Abstract
We investigate the phenomenon of stochastic bursting in a noisy excitable unit with multiple weak delay feedbacks, by virtue of a directed tree lattice model. We find statistical properties of the appearing sequence of spikes and expressions for the power spectral density. This simple model is extended to a network of three units with delayed coupling of a star type. We find the power spectral density of each unit and the cross-spectral density between any two units. The basic assumptions behind the analytical approach are the separation of timescales, allowing for a description of the spike train as a point process, and weakness of coupling, allowing for a representation of the action of overlapped spikes via the sum of the one-spike excitation probabilities.
Highlights
Bursting, which plays an important role in neuronal communication and synchronization, refers to a dynamical state where a neuron repeatedly fires a relatively regular finite sequence spikes; bursts are separated by epochs where the neuron is in a resting state (Izhikevich 2000; Coombes and Bressloff 2005)
In our previous works (Zheng and Pikovsky 2018, 2019), we have demonstrated that a coherent spike pattern, which we call stochastic bursting, can appear in simple excitable units due to the combined effect of time-delayed feedback and noise
We described stochastic bursting statistically in the case of a single excitable unit (Zheng and Pikovsky 2018) and for networks of unidirectionally delay-coupled units (Zheng and Pikovsky 2019)
Summary
Bursting, which plays an important role in neuronal communication and synchronization, refers to a dynamical state where a neuron repeatedly fires a relatively regular finite sequence spikes; bursts are separated by epochs where the neuron is in a resting state (Izhikevich 2000; Coombes and Bressloff 2005). We described stochastic bursting statistically in the case of a single excitable unit (Zheng and Pikovsky 2018) and for networks of unidirectionally delay-coupled units (Zheng and Pikovsky 2019) What these two cases have in common, is that any two delay-induced kicks do not overlap. This allowed for a full statistical description of the bursting as a point process, where the only parameters are the spontaneous rate of excitation and the probability to excite a follower. The point process model is an idealization based on the timescale separation: It is assumed that the characteristic duration of Biological Cybernetics It becomes more challenging when neurons have multiple feedback or more complex coupling topology, where two or more delay-induced spikes could overlap. The probability current across threshold yields the rate of spontaneous spike excitations: λ=C
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