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.

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.

Reviewer: Klaus R. Schneider (Berlin)

##### 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 |