Spike-phase coupling as an order parameter in a leaky integrate-and-fire model

It is known that the leaky integrate-and-fire neural model shows a transition from irregular to synchronous firing by increasing the coupling between the neurons. However, a quantitative characterization of this order-disorder transition, that is, the determination of the order of transition and also the critical exponents in the case of continuous transition, is not entirely known. In this work, we consider a network of N excitatory neurons with local connections, residing on a square lattice with periodic boundary conditions. The cooperation between neurons K plays the role of the control parameter that generates criticality when the critical cooperation strength K_{c} is adopted. We introduce the population-averaged voltage (PAV) as a representative value of the network's cooperative activity. Then, we show that the coupling between the timing of spikes and the phase of temporal fluctuations of PAV defined as m resorts to identify a Kuramoto order parameter. By increasing K, we find a continuous transition from irregular spiking to a phase-locked state at the critical point K_{c}. We deploy the finite-size scaling analysis to calculate the critical exponents of this transition. To explore the formal indicator of criticality, we study the neuronal avalanches profile at this critical point and find a scaling behavior with the exponents in a fair agreement with the experimental values both in vivo and in vitro.