Dynamics of large cooperative pulsed-coupled networks

Eleonora Catsigeras

doi:10.3934/jdg.2014.1.255

Article Contents

2014, Volume 1, Issue 2: 255-281. Doi: 10.3934/jdg.2014.1.255

This issue Previous Article Approachability, regret and calibration: Implications and equivalences Next Article A strategic market game approach for the private provision of public goods

Dynamics of large cooperative pulsed-coupled networks

Eleonora Catsigeras^1,

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

Received: January 31, 2013

Revised: November 30, 2013

Published: March 2014

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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.

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Mathematics Subject Classification: Primary: 37NXX, 92B20; Secondary: 34D06, 05C82, 94A17, 92B25.

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