Dependence of spontaneous neuronal firing and depolarisation block on astroglial membrane transport mechanisms

Abstract

Exposed to a sufficiently high extracellular potassium concentration ([K⁺]_o), the neuron can fire spontaneous discharges or even become inactivated due to membrane depolarisation (‘depolarisation block’). Since these phenomena likely are related to the maintenance and propagation of seizure discharges, it is of considerable importance to understand the conditions under which excess [K⁺]_o causes them. To address the putative effect of glial buffering on neuronal activity under elevated [K⁺]_o conditions, we combined a recently developed dynamical model of glial membrane ion and water transport with a Hodgkin–Huxley type neuron model. In this interconnected glia-neuron model we investigated the effects of natural heterogeneity or pathological changes in glial membrane transporter density by considering a large set of models with different, yet empirically plausible, sets of model parameters. We observed both the high [K⁺]_o-induced duration of spontaneous neuronal firing and the prevalence of depolarisation block to increase when reducing the magnitudes of the glial transport mechanisms. Further, in some parameter regions an oscillatory bursting spiking pattern due to the dynamical coupling of neurons and glia was observed. Bifurcation analyses of the neuron model and of a simplified version of the neuron-glia model revealed further insights about the underlying mechanism behind these phenomena. The above insights emphasise the importance of combining neuron models with detailed astroglial models when addressing phenomena suspected to be influenced by the astroglia-neuron interaction. To facilitate the use of our neuron-glia model, a CellML version of it is made publicly available.

Appendices

Appendix A: Choice of parameters and initial values

Most parameters of the neuron model were set to values equal or in close proximity to values used in Kager et al. (2000) except J _{NaKATPase,n,max} and g _leak,K which were estimated by requiring that the net fluxes of sodium and potassium across the neuronal membrane be zero under baseline conditions. Initial values for the gating variables of the neuron model (given in Table 3) were selected as steady-state values of the corresponding model equations.

Initial values for [K⁺]_n, [K⁺]_o, [K⁺]_g, [Na⁺]_n, [Na⁺]_o, [Na⁺]_g, and [HCO$^-_3$]_o, the glial membrane potential at baseline $V_{\mathrm{m}}^{\mathrm{(g)}}$, the initial glia volume to area ratio w _g and the initial ratio of glial to ECS volume v _g/v _o = w _g/w _o, and values for the parameters K _m,Na, K _m,K, g _Na, g _Cl, g _NKCC1, g _NBC and L _p were all obtained from available experimental data (see Østby et al. 2009a, for details). Parameters g _K, J _{NaKATPase,g,max}, X _g and ρ and initial values of ion concentrations [Cl−]_g, [Cl−]_o, [HCO$^-_3$]_g for which no empirical estimates were available were derived from the model equations (3a)–(5) to make certain that the model is in a steady-state at baseline conditions.

Since the empirically determined quantities (parameter values and concentrations) come from various studies and different experimental settings, we chose for the simulations reported in Section 4 an approach where the value for each of the selected quantities was randomly drawn from a uniform distribution around the set point value. For these ‘empirically plausible’ parameter sets the values for the baseline ion concentrations were drawn at random from the interval (0.9[S]_g,o,1.1[S]_g,o), where [S]_g,o is the empirical baseline ion concentration of ion S (specified in Table 3). The initial glia volume to area ratio was sampled at random from the interval (1/13 μm, 1/27 μm). The remaining parameters were sampled at random from (0.5P, 1.5P), where P denotes the mean empirical value of any of the above parameters (given in Table 4). Given the randomly sampled values of parameters and initial ion concentrations, values of g _K, J _{NaKATPase,g,max}, X _g and ρ and initial values of ion concentrations [Cl−]_g, [Cl−]_o, [HCO$^-_3$]_g were estimated by, as above, requiring that the model is in a steady-state at baseline conditions.

Appendix B: Model translation

Although mathematical modelling has been identified as a valuable method for analysing large amounts of experimental data, unfortunately, inaccuracies often arise with the current method of mathematical model publication (Hunter et al. 2002; Lloyd et al. 2004). Problems stem from the fact that models are developed and simulated in computer code, but require translation into text and equations for publication in journals. Replicating published results, or further developing a published model, is frequently impeded due to errors introduced during the publishing process such as typographical errors, missing parameters, or equations. Further, even when the model source code is made freely available, the code is often specific to a particular computer platform, or is incompatible with other modelling architectures. CellML is an XML-based modelling language which provides an unambiguous method of defining models of biological processes (Lloyd et al. 2004). It has been developed as a potential solution to the problems associated with publishing and implementing a mathematical model. The current model has been translated into CellML and the code is freely available for download from http://models.cellml.org/e/2e. Model simulations can be run using the Physiome CellML Environment (PCEnv) or Cellular Open Resource (COR), two open source tools which can be downloaded from http://www.cellml.org/downloads/pcenv/ and http://cor.physiol.ox.ac.uk/, respectively.