Phase transitions and critical behavior are crucial issues both in theoretical and experimental neuroscience. We report analytic and computational results about phase transitions and self-organized criticality (SOC) in networks with general stochastic neurons. The stochastic neuron has a firing probability given by a smooth monotonic function Φ(V) of the membrane potential V, rather than a sharp firing threshold. We find that such networks can operate in several dynamic regimes (phases) depending on the average synaptic weight and the shape of the firing function Φ. In particular, we encounter both continuous and discontinuous phase transitions to absorbing states. At the continuous transition critical boundary, neuronal avalanches occur whose distributions of size and duration are given by power laws, as observed in biological neural networks. We also propose and test a new mechanism to produce SOC: the use of dynamic neuronal gains – a form of short-term plasticity probably located at the axon initial segment (AIS) – instead of depressing synapses at the dendrites (as previously studied in the literature). The new self-organization mechanism produces a slightly supercritical state, that we called SOSC, in accord to some intuitions of Alan Turing.

Recent experiments suggested that homeostatic regulation of synaptic balance leads the visual system to recover and maintain a regime of power-law avalanches. Here we study an excitatory/inhibitory (E/I) mean-field neuronal network that has a critical point with power-law avalanches and synaptic balance. When short term depression in inhibitory synapses and firing threshold adaptation are added, the system hovers around the critical point. This homeostatically self-organized quasi-critical (SOqC) dynamics generates E/I synaptic current cancellation in fast time scales, causing fluctuation-driven asynchronous-irregular (AI) firing. We present the full phase diagram of the model without adaptation varying external input versus synaptic coupling.This system has a rich dynamical repertoire of spiking patterns: synchronous regular (SR), asynchronous regular (AR), synchronous irregular (SI), slow oscillations (SO) and AI. It also presents dynamic balance of synaptic currents, since inhibitory currents try and compensate excitatory currents over time, resulting in both of them scaling linearly with external input.Our model thus unifies two different perspectives on cortical spontaneous activity: both critical avalanches and fluctuation-driven AI firing arise from SOqC homeostatic adaptation, and are indeed two sides of the same coin.

Networks of stochastic spiking neurons are interesting models in the area of theoretical neuroscience, presenting both continuous and discontinuous phase transitions. Here, we study fully-connected networks analytically, numerically and by computational simulations. The neurons have dynamic gains that enable the network to converge to a stationary slightly supercritical state (self-organized supercriticality (SOSC)) in the presence of the continuous transition. We show that SOSC, which presents power laws for neuronal avalanches plus some large events, is robust as a function of the main parameter of the neuronal gain dynamics. We discuss the possible applications of the idea of SOSC to biological phenomena like epilepsy and Dragon-king avalanches. We also find that neuronal gains can produce collective oscillations that coexist with neuronal avalanches.