Fig. 4.

Bifurcation analysis explains termination of seizure in a single-cell model. A: bifurcation diagram depicts minimum (blue) and maximum (green) V_d values plotted against the bifurcation parameter [K⁺]_o. [Na⁺]_i was fixed at 20 mM, [K⁺]_i and [Na⁺]_o were fixed at 130 mM, and [Cl⁻]_i was fixed at 10 mM; solid and dashed branches correspond to increasing or decreasing [K⁺]_o, respectively. B: voltage traces of a single neuron for selected values of [K⁺]_o ([Na⁺]_i = 20 mM and [Cl⁻]_i = 10 mM). Red and black dots represent local voltage minima and maxima, respectively (which were used in A). ε = 0.5 was used to illustrate the activity in the bistable region ([K⁺]_o = 7.5 mM plot). Top, left: the neuron displayed low-frequency spiking activity. Bottom, right: each burst consisted of high-frequency oscillations, followed by a depolarized block and a prolonged hyperpolarization. The green and blue markers indicate the global maxima and minima, whereas red and black indicate the local maxima and minima. C: calcium ion concentration ([Ca²⁺]-V_d) phase-space projection, showing 1-dimensional (1D) quiescent (green line, M_eq) and a 2D tonic-spiking (attracting yellow and blue surfaces, M_lc) manifold for [K⁺]_o = 75 mM, ([Na⁺]_i = 20 mM and [Cl⁻]_i = 10 mM), which is spanned, respectively, by stable and unstable periodic orbits of the model. The neuron exhibited bistability of coexisting spiking (blue trajectory, marked as S) and bursting (black trajectory) regimes. The yellow part of the 2D manifold (M_lc) is a local attraction basin for tonic spiking. Initial conditions were chosen on the blue part of M_lc, unstable for tonic oscillations, resulting in the onset of the bursting activity (black trajectory). Solid and dashed segments of the 1D quiescent manifold (green line, M_eq) correspond to stable and unstable equilibria (quiescent states). The solid red line (<V_d>) represents the effective potential curve made by averaging variables of periodic orbits spanning the 2D manifold M_lc; its end points identify the underlying bifurcations: subcritical Andronov-Hopf (AH) and homoclinic (at point X). The solid, orange line ([Ca²⁺]_i′ = 0) is the slow nullcline; [Ca²⁺]_i decreases (increases) below (above) this line. The red sphere indicates the location of an unstable equilibrium state at the intersection of M_eq with this slow nullcline. Points Y, K, and Z indicate the hyperpolarized knee point on M_eq and that of <V_d> and the intersection of the curve <V_d> with the slow nullcline [Ca²⁺]_i′ = 0, respectively. Inset shows the 3D phase space, where m_INa is the activation variable of the voltage-sensitive sodium current. D: (Ca²⁺-V_d) phase-space projection showing the critical manifolds for [K⁺]_o = 90 mM ([Na⁺]_i = 20 mM and [Cl⁻]_i = 10 mM); other descriptions are same as for C. E: bifurcation diagram of the quiescent states in the Na⁺-K⁺ space. Solid and dashed lines indicate saddle-node (SN) and AH bifurcations, respectively, for different strengths of the Na⁺/K⁺ pump current. F: the arrangement of the critical manifolds near termination of seizure ([K⁺]_o = 90 mM; [Na⁺]_i = 20 mM in equation for reversal potential and [Na⁺]_i = 22 mM in equation for Na⁺/K⁺ pump current; and [Cl⁻]_i = 10 mM). Black line indicates bursting activity. The blue line (S) on the 2D manifold M_lc indicates the tonic-spiking solution that is unreachable from physiological initial conditions; other descriptions are the same as in C.