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

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 > 0 representing the distance from a heteroclinic bifurcation point. In the → 0 limit, 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. keywords: (MSC database)• 34C05 Location of integral curves, singular points, limit cycles • 37C27 Periodic orbits of vector fields and flows • 37C29 Homoclinic and heteroclinic orbits • 37G15 Bifurcations of limit cycles and periodic orbits • 70K05 Phase plane analysis, limit cycles • 70K44 Homoclinic and heteroclinic trajectories Movies:•

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.…”

“…Instead, trajectories near Γ het traverse the same neighborhood of phase space with progressively longer and longer intervals required to pass each saddle point in turn. Because there is no finite period, the phase and the asymptotic phase cannot be defined; see [24] for a discussion of the phase reduction problem for the deterministic case, and [25] for the stochastic case. A spiral sink system possesses a stable equilibrium point for which the Jacobian matrix has a complex conjugate pair of eigenvalues.…”

“…Synchronization and entrainment of deterministic oscillators may be analyzed by finding a coordinate transformation to a one-dimensional phase variable defined in terms of the asymptotic phase of a limit cycle [8][9][10][11]. The classical approach breaks down in several cases of interest including: (a) limit-cycle systems endowed with noise [1,[12][13][14][15][16][17][18], (b) spiral-sink systems with noise-sustained oscillations, also known as quasicycles [19][20][21][22][23], and (c) systems with an attracting heteroclinic cycle [24][25][26].…”

Asymptotic Phase for Stochastic Oscillators

“…n − 1, t|v, n, s) + β(v )(n + 1)ρ(v , n + 1, t|v, n, s)(10) − ∂ ∂s ρ(v , n , t|v, n, s)= L † v [ρ] = f (v, n) ∂ρ ∂v + α(v) (N tot − n) {ρ(v ,n , t|v, n + 1, s) − ρ(v , n , t|v, n, s)} +β(v)n {ρ(v , n , t|v, n − 1, s) − ρ(v , n , t|v, n, s)} (11) (color online) Trajectory, nullclines, eigenvalues of the backward operator, and asymptotic phase lines for the persistent-sodium-potassium model. (A) Sample trajectory (thin black line) for the (V, N ) process for Ntot = 100 channels, and nullclines for the deterministic v (thick grey line) and n (thick black line) dynamics.…”

A study on limit cycle and non-homoclinic orbits for FitzHugh-Nagumo System

“…In 13 , we studied the system (1.2) with all parameters not zeros and proved the uniqueness of limit cycle. There are many articles in the field of limit cycles and homoclinic orbits for example see 20,21,22 .…”