We analyze the dynamics of a large network of coupled quadratic integrate-and-fire neurons, which represent the canonical model for class I neurons near the spiking threshold. The network is heterogeneous in that it includes both inherently spiking and excitable neurons. The coupling is global via synapses that take into account the finite width of synaptic pulses. Using a recently developed reduction method based on the Lorentzian ansatz, we derive a closed system of equations for the neuron's firing rate and the mean membrane potential, which are exact in the infinite-size limit. The bifurcation analysis of the reduced equations reveals a rich scenario of asymptotic behavior, the most interesting of which is the macroscopic limit-cycle oscillations. It is shown that the finite width of synaptic pulses is a necessary condition for the existence of such oscillations. The robustness of the oscillations against aging damage, which transforms spiking neurons into nonspiking neurons, is analyzed. The validity of the reduced equations is confirmed by comparing their solutions with the solutions of microscopic equations for the finite-size networks.

No abstract

The act-and-wait control algorithm is proposed to suppress synchrony in globally coupled oscillatory networks in the situation when the simultaneous registration and stimulation of the system is not possible. The algorithm involves the periodic repetition of the registration (wait) and stimulation (act) stages, such that in the first stage the mean field of the free system is recorded in a memory and in the second stage the system is stimulated with the recorded signal. A modified version of the algorithm that takes into account the charge-balanced requirement is considered as well. The efficiency of our algorithm is demonstrated analytically and numerically for globally coupled Landau-Stuart oscillators and synaptically all-to-all coupled FitzHugh-Nagumo as well as Hodgkin-Huxley neurons.

We consider the FitzHugh-Nagumo model axon under action of a homogeneous high-frequency stimulation (HFS) current. Using a multiple scale method and a geometrical singular perturbation theory, we derive analytically the main characteristics of the traveling pulse. We show that the effect of HFS on the axon is determined by a parameter proportional to the ratio of the amplitude to the frequency of the stimulation current. When this parameter is increased, the pulse slows down and shrinks. At some threshold value, the pulse stops and its width becomes zero. The HFS prevents the pulse propagation when the parameter exceeds the threshold value. The analytical results are confirmed by numerical experiments performed with the original system of partial differential equations.

No abstract