Relaxation self-oscillations in neuron systems. I
As a model for the electrical activity of a single neuron, the authors consider the scalar differential-delay equation \[ {du\over dt}= \lambda[-1+\alpha f(u(t- 1))-\beta g(u(t))]\,u(t),\tag{\(*\)} \] where the parameter \(\lambda\) is sufficiently large (that means that \((*)\) is a singularly perturbed equation), the positive parameters \(\alpha\) and \(\beta\) satisfy the condition \(\alpha> 1+\beta\), and the continuously differentiable functions \(f\) and \(g\) satisfy the relations \[ \begin{aligned} f(0)= g(0)= 1,\quad 0<\beta g(u)+ 1<\alpha\quad &\text{for }u\geq 0,\\ f(u),\;g(u),\;uf'(u),\;ug'(u)= O\Biggl({1\over u}\Biggr)\quad &\text{as }u\to +\infty.\end{aligned} \] They prove that under these assumptions equation \((*)\) has for sufficiently large \(\lambda\) an exponentially stable cycle \(u= u_*(t,\lambda)\) with period \(T_*(\lambda)\) obeying \[ \lim_{\lambda\to+\infty} T_*(\lambda)= \alpha+ 1+(\beta+ 1)/(\alpha- \beta- 1), \] moreover, the spiking behavior of \(u_*(b,\lambda)\) is described.
scientific article; zbMATH DE number 5963591
scientific article