On born of in-phase limit cycle in ensemble of excitatory coupled FitzHugh-Nagumo elements

Alexander G Korotkov,Tatiana A Levanova

doi:10.15507/2079-6900.20.201803.273-281

Abstract

We proposed and studied numerically efficient phenomenological model of ensemble of two FitzHugh-Nagumo neuron-like elements that are coupled by symmetric synaptic excitatory coupling. This coupling is defined by function that depends on phase of active element and that is smooth approximation of rectangular impulse function. Above-mentioned coupling depends on three parameters that define the beginning of element activation, the duration of the activation and the coupling strength. We show analytically that in the phase space of the model there exists stable in-phase limit cycle that corresponds to regular oscillations with equal phases and frequencies of elements. It is proved that this limit cycle is a result of supercritical Andronov-Hopf bifurcation. The chart of activity regimes is depicted on the plane of parameters that define beginning and duration of activation. The boundaries of bifurcations that lead to birth of this cycle are found.

Highlights

We proposed and studied numerically ecient phenomenological model of ensemble of two FitzHugh-Nagumo neuron-like elements that are coupled by symmetric synaptic excitatory coupling
This coupling is dened by function that depends on phase of active element and that is smooth approximation of rectangular impulse function
We show analytically that in the phase space of the model there exists stable in-phase limit cycle that corresponds to regular oscillations with equal phases and frequencies of elements

Summary

Ñîñòîÿíèÿ ðàâíîâåñèÿ â ñèñòåìå îïðåäåëÿþòñÿ èç ñîîòíîøåíèé

Â ñîñòîÿíèè ðàâíîâåñèÿ O(a, y0, a, y0) äëÿ ÷àñòíûõ ïðîèçâîäíûõ ôóíêöèè ñâÿçè I (φ) âûïîëíÿþòñÿ ñëåäóþùèå ñîîòíîøåíèÿ: Ix1 (a, y0) = Ix2 (a, y0) = Ix, Iy1 (a, y0) = Iy2 (a, y0) = Iy. Òîãäà õàðàêòåðèñòè÷åñêîå óðàâíåíèå (3.2) ìîæåò áûòü ïðåîáðàçîâàíî ê âèäó a2 ). 0, ïîëó÷èì äâà óñëîâèÿ, îïðåäåëÿþùèõ áèôóðêàöèè Àíäðîíîâà-Õîïôà: Ix = a2 − 1, a (3.8). Äëÿ ñèíôàçíîãî ïðåäåëüíîãî öèêëà âûïîëíÿþòñÿ óñëîâèÿ x1(t) = x2(t) = x(t) è y1(t) =. Íåÿâíî çàäàííîé ñîîòíîøåíèåì (3.10): dy = 1 − x2 + Ix. Â èòîãå êðèâàÿ áèôóðêàöèè ðîæäåíèÿ öèêëà íàõîäèòñÿ èç óñëîdx. ×òîáû èç ýòèõ æå ñîîáðàæåíèé ïîëó÷èòü óñëîâèå (3.9), íóæíî çàìåòèòü, ÷òî äëÿ ôóíêöèè ñâÿçè I(φ; α, β) âûïîëíÿåòñÿ ðàâåíñòâî I(φ; α, β) ≈ g − I(φ; β, α + 2π) (ïðè÷åì âûïîëíÿåòñÿ ðàâåíñòâî lim (I(φ; α, β) + I(φ; β, α + 2π)) = g).

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