# The Solution of the Second Peskin Conjecture and Developments

• M. U. Akhmet
Chapter
Part of the Nonlinear Systems and Complexity book series (NSCH, volume 14)

## Abstract

The integrate-and-fire cardiac pacemaker model of pulse-coupled oscillators was introduced by C. Peskin. Because of the pacemaker’s function, two famous synchronization conjectures for identical and nonidentical oscillators were formulated. The first of Peskin’s conjectures was solved in the paper (J. Phys. A 21:L699–L705, 1988) by S. Strogatz and R. Mirollo. The second conjecture was solved in the paper by Akhmet (Nonlinear Stud. 18:313–327, 2011). There are still many issues related to the nature and types of couplings. The couplings may be impulsive, continuous, delayed, or advanced, and oscillators may be locally or globally connected. Consequently, it is reasonable to consider various ways of synchronization if one wants the biological and mathematical analyses to interact productively. We investigate the integrate-and-fire model in both cases—one with identical and another with not-quite-identical oscillators. A combination of continuous and pulse couplings that sustain the firing in unison is carefully constructed. Moreover, we obtain conditions on the parameters of continuous couplings that make possible a rigorous mathematical investigation of the problem. The technique developed for differential equations with discontinuities at nonfixed moments (Akhmet, Principles of Discontinuous Dynamical Systems, Springer, New York, 2010) and a special continuous map form the basis of the analysis. We consider Peskin’s model of the cardiac pacemaker with delayed pulse couplings as well as with continuous couplings. Sufficient conditions for the synchronization of identical and nonidentical oscillators are obtained. The bifurcation of periodic motion is observed. The results are demonstrated with numerical simulations.

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