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A rigorous stochastic theory for spike pattern formation in recurrent neural networks with arbitrary connection topologies

Cortical networks exhibit synchronized activity which often occurs in spontaneous events in the form of spike avalanches. Since synchronization has been causally linked to central aspects of brain function such as selective signal processing and integration of stimulus information, participating in an avalanche is a form of a transient synchrony which temporarily creates neural assemblies and hence might especially be useful for implementing flexible information processing. For understanding how assembly formation supports neural computation, it is therefore essential to establish a comprehensive theory of how network structure and dynamics interact to generate specific avalanche patterns and sequences. Here we derive exact avalanche distributions for a finite network of recurrently coupled spiking neurons with arbitrary non-negative interaction weights, which is made possible by formally mapping the model dynamics to a linear, random dynamical system on the $N$-torus and by exploiting self-similarities inherent in the phase space. We introduce the notion of relative unique ergodicity and show that this property is guaranteed if the system is driven by a time-invariant Bernoulli process. This approach allows us not only to provide closed-form analytical expressions for avalanche size, but also to determine the detailed set(s) of units firing in an avalanche (i.e., the avalanche assembly). The underlying dependence between network structure and dynamics is made transparent by expressing the distribution of avalanche assemblies in terms of the induced graph Laplacian. We explore analytical consequences of this dependence and provide illustrating examples.