In the latter case the system contains a moving fixed point. The driving, in that regard, could be informational, as for instance in the visual tracking of an object. (Inclusion of the informational drive into the dynamics renders the latter autonomous, that is, provides for a closed description.) Invariant flows, however, do not rule out informational coupling. Information could enter unexpectedly and require “on-line” behavioral modifications. In such cases adjustments are typically made as early as ⬃150 ms following change in target position (Brenner & Smeets, 1997, 2003; Prablanc & Martin, 1992; Saunders & Knill, 2003). Moreover, the longstanding notion that the ballistic phase of movement is immune to sudden changes in the task environment has been invalidated in various recent experiments (Grierson & Elliott, 2009; Grierson, Lyons, & Elliott, 2011). Similar adjustments are present in the case of rhythmic behavior. Under the coexistence of distance and accuracy constraints, adaptations to sudden modifications in task constraints depend on the nature of the informational change. Alleviation of task difficulty in a (repetitive) Fitts’s task via sudden target distance decrease or target width increase are adjusted to relatively slowly (from the second cycle following the

perturbation onward) via adjustments of the entire kinematic pattern. Conversely, reducing task difficulty (via target width or distance manipulation) evokes an immediate lengthening of the deceleration phase followed by slower adjustments similar to those observed when increasing task difficulty. These findings were interpreted in terms of two concurrent processes at different time scales (Fernandez, Warren, & Bootsma, 2006; for similarities in a discrete Fitts’s task, see Cheng, De Grosbois, Smirl, Heath, & Binsted, 2011; Heath, Hodges, Chua, & Elliott, 1998). The slow adaptation can be understood in terms of a parameterization of the functional dynamics (cf. Saltzman & Munhall, 1992), whereas the fast adaptation can be understood in terms of either a switch to the driven fixed-point (i.e., time-varying flow) mechanism or an additional time-dependent (i.e., nonautonomous) impact to the (existing) dynamics. When the task-relevant information is invariant during task performance, the observable (movement) dynamics are molded by the information, as is the case in Fitts’s task. In this case, the information constraining the task affects the flow gradually (Mottet & Bootsma, 1999) even though it may evoke a bifurcation (Huys, Fernandez, et al., 2010). The coupling-induced deformation of the phase flow is often present at particular regions of the phase space (generally around the position extrema), which is referred to as anchoring (Beek, 1989; Byblow, Carson, & Goodman, 1994; Jirsa, Fink, Foo, & Kelso, 2000; Roerdink, Ophoff, Peper, & Beek, 2008; Roerdink, Peper, & Beek, 2005). The so-called anchor points allegedly contain important task-specific information for the organization of (rhythmic) behavior (Beek, 1989) and are characterized by a reduced (spatial and/or temporal) variability.9 Anchor points may be imposed through the presentation of information (Byblow et al., 1994, 1995; Fink, Jirsa, Foo, & Kelso, 2000) or by constraining gaze to particular regions (Roerdink et al., 2008). Alternatively, these points may be actively created through, for instance, specific gaze patterns as in rhythmic visuomotor coordination (Roerdink et al., 2005). In that regard, in real-world tasks like juggling, the hands’ oscillator dynamics is coupled to the balls’ trajectories (Beek, 1989; van Santvoord & Beek, 1994), and gaze is directed toward the balls in a experience-dependent fashion (Huys & Beek, 2002; Huys, Daffertshofer, & Beek, 2004). That is, the search for information is directed at specific locations and is modified by experience. A similar coupling underlies one-handed catching (Amazeen, Amazeen, & Beek, 2001; Amazeen, Amazeen, Post, & Beek, 1999). The phenomenon of anchoring, and the increased stabilization of coordination patterns in the presence of (external) information, more generally, was modeled by Jirsa et al. (2000). Their theoretical work indicated that the effect of external driving is better accounted for by multiplicative coupling (i.e., the driving directly affects the system’s state variables) than by additive coupling (see also Kudo, Park, Kay, & Turvey, 2006). In the anchoring examples, information quantitatively changed the flow. Information may also drive the system through a bifurcation involving the switch from rhythmic to discrete movements as in Fitts’s task (Huys, Studenka, et al., 2010) and in ball bouncing (Ronsse, Wei, & Sternad, 2010). In the latter study, partici9 Anchoring can also be of a musculoskeletal origin independent of informational impacts (Roerdink, Ophoff, et al., 2008).



