Singularly perturbed phase response curves

Abstract

In this paper we propose a novel geometric method, based on singular perturbations, to approximate isochrones of relaxation oscillators and predict the qualitative shape of their (finite) phase response curve. This approach complements the infinitesimal phase response curve approach to relaxation oscillators and overcomes its limitations near the singular limit. We illustrate the power of the methodology by deriving semi-analytic formula for the (finite) phase response curve of generic planar relaxation oscillators to impulses and square pulses of finite duration and verify its goodness numerically on the FitzHugh-Nagumo model.

Figures (7)

• Figure 1: Geometry of relaxation oscillators. The critical manifold $\mathcal{S}^0$ is a S-shaped curve. Under some technical assumptions Krupa:2001ez, the singular system \ref{['eq:relax_fast_syst']} admits a singular periodic orbit $\gamma^0$ defined as the union of two pieces of the critical manifold associated with a slow evolution (green solid lines) and two critical fibers associated with jumps (green dashed lines).
• Figure 1: Qualitative trade-off between the infinitesimal approximation and the singular approximation as a function of the time-scale separation $\epsilon$.
• Figure 1: Phase response curves for excitatory impulses: singular geometric prediction (solid line) and numerical simulations (dots). (Parameter values: $a = 0.7$, $b = 0.8$, $I = 1$, $|\alpha| = 1.5$.)
• Figure 2: Geometric construction of (asymptotic) singular phase map. The phase map associates with each point on the periodic orbit a phase which corresponds to the normalized time $\omega_{\text{s}}^0\,\Delta\tau$ required to reach this point from the reference position $(x_-,z_-)$. For points on the lower branch, it is convenient to measure the normalized time from $(x_+,z_+)$ and to add the phase $\theta_+ \mathrel{:=} \omega_{\text{s}}^0\,\Delta\tau_{+}$. Because all points on a same vertical ray (in the bistable region) and converging to the same branch instantaneously jump on the branch in the singular limit, the asymptotic phase map associates them with the same asymptotic phase. In addition, other vertical lines (outside the bistable region) are associated with the same phase because these points converge in the same $\Delta\tau \pmod{T_{\text{s}}^0}$ to $(x_+,z_+)$.
• Figure 2: Phase response curves for excitatory pulses of finite duration: singular geometric prediction (solid line) and numerical simulations (dots). (Parameter values: $a = 0.7$, $b = 0.8$, $I = 1$, $|\bar{u}| = 0.25$, $\Delta = 0.1\,T$.)

Theorems & Definitions (4)

• Remark 1
• Remark 2
• Remark 3
• Remark 4