Fig. 7.

Bifurcation analysis reveals mechanisms of seizure initiation in a single-cell model. A: bifurcation diagram depicts minimum (blue) and the maximum (green) V_d values plotted against the bifurcation parameter [K⁺]_o in the model with 50% Na⁺/K⁺ pump-current reduction (the ion concentrations were fixed at [Na⁺]_i = 20 mM and [Cl⁻]_i = 10 mM). Solid and dashed branches correspond to increasing and decreasing [K⁺]_o, respectively. Red and black dots denote local minima and maxima of voltage, respectively. Note that transition from silent state to bursting now occurs for very low levels of [K⁺]_o ([K⁺]_o ∼ 2.6 mM). B: characteristic voltage traces of a single neuron for few selected values of [K⁺]_o ([Na⁺]_i = 20 mM and [Cl⁻]_i = 10 mM) and reduced Na⁺/K⁺ pump current illustrate different dynamical states: periodic, quasiperiodic, bursting. Top, right: slow-frequency modulation of the spiking activity. Bottom: low-frequency spikes whose frequency progressively increased before transition to depolarization block. C: (Ca²⁺-V_d) phase space depicting the arrangement of the 1D quiescent (green line, M_eq) and 2D tonic-spiking (yellow, M_lc) manifolds for [K⁺]_o = 75 mM in conditions of Na⁺/K⁺ pump-current reduction. For [K⁺]_o = 7.5 mM, only the periodic orbit or torus (blue curve) solutions were stable. Solid and dashed segments of the 1D manifold represent stable and unstable equilibria (quiescent states), respectively. The neuron exhibits a monostable, tonic-spiking activity, corresponding to the blue trajectory (S) near the fold on M_lc. Note that the intersection point (Z) of the <V_d> curve of effective potentials with the slow nullcline [Ca²⁺]_i′ = 0 is below its knee point K and that the homoclinic point X coincides with the knee point Y on the manifold M_eq, thus indicating homoclinic SN bifurcation [SN on invariant circle (SNIC)]; other descriptions are the same as in Fig 4, C. D: rearrangement of the critical manifolds supports bursting for [K⁺]_o = 90 mM in conditions of 50% Na⁺/K⁺ pump-current reduction. The bursting trajectory (black) repeatedly switches between the hyperpolarized quiescent branch of M_eq and the 2D tonic-spiking manifold (M_lc) glued to the knee point (Y) through a SNIC bifurcation. Note that the knee point K of the effective potential curve <V_d> moved above its intersection point K with the slow nullcline [Ca²⁺]_i′ = 0 to promote persistent bursting.