pants bounced a ball on a racket (to a constant height) in a virtual setup. The ball flights were manipulated by reducing gravity (up to about a factor 10) so that the required period of hitting the ball was greatly increased. Under normal and moderately reduced gravity, the racket movement was clearly rhythmic. As gravity was further reduced, it became discrete. In other words, the features of the ball flight defined which motor class was used to comply with the task demands. To conclude, information modifies the dynamics underlying motor behavior. Indeed, it is a basic tenet of coordination dynamics that information acts in the same space in which the functional modes are defined (Jirsa & Kelso, 2004; Kelso, 1995). In the case of rhythmic behavior, the impact of information has been explicitly studied and modeled in coordination dynamics. For the nonrhythmic cases, this has been far less (if at all) the case. We contend, however, that the principle of operation is invariant for motor class: Information—whether changing on faster or slower time scales than that of the behavior— deforms the flow (either by coupling information to the system’s state variables or by adjusting its parameters). The flow topology, in turn, may be conserved (indicative of weak coupling) or not.

Trajectory Formation and Control We reviewed experimental findings that are consistent with the idea that structured flows underlie trajectory information. Across studies, we found evidence consistent with distinct trajectory generating mechanisms, involving (a) limit cycle dynamics (fully autonomous), (b) fixed points and separatrices (nonautonomous but requiring quasi-instantaneous impact only), and (c) a driven fixed point (nonautonomous). Trajectory formation corresponding to some of these classes, in particular, limit cycles, is well established (see above). Also, movement generation involving a moving equilibrium (or fixed) point is at the heart of EP theory. EP shifts are assumed to occur gradually rather than in a stepwise fashion (Bizzi, Accornero, Chapple, & Hogan, 1984; Feldman, Adamovich, Ostry, & Flanagan, 1990). Some controversy exists, however, in whether the shifts end at movement onset or around peak velocity (see Ghafouri & Feldman, 2001, and references therein). For ballistic movements, intermittent EP control results in fastest movement (Kistemaker et al., 2006), although fast movements (with overshoot) can also be obtained by making the damping (in a linear mass-spring model) relative to the velocity of the EP (de Lussanet, Smeets, & Brenner, 2002). From our present perspective, the precise timing and duration of the EP shift is of secondary importance only, the fact that the EP shift entails a switch of EP location is pertinent—it establishes the EP mechanism as one class (corresponding to the driven fixed point; cf. Won & Hogan, 1995) among other classes. Next to these two well-documented classes for trajectory formation, the monostable class (i.e., including a stable fixed point and separatrix) is novel—the proposition that discrete movements may belong to the class of excitable systems is quite recent (Jirsa & Kelso, 2005). Another potential class of excitable systems is constituted by a phase flow involving two (or more) stable fixed points separated by a separatrix (the bistable regime, used in the handwriting architecture). The monostable and bistable dynamics form distinct classes—they are not topologically equivalent— but share a dependency for trajectory initiation on external impact.

