Neural mass modeling of slow-fast dynamics of seizure initiation and abortion
Fig 5
Geometrical analyses of a constant input.
Constant input is applied to SOM+ interneurons (a), to pyramidal cells (b) and to both SOM+ interneurons and pyramidal cells (c). Left panels show the projection of the nullsurfaces, critical slow manifold and the orbit of the reduced model (7). Right panels show the LFP signal of the full system (3) subject to the constant inputs analyzed on the left. All parameters are as given in Table 1, except B = 15. (a) The y5-nullsurface Θ (blue surface for kP = 0), and y7-nullsurface Σ (red surface for kSOM = −1, black surface for kSOM = 0, green surface for kSOM = 1) are projected on the (v2, y5, v0)-space. The blue curve L0 (stable on the bold, unstable on the dashed) is on the intersection between Θ and the {y5 = 0}-hyperplane. The black and red orbits are the solutions of the system for kSOM = 0 and kSOM = −1, respectively. For kSOM = 1, the solution approaches to the green stable equilibrium point on the intersection between Σ(kSOM = 1) and L0. Panel (a1) shows the time series for kSOM = {0, 1}, and panel (a2) for kSOM = −1. (b) The y5-nullsurface Θ (red surface for kP = 1, green surface for kP = −1), and y7-nullsurface Σ (black surface for kSOM = 0) are projected on the (v2, y5, v0)-space. The red curve L0 (stable on the bold, unstable on the dashed) is on the intersection between Θ(kP = 1) and the {y5 = 0}-hyperplane. The green curve L0 (stable on the bold, unstable on the dashed) is on the intersection between Θ(kP = −1) and the y5 = 0 hyperplane. The green and red orbits are the solutions of the system for kP = 1 and kP = −1, respectively. Panel (b1) shows time series for kP = 1, and panel (b2) for kP = −1. (c) The y5-nullsurface Θ (red surface for kP = 1) and y7-nullsurface Σ (green surface for kSOM = 1, blue surface for kSOM = 2) are projected on the (v2, y5, v0)-space. The red curve L0 (stable on the bold, unstable on the dashed) is on the intersection between Θ(kP = 1) and the y5 = 0 hyperplane. The green curve L0 (stable on the bold, unstable on the dashed) is on the intersection between Θ(kP = 1) and the {y5 = 0}-hyperplane. The green orbit is the solution of the system for (kP, kSOM) = (1,1). For (kP, kSOM) = (1,2) the solution approaches to the cyan stable equilibrium point on the intersection between Σ(kSOM = 2) and L0. Panel (c1) shows time series for (kP, kSOM) = (1,1), and panel (c2) for (kP, kSOM) = (1,2).