Refractory period in network models of excitable nodes: self-sustaining stable dynamics, extended scaling region and oscillatory behavior

Abstract

Networks of excitable nodes have recently attracted much attention particularly in regards to neuronal dynamics, where criticality has been argued to be a fundamental property. Refractory behavior, which limits the excitability of neurons is thought to be an important dynamical property. We therefore consider a simple model of excitable nodes which is known to exhibit a transition to instability at a critical point (λ = 1), and introduce refractory period into its dynamics. We use mean-field analytical calculations as well as numerical simulations to calculate the activity dependent branching ratio that is useful to characterize the behavior of critical systems. We also define avalanches and calculate probability distribution of their size and duration. We find that in the presence of refractory period the dynamics stabilizes while various parameter regimes become accessible. A sub-critical regime with λ < 1.0, a standard critical behavior with exponents close to critical branching process for λ = 1, a regime with 1 < λ < 2 that exhibits an interesting scaling behavior, and an oscillating regime with λ > 2.0. We have therefore shown that refractory behavior leads to a wide range of scaling as well as periodic behavior which are relevant to real neuronal dynamics.

Spike-Timing-Dependent Plasticity With Axonal Delay Tunes Networks of Izhikevich Neurons to the Edge of Synchronization Transition With Scale-Free Avalanches

“…However, in some recent studies, it has been indicated that the brain is maintained near a synchronization transition (Poil et al, 2012; Gautam et al, 2015; di Santo et al, 2018; Dalla Porta and Copelli, 2019; Fontenele et al, 2019). We note that some authors have also argued for the existence of the extended critical region similar to that of “Griffiths phase” (Munoz et al, 2010; Moretti and Munoz, 2013; Odor et al, 2015; Moosavi et al, 2017). However, such critical regions also typically occur near the absorbing phase transition where the system transitions from an inactive phase to an active phase.…”

“…Therefore, one expects b ( M ) < 1 to generally indicate subcritical behavior, while b ( M ) > 1 to indicate supercritical behavior. In fact, b ( M ) has been used to ascertain criticality in a wide range of systems including sandpile models of SOC or solar flares (Martin et al, 2010) as well as neural networks (Larremore et al, 2014; Moosavi and Montakhab, 2014; Moosavi et al, 2017).…”

Beta-Rhythm Oscillations and Synchronization Transition in Network Models of Izhikevich Neurons: Effect of Topology and Synaptic Type

“…Specifically, we will focus on the order of the emerging phase transition for various network structures and different synaptic interactions. It is believed that normal brain activity requires it to be close to a phase boundary (a critical point) which consequently provide access to both synchronous and asynchronous oscillations with small change in the input (Beggs and Timme, 2012 ; Hesse and Gross, 2014 ; Moosavi et al, 2017 ). Hence, it is important to know whether the emerging synchronization transition is continuous or abrupt.…”

Damped oscillations of the probability of random events followed by absolute refractory period: exact analytical results

“…In turn, a wide spectrum of network-based cellular automata models, which typically constitute randomly-connected simple excitable units with nonlinear interaction, has been elaborated for large-scale numerical and/or analytically-tractable studies of critical collective behavior via avalanches, emergent synchronization and global oscillations of neuronal activity [3537,53] (some earlier ndings were reviewed in [54]). In addition to the foregoing models, some relevant models, where interference between the absolute refractory period and stochastic event occurrence may be essential, are the two-state unit model [55], the threestate unit model [5658] (see also [59,60]), the DeVille-Peskin multi-state model [61], the cortical branching model [62], and others [6365].…”