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Self-excited relaxation oscillations in networks of impulse neurons. (English. Russian original) Zbl 1361.34093
Russ. Math. Surv. 70, No. 3, 383-452 (2015); translation from Usp. Mat. Nauk. 70, No. 3, 3-76 (2015).
The first part of the paper presents some known ODE-models describing the activity of a neuron: Hodgkin-Huxley, FitzHugh-Nagumo, FitzHugh-Rinzel, Morris-Lecar, Hindmarsh-Rose which are in general singularly perturbed systems. For networks of neurons the Hopfields models are described: systems of ODEs and systems of differential-delay equations. Finally, the authors propose a new phenomenological model of a neuron which consists of a scalar differential-delay equation with one or two delays.
In the second part of the paper, the authors study the one-delay model ${du\over dt}= \lambda f(u(t-1))\,u.\tag{1}$ Under certain assumptions on $$f$$ they prove that (1) has a stable relaxation cycle $$u_*(t,\lambda)$$ for $$\lambda\gg 1$$. Part 3 is devoted to networks of coupled neurons described by (1): ${du_j\over dt}= d(u_{j+1}-2u_j+ u_{j-1})+\lambda f(u_j(t-1))\,u_j\tag{2}$ with $$j= 1,\dots,m$$, $$u_0= u_1$$, $$u_{m+1}= u_m$$, $$\lambda\gg 1$$, $$d>0$$. The authors prove that (2) has at least $$m$$ exponentially orbitally stable periodic solutions.
In the last part of the paper, the authors investigate the two-delays model ${du\over dt}= \lambda[f(u(t-h))-g(u(t-1))]\,u.\tag{3}$ They prove that to any integer $$N$$ the parameters involved in (3) can be chosen in such a way that (3) has at least $$N$$ exponentially orbitally stable periodic solutions.

##### MSC:
 34K60 Qualitative investigation and simulation of models involving functional-differential equations 34K26 Singular perturbations of functional-differential equations 92C20 Neural biology 34K13 Periodic solutions to functional-differential equations 34K20 Stability theory of functional-differential equations
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