A simple model considering spiking probability during extracellular axon stimulation

Abstract

The spiking probability of an electrically stimulated axon as a function of stimulus amplitude increases in a sigmoidal dependency from 0 to 1. However, most computer simulation studies for neuroprosthetic applications calculate thresholds for neural targets with a deterministic model and by reducing the sigmoid curve to a step function, they miss an important information about the control signal, namely how the spiking efficiency increases with stimulus intensity. Here, this spiking efficiency is taken into account in a compartment model of the Hodgkin Huxley type where a noise current is added in every compartment with an active membrane. A key parameter of the model is a common factor knoise which defines the ion current fluctuations across the cell membrane for every compartment by its maximum sodium ion conductance. In the standard model Gaussian signals are changed every 2.5 μs as a compromise of accuracy and computational costs. Additionally, a formula for other noise transmission times is presented and numerically tested. Spiking probability as a function of stimulus intensity can be approximated by the cumulative distribution function of the normal distribution with RS = σ/μ. Relative spread RS, introduced by Verveen, is a measure for the spread (normalized by the threshold intensity μ), that decreases inversely with axon diameter. Dynamic range, a related measure used in neuroprosthetic studies, defines the intensity range between 10% and 90% spiking probability. We show that (i) the dynamic range normalized by threshold is 2.56 times RS, (ii) RS increases with electrode—axon distance and (iii) we present knoise values for myelinated and unmyelinated axon models in agreement with recoded RS data. The presented method is applicable for other membrane models and can be extended to whole neurons that are described by multi-compartment models.