Analysis of biological integrate-and-fire oscillators

Abstract

We consider discontinuous dynamics of integrate-and-fire models, which consist of pulse-coupled biological oscillators. A thoroughly constructed map is in the basis of the analysis. Synchronization of non-identical oscillators is investigated. Significant advances for the solution of second Peskin's conjecture have been made. Examples with numerical simulations are given to validate the theoretical results. Perspectives are discussed.

Synchronization of the Cardiac Pacemaker Model with Delayed Pulse-coupling

“…The cardiac pacemaker model of identical and non-identical oscillators with delayed pulse-couplings is investigated in the paper. We apply the method of investigation proposed in [15], which is based on a specially defined map. The map is, in fact a Poincaré map if one considers the identity of oscillators.…”

“…The map is, in fact a Poincaré map if one considers the identity of oscillators. Sufficient conditions are found such that delay involvement in the Peskin's model does not change the synchronization result for identical and non-identical oscillators [1,2,15]. The result has a biological sense, since retardation is often presents in biological processes and if one proves that a phenomenon preserves even with delays, that makes us more confident that the model is adequate to the reality.…”

“…Moreover, the method of treatment of models with delay can be useful for neural networks and earthquake faults [4,5,29,30,31] analysis. All proved assertions can be easily specified with τ = 0, to obtain synchronization of the Peskin's model for identical [1,2] and nonidentical [15] oscillators. In particular, from the theorem of Section 2 one can obtain the result of Example 2.2 of [15].…”

“…All proved assertions can be easily specified with τ = 0, to obtain synchronization of the Peskin's model for identical [1,2] and nonidentical [15] oscillators. In particular, from the theorem of Section 2 one can obtain the result of Example 2.2 of [15].…”

“…In paper [15] we have introduced a new method of investigation of biological oscillators. The method seems to be universal to analyze quite identical integrate-and-fire oscillators.…”