Extreme phase sensitivity in systems with fractal isochrons

TL;DR

This paper explores the fractal nature of isochrons in certain periodic systems, revealing how their complex structure influences phase sensitivity and response, with implications for neuronal models.

Contribution

It introduces a global phase sensitivity measure linked to the fractal dimension of isochrons, applicable to both continuous and discrete systems, and demonstrates its relevance in neuronal dynamics.

Findings

Isochrons can have high fractal dimensions affecting phase sensitivity.

The phase sensitivity coefficient is invariant and related to capacity dimension.

Elliptic bursting neurons exhibit high fractal isochrons and sensitive phase responses.

Abstract

Sensitivity to initial conditions is usually associated with chaotic dynamics and strange attractors. However, even systems with (quasi)periodic dynamics can exhibit it. In this context we report on the fractal properties of the isochrons of some continuous-time asymptotically periodic systems. We define a global measure of phase sensitivity that we call the phase sensitivity coefficient and show that it is an invariant of the system related to the capacity dimension of the isochrons. Similar results are also obtained with discrete-time systems. As an illustration of the framework, we compute the phase sensitivity coefficient for popular models of bursting neurons, suggesting that some elliptic bursting neurons are characterized by isochrons of high fractal dimensions and exhibit a very sensitive (unreliable) phase response.