A Threshold Equation for Action Potential Initiation

Abstract

In central neurons, the threshold for spike initiation can depend on the stimulus and varies between cells and between recording sites in a given cell, but it is unclear what mechanisms underlie this variability. Properties of ionic channels are likely to play a role in threshold modulation. We examined in models the influence of Na channel activation, inactivation, slow voltage-gated channels and synaptic conductances on spike threshold. We propose a threshold equation which quantifies the contribution of all these mechanisms. It provides an instantaneous time-varying value of the threshold, which applies to neurons with fluctuating inputs. We deduce a differential equation for the threshold, similar to the equations of gating variables in the Hodgkin-Huxley formalism, which describes how the spike threshold varies with the membrane potential, depending on channel properties. We find that spike threshold depends logarithmically on Na channel density, and that Na channel inactivation and K channels can dynamically modulate it in an adaptive way: the threshold increases with membrane potential and after every action potential. Our equation was validated with simulations of a previously published multicompartemental model of spike initiation. Finally, we observed that threshold variability in models depends crucially on the shape of the Na activation function near spike initiation (about −55 mV), while its parameters are adjusted near half-activation voltage (about −30 mV), which might explain why many models exhibit little threshold variability, contrary to experimental observations. We conclude that ionic channels can account for large variations in spike threshold.

Author Summary

Neurons communicate primarily with stereotypical electrical impulses, action potentials, which are fired when a threshold level of excitation is reached. This threshold varies between cells and over time as a function of previous stimulations, which has major functional implications on the integrative properties of neurons. Ionic channels are thought to play a central role in this modulation but the precise relationship between their properties and the threshold is unclear. We examined this relationship in biophysical models and derived a formula which quantifies the contribution of various mechanisms. The originality of our approach is that it provides an instantaneous time-varying value for the threshold, which applies to the highly fluctuating regimes characterizing neurons in vivo. In particular, two known ionic mechanisms were found to make the threshold adapt to the membrane potential, thus providing the cell with a form of gain control.