Nonlinear Dynamics in Human Behavior

This chapter is intended as an introduction or tutorial to nonlinear dynamical systems in one and two dimensions with an emphasis on keeping the mathematics as elementary as possible. By its nature such an approach does not have the mathematical rigor that can be found in most textbooks dealing with this topic. On the other hand it may allow readers with a less extensive background in math to develop an intuitive understanding of the rich variety of phenomena that can be described and modeled by nonlinear dynamical systems. Even though this chapter does not deal explicitly with applications – except for the modeling of human limb movements with nonlinear oscillators in the last section – it nevertheless provides the basic concepts and modeling strategies all applications are build upon. The chapter is divided into two major parts that deal with one- and two-dimensional systems, respectively. Main emphasis is put on the dynamical features that can be obtained from graphs in phase space and plots of the potential landscape, rather than equations and their solutions. After discussing linear systems in both sections, we apply the knowledge gained to their nonlinear counterparts and introduce the concepts of stability and multistability, bifurcation types and hysteresis, hetero- and homoclinic orbits as well as limit cycles, and elaborate on the role of nonlinear terms in oscillators.

The search for a mathematical framework for describing motor behavior has a long but checkered history. Most studies have focused on recurrent, deterministic features of behavior. The use of dynamical systems to account for the qualitative features of end-effector trajectories of limb oscillations gained momentum in the last twenty-five years or so. There, salient characteristics of human movement served as guidelines for model developments. For instance, trajectories of limb cycling describing a bounded area in the position-velocity or phase plane may be interpreted as indicative of a limit cycle attractor, at least when modeling efforts are restricted to identifying deterministic forms, thereby disregarding variability. By using averaging methods, which are typically applied for first-order analyses of nonlinear oscillators, e.g., harmonic balance, Kay et al (1987, 1991) derived second-order nonlinear differential equations that mimic experimentally observed amplitude-frequency relations and phase response characteristics of rhythmic finger and wrist movements. The self-sustaining oscillators include weak dissipative nonlinearities that stabilize the underlying limit cycle, cause a drop of amplitude and an increase in peak velocity with increasing movement tempo or frequency.

The early 1980s saw the development of a new perspective on motor control inspired by theories of self-organization and dynamical systems theory. Its first efforts were directed at the investigation of rhythmic movements in terms of two-dimensional (autonomous) limit cycle oscillators. The corresponding studies are characterized by the development of detailed and generally task-specific models, which have resulted in a detailed documentation of the relation between oscillator properties and task requirements. The study of discrete movements for a long time received far less attention; its conjunctional theoretical and empirical investigation has only recently set off and is characterized by an explicit focus on phase flows and topologies therein.

This chapter focuses on motor coordination between similar as well as different classes of movements from a phase flow perspective. Most studies on coordination dynamics are concerned with coordination of rhythmic movements. This constraint enables the modeler to describe the interaction between the oscillating movement components by a phase description, and its dynamics by a potential function. However, potential functions are extremely limited and exist only in a limit number of cases. In contrast, dynamical systems can be unambiguously described through their phase flows. The present chapter elaborates on coordination dynamics from the phase flow perspective and sheds new light on the meaning of biological coupling. The phase flow deformations of coupled systems may be understood using the notion of convergence and divergence of the phase space trajectories and aid in explaining the mechanisms of trajectory formation and the interaction (coupling) between arbitrary movements.

The issue of how humans and animals perform accurate movements has been addressed in various ways. Although this book is promoting concepts stemming from dynamical systems theory, other approaches have contributed to the understanding of movement as well. Among others, the equilibrium point theory and the computational theory deserve to be listed for their contribution to this field of research called motor control. In this chapter, using single-joint rhythmic movement as an example, I will start first emphasizing the respective contributions and drawbacks of each approach. Then I will address the issue of parameter selection. Indeed, despite diverging opinions about the possible nature of control parameter(s), all three approaches must deal with the problem of how adequate parameter(s) to achieve a desired movement are selected. At the end of this chapter, I will expose how the concept of internal model may offer a solution to this problem.

The history of research on speech perception and speech production is replete with examples of nonlinearities between articulation and acoustics, and between acoustics and perception. These nonlinearities are useful for communication. They allow 1) adequate production of speech sounds and words despite people having different vocal tracts with different resonance capabilities, and 2) adequate word recognition despite variation in the acoustic signal across speakers, emphasis, background noise, etc. Yet context and the listener’s expectancies often strongly influence what is perceived; perception is dynamic, influenced by multiple factors that change slowly or quickly as speech goes on. In this chapter we present a selected history of demonstrations of nonlinearities in speech and attempt to exploit the nonlinearities in order to uncover the dynamics of both perception and production of speech.

