[Paper Review] Singularly perturbed phase response curves

Pierre Sacré, Alessio Franci|arXiv (Cornell University)|Jun 5, 2015

Nonlinear Dynamics and Pattern Formation19 references3 citations

TL;DR

This paper introduces a geometric method based on singular perturbation theory to predict the finite phase response curve (PRC) of relaxation oscillators in the singular limit, overcoming the limitations of the infinitesimal PRC approach. By analyzing isochrones and the global structure of the oscillator's phase space, the method provides semi-analytic formulas that converge to the true PRC as time-scale separation increases, validated numerically on the FitzHugh-Nagumo model for impulses and square pulses.

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.

Motivation & Objective

• To address the failure of the infinitesimal phase response curve (PRC) approach in the singular limit of relaxation oscillators with finite-amplitude inputs.
• To develop a geometric framework for predicting the qualitative shape of finite PRCs using isochrones and singular perturbation theory.
• To provide a semi-analytic method that improves in accuracy as time-scale separation increases, unlike the infinitesimal approach.
• To extend phase response curve analysis beyond linear approximations to capture nonlinear, finite-input effects in fast-slow systems.
• To lay the foundation for a geometric theory of finite PRCs in complex oscillators such as bursters.

Proposed method

• The method uses singular perturbation theory to analyze the fast-slow dynamics of planar relaxation oscillators, decomposing the system into layer and reduced dynamics in the singular limit (ε → 0).
• It constructs the singular periodic orbit γ⁰ as the union of slow segments on the critical manifold S⁰ and fast jumps along critical fibers.
• The isochrones are approximated geometrically by tracking the asymptotic phase map using the structure of the singular orbit and the slow flow on S⁰.
• The finite PRC is predicted by determining how perturbations shift the phase along the singular orbit, based on the location of the perturbation relative to the slow and fast segments.
• The approach derives piecewise laws for phase shifts, distinguishing regions where perturbations either remain on the same branch or trigger a jump to the opposite branch.
• The method is applied to the FitzHugh-Nagumo model, yielding semi-analytic expressions for PRCs under impulses and square pulses of finite duration.

Experimental results

Research questions

• RQ1 How can the finite phase response curve of a relaxation oscillator be predicted in the singular limit, beyond the scope of infinitesimal perturbation theory?
• RQ2 What geometric features of the phase space—such as isochrones and the singular periodic orbit—determine the shape of the finite PRC?
• RQ3 Why does the infinitesimal PRC fail to capture finite-amplitude input effects as time-scale separation increases?
• RQ4 How do the locations of perturbations along the periodic orbit affect phase shifts, and can these be captured by a piecewise analytic law?
• RQ5 Can the proposed geometric method accurately predict PRCs for both impulses and finite-duration pulses in fast-slow systems?

Key findings

• The geometric prediction of the finite PRC matches numerical simulations extremely well for small ε, particularly for impulses and square pulses.
• For small ε, the predicted PRC captures both the magnitude and location of phase advances and delays, with errors decreasing as ε → 0.
• The method correctly identifies two distinct regions in the PRC for square pulses: one where phase advances occur due to perturbations near the upper fold, and another where delays occur near the lower fold.
• The PRC for impulses is non-zero only near the upper fold region, where excitatory inputs advance the onset of the upper branch, with phase advance decreasing monotonically to zero.
• For larger ε, the prediction remains qualitatively accurate, capturing the main features of the PRC, though it underestimates small phase shifts near the upper fold.
• The method reveals that finite pulses can delay the termination of the upper branch, a phenomenon not captured by impulse-based PRCs, highlighting the importance of input duration in phase response.