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Their existence predicts the implication of a movement timing and initiation mechanism, linking it to the notion of “central timers” or “clocks” (Wing & Kristofferson, 1973a, 1973b). The implication of such timers in some but not all movements resonates with the notion of emergent timing and event timing as developed by Zelaznik and colleagues (Robertson et al., 1999; Zelaznik, Spencer, & Doffin, 2000; Zelaznik et al., 2002, 2005; see also Delignières et al., 2004, 2008). This distinction was initially developed based on the presence (or not) of correlations across tapping and circle drawing performances at various frequencies. In subsequent brain imaging (Spencer, Verstynen, Brett, & Ivry, 2007) and patient (Spencer et al., 2003) studies evidence was found for the implication of the cerebellum in event timing tasks but not in emergent timing tasks. Moreover, the (negative) correlation between subsequent taps (an event timing task) as found and predicted by the Wing–Kristofferson model (Wing & Kristofferson, 1973a, 1973b) is absent in emergent timing finger oscillations (Torre & Delignières, 2008) as predicted by theoretical analysis (Daffertshofer, 1998). Note, however, that whereas the implication of timers at first blush renders trajectory formation nonautonomous, an autonomous description may be achieved if the timers can be incorporated in the phase flows (as in anchoring; see above) or if it can be embedded into a larger functional organization (as in the handwriting architecture). Central to our framework is the focus on phase flow structure, that is, the form of dynamical objects associated with distinct behaviors. As exemplified in the handwriting architecture, the behavioral output is scalable, while its spatiotemporal structure remains invariant—structure and metrics were controlled separately. The notion of structural invariance in the face of metric changes has been discussed in the literature along various lines. For instance, Bernstein (1967, pp. 42–50), in discussing the spatial aspects of movements, dissociated a goalrelated (or upper) level of control that is in charge of the ensemble of qualitative space configurations and of the form of movements from the metric (or lower) level associated with quantitative variations therein. Accordingly, the former is indifferent to its muscular realization. The GMP concept dissociates invariant, structural features, such as the relative timing between muscles, from metric ones (cf. R. A. Schmidt, 1975; R. A. Schmidt & Lee, 2005). Coordination dynamics also stresses relational invariance (in, for example, the coordination among effectors, effectors and environment, or individuals), primarily focusing on the temporal organization of actions, but rather than viewing it as imposed onto movements in an a priori fashion, holds that it results from the perceptual–motor system’s functional organization (cf. Kelso, 1981, 1997; see also Beek, 1992). The “invariance under scaling” under Bernstein’s view and that of proponents of coordination dynamics are not equivalent; Bernstein’s discussion deals with the spatial organization of movement, while the notion of relational invariance in coordination dynamics concerns the (spatio)temporal organization of movement. However, we believe that both would ascribe to the idea that task goals (spatiotemporal structure; for instance, writing an e) are somehow encoded in a musculatureindependent abstract fashion, and that the metrical scaling of the realization with a given effector system does not affect the relative timing among the contributing muscles. Anyway, the (control) dissociation between structure and metrics is consis-


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tent with an experience we all have, namely, that it is much easier to “learn” to perform a given movement pattern with a different amplitude than to learn a novel pattern. In the laboratory, evidence supporting this dissociation has been found in terms of differential reaction times to action choices among metric variations and movement pattern variations (the first ones being faster than the second ones; Quinn & Sherwood, 1983). Also, reducing feedback frequency during practice facilitates GMP accuracy but deteriorates parameterization (Wulf & Schmidt, 1996; Wulf, Schmidt, & Deubel, 1993). Regardless, the summarized research has provided strong evidence that the respective movements can be conceived in terms of their corresponding dynamical structures and phase flows. Following this evidence, we hypothesize that the organization of the motor system (embedded in a natural environment) favors control (or “codes for”) in terms of structured phase flows rather than individual trajectories (for autonomous movements), which we view as singular instantiations of the underlying control structure (see also Saltzman & Kelso, 1987). In other words, control is not geared toward the execution of a specific trajectory but rather to instantiate a dynamic that favors goal achievement including trajectory formation, stability and robustness of trajectory formation, as well as redundancy and compensation. An interesting and unexpected observation that is consistent with this hypothesis comes from the work of Graziano and colleagues (Graziano, Taylor, & Moore, 2002; see also Aflalo & Graziano, 2006; Graziano & Aflalo, 2007). In that regard, while the primate motor cortices are held to contain a “body map” that is used for movement control, many questions about the somatotopic organization remain unanswered. In order to investigate this map, researchers usually use brief (⬍50 ms) electrical stimulation evoking muscle twitch. Graziano and colleagues, however, administered long (500 ms) stimulation trains to monkey motor cortices, effectively stimulating relatively large networks. These stimulations evoked a repertoire of movements including hand-to-mouth (feeding) movements and protective movements. Surprisingly, for a given stimulation location, the same final position was reached regardless of the initial hand position. This finding is not easily reconciled with trajectory control but fits well with SFM control. SFM-based control not only guarantees robustness in the face of perturbations but also reduces the computational burden because the (exact) initial state hardly matters.10 A resulting trajectory, then, is (merely) a singular instantiation (or expression) of the underlying control structure.

