# American Institute of Mathematical Sciences

April  2014, 1(2): 255-281. doi: 10.3934/jdg.2014.1.255

## Dynamics of large cooperative pulsed-coupled networks

 1 Instituto de Matemática y Estadística Rafael Laguardia, Universidad de la República, Av. Herrera y Reissig 565, C.P.11300, Montevideo

Received  February 2013 Revised  December 2013 Published  March 2014

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.
Citation: Eleonora Catsigeras. Dynamics of large cooperative pulsed-coupled networks. Journal of Dynamics & Games, 2014, 1 (2) : 255-281. doi: 10.3934/jdg.2014.1.255
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##### References:
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