Steady-state solutions in nonlocal neuronal networks
The article investigates the scalar integro-differential equation \[ {\partial u \over \partial t} + u = (\alpha - \gamma u) \int_\mathbb R K(x-y)H(u(y,t)-\theta) \,dy + (\beta - \delta u) \int_\mathbb R K(x-y)H(u(y,t)-\Theta) \,dy \] and the nonlinear system \[ \begin{aligned} {\partial u \over \partial t} + u + w =& (\alpha - \gamma u) \int_\mathbb R K(x-y)H(u(y,t)-\theta) \,dy \\ & + (\beta - \delta u) \int_\mathbb R K(x-y)H(u(y,t)-\Theta) \,dy, \\ {\partial w \over \partial t} =& \varepsilon (u - \tau w), \end{aligned} \] where \(u(x, t)\) represents the membrane potential of a neuron, and \(w\) represents the leaking current processes on a slow time scale. The constants \(\theta\) and \(\Theta\) are the thresholds for synaptic coupling. The firing rate \(H\) is the Heaviside step function and the kernel \(K\) is either an even probability function with exponential decay at infinity or a ``Mexican hat'' type function. First the steady state solutions for the scalar equation with only the first integral on the right hand side are investigated. Depending on the parameters \(\gamma\), \(\alpha\) and \(\theta\) these solutions can be unique, monotone and spectrally stable. Similar results hold for the simplified system involving \(w\). Then also the solutions for the full problem are derived and their spectral properties are computed.
scientific article; zbMATH DE number 5011375