Approximating network dynamics: Some open problems

Winfried Just

doi:10.3934/dcdsb.2018188

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2018, Volume 23, Issue 5: 1917-1930. Doi: 10.3934/dcdsb.2018188

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Approximating network dynamics: Some open problems

Winfried Just

Department of Mathematics, Ohio University, Athens, OH 45701, USA

Received: April 2017

Revised: June 2017

Early access: May 2018

Published: May 2018

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Abstract

Discrete-time finite-state dynamical systems on networks are often conceived as tractable approximations to more detailed ODE-based models of natural systems. Here we review research on a class of such discrete models $N$ that approximate certain ODE models $M$ of mathematical neuroscience. In particular, we outline several open problems on the dynamics of the models $N$ themselves, as well as on structural features of ODE models $M$ that allow for the construction of discrete approximations $N$ whose predictions will be consistent with those of $M$.

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Mathematics Subject Classification: Primary: 05C80, 37C50; Secondary: 37N25, 92B20.

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