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

Abstract

Many events are followed by an absolute refractory state, when for some time after the event a repetition of a similar event is impossible. If uniform events, each of which is followed by the same period of absolute refractoriness, occur randomly, as in the Bernoulli scheme, then the event probability as a function of time can exhibit damped transient oscillations caused by a specific initial condition. Here we give an exact analytical description of the oscillations, with a focus on application within neuroscience. The resulting formulas stand out for their relative simplicity, enabling analytical calculation of the damping coefficients for the second and third peaks of the event probability.

Emergent population activity in metric-free and metric networks of neurons with stochastic spontaneous spikes and dynamic synapses

“…1 in Ref. [65]) with the period determined by τ ref . One can imagine that these transient oscillations could lead to sustained ones due to the small but nonlinear influence of synaptic coupling.…”

“…is the sequence of natural numbers, t 0 = 0, and t (i,j) sp is the moment when i-th neuron generates its j-th spike. In the limiting case t avg = t for N disconnected, independently spiking neurons with the same initial condition, one can obtain an exact linear recurrent sequence for A k analytically (with n ref = τ ref / t) [65]:…”

“…the neuron emits a spike (provided that this neuron has not emitted spike a little earlier and is not in the refractory state). The presence of the absolute refractory period τ ref decreases the probability of spontaneous spike generation per Δ t to the average value [65] which determines the average spiking rate .…”

“…The normalized spiking activity of the neuronal network is defined as a ratio of the number of neurons that emitted spikes during some averaging interval t avg to the total number N of network neurons: where t k = kt avg , k = 1, 2,… is the sequence of natural numbers, t 0 = 0, and is the moment when i -th neuron generates its j -th spike. In the limiting case t avg = Δ t for N disconnected, independently spiking neurons with the same initial condition, one can obtain an exact linear recurrent sequence for A k analytically (with n ref = τ ref / Δ t ) [65]: Here A k has an accurate meaning of spike generation probability for a statistical ensemble of N neurons at k -th time step. At k → ∞ this converges to A ∞ = p sp / (1 + p sp n ref ), i.e.…”

“…Note that Eq. (6) describes transient damped oscillations in the general case [65].…”