Abstract

The classical Hodgkin-Huxley (HH) model neglects the time-dependence of ion concentrations in spiking dynamics. The dynamics is therefore limited to a time scale of milliseconds, which is determined by the membrane capacitance multiplied by the resistance of the ion channels, and by the gating time constants. We study slow dynamics in an extended HH framework that includes time-dependent ion concentrations, pumps, and buffers. Fluxes across the neuronal membrane change intra- and extracellular ion concentrations, whereby the latter can also change through contact to reservoirs in the surroundings. Ion gain and loss of the system is identified as a bifurcation parameter whose essential importance was not realized in earlier studies. Our systematic study of the bifurcation structure and thus the phase space structure helps to understand activation and inhibition of a new excitability in ion homeostasis which emerges in such extended models. Also modulatory mechanisms that regulate the spiking rate can be explained by bifurcations. The dynamics on three distinct slow times scales is determined by the cell volume-to-surface-area ratio and the membrane permeability (seconds), the buffer time constants (tens of seconds), and the slower backward buffering (minutes to hours). The modulatory dynamics and the newly emerging excitable dynamics corresponds to pathological conditions observed in epileptiform burst activity, and spreading depression in migraine aura and stroke, respectively.

Highlights

  • In this paper we study ion dynamics in ion-based neuron models

  • While spiking activity is in the order of milliseconds, the time scales of ion dynamics range from seconds to minutes and even hours depending on the process

  • The results are presented in three parts that describe (i) the stability of closed models, where we treat the change K~e of the potassium content as a bifurcation parameter, (ii) open models, i.e., K~e becomes a dynamical variable, with glial buffering and (iii) oscillations in ion concentrations in open models for bath coupling with the bath concentration Kbath as a bifurcation parameter

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Summary

Introduction

In this paper we study ion dynamics in ion-based neuron models. In comparison to classical HH type membrane models this introduces dynamics on much slower time scales. Phase space excursions with large changes in the ionic variables establish a new type of ionic excitability as observed in cortical spreading depression (SD) during stroke and in migraine with aura [1,2]. Such newly emerging dynamics can be understood from the phase space structure of the ion-based models. In 1978 Tuckwell and Miura developed a SD model that is amenable to a more direct interpretation in terms of biophysical quantities [7] It contains ion movements across the neural membrane and ion diffusion in the ECS.

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