Phase Flows, Invariance, and the Dynamical Repertoire Throughout the article, we reiterated that phase flows are invariant on the time scale of the movement. Apart from their mathematical definition, how does one conceive of phase flows, the abstract structures that remain invariant during an event that by definition signifies change? Importantly, the invariance is defined relative to a particular (and limited) duration, namely that of a functional act. Changes on slower time scales such as development, aging, learning, adaptation, etc., can be considered constant on the time scale of action. Such changes can be cast in terms of parameter dynamics and graph dynamics, with

the latter on typically being slower than the former (Saltzman & Munhall, 1992). Parameter dynamics typically refers to changes in the dynamics pertaining to the functional modes due to learning and development (next to task-induced reparameterization, as discussed above). Graph dynamics pertains to changes in the system’s composition and connectivity. These dynamics, too, are involved in learning and development but on time scales slower than those corresponding to the parameter dynamics. Both types may drive an organism through bifurcations, and thus modify the set of behaviors it is capable of performing or how it achieves a given function. Regardless, on the time scale of performance the corresponding constraints act as boundary conditions within which action unfolds. Similarly, changes occurring at faster time scales, such as enzyme turnover in reaction processes (associated time constants of ⬃10⫺6 to 10⫺3 s; Tsong, 1990; Tsong & Astumian, 1988), and other (fast) cell metabolism processes do not affect the phase flow (at least, in isolation). Theoretical work shows that population dynamics of neurons may give rise to SFM through systematic symmetry breaking of the network connectivity (Pillai, 2008; Pillai & Jirsa 2007). The same dependency for the emergence of structured flows on network asymmetry holds for networks of excitatory and inhibitory neurons (Jirsa et al., 2012; Woodman & Jirsa, 2013). Functional modes arise through the network interactions as correlated neural activity, which is invariant during a functional event. As an end-effector dynamics evolves in time tracing out a trajectory along the SFM, the neuronal population activity evolves systematically, but each neuron not necessarily following the same dynamics as prescribed by the SFM. Consistent with these results, in real biological systems the connection strength among neurons strongly influences the motor output pattern (at least in lobster stomatogastric ganglion; Selverston, Elson, Rabinovich, Huerta, & Abarbanel, 1998). As an ensemble, the set of all SFM can be conceived of as the range of fundamental behaviors, out of which more complex behaviors can be composed. In other words, it defines the dynamic repertoire. In that regard, a different line of research has developed a related comprehension of the notion of dynamic repertoire of human function. A large body of experimental work (see Raichle 2011, for review) has demonstrated that in the absence of any task (i.e., the so-called resting state), the brain’s resting state dynamics exhibits spontaneous coherent intermittent network activations, organized into various robust resting state networks (RSNs). These RSNs show a strong overlap with the activation of network patterns known from task conditions and can be identified in functional terms including visual, auditory, sensorimotor, saliency, attentional, and default mode network. The latter is a well-known network that comprises areas that get activated in absence of any task and deactivated during task conditions. Presently existing empirical estimates suggest the existence of about six to 10 resting state networks. Theoretical research hypothesized that the brain’s resting state dynamics operates close to criticality (see Deco et al., 2011, for a review), that is, within proximity of a bifurcation 10 A degree of independence of initial conditions is also present in the EP approach (cf. Feldman & Levin, 1995).



