Nonlinear Dynamics of Networks
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The role of network topology on the dynamic range of coupled excitable systems
Juan Restrepo
University of Colorado at Boulder

Abstract:
We study the effect of network structure on the dynamical response of networks of coupled discretestate excitable elements. Such systems have been used as models for the dynamics of some human sensory neuronal networks and neuron cultures. An important characteristic of these systems is the "dynamic range", the range of stimulus strength over which the network response varies significantly. There has been recent interest in explaining the large dynamic range of brain tissue. It has been previously argued that the dynamic range is maximized when the product of mean network degree and transmission probability is one. We show more generally that, for a large class of networks, the dynamic range is maximized when the largest eigenvalue of the connectivity matrix A is one. The entries of the connectivity matrix A_{ij} are the probability of excitation transmission from node j to node i. In addition, by studying a nonlinear model of the excitable network dynamics, we find analytical expressions for the dynamic range in terms of eigenvectors of the matrix Q. Using these approximations, we study analytically and numerically the effect of network topology on the optimum dynamic range. These results provide a better understanding of the sources of enhanced dynamic range of neuronal networks and may be applicable to the study of other systems that can be modeled as a network of coupled excitable elements (e.g., some epidemic models). 
