Hopf Bifurcations in Complex Multiagent Activity: The Signature of Discrete to Rhythmic Behavioral Transitions

Abstract

Most human actions are composed of two fundamental movement types, discrete and rhythmic movements. These movement types, or primitives, are analogous to the two elemental behaviors of nonlinear dynamical systems, namely, fixed-point and limit cycle behavior, respectively. Furthermore, there is now a growing body of research demonstrating how various human actions and behaviors can be effectively modeled and understood using a small set of low-dimensional, fixed-point and limit cycle dynamical systems (differential equations). Here, we provide an overview of these dynamical motorprimitives and detail recent research demonstrating how these dynamical primitives can be used to model the task dynamics of complex multiagent behavior. More specifically, we review how a task-dynamic model of multiagent shepherding behavior, composed of rudimentary fixed-point and limit cycle dynamical primitives, can not only effectively model the behavior of cooperating human co-actors, but also reveals how the discovery and intentional use of optimal behavioral coordination during task learning is marked by a spontaneous, self-organized transition between fixed-point and limit cycle dynamics (i.e., via a Hopf bifurcation).

Task dynamics define the contextual emergence of human corralling behaviors

“…When ζ >1, the system will approach the attractor slowly. When ζ = 1, the system will approach the attractor with the least amount of time that results in at most one overshoot [ 68 ]. By varying ζ to define b , it is possible to assess the effect of underdamped or overdamped dynamics on the resultant patterns of behavior during the corralling task as a function of ε .…”

Navigational Behavior of Humans and Deep Reinforcement Learning Agents

“…Central to the approach is both identifying and then defining the self-organizing physical and informational constraints and couplings that underly the emergence of stable and effective human perceptual-motor behavior in the form of a non-linear dynamical system. Of particular relevance here is that, although identifying and defining such non-linear dynamical models may at first seem rather difficult, there is now a substantial body of research demonstrating how human perceptual-motor behavior can be modeled (and perhaps even derived) using a simple set of dynamical motor primitives (Haken et al, 1985 ; Kay et al, 1987 ; Schaal et al, 2005 ; Warren, 2006 ; Ijspeert et al, 2013 ; Richardson et al, 2015 ; Amazeen, 2018 ; Patil et al, 2020 ). Specifically, these dynamical motor primitives correspond to the fundamental properties of non-linear dynamical systems, namely (i) point-attractor dynamics and (ii) limit-cycle dynamics, with the former capable of capturing discrete movements or actions (e.g., tapping a key, passing, or throwing a ball) by means of environmentally coupled damped mass-spring functions, and the latter capable of capturing rhythmic movements (e.g., hammering, walking) by means of forced (driven) damped-mass spring systems or non-linear self-sustained oscillators (e.g., Rayleigh or van der Pol oscillator).…”

Countering the Deleterious Effects of Electromagnetic Pulse

“…Timing circuits (for example those in Figure 2) (Timer Circuits: Digital C, 2021; Sekikawa et al, 2011;Van der Pol oscillator La, 2021) utilize capacitors as their basis, where the time constant of the capacitor is the product of the resistance and capacitance producing predictable capacitor charge and discharge curves for both voltage and current, and their governing equations are highly nonlinear and are often described as "chaos" as depicted in Figure 1B, where the bifurcation or transition from a fixed-point to limit cycle behavior is called a Hopf bifurcation (Patil et al, 2020). Cooper et al (2017) introduced adaptive notions of controlling chaos [with respect to the van der Pol oscillator in Eq.…”

“…2, and this manuscript evaluates the use of those methods to provide resiliency to the deleterious effects of electromagnetic pulse induced voltages. The van der Pol oscillator is applicable to several fields of endeavors including brain science (Patil et al, 2020) and even dynamics of phase synchronization between solar polar magnetic fields (Savostianov et al, 2020)], so the proposed methods in this manuscript become generalizable. Using the proposed methods, the goal is to force the system to respond with a perfect circle in phase space rather than the chaotic limit cycle displayed in Figure 1B, despite the imposition of a large, rapid voltage spike from an electromagnetic pulse.…”