Stable relaxation cycle in a bilocal neuron model
The authors assume that the membrane potential \(u\) of a single neuron can be modeled by the delay differential equation \[ \frac{du}{dt}=\lambda f(u(t-1))u, \] where \(\lambda>0\) is a large parameter and \(f:\mathbb{R}\to\mathbb{R}\) is a twice continuously differentiable function with the properties \[ f(0)=1,\ f(u)=-a+O\left(\frac 1n\right),\ uf'(u)=O\left(\frac 1n\right),\ u^2f''(u)=O\left(\frac 1n\right) \] as \(n\to \infty\), where \(a\) is a positive constant. They consider the system \[ \begin{aligned}\frac{du_1}{dt}&=d(u_2-u_1)+\lambda f(u_1(b-1)) u_1, \\ \frac{du_2}{dt}&=d(u_1-u_2)+\lambda f(u_2(t-1))u_2\end {aligned}\tag{\(*\)} \] under the conditions \begin{itemize} \item[(H)] \(d=e^{-b\lambda}\), \(b>0\), \(\lambda\gg 1\), \(a>1\), \(1+\frac 1a<b<1+\frac 12\left(a+\frac 1a\right)\). \end{itemize} Their main results read Theorem: Under the hypothesis (H), system (\(*\)) has the \(2h\)-periodic solution \[ (u_{1p}(t),u_{2p}(t))= (u_p(t,\lambda),u_p(t-h(\lambda),\lambda)), \] where \(u_p(t,\lambda)\) satisfying \(u_p(0,\lambda)\equiv 1\) is the \(2h\)-periodic solution of the delay differential equation \[ \frac{du}{dt}=d(u(t-\lambda)-u)+\lambda f(u(t-1))u. \] \((u_{1p}(t),u_{2p}(t))\) is exponentially orbitally stable. Additionally, the asymptotic properties of \(h(\lambda)\) as \(\lambda\to\infty\) are studied.
scientific article; zbMATH DE number 7043428