Growing critical: Self-organized criticality in a developing neural system
A variety of neural systems generate neuronal avalanches: bursts of activity with characteristics typical for critical dynamics. A hallmark of such systems is the variability of the activity, which may be functionally relevant: An individual avalanche can involve any number of neurons, from one to the entire network; avalanches of all sizes occur with non-negligible frequency. A possible explanation for the occurrence of neuronal avalanches is that the underlying networks organize themselves into a critical state. Here we propose a simple spiking model for a developing neural system. It shows how networks may “grow into” the critical state during development. Consistent with earlier network growth models, the extents of neurites are represented by discs, with synaptic coupling strengths proportional to the discs’ overlap; neurons are modeled as inhomogeneous, coupled Poisson processes. Inputs increase the Poisson spike rate of a neuron. Neurites grow if a neuron is silent and shrink when spikes are generated. Both processes balance at some spike rate, the network becomes stationary. Our analysis and numerical simulations show that the larger the stationary state’s spike rate is compared to the spontaneous one of isolated neurons, the nearer the stationary network is to the critical point. The avalanche size distribution approaches a power law distribution with exponent -3/2. We also derive the avalanche duration distribution analytically and show that its tail approaches a power law with exponent -2. Both exponents have been reported in experiments. Refractoriness of neurons can be included, but tends to drive the system away from criticality. In the stationary state, our model dynamics is that of a Markovian Hawkes process, which is used in life sciences, finance, social sciences and geophysics to model clustering phenomena. In particular our derivation of the avalanche duration distribution may thus prove useful in other scientific fields as well.