The global geometry of phase-resetting surfaces: the role of critical level sets and isochrons
Peter Langfield, Bernd Krauskopf, Kyoung Hyun Lee, and Hinke M. Osinga
Abstract
Given a system with a stable oscillation represented by an underlying attracting periodic orbit Γ, what is its reaction to perturbations in a fixed direction d for any amplitude A of the perturbation? Such phase resets are often expressed by the phase transition curve (PTC) that, for fixed A ≥ 0, relates the `old' or `original' phase ϑo in the cycle, at which the perturbation is applied, to the `new' phase ϑn one observes after relaxation back to the stable oscillation. It is well known that the nature of the PTC changes with A, and this can be represented by the phase-resetting surface, defined as the graph of the function Pd(ϑo, A) that assigns ϑn to every combination of (ϑo, A)-pairs for which the associated phase reset in the direction d returns to Γ.
We study the phase-resetting surface graph(Pd(ϑo, A)) by considering the properties of its level sets. This work is motivated by sketches from the 1970s of such level sets for the four-dimensional Hodgkin–Huxley model; Eric Best drew these sketches by hand, after performing numerical integrations over a grid of perturbations in the (ϑo, A)-plane. Best accurately captured the general arrangement of level sets, but was not able to resolve them in certain regions. By making use of a numerical setup based on the formulation of suitable boundary value problems, we are able to complete Best's sketch via the computation of many level sets of the phase-resetting surface directly as curves in the (ϑo, A)-plane.
The theoretical underpinning for this approach is the realization that the level sets are the intersection sets of a half-cylinder, comprising translations of Γ in the direction d, with the isochrons of Γ. The isochrons are submanifolds of codimension one consisting of all points that approach Γ with a given phase, and they foliate the basin of attraction of Γ. With our geometric approach, we identify critical level sets and associated isochrons characterized by certain tangencies with the two-dimensional half-cylinder. We illustrate with two examples of planar vector fields that critical level sets are generally a geometric necessity and, hence, do not merely arise because the Hodgkin–Huxley model is of higher dimension.
The work presented here also demonstrates, from a more general perspective, that computing (level sets of) graph(Pd(ϑo, A)) for one or several suitable directions d is an effective way to investigate foliations of basins of attraction by isochrons of dimensions two or higher.
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