Designing responsive pattern generators: stable heteroclinic channel cycles for modeling and control

Abstract

A striking feature of biological pattern generators is their ability to respond immediately to multisensory perturbations by modulating the dwell time at a particular phase of oscillation, which can vary force output, range of motion, or other characteristics of a physical system. Stable heteroclinic channels (SHCs) are a dynamical architecture that can provide such responsiveness to artificial devices such as robots. SHCs are composed of sequences of saddle equilibrium points, which yields exquisite sensitivity. The strength of the vector fields in the neighborhood of these equilibria determines the responsiveness to perturbations and how long trajectories dwell in the vicinity of a saddle. For SHC cycles, the addition of stochastic noise results in oscillation with a regular mean period. In this paper, we parameterize noise-driven Lotka-Volterra SHC cycles such that each saddle can be independently designed to have a desired mean sub-period. The first step in the design process is an analytic approximation, which results in mean sub-periods that are within 2% of the specified sub-period for a typical parameter set. Further, after measuring the resultant sub-periods over sufficient numbers of cycles, the magnitude of the noise can be adjusted to control the mean period with accuracy close to that of the integration step size. With these relationships, SHCs can be more easily employed in engineering and modeling applications. For applications that require smooth state transitions, this parameterization permits each state's distribution of periods to be independently specified. Moreover, for modeling context-dependent behaviors, continuously varying inputs in each state dimension can rapidly precipitate transitions to alter frequency and phase.

Phase descriptions of a multidimensional Ornstein-Uhlenbeck process

“…Transient oscillatory activity may arise in deterministic models that do not possess limit cycles, for example spiral sink systems and stable heteroclinic cycles. A deterministic dynamical systems has a stable heteroclinic cycle if there is a closed attracting set Γ het composed of trajectories connecting a repeating sequence of saddle equilibrium points [24,[33][34][35][36]. In this situation, there is no periodic trajectory with a finite period.…”

Heteroclinic switching between chimeras

“…Thus, sequential switching between distinct localized dynamics has been associated with neural computation [6][7][8][9]; sequential dynamics in the hippocampus in the absence of external input [10] are a striking example. Most efforts to understand switching dynamics between localized frequency synchrony patterns rely on averaged models which neglect the contributions of individual oscillators to the network dynamics [11][12][13][14][15] or are statistical [16]. For finite networks, however, the dynamics of individual oscillators cannot be neglected.…”

Control for Multifunctionality: Bioinspired Control Based on Feeding in Aplysia californica

“…Though the nodes do not map precisely to known neural connectivity, their dynamics can be simulated rapidly, connected to basic kinematic models of the periphery, and respond to changes in sensory inputs, such as the load on the seaweed. Furthermore, stable heteroclinic channel controllers have been successfully translated to robotic applications [59,148]. However, such models do not provide insight into the detailed neural mechanisms underlying multifunctional control.…”