Perceptual stability is ubiquitous in our everyday lives. Objects in the world may look somewhat different as the perceiver’s viewpoint changes, but it is rare that their essential stability is lost and qualitatively different objects are perceived. In this chapter we examine the source of this stability based on the principle that perceptual experience is embodied in the neural activation of ensembles of detectors that respond selectively to the attributes of visual objects. Perceptual stability thereby depends on processes that stabilize neural activation. These include biophysical processes that stabilize the activation of individual neurons, and processes entailing excitatory and inhibitory interactions among ensembles of stimulated detectors that create the "detection instabilities" that ensure perceptual stability for near threshold stimulus attributes. It is shown for stimuli with two possible perceptual states that these stabilization processes are sufficient to account for spontaneous switching between percepts that differ in relative stability, and for the hysteresis observed when attribute values are continually increased or decreased.

The responsiveness of the visual system to changes in stimulation has been the focus of psychophysical, neurophysiological, and theoretical analyses of perception. Much less attention has been given to the role of persistence, the effect of the visual system’s response to previous visual events (its prior state) on its response to the current visual input. Perceiving an object can facilitate its continued perception when a passing shadow briefly degrades its visibility, when attention is momentarily distracted by another object, when the eyes blink, or when a random fluctuation within the visual system potentially favors an alternative percept. Having perceived an object’s shape from one viewpoint can facilitate its continued perception despite changes in viewpoint that distort its retinal projection, potentially creating a non-veridical percept. These examples highlight the importance of the visual system’s prior state, not just for perceptual stability, but also for perceptual selection; i.e., for the determination of which among two or more alternatives is realized in perceptual experience.

In this essay we discuss three neural properties that form a sufficient basis for a theory of perceptual dynamics that addresses the relationship between persistence, responsiveness to changes in stimulation, and selection. These neural properties are: 1) Individual neurons have the intrinsic ability to stabilize their activation state. 2) Neurons responsive to sensory information (i.e., detectors) are organized into ensembles whose members respond preferentially to different values of the same attribute (e.g., motion direction). Members of such ensembles have overlapping tuning functions; i.e., a detector responding optimally to one stimulus value will also respond, though less strongly, to similar attribute values. 3) The activation levels of a detector affects and is affected by nonlinear excitatory and inhibitory interactions with other detectors.

On this basis, we examine the persistence of steady-state detector activation despite the presence of random perturbations, the effect of neural stabilization on a detector’s response to stimulation, the crucial role of “detection instabilities” in minimizing perceptual instability and uncertainty for near-threshold stimuli, and the importance of differences in the rate-of-change in activation for perceptual selection. Finally, we demonstrate that the signature features of perceptual dynamics, spontaneous switching between percepts differing in relative stability, and hysteresis, follow from the same three neural properties.

Simple and intriguing examples for nonlinear dynamics in visual perception are presented by means of optical illusions. Well known visual effects such as the temporal perception of ambiguous figures, autostereograms, and moving patterns are presented and interpreted from the perspective of nonlinear dynamics. Furthermore new results on the interdependency between the perception of colour and motion are presented, including an explanation of the classic "Fluttering Hearts effect" and the new "Leaning Tower of Pisa effect" which is responsible for a perceptual shift of rotating coloured areas.

Music is a form of communication that relies on highly structured temporal sequences comparable in complexity to language. Music is found among all human cultures, and musical languages vary across cultures with learning. Tonality – a set of stability and attraction relationships perceived among musical frequencies – is a universal feature of music, found in virtually every musical culture. In this chapter, a new theory of central auditory processing and development is proposed, and its implications for tonal cognition and perception are explored. A simple model is put forward, based on knowledge of auditory organization and general neurodynamic principles. The model is simplified as compared to the organization and dynamics of the real auditory system, nevertheless it makes realistic predictions about neurodynamics. The analysis predicts the existence of natural resonances, the potential for tonal language learning, the perceptual categorization of intervals, and most importantly, relative stability and attraction relationships among musical tones. This approach suggests that high-level music cognition and perception may arise from the interaction of acoustic signals with the dynamics of the auditory system. Musical universals are predicted by intrinsic neurodynamics that provide a direct link to neurophysiology, and Hebbian synaptic modification could explain how different tonal languages are established.