6. Stochastic Partial Differential Equations in Neurobiology: Linear and Nonlinear Models for Spiking Neurons

Stochastic differential equation (SDE) models of nerve cells for the most part neglect the spatial dimension. Including the latter leads to stochastic partial differential equations (SPDEs) which allow for the inclusion of important variations in the densities of ion channels. In the first part of this work, we briefly consider representations of neuronal anatomy in the context of linear SPDE models on line segments with one and two components. Such models are reviewed and analytical methods illustrated for finding solutions as series of Ornstein–Uhlenbeck processes. However, only nonlinear models exhibit natural spike thresholds and admit traveling wave solutions, so the rest of the article is concerned with spatial versions of the two most studied nonlinear models, the Hodgkin–Huxley system and the FitzHugh–Nagumo approximation. The ion currents underlying neuronal spiking are first discussed and a general nonlinear SPDE model is presented. Guided by recent results for noise-induced inhibition of spiking in the corresponding system of ordinary differential equations, in the spatial Hodgkin–Huxley model, excitation is applied over a small region and the spiking activity observed as a function of mean stimulus strength with a view to finding the critical values for repetitive firing. During spiking near those critical values, noise of increasing amplitudes is applied over the whole neuron and over restricted regions. Minima have been found in the spike counts which parallel results for the point model and which have been termed inverse stochastic resonance. A stochastic FitzHugh–Nagumo system is also described and results given for the probability of transmission along a neuron in the presence of noise.