Noise-induced phase transitions in neuronal networks

Using an exactly solvable cortical model of a neuronal network, we show that, by increasing the intensity of shot noise (flow of random spikes bombarding neurons), the network undergoes first-and second-order non-equilibrium phase transitions. We study the nature of the transitions, bursts and avalanches of neuronal activity. Saddle-node and supercritical Hopf bifurcations are the mechanisms of emergence of sustained network oscillations. We show that the network stimulated by shot noise behaves similar to the Morris-Lecar model of a biological neuron stimulated by an applied current. In the brain, interactions among neurons lead to diverse collective phenomena such as phase transitions, self-organization, avalanches, and brain rhythms [1, 2]. A non-equilibrium second-order phase transition was observed in human bimanual coordination [3–5]. Stimulation of living neural networks by electric fields causes a first-order phase transition [6]. There are evidences that brain rhythms, epileptic seizures, and the ultraslow oscillations of BOLD fMRI patterns also emerge as a result of non-equilibrium phase transitions [7]. The Hopf and saddle-node bifurcations are generic mechanisms for the emergence of oscillations in nonlinear differential equation models [8]. These mechanisms were found in the Morris-Lecar model of a biological neuron [9, 10]. In the context of brain rhythms, the Hopf bifurcations were discussed within mean-field cortical models [7] and for networks of randomly connected integrate-and-fire neurons [11–14]. At the present time, understanding of nature of collective phenomena in the brain is elusive. For a complete description of a phase transition it is not enough to identify the bifurcation. It is also necessary to find critical phenomena that accompany it. In statistical physics, exactly solved models largely help us to understand phase transitions and critical phenomena [15]. In the present paper, we propose an exactly solvable cortical model with stochastic excitatory and inhibitory neurons stimulated by shot noise (a flow of random spikes bombarding neurons). We show that shot noise stimulates first-and second-order non-equilibrium phase transitions in collective dynamics of neuronal networks. The first-order phase transition occurs as a discontinuous transition from low to high neuronal activity. Avalanches precede the transition. The saddle-node and supercritical Hopf bifurcations are the mechanism of emergence of sustained network oscillations. We find the order parameter for the continuous phase transitions and critical phenomena that accompany them in activity fluctuations. We show that the model exhibits collective excitability similar to excitability of the Morris-Lecar neuron stimulated by an applied current. Model. We study a cortical model …