Non-trivial dynamics in a model of glial membrane voltage driven by open potassium pores

Abstract

Despite the molecular evidence that a nearly linear steady-state current-voltage relationship in mammalian astrocytes reflects a total current resulting from more than one differentially regulated K⁺ conductance, detailed ordinary differential equation (ODE) models of membrane voltage V_m are still lacking. Various experimental results reporting altered rectification of the major Kir currents in glia, dominated by Kir4.1, have motivated us to develop a detailed model of V_m dynamics incorporating the weaker potassium K2P-TREK1 current in addition to Kir4.1, and study the stability of the resting state V_r. The main question is whether, with the loss of monotonicity in glial I-V curve resulting from altered Kir rectification, the nominal resting state V_r remains stable, and the cell retains the trivial, potassium electrode behavior with V_m after E_K. The minimal two-dimensional model of V_m near V_r showed that an N-shape deformed Kir I-V curve induces multistability of V_m in a model that incorporates K2P activation kinetics, and nonspecific K⁺ leak currents. More specifically, an asymmetrical, nonlinear decrease of outward Kir4.1 conductance, turning the channels into inward rectifiers, introduces instability of V_r. That happens through a robust bifurcation giving birth to a second, more depolarized stable resting state V_dr > -10 mV. Realistic recordings from electrographic seizures were used to perturb the model. Simulations of the model perturbed by constant current through gap junctions and seizure-like discharges as local field potentials led to depolarization and switching of V_m between the two stable states, in a downstate-upstate manner. In the event of prolonged depolarizations near V_dr, such catastrophic instability would affect all aspects of the glial function, from metabolic support to membrane transport, and practically all neuromodulatory roles assigned to glia.