Phase Resetting in an Asymptotically Phaseless System: On the Phase Response of Limit Cycles Verging on a Heteroclinic Orbit

TL;DR

This paper investigates how proximity to heteroclinic cycles affects the phase response of limit cycles, revealing phase-specific sensitivities and divergence behaviors in both smooth and piecewise linear systems near bifurcation points.

Contribution

It provides exact expressions for the infinitesimal phase response curve in a piecewise linear system near heteroclinic bifurcation, and demonstrates phase-dependent divergence or convergence in response to perturbations.

Findings

Perturbations parallel to unstable eigenvectors cause divergence as the bifurcation is approached.

Perturbations parallel to stable eigenvectors can cause divergence or convergence depending on phase.

Numerical evidence shows phase-specific sensitivity in smooth systems near heteroclinic cycles.

Abstract

Rhythmic behaviors in neural systems often combine features of limit cycle dynamics (stability and periodicity) with features of near heteroclinic or near homoclinic cycle dynamics (extended dwell times in localized regions of phase space). Proximity of a limit cycle to one or more saddle equilibria can have a profound effect on the timing of trajectory components and response to both fast and slow perturbations, providing a possible mechanism for adaptive control of rhythmic motions. Reyn showed that for a planar dynamical system with a stable heteroclinic cycle (or separatrix polygon), small perturbations satisfying a net inflow condition will generically give rise to a stable limit cycle (Reyn, 1980; Guckenheimer and Holmes, 1983). Here we consider the asymptotic behavior of the infinitesimal phase response curve (iPRC) for examples of two systems satisfying Reyn's inflow criterion, (i) a smooth system with a chain of four hyperbolic saddle points and (ii) a piecewise linear system corresponding to local linearization of the smooth system about its saddle points. For system (ii), we obtain exact expressions for the limit cycle and the iPRC as a function of a parameter, mu>0, representing the distance from a heteroclinic bifurcation point. In the limit, as mu approaches zero, we find that perturbations parallel to the unstable eigenvector direction in a piecewise linear region lead to divergent phase response, as previously observed (Brown, Moehlis and Holmes (2004), Neural Computation). In contrast to previous work, we find that perturbations parallel to the stable eigenvector direction can lead to either divergent or convergent phase response, depending on the phase at which the perturbation occurs. In the smooth system (i), we show numerical evidence of qualitatively similar phase specific sensitivity to perturbation.