Epileptor’s fixed points and bursting trajectory.

A: Fixed points manifold for x1. For x1 x1 ≥ 0 we have one fixed point when and two solutions for smaller values. The lower branch of fixed points is the rest or interictal state. The upper branch is stable for values of and unstable otherwise. SN+ occurs for , SN− for , SN0 for μ = −1 and the Hopf curve for . B: Trajectory of the Epileptor’s fast subsystem (x1, x2) plotted against the parameter μ. It can be observed the hysteresis-loop caused by the two SN curves.</p

Code.

The folder contains the Matlab code used to generate the figures. (ZIP)</p

Hysteresis-loop bursting in the DTB bursting model.

A: This is one portion of the unfolding in which SN/SH bursting can be placed, together with other classes [45]. Saddle-Node (SN) and supercritical Hopf (SupH) curves meet at the Takens-Bogdanov point TB. Bifurcation curves partition the map in five regions with different state space configurations (Roman Numerals). When this map is used for the fast subsystem of a fast-slow bursters with a hysteresis-loop mechanism for the slow variable, possible classes in the map are: SN/SH, SN/SupH, SupH/SH and SupH/SupH plus SN/SN where the system alternates between the two stable fixed points [45]. When more than one fixed point exist, the resting (or inter ictal) state is the one on the right, the other one we call ‘active rest’. The resting state corresponds to the upper branch of fixed points in panel B. B: Typical bifurcation diagram for the SN/SH class. When the system is at rest, z increases until the fixed point destabilizes through the SN bifurcation and the system jumps into the stable limit cycle. Now that the system is far from rest, z decreases until the limit cycle destabilizes through a SH bifurcation and the system jumps back to rest. If the destabilization of the fixed point/limit cycle is obtained through a different bifurcation, we will have a different onset/offset class and a different appearance of the burster’s timeseries.</p

Epileptor’s main mechanisms.

The Epileptor field potential (purple), is a combination of the activity of one fast variable (blue) and an intermediate one (magenta). The fast and slow variables dynamics and feedback among them constitute the core of the model (blue), while the intermediate variables (magenta) modulate the fast activity. All the simulations in this work are performed without noise.</p

Role of <i>x</i>₀ on the path.

A: A sketch of how, in hysteresis-loop bursting, x0 changes the position of the z-nullcline with regards to the resting state branch. B: When the intermediate subsystem is not allowed to oscillate (Irest2 = 0), different values of x0 do not change the path but only affect the velocities at which the slow variable evolves when the fast ones are in the interictal or ictal states. C: Allowing for intermediate oscillations to modulate the path (Irest2 > 0), changing x0 modifies the amount of spikes in the path (more spikes when the slow variable evolves slower while in the ictal state).</p

New classes of bursting identified in the Epileptor model.

Path on the map (left) and timeseries (right) for the new classes SupH/SH (A) and SupH/SupH (B). Maps represent the amplitude of the limit cycle.</p

Paths on the map.

Paths followed by the full Epileptor model in the map for four different conditions (left column), and the related timeseries (right column). All parameters are kept the same except for those specified. Both on maps and in timeseries, a star/triangle approximately marks the onset/offset of oscillations in the fast subsystem, when present. These types of Epileptor behaviors have been described in the literature.</p

Topological equivalence between bifurcation diagrams of the fast subsystems of the Epileptor and of the DBT bursting model.

In the left panels we show the portion of bifurcation diagram of the DTB bursting model (that is the unfolding of the DTB singularity) in which SN/SH bursting occurs, and the behavior of amplitude and frequency of the limit cycle. In the right panels, the same for the Epileptor’s fast subsystem. In the latter case the presence of a SH bifurcation stemming from the TB point can be inferred by the behavior of the frequency (Hz) of the limit cycle identified through simulations, which scales down to zero. Roman Numerals refer to the configurations as in Fig 1.</p

Simulated seizure-like event in the absence of glutamatergic and GABAergic synaptic couplings.

<p>Raster plots 1 & 2 display spikes of population 1 and 2 neurons, with activation threshold at 0 mV; third trace is the mean of the two populations with 80% contribution of excitatory ensembles and 20% contribution of inhibitory ensembles; slow permittivity variable is the z variable from <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004209#pcbi.1004209.e013" target="_blank">Eq 3</a> (in arbitrary unit), mainly affected by excitatory cells, exerting a inhibitory effect on inhibitory cells upon exogenous factors; bottom plot is experimental data. Parameters: C_E = 0.8; x₀ = 3.0;W^max = 0.2; G_s^i,j = 0; G_s^i,i = 0; r = 0.000004.</p

Parameter space during status epilepticus (SE).

<p>Color code represents the degree of synchronization within neural populations. Roman numbers indicate the phases of the SE corresponding to the observed dynamics following <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004209#pcbi.1004209.g002" target="_blank">Fig 2</a>. Arrows depict the trajectory of the sequence of theses phases during the whole SE event. Inner x- and y-axis are inter (G_s^i,j) and intra (G_s^i,i) population synaptic coupling strength. Outer x- and y-axes are gap junction coupling strength and excitability, respectively. Axis values are given for indicative purposes and do not portray biophysical units.</p