Supplementary Figure 5: Modeling the transition to seizure.

(a) Transitions between interictal state and seizure were modeled as a slow-fast process, where the slow variable represents a change in excitability and the fast variable is a mean population firing rate. For extremely low or high values of excitability only one equilibrium state exists; either the interictal (low firing) or the seizure (high-firing) state, respectively. For intermediate values of excitability, the system has two equilibrium states separated by unstable fixed (tipping) points. (b) If the excitability is dynamically changing, then the system periodically oscillates between these two states and in a phase portrait it has a character of a limit cycle. If the interictal excitability crosses the catastrophic bifurcation F₁ the system undergoes a rapid and large shift to a seizure regime. The reverse situation occurs if the excitability during seizure reaches the second catastrophic bifurcation F₂ when the seizure suddenly terminates. (c) The corresponding time series demonstrates a repeated transition between the contrasting dynamical regimes, i.e. interictal and seizure states.