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Abstract

Neural networks with long-range connectivity are known to display critical
behavior including power-law activity fluctuations. It has been shown earlier by
the authors that precisely specified connections strengths are sufficient to
produce this behavior. The predictions of the model have been observed experimentally both in neural cultures and in slices. In a more realistic model which includes synaptic dynamics on short time scales, the system regulates itself to the critical point. In the present contribution we proceed a further step by imposing a slow adaptive dynamics to the network which achieves criticality
by a learning process. The learning rule is presented on a firm mathematical basis within the theory of branching processes, which have have been used in similar contexts as an approximation of the true systems dynamics. Branching processes are Markovian, i.e. may be appropriate for feed-forward networks, and model only how many neurons are active at a time. For instance, these idealized processes have been used to model threshold systems with recurrent connections and which exhibit correlations even without affecting a threshold crossing in the target of the connection. We prove the asymptotic equivalence between avalanches in the dynamics of globally connected networks of spiking neurons and avalanches generated by a Galton-Watson branching process. We use the established equivalence to explain why the critical exponent assumes the value -3/2 and how it results from specific coupling strengths. In the present case we are able to ensure rigorously the validity of the approximation of the network dynamics by a branching process such that the learning rule is automatically eligible for the purpose of guiding the neural network towards the edge of criticality. The theoretically predicted behavior is confirmed in all aspects by
numerical simulations.