Membrane capacitive memory alters spiking in neurons described by the fractional-order Hodgkin-Huxley model.

Seth H Weinberg,Thomas Wennekers

doi:10.1371/journal.pone.0126629

Abstract

Excitable cells and cell membranes are often modeled by the simple yet elegant parallel resistor-capacitor circuit. However, studies have shown that the passive properties of membranes may be more appropriately modeled with a non-ideal capacitor, in which the current-voltage relationship is given by a fractional-order derivative. Fractional-order membrane potential dynamics introduce capacitive memory effects, i.e., dynamics are influenced by a weighted sum of the membrane potential prior history. However, it is not clear to what extent fractional-order dynamics may alter the properties of active excitable cells. In this study, we investigate the spiking properties of the neuronal membrane patch, nerve axon, and neural networks described by the fractional-order Hodgkin-Huxley neuron model. We find that in the membrane patch model, as fractional-order decreases, i.e., a greater influence of membrane potential memory, peak sodium and potassium currents are altered, and spike frequency and amplitude are generally reduced. In the nerve axon, the velocity of spike propagation increases as fractional-order decreases, while in a neural network, electrical activity is more likely to cease for smaller fractional-order. Importantly, we demonstrate that the modulation of the peak ionic currents that occurs for reduced fractional-order alone fails to reproduce many of the key alterations in spiking properties, suggesting that membrane capacitive memory and fractional-order membrane potential dynamics are important and necessary to reproduce neuronal electrical activity.

Highlights

The properties of excitable cells and cell membranes have been studied for over a century, dating back to the studies of Weiss [1], Lapicque [2], and Nernst [3] in the early 1900s
We investigate the spiking properties of the neuronal membrane patch, nerve axon, and neural networks described by the fractional-order Hodgkin-Huxley neuron model
We investigate the properties of fractional-order neuron models and characterize how the fractional-order α influences the properties of the neuron, the spatially-distributed doi:10.1371/journal.pone.0126629.g011

Summary

Introduction

The properties of excitable cells and cell membranes have been studied for over a century, dating back to the studies of Weiss [1], Lapicque [2], and Nernst [3] in the early 1900s. Irheo Ea;1ðÀðd=tÞaÞ ð9Þ and, importantly, yield a strength-duration relation that follows a general power law for short duration stimuli and asymptotic behavior for long duration stimuli (Fig 1A). Despite these early studies which suggest that a non-ideal (or fractional-order) capacitive element may be more appropriate to represent passive membrane dynamics, essentially all excitable cell models assume an ideal capacitor, i.e., α = 1. While it is clear that fractional-order membrane potential Vm dynamics will alter the passive response to sub-threshold stimuli, it is not obvious if fractionalorder dynamics will alter the properties of the active neuron and response to super-threshold stimuli, given the bi-directional coupling between Vm and ionic currents.

Background on fractional calculus

Numerical methods

Summary of main findings

Limitations