The influence of sodium and potassium dynamics on excitability, seizures, and the stability of persistent states: I. Single neuron dynamics

Abstract

In these companion papers, we study how the interrelated dynamics of sodium and potassium affect the excitability of neurons, the occurrence of seizures, and the stability of persistent states of activity. In this first paper, we construct a mathematical model consisting of a single conductance-based neuron together with intra- and extracellular ion concentration dynamics. We formulate a reduction of this model that permits a detailed bifurcation analysis, and show that the reduced model is a reasonable approximation of the full model. We find that competition between intrinsic neuronal currents, sodium-potassium pumps, glia, and diffusion can produce very slow and large-amplitude oscillations in ion concentrations similar to what is seen physiologically in seizures. Using the reduced model, we identify the dynamical mechanisms that give rise to these phenomena. These models reveal several experimentally testable predictions. Our work emphasizes the critical role of ion concentration homeostasis in the proper functioning of neurons, and points to important fundamental processes that may underlie pathological states such as epilepsy.

Fig. 1

Comparison of the reduced model to the full spiking model. The top plot shows the membrane voltage for the neuron in the full model. The middle traces show [K]_o for both the full model (solid line) and the reduced model (dashed line). The bottom traces show [Na]_i with the same convention. All data were integrated with an elevated bath potassium concentration at k̄_o,∞ = 2.0, with all other parameters set to their normal values

Fig. 2

The bifurcation diagram for [K]_o as a function of the bath potassium concentration k̄_o,∞, revealing a region of oscillatory behavior. All other parameters were set equal to their normal values. Triangles represent equilibria (i.e., steady states), and circles represent periodic orbits (i.e., oscillatory behavior); stability is denoted by solid (stable) and open (unstable) symbols

Fig. 3

Bifurcation diagrams for [K]_o as a function of a the normalized pump strength, b the diffusion coefficient, and c the glial strength. All plots were produced with an elevated bath potassium concentration at k̄_o,∞ = 2.0

Fig. 4

A two-dimensional bifurcation diagram showing the boundaries of the region of oscillation (RO) as a function of the diffusion coefficient and the glial strength. The black curves denote Hopf bifurcations; within this region, the ion concentrations exhibit oscillatory behavior. The dashed lines correspond to the one-dimensional bifurcation diagrams in Figs. 3b, c and 5 (see text). Examples of the dynamics of the full model, obtained at parameter values corresponding to the numbered points, are shown in Fig. 6

Fig. 5

The one-dimensional bifurcation diagram for [K]_o as a function of the normalized diffusion coefficient ε̄ for k̄_o,∞ = 2.0 and Ḡ_glia = 1.75

Fig. 6

a–f Examples of the dynamics of the full model, obtained at parameter values corresponding to the numbered points in Fig. 4. The top trace shows the membrane voltage and the lower traces show [K]_o (solid trace) and [Na]_i (dashed trace) on the same time scale

Fig. 7

The effect of changing the bath concentration on the location of the region of oscillation (RO) is illustrated for ρ̄ = 1 and various values of the bath potassium concentration k_o,∞. The square represents normal values of the diffusion and glial strength. a The RO is seen to appear and move to the right as the bath potassium concentration is increased from k̄_o,∞ = 1.77 (grey curve) to k̄_o,∞; = 1.9 (black curve), where it intersects the square (compare the left bifurcation in Fig. 2). b The square lies within the RO for k̄_o,∞ = 2.0. c At k̄_o,∞ = 2.1, the RO has moved further to the right, and the square is close to the left boundary

Fig. 8

Other bursting patterns. Traces similar to those in Fig. 6 obtained with the full model for a k̄_o,∞ = 6.0, Ḡ_glia = 0.1, and ε̄ = 0.4; and b same, but with k̄_o,∞ reduced slightly. The quiescent states in (a) correspond to depolarization block; see text for further description