Journal of Dynamics and Games (JDG)

Dynamics of large cooperative pulsed-coupled networks

Pages: 255 - 281, Volume 1, Issue 2, April 2014      doi:10.3934/jdg.2014.1.255

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Eleonora Catsigeras - Instituto de Matemática y Estadística Rafael Laguardia, Universidad de la República, Av. Herrera y Reissig 565, C.P.11300, Montevideo, Uruguay (email)

Abstract: We study the deterministic dynamics of networks ${\mathcal N}$ composed by $m$ non identical, mutually pulse-coupled cells. We assume weighted, asymmetric and positive (cooperative) interactions among the cells, and arbitrarily large values of $m$. We consider two cases of the network's graph: the complete graph, and the existence of a large core (i.e. a large complete subgraph). First, we prove that the system periodically eventually synchronizes with a natural "spiking period" $p \geq 1$, and that if the cells are mutually structurally identical or similar, then the synchronization is complete ($p= 1$) . Second, we prove that the amount of information $H$ that ${\mathcal N}$ generates or processes, equals $\log p$. Therefore, if ${\mathcal N}$ completely synchronizes, the information is null. Finally, we prove that ${\mathcal N}$ protects the cells from their risk of death.

Keywords:  Pulse-coupled networks, impulsive ODE, cooperative evolutive game, synchronization, information.
Mathematics Subject Classification:  Primary: 37NXX, 92B20; Secondary: 34D06, 05C82, 94A17, 92B25.

Received: February 2013;      Revised: December 2013;      Available Online: March 2014.