Steady-state solutions in nonlocal neuronal networks
type: Item
instance of: scholarly article
title: Steady-state solutions in nonlocal neuronal networks
zbMATH Open document ID: 1087.35013
author: Linghai Zhang
published in: Dynamics of Partial Differential Equations
publication date: 2006-03-10T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 35B35
Mathematics Subject Classification ID: 35R10
Mathematics Subject Classification ID: 45K05
Mathematics Subject Classification ID: 92C20
zbMATH DE Number: 5011375
zbMATH Keywords: existence
zbMATH Keywords: stability
zbMATH Keywords: bifurcation
reviewed by: Alois Steindl
MaRDI profile type: Publication
DOI: 10.4310/DPDE.2005.V2.N1.A4
description: scientific article; zbMATH DE number 5011375