The Gesture as an Autonomous Nonlinear Dynamical System

We propose a theory of how the speech gesture determines change in a functionally relevant variable of vocal tract state (e.g., constriction degree). A core postulate of the theory is that the gesture determines how the variable evolves in time independent of any executive timekeeper. That is, the theory involves intrinsic timing of speech gestures. We compare the theory against others in which an executive timekeeper determines change in vocal tract state. Theories that employ an executive timekeeper have been proposed to correct for disparities between theoretically predicted and experimentally observed velocity profiles. Such theories of extrinsic timing make the gesture a nonautonomous dynamical system. For a nonautonomous dynamical system, the change in state depends not just on the state but also on time. We show that this nonautonomous extension makes surprisingly weak kinematic predictions both qualitatively and quantitatively. We propose instead that the gesture is a theoretically simpler nonlinear autonomous dynamical system. For the proposed nonlinear autonomous dynamical system, the change in state depends nonlinearly on the state and does not depend on time. This new theory provides formal expression to the notion of intrinsic timing. Furthermore, it predicts experimentally observed relations among kinematic variables. Extrinsic timing of speech gestures has been proposed to correct for disparities between theoretically predicted and experimentally observed speech kinematics (Byrd & Saltzman, 1998, 2003; Kr€oger, Schr€oder, & Opgen-Rhein, 1995). Extrinsic timing makes the speech gesture a nonautonomous dynamical system. This means that the state of the system does not fully determine how it evolves and that the system is further influenced by a process which depends explicitly on time. This process is external to the gesture but possibly internal to the phonological system. This time-dependent process influences how the gesture evolves, but the gesture does not in turn influence this process. Thus, the time-dependent process plays the role of an external timekeeper, governing the evolution of the gesture as a sort of clock. Nonautonomy has been argued to be necessary in order to correct for disparities between theoretically predicted and experimentally observed speech kinematics. However, CONTACT Tanner Sorensen [email protected] University of Southern California, 3740 McClintock Avenue, EEB 400, Los Angeles, CA 90089-2564; Adamantios Gafos [email protected] Universit€at Potsdam, Department Linguistik, Haus 14, Karl-Liebknecht-Straße 24-25, Potsdam, Germany. © 2016 Taylor & Francis Group, LLC ECOLOGICAL PSYCHOLOGY 2016, VOL. 28, NO. 4, 188–215 http://dx.doi.org/10.1080/10407413.2016.1230368 D ow nl oa de d by [ U SC U ni ve rs ity o f So ut he rn C al if or ni a] a t 1 1: 24 0 9 A ug us t 2 01 7 nonautonomy is undesirable for theoretical reasons. For one thing, nonautonomy severely constrains analytical options. A deeper conceptual problem with nonautonomy is that it implies a regress of control structures. If an external timekeeper influences the evolution of the gesture, then we ought to identify the timekeeper of the timekeeper, and so on ad infinitum (Beek, Turvey, & Schmidt, 1992, p. 69). Conversely, autonomy posits no external timekeeper, thus halting the regress of control structures. We argue that, despite making the gestural dynamics more complex, the kinematic predictions of the nonautonomous extension are surprisingly weak both qualitatively and quantitatively. We propose a revised nonlinear dynamical system which maintains that the gesture is an autonomous dynamical system. The system is a model of intragestural dynamics, that is, of the single speech gesture. The system does not determine intergestural dynamics, that is, how gestures are coordinated. The following two sections distinguish autonomous from nonautonomous and nonlinear from linear theories of the speech gesture. We propose that the gesture is a nonlinear autonomous dynamical system. We lay out the predictions of the proposed dynamical system and qualitatively evaluate these predictions. Using the University of Wisconsin X-ray microbeam database (Westbury, Milenkovic, Weismer, & Kent, 1990), we then quantitatively evaluate the fit of the proposed dynamical system to the movements of tongue dorsum raising and lowering. Finally, we illustrate how, in an isochronous speech task (e.g., “bapabapa ...”), where the proposed revised dynamical system is driven by a periodic external force, trajectories become aperiodic. Phase portraits and Hooke diagrams of the proposed driven nonlinear system are consistent with empirical observations. In sum, we illustrate the revised system’s fit to the kinematics in both noncyclic speech and cyclic tasks (i.e., regularly timed speech with a metronome). The proposed nonlinear dynamical system gives both formal expression and empirical justification to Carol Fowler’s vision of intrinsic timing at the level of a single gestural event. It restores the autonomy postulate and offers a basis for moving forward with that vision. Autonomous versus nonautonomous Theoretical work suggests that the gesture has intrinsic timing (Fowler, 1980). This means that the spatial and temporal extent of the gesture is determined by the gesture itself (i.e., by the state of the system and the system parameters), not by any system external to the gesture. In contrast, an extrinsic timing theory of the gesture would mean that the spatial and temporal extent of the gesture is influenced by systems that are external to the gesture. This section defines terms and introduces the autonomous versus nonautonomous distinction as it has been applied to gestures. A dynamical system is a formal model that expresses a rule for how the state of a system changes in time. The state is the minimal set of variables required to predict how the system evolves. The set of all possible states is the phase space. For one-dimensional systems, where there is only one state variable x, the phase space is the line of real numbers. Change in the state of the system is due to presumed causes, represented by the force field f x ð Þ of the dynamical system. As the state of the system changes, it traces out a path in phase space called a trajectory. ECOLOGICAL PSYCHOLOGY 189 D ow nl oa de d by [ U SC U ni ve rs ity o f So ut he rn C al if or ni a] a t 1 1: 24 0 9 A ug us t 2 01 7 Autonomous dynamical systems have the form of Equation (1). Specifically, this equation states that at any time instant t the rate of change _ x D dx6 dt of x is a function f x ð Þ, which depends only on the state x and not on time t.