point.11 This hypothesis has received more and more support since its first articulation by Ghosh, Rho, McIntosh, Kötter, and Jirsa (2008). For instance, Deco and Jirsa (2012) demonstrated quantitatively that the best correspondence between experimental and brain network modeled data is obtained when the network is subcritical (just below the instability threshold). To understand the purpose of this criticality, Ghosh et al. (2008, abstract) suggested, “The combination of anatomical structure and time delays creates a space–time structure, in which the neural noise enables the brain to explore various functional configurations representing its dynamic repertoire.” In order to enable this exploration, criticality is essential, since otherwise the brain remains stuck in one of its attractor RSN. In computational studies, the resting state networks have been conceptualized as attractors defined in terms of networks of local oscillatory activity (Deco, Jirsa, McIntosh, Sporns, & Kötter, 2009; Ghosh et al., 2008; Honey, Kötter, Breakspear, & Sporns, 2007) or, more recently, of the firing rates of clusters of neurons (Deco & Jirsa, 2012). The bifurcation parameter is given by the brain’s global connectivity structure (i.e., the coupling strength among clusters). Below the bifurcation point, that is, when the system is in its ground state, the dynamical repertoire’s attractors are latently present (so-called ghost attractors). Driven by noise, the resting state brain explores its dynamical repertoire in a manner that is sculpted by the influence of the ghost attractors (Ghosh et al., 2008). This exploration has also been portrayed in terms of competition between resting state networks (Deco & Corbetta, 2011). By hypothesis, little task-relevant stimulation is required only when the system is in its ground state to settle into a task-relevant attractor through (minor) changes in the brain’s connectivity structure. The picture that emerges, then, is that functional modes, defined in terms of structured phase flows at the behavioral level of observation, reflect the functional lowdimensional expressions of activity in specific, temporarily invariant brain network configurations. Multiple such (attractor) networks exist latently, the ensemble of which defines an organism’s dynamical repertoire. These latently present networks, and activity therein, most likely reflect the intrinsic dynamics of coupled neuronal clusters shaped through evolution, development, and learning (Deco & Corbetta, 2011; C. M. Lewis, Baldassarre, Committeri, Romani, & Corbetta, 2009).

Functional Architectures and Sequential Behavior The skeleton of functional architectures, as we formulated it, contains a dynamical repertoire and a competition mechanism that hierarchically relate to each other through differences in their respective time scales of operation. In contrast to most existing hierarchical schemes (see above), however, we have posited a dynamical repertoire containing functional patterns defined as dynamical objects, and explicitly link the time scales at which various processes operate to their functional significance for the emergence of sequential behavioral patterns. (The frameworks discussed above typically contain one of these features but not both.) With these ingredients, numerous sequential behaviors can be performed, and a given behavior can be performed in multiple ways (Perdikis et al., 2011a). While the notion of competition in behavior has a longer history in the perceptual domain (Ditzinger, 2011; Mazurek, 2003) and decision making (Churchland, Kiani, & Shadlen, 2008; Ratcliff & McKoon, 2008), recent studies have

