# American Institute of Mathematical Sciences

July  2012, 17(5): 1365-1381. doi: 10.3934/dcdsb.2012.17.1365

## Digraphs vs. dynamics in discrete models of neuronal networks

 1 Department of Mathematical Sciences, Indiana University-Purdue University Indianapolis, IN 46202, United States 2 Department of Mathematics, Ohio University, OH 45701, United States

Received  January 2011 Revised  December 2011 Published  March 2012

It has recently been shown that discrete-time finite-state models can reliably reproduce the ordinary differential equation (ODE) dynamics of certain neuronal networks. We study which dynamics are possible in these discrete models for certain types of network connectivities. In particular we are interested in the number of different attractors and bounds on the lengths of attractors and transients. We completely characterize these properties for cyclic connectivities and derive additional results on the lengths of attractors in more general classes of networks.
Citation: Sungwoo Ahn, Winfried Just. Digraphs vs. dynamics in discrete models of neuronal networks. Discrete & Continuous Dynamical Systems - B, 2012, 17 (5) : 1365-1381. doi: 10.3934/dcdsb.2012.17.1365
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