Avalanches in a Stochastic Model of Spiking Neurons

Neuronal avalanches are a form of spontaneous activity widely observed in cortical slices and other types of nervous tissue, both in vivo and in vitro. They are characterized by irregular, isolated population bursts when many neurons fire together, where the number of spikes per burst obeys a power law distribution. We simulate, using the Gillespie algorithm, a model of neuronal avalanches based on stochastic single neurons. The network consists of excitatory and inhibitory neurons, first with all-to-all connectivity and later with random sparse connectivity. Analyzing our model using the system size expansion, we show that the model obeys the standard Wilson-Cowan equations for large network sizes ( neurons). When excitation and inhibition are closely balanced, networks of thousands of neurons exhibit irregular synchronous activity, including the characteristic power law distribution of avalanche size. We show that these avalanches are due to the balanced network having weakly stable functionally feedforward dynamics, which amplifies some small fluctuations into the large population bursts. Balanced networks are thought to underlie a variety of observed network behaviours and have useful computational properties, such as responding quickly to changes in input. Thus, the appearance of avalanches in such functionally feedforward networks indicates that avalanches may be a simple consequence of a widely present network structure, when neuron dynamics are noisy. An important implication is that a network need not be “critical” for the production of avalanches, so experimentally observed power laws in burst size may be a signature of noisy functionally feedforward structure rather than of, for example, self-organized criticality.Author Summary: Networks of neurons display a broad variety of behavior that nonetheless can often be described in very simple statistical terms. Here we explain the basis of one particularly striking statistical rule: that in many systems, the likelihood that groups of neurons burst, or fire together, is linked to the number of neurons involved, or size of the burst, by a power law. The wide-spread presence of these so-called avalanches has been taken to mean that neuronal networks in general operate near criticality, the boundary between two different global behaviors. We model these neuronal avalanches within the context of a network of noisy excitatory and inhibitory neurons interconnected by several different connection rules. We find that neuronal avalanches arise in our model only when excitatory and inhibitory connections are balanced in such a way that small fluctuations in the difference of population activities feed forward into large fluctuations in the sum of activities, creating avalanches. In contrast with the notion that the ubiquity of neuronal avalanches implies that neuronal networks operate near criticality, our work shows that avalanches are ubiquitous because they arise naturally from a network structure, the noisy balanced network, which underlies a wide variety of models.