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shown clear effects of competition in the motor domain (Cui & Andersen, 2011; McKinstry, Dale, & Spivey, 2008; Oliveira, Diedrichsen, Verstynen, Duque, & Ivry, 2010; Pastor-Bernier & Cisek, 2011; Rushworth, Walton, Kennerley, & Bannerman, 2004). Interestingly, novel findings suggest that the traditional view of decision making and action planning as distinct serial processes (rather than an integrated process) may be incorrect (Cui & Andersen, 2011). This suggestion is fully consistent with the present framework, which portrays decision making and action planning as two sides of the same coin: shifting the weighting of the competition parameters toward one winner (decision making), simultaneously establishing the corresponding functional mode as the one to be expressed (action planning). (This notion is independent of nature of the competition.) Along these lines, Freeman, Ambady, Midgley, and Holcomb (2011), using electroencephalography in the context of sex categorization, reported that the motor cortex initiated response preparation before sex categorization had finished. That is, the perception and action appeared as simultaneous parallel processes rather than sequential ones. According to a notion consistent herewith, the “affordance competition hypothesis,” (visual) information biases the competition among multiple potential actions (Cisek, 2007). Thus, rather than affecting a given motor solution in terms of adjustments of the phase flow as discussed above, information often determines which solution is appropriate in the first place through a competition mechanism. The same notion is being put forward by the previously mentioned resting state studies, in which criticality (proximity to instability) plays a fundamental role. Beyond the exploration of the brain’s dynamic repertoire, criticality actually guarantees that a small shift in the parameters is sufficient to bias the competition among the patterns toward a winner. The notion of motor equivalence, under established views, points to the observation that a given motor task can be performed with different end-effectors. The possibility thereof is typically believed to be due to the nervous system encoding for action in abstract terms rather than in terms of specific commands to a given set of muscles (Kelso, 1995; R. A. Schmidt & Lee, 2005; Wing, 2000). In defining functional modes in terms of abstract dynamical objects, the present framework is on a par with this view. Indeed, as the examples in Figure 3 show, a given structured flow may arise on manifolds with a different geometry (plane, sphere, etc.). In other words, the same functional dynamics may emerge in different (work)spaces. Our insights gained through the examination of distinct architectures, however, predict an alternative route for motor equivalence, since a given behavioral solution can (also) be reached via distinct architectural schemes that are set apart through time scale separation. That is, behavioral patterns can be accomplished not only through the implication of a different set of neuromuscular linkages, but also through the utilization of distinct architectures utilizing (at least partly) different functional modes 11 The idea that the brain is operating near instability has previously been deduced from phase transitions in the brain during behavioral paradigms (cf. Daffertshofer et al., 2005; Fuchs, Kelso, & Haken, 1992; Jirsa, Friedrich, Haken, & Kelso, 1994; Kelso, 1995). In behavioral studies, however, the observed transitions pertain to the synergy (coordinative structure). Given the (bidirectional) information flows between brain and periphery, it cannot be assured that the brain operates near criticality in the absence of (sensori)motor tasks.


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(engaging the same or different neuromuscular linkages), as exemplified in Perdikis et al. (2011a). That is, different control structures may accomplish the same behavioral goal. Because the different control structures are likely to draw on distinct neural network components, they link this prediction to the concept of degeneracy (cf. Edelman & Gally, 2001; Tononi, Sporns, & Edelman, 1999). Two further predictions follow naturally from the theoretical framework outlined in this article. First, if complex sequential behavior arises from the sewing together of functional modes, then deterioration or even the breakdown of meaningful behavior may be observed at the level of the building blocks but also at the competition level. Evidence consistent therewith are so-called actions slips, that is, doing the “wrong thing” in the “right way” (Botvinick & Bylsma, 2005; Norman, 1981; Reason & Mycielska, 1982; see also Goldstein et al., 2007, for the “slip of the tongue” phenomenon). Also, in rats it has been observed that impairments of the corpus striatum have a degenerative effect of action sequencing in grooming while individual movement components remain largely intact (Berridge & Whishaw, 1992; Cromwell, Berridge, Drago, & Levine, 1998). Second, given a dynamical repertoire consisting of functional modes, and a (linear, or otherwise) summation of the modes, it follows that the phase flow during a transition depends on the two modes involved. That is, the transition from mode B to mode A is different than from mode C to mode A. Consequently, sequential behavior should contain context dependency. Several observations perfectly fit this prediction. For instance, how a given letter is written depends on the letter preceding and following it (Rosenbaum, 1991; for similar results in typing, see Viviani & Terzuolo, 1980). Similarly, in speech so-called coarticulation, which refers to the influence of adjacent (and even near-adjacent) gestures on the current one (see Saltzman & Munhall, 1989; Saltzman et al., 2008, and references therein), is the rule rather than the exception. Context dependency apparently abounds. Finally, in order to produce meaningful writing, the word’s letter order was set by hand as a first step via the Cj and Lj parameters (cf. Bullock, 2004; Grossberg, 1978). However, these parameters are analogue real-valued dynamical quantities that receive feedback as a function of the architecture’s “output” as well as from the (active) modes (in our handwriting example). That is, they are quite unlike symbolic representations even though the letter sequence is somehow set at the outset. (In fact, sequencing errors may occur when Cj and Lj have similar values for some pair of j indexes, or due to the feedback and noise.) No models exist, to our best knowledge, that are able to evade this problem. The dynamics of a sequence’s parameters, however, could in principle be acquired in the spirit of graph dynamics (Saltzman & Munhall, 1992; see above). In this sense, when viewed over appropriate time scales, the resulting behavior may still be truly said to be emergent.

