A closed form equation for extracellular field at a point for time series simulation in diffuse structures

Since seminal studies of a neuron membrane potential were summarized in a Hodgkin-Huxley model it be-came possible to study the relationship between morphology and electrical signalling of a neuron, to model larger-scale networks of neurons and to understand the shape of an action potential recorded extracellularly. New challenges have also accumulated over time: one of them is that the data-analytic tools used in in-vivo large population recordings are mathematically distinct from ’forward’ models. A step in overcoming this is to simplify the computation of an extracellular potential from the modeled membrane potential. It turns out that early engineering work modeled the properties of an electrode with an electric-circuit which can be matched with an appropriate version of the Hodgkin-Huxley circuit. The circuit model is used to derive a stochastic differential equation with a parameter encoding the distance between the electrode and the membrane – a crucial feature of a standard LFP model currently used for approximating extra-cellular potential. This equation helps to quickly prototype multi-channel recordings with modern electrodes in relation to methods of population activity analyses. Numerical examples are shown