Neural mass modeling of slow-fast dynamics of seizure initiation and abortion
Fig 3
Bursting orbit of system (3).
(a) Solution of (3) (blue orbit) and L0 (red curve) on the critical surface S0(green surface) projected on the (v0, v2, v3)-space. Single-headed, double-headed and triple-headed arrows indicate the flow direction during superslow, slow and fast time-scales, respectively. LP denotes limit point bifurcation. The L0 curve changes stability at two limit points, LP1 and LP2 (red dots). The middle branch of the L0 curve between these limit points is unstable (dashed). (b) Time course of the variables (v3, v0, v2) of the orbit plotted in panel (a). (c) Solution of (3) projected on the bifurcation diagram (black curve) of (4) for ε = 0 where v2 is threaded as a parameter. Arrows show the direction of the flow with respective time-scales. Bold and dashed lines correspond to stable and unstable solutions, respectively. H donates a Hopf bifurcation, LP a limit point bifurcation. The equilibrium points along the black Z-shaped curve are unstable on the middle branch of the curve, between LP1 at and LP2 at
(black dots), and on the upper branch between H1 at
and H2 at
(green dots). The amplitude of the stable limit cycles is bounded by the green continuous curves connecting the H1 and H2 in the ε = 0 limit.