Discrete and Continuous Dynamical Systems - Series B (DCDS-B)

Digraphs vs. dynamics in discrete models of neuronal networks

Pages: 1365 - 1381, Volume 17, Issue 5, July 2012      doi:10.3934/dcdsb.2012.17.1365

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Sungwoo Ahn - Department of Mathematical Sciences, Indiana University-Purdue University Indianapolis, IN 46202, United States (email)
Winfried Just - Department of Mathematics, Ohio University, OH 45701, United States (email)

Abstract: 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.

Keywords:  Discrete dynamical system, neuronal network, attractor, transient, directed cycle.
Mathematics Subject Classification:  05C20, 05C82, 37F20, 92C20, 92C42.

Received: January 2011;      Revised: December 2011;      Available Online: March 2012.