Limitations and Outlook We outlined our perspective on the generation of complex behaviors using SFM. Obviously, however, there are also issues that are not (yet) covered by our framework and others that may turn out to be incorrect. For instance, we have set dynamical form, that is, SFM, on the center stage and believe this to be a key

component of behavior. However, all behaviors also have metric properties, and sometimes these are important for goal achievement. Using the handwriting architecture, we have shown that at least some and likely many or all behaviors are scalable. The implementation thereof was, however, engineered such that the SFM comprising the dynamic repertoire were artificial in the sense that no attempt was made to derive them from experimental data. A challenging extension of the present architecture would include behavioral scalability in an autonomous fashion, which for most cases will imply informational coupling with the environment. Also, we have defined classes on the bases of mathematical topologies. While this appears as a good starting point—it avoids ambiguous and/or overlapping classes—it is still an open question whether behavior and cognition are organized in terms of mathematical topologies or whether the constraints may have to be loosened (Petitot, 1995). We have chosen to outline our framework in the context of motor behavior. The framework’s abstract nature, however, readily allows for transfer to other domains, including pattern recognition, music, language—its perception and production, and other forms of cognition. A methodological difficulty with perception and cognition in general, however, is that it is far from trivial how to retrieve the dynamics underlying perceptual and cognitive processes. Through speech, however, cognitive processes are coupled to motor behavior that more easily allow for the identification of behavioral units (cf. Goldstein et al., 2007), and in fact, especially speech errors have been investigated as a window into phonological encoding (Meyer, 1992; Shattuck-Hufnagel, 1992). That is, language provides the potential to transfer the present framework to perception and cognition. Furthermore, we illustrated a pattern recognition architecture using functional modes as (part of) an observer’s “recognizable objects or patterns” or generative models in the parlance of the active inference framework. Accordingly, both action and perception are cast as pure inference problems. Under the assumption that one could regard SFM as predicting both the proprioceptive and exteroceptive consequences of action, and because Bayesian inversion of generative models corresponds to perception or recognition, the predictions (SFM) can be used to form prediction errors and recognize the visual consequences of movement through predictive coding implementations of active inference. The dual generation of proprioceptive and exteroceptive predictions has been used to simulate handwriting and its recognition to model the mirror neuron system in this spirit (Friston et al., 2011). Thus, active inference can (in principle) exploit SFM for both action and perception. The notion that distinct architectures may realize a given solution using different functional modes (at least partly) may provide a novel window into chunking, the putting together of multiple elements (behavioral, cognitive) into a single structure. For instance, novel developments in robotics (Ijspeert, Nakanishi, & Schaal, 2003) and engineering (Shevchenko et al., 2009) allow for the learning of more complex behaviors from simpler ones. In juggling, per example, the impact of ball– hand contact at the catch is a clearly identifiable event in beginning jugglers, which vanishes with increasing expertise (Huys et al., 2004). This suggests that the juggle of novices constitutes a sequence of discrete acts, while experts have integrated catching and throwing into a single rhythmic act. Consistent therewith is the



suggestion that emergent timing may well require long practice (Studenka & Zelaznik, 2008). In development, before infants can successfully reach for an object, the reach appears to consist of multiple submovements. Over time, the number of submovements diminishes (Thelen, 1998). Interestingly, babies reveal large individual differences in their reaching style, and although these differences decrease as the reaching becomes more stable, their individual style remains present to some degree. This observation suggests that the newly acquired reach is (indeed) acquired via the integration of existing submovements. In other words, applications of the present framework to learning and development are multiple and may shed a novel light on a variety of observations in those domains. All in all, while still in its infancy, we believe that SFM and functional architectures provide a promising framework with the potential to integrate novel research foci and long-standing notions within a single framework.

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Received June 19, 2013 Revision received January 28, 2014 Accepted January 29, 2014 䡲

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