^{1,a)}, Bernd Krauskopf

^{1,b)}and Hinke M. Osinga

^{1,c)}

### Abstract

We consider the FitzHugh-Nagumo model, an example of a system with two time scales for which Winfree was unable to determine the overall structure of the isochrons. An isochron is the set of all points in the basin of an attracting periodic orbit that converge to this periodic orbit with the same asymptotic phase. We compute the isochrons as one-dimensional parametrised curves with a method based on the continuation of suitable two-point boundary value problems. This allows us to present in detail the geometry of how the basin of attraction is foliated by isochrons. They exhibit extreme sensitivity and feature sharp turns, which is why Winfree had difficulties finding them. We observe that the sharp turns and sensitivity of the isochrons are associated with the slow-fast nature of the FitzHugh-Nagumo system; more specifically, it occurs near its repelling (unstable) slow manifold.

We thank Bard Ermentrout, John Guckenheimer, Bruce Peckham, and Jonathan Rubin for stimulating discussions during their visits to The University of Auckland. P.L. would like to thank Eusebius Doedel for the fruitful discussions and hospitality while visiting Concordia University in Montreal, Jeff Moehlis for references to his work in this area, and to Alexandre Mauroy for explaining the details of Refs. 24 and 25.

I. INTRODUCTION A. Background and notation B. Properties of isochrons C. Isochrons for a simple example D. Organisation of this paper II. COMPUTATION OF ISOCHRONS A. Overview of methods for computing isochrons B. Isochrons computed with numerical continuation of a BVP C. Illustration of the method III. WINFREEâ€™S ISOCHRONS OF THE FITZHUGH-NAGUMO MODEL IV. GEOMETRIC PROPERTIES OF WINFREE'S ISOCHRONS A. The slow manifold B. Computing the slow manifold C. The role of the slow manifold V. DISCUSSION

### Key Topics

- Manifolds
- 42.0
- Eigenvalues
- 17.0
- Oscillators
- 12.0
- Phase space methods
- 11.0
- Boundary value problems
- 9.0

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