Pulse-coupled decentral synchronization

Synchronous firing of biological oscillators is explained hy a mathematical model developed by Mirollo and Strogatz [SIAM J. Appl. Math., 50 (1990), pp. 1645-1662]. Exactly the same principles may be carried over to achieve decentral synchronization in time-division multiple access systems. However, some extensions of the basic model are necessary in view of technical implementations. This paper deals with synchronization ill finite time for two oscillators and convergence to synchrony except on a set of initial conditions of measure zero for cliques of oscillators, particularly for nondifferentiable (e.g., piecewise linear) state functions. Moreover, for oscillator pairs we prove sychronization even if there is a certain delay between transmission and reception of pulses. A simulation tool, named ImDeSy, is developed to investigate general networks of oscillators with arbitrary topologies and more refined reaction strategies. This is demonstrated in two extensive examples.