Critical Dynamics in Complex Excitable Networks

We study the effect of network structure on the dynamical response of networks of coupled discrete-state excitable elements to two distinct types of stimulus. First, we consider networks which are stochastically stimulated by an external source. Such systems have been used as toy models for the dynamics of some human sensory neuronal networks and neuron cultures. The collective dynamics of such systems depends on the topology of the connections in the network. Here we develop a general theoretical approach to study the effects of network topology on dynamic range, which quantifies the range of stimulus intensities resulting in distinguishable network responses. We find that the largest eigenvalue of the weighted network adjacency matrix governs the network dynamic range. Specifically, a largest eigenvalue equal to one corresponds to a critical regime with maximum dynamic range. This result appears to hold for random, all-to-all, and scale free topologies, and is robust to the inclusion of time delays and refractory states. We gain deeper insight into the effects of network topology using a nonlinear analysis in terms of additional spectral properties of the adjacency matrix. We find that homogeneous networks can reach a higher dynamic range than those with heterogeneous topology. Second, we consider networks stimulated only once at a single node, with dynamics allowed to evolve without additional stimulus. Each realization of such a process will create a cascade of activity of varying duration and size. We analyze the distributions of cascade size and duration in complex networks resulting from a single nodal excitation, finding that when the largest eigenvalue is equal to one, so-called ``critical avalanches'' are power-law distributed in size, with exponent -3/2, and power-law distributed in duration, with exponent -2. We employ techniques from dynamical systems to recover the differences among avalanches started at different network nodes, also deriving distributions for avalanches in subcritical and supercritical regimes.

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