Avalanches in a Stochastic Model of Spiking Neurons

Abstract

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.

Landau–Ginzburg theory of cortex dynamics: Scale-free avalanches emerge at the edge of synchronization

“…(59,60). We adopt this well-established model and, for simplicity, keep only the leading terms in a power-series expansion, and rename the constants, yielding the deterministic part of Eq.…”

Power-law statistics and universal scaling in the absence of criticality

“…Newman and colleagues showed that random walks generate several statistics scaling as power-laws [3], Yule introduced a process with broad applications, particularly in evolution, naturally associated to power-law distributions, and Takayasu and collaborators [12][13][14] showed that systems with aggregation and injection naturally generate clusters of size scaling as a power-law with exponent −3/2. In neuroscience, Benayoun, Wallace and Cowan have shown that neuronal networks models in a regime of balance of excitation and inhibition also provide power-law scalings of avalanches [15]. All these mechanisms are independent of any phase transition and arise away from criticality from a particular way of considering a random process.…”

Slow synaptic dynamics in a network: From exponential to power-law forgetting

“…All synapses are therefore excitatory for b > 0, and inhibitory for b < 0. The kind of collective behavior discussed here, either along the critical manifold of at its tricritical endpoint, is therefore different in nature from the chaotic dynamical features which have been emphasized in balanced networks [50][51][52], where excitatory and inhibitory effects balance each other on average.…”