Threshold, excitability and isochrones in the Bonhoeffer–van der Pol system

Abstract

Some new insight is obtained for the structure of the Bonhoeffer-van der Pol system. The problems of excitability and threshold are discussed for all three types of the system classified according to the existing attractors: a focus only, a limit cycle only and a limit cycle together with a focus. These problems can be treated by the T-repellers and the T-attractors of the system which are mutually reciprocal under time inversion. The threshold depends on the structure of the T-repeller (unstable part of integral manifold). This structure is then used to understand the behavior and the properties of the two different types of isochrones: Winfree isochrones (W-isochrones) and regular isochrones. Winfree's description of a W-isochrone is extended to excitable systems. Both W-isochrones and regular isochrones are calculated for the Bonhoeffer-van der Pol system in its limit cycle and excitable regimes. The important role of the T-repeller as an asymptotic limit for both types of isochrones is manifested. (c) 1999 American Institute of Physics.

Period-adding bifurcations and chaos in a periodically stimulated excitable neural relaxation oscillator

“…In this case the system spends all of its time on the branches S = 0 and S = 1. Following the work of Rabinovitch [24,25] and later work by Ichinose et al [26] and Yoshino et al [27] we define an extended isochron as a set of states synchronously approaching to an asymptotically stable fixed point. The isochronal co-ordinate, τ (w, S), with origin at (v, w) = ((1 + α)/2, w 2 ) is considered to be…”

“…Importantly, it has been established that some of the techniques for dealing with oscillatory systems may be taken over to the excitable regime. Notably work by Rabinovitch et al extends the concept of isochronal coordinates to excitable systems with specific application to the forced Bonhoeffer-van der Pol oscillator in its excited mode [24,25]. This has been extended to the case of neural systems by Ichinose et al [26] and Yoshino et al [27].…”

Converging to and escaping from the global equilibrium: Isostables and optimal control

“…(Note that a similar notion was introduced in [10] in the particular case of slow-fast systems.) In this section, we review the concept of isostables for both linear and nonlinear systems and we highlight their relationship with the so-called Koopman operator.…”

“…In particular, we show that the relevant end cost function for the problem-to be maximized when the control is switched off-is based on the notion of isostables. Introduced in our recent work [5] (see also [10] for slow-fast systems), the isostables are sets of the state space that capture the asymptotic behavior of the uncontrolled system. They provide a unique and rigorous measure of how far-with respect to time-the trajectory is from the equilibrium.…”

Changes in the dynamical behavior of nonlinear systems induced by noise

“…(4.80) for I "0.2 has no repellers and only a single attractor: the stable singular point. However, due to their similarity to an ordinary repeller and attractor they have been called [114] a transient repeller and transient attractor, respectively. We will follow this convention.…”