Excitable scale free networks

Abstract

When a simple excitable system is continuously stimulated by a Poissonian external source, the response function (mean activity versus stimulus rate) generally shows a linear saturating shape. This is experimentally verified in some classes of sensory neurons, which accordingly present a small dynamic range (defined as the interval of stimulus intensity which can be appropriately coded by the mean activity of the excitable element), usually about one or two decades only. The brain, on the other hand, can handle a significantly broader range of stimulus intensity, and a collective phenomenon involving the interaction among excitable neurons has been suggested to account for the enhancement of the dynamic range. Since the role of the pattern of such interactions is still unclear, here we investigate the performance of a scale-free (SF) network topology in this dynamic range problem. Specifically, we study the transfer function of disordered SF networks of excitable Greenberg-Hastings cellular automata. We observe that the dynamic range is maximum when the coupling among the elements is critical, corroborating a general reasoning recently proposed. Although the maximum dynamic range yielded by general SF networks is slightly worse than that of random networks, for special SF networks which lack loops the enhancement of the dynamic range can be dramatic, reaching nearly five decades. In order to understand the role of loops on the transfer function we propose a simple model in which the density of loops in the network can be gradually increased, and show that this is accompanied by a gradual decrease of dynamic range.

Statistical physics approach to dendritic computation: The excitable-wave mean-field approximation

“…This signal compression property, or enhancement of dynamic range, is a general property of excitable media and has proven very robust against variations in the topology of the medium and the level of modeling, from cellular automata to compartmental conductance-based models [21][22][23][24][25][26][27][28][29][30][31][32][33]. Furthermore, the idea that dynamic range can be enhanced in neuronal excitable media has received support from experiments in very different setups [34,35], which again suggests that the phenomenon is robust.…”

How to enhance the dynamic range of excitatory-inhibitory excitable networks

“…It was shown that such networks have their sensitivity and dynamic range maximized at the critical point of a nonequilibrium phase transition [3]. Later, models on other diverse networks, including those with scale-free, degree-correlated, and assortative topologies, were discussed [17,18]. More recently, a general theoretical approach to study the effects of network topology on dynamic range was presented by Larremore, Shew, and Restrepo [19,20].…”

Criticality at Work: How Do Critical Networks Respond to Stimuli?

“…The ability to process stimulus intensities across several orders of magnitude is maximal at the point where stimuli are amplified as much as possible, but not so much as to engage the network in stable self-sustained activity. That result has been generalized in a number of models, including excitable networks with scale-free topology [6,7], with signal integration where discontinuous transitions are possible [39], or with an interplay between excitatory and inhibitory model neurons [40]. In fact, the mechanism does not even need to involve excitable waves, appearing also in a model of olfactory processing where a disinhibition transition involving inhibitory units takes place [41].…”

“…It is important to emphasize that some of the simplifications employed in this model cannot be taken for granted. The hand-waving argument for = 1 being the critical value of the coupling parameter, for instance, fails if the topology of the network is more structured than a simple Erdős-Rényi random graph [6,7]. The extension of the above results to more general topologies was put forward by Restrepo and collaborators [8,9].…”