Fig 1.
Phase space for relaxation oscillators.
The slow manifold, , is a S-shaped curve having two stable branches
(solid red line) and one repelling
(dashed red line) (see Eq (4)). Stable and unstable branches of
are separated by the fold points
. A given point, p, (see A) will quickly converge to the attracting branch of the slow manifold
(see B). Then, it evolves along
following (6) until reaching the fold point
(see C) where it traverses fast to the other branch
(see D). Then, following again the slow dynamics, the trajectory approaches
(see E) where it goes back to
(see F). Therefore, the system (1) in the singular limit (ϵ → 0) admits a singular periodic orbit Γϵ (in blue) generating relaxation oscillations.
Fig 2.
The antagonistic effect of IEDs on the transition to seizure.
Panels A and B show, in red, the v-nullcline whose stable branches correspond to the stable low and high activity states of the system. The unstable part of the v-nullcline (dashed red line) separates the basin of attraction of both branches. As was illustrated in [22], Figure 5, whether the pulses make the system cross the unstable part of the v-nullcline determines the opposite nature of IEDs. For a random train with amplitude A = 0.25 and the system goes to seizure (panel A). By contrast, for a random train with amplitude A = 0.5 and
the system avoids the seizure state (panel B). By plotting the change in seizure rate Δ as a function of both the amplitude, A, and the mean inter-perturbation interval, Ts (panel C), we can distinguish between pro-convulsive regimes (yellow and white areas) in which the transition is potentiated, and preventive regimes (red and black areas) in which the transition is delayed or completely suppressed. We refer the reader to Materials and Methods section for the specific details about the computation of panel C.
Fig 3.
Isochrons and PRCs for the phenomenor model (7).
Panel (A) shows the limit cycle Γpheno, 16 equispaced isochrons, the v and a nullclines (dashed black and green curves, respectively) and the fixed point, P, at their intersection. The distribution of isochrons clarifies the time dependency of perturbations: as panel (B) shows, a pulse of amplitude A applied at a time t1 (t2) causes a negative (positive) phase shift, delaying (promoting) the transition to seizure. This time dependency can be directly inferred from panel (A): a pulse of amplitude A applied at a point on the cycle z1 = γ(θ1) = γ(t1/T) (z2 = γ(θ2) = γ(t2/T)) displaces the trajectory to a point (
). Since
(
) the perturbation causes a phase shift
(
) delaying (advancing) the phase of the oscillator. The panel (C) shows the PRCs for the phenomenor for positive voltage pulses of different amplitudes summarising the timing (phasic) effect of a given perturbation.
Fig 4.
The slow vector field shapes the isochrons for relaxation oscillators.
In the limit ϵ → 0 isochrons are lines of y constant denoted by . However, since ϵ ≠ 0 but small, the isochrons are
perturbations of
. As we show in the right panel, the sign of the
corrections depends on the difference of speeds between the converging point
and the base point z during the convergence time th. In this case, to approach Γϵ,
has to cross layers of x whose values are smaller than the ones surrounding Γϵ. For this reason
travels slower than z. Since
and z have to meet after a time th at the same point on Γϵ, but
travels slower than z, then
needs to travel a short distance. This determines the sign of the
correction. Furthermore, if the slow vector field is monotonous along the fast direction, the farther the point
, the slower (faster) it travels, so the slope of the isochrons will have the same sign for all the points
satisfying fast convergence, thus determining the effect of perturbations in the fast direction.
Fig 5.
Relationship between the curvature of isochrons and the values for the slow vector field.
For the phenomenological epilepsy model (7) with the set of parameters in (8) the figure shows: (A) Limit cycle Γpheno and its isochrons
(left). (B) Values of the slow vector field (corresponding to
in (7)) for points
.
Fig 6.
PRC of pulses A > 0 for relaxation oscillators.
Next we sketch the PRCs for pulses of amplitude A > 0 for the case . For phases θ < θ*, where θ* satisfies
, due to the slope of isochrons the effect of the pulses will be to delay trajectories. Since isochrons approach the unstable point P through
, the closer the phase θ to θ*, the larger the bending of the isochrons and thus the larger the corresponding delay value. For phases
, there is an advancement proportional to the fraction of cycle skipped. This prevalence of advancements is also seen for points in the upper branch. For phases in a neighbourhood of the fold point
, we expect a transition between advancement and delays not drawn because our analysis is only valid for normally hyperbolic points.
Fig 7.
Mechanism preventing the emergence of seizures.
To suppress the original oscillation and keep the system in the lower branch of Γϵ the amplitude A of the pulse has to be large enough so besides causing a delay Δθ, it displaces trajectories above enough the slow-nullcline so the distance travelled in the negative direction overcomes the distance travelled in the positive direction, thus causing a negative net displacement. The locking appears by repeating this mechanism after Ts = TΔθ intervals so the new pulse always hits the system at the same initial point.
Fig 8.
Response of perturbations for the phenomenor.
We plot the change in the seizure rate Δ for a random train of pulses following a Gaussian distribution of mean time Ts and standard deviation σ denoted as . Panels (A) and (B) correspond to the deterministic periodic case
and to the random case
, respectively. For panel (A) we plot a purple solid line corresponding to the bifurcation of the phase map (25). We plot the same curve as a dashed purple curve in panel (B) illustrating the resilience of the deterministic phase-locked states to noise. For the specific details about the numerical computation of this Figure, we refer the reader to Materials and Methods section.
Table 1.
Different parameters for the reduced 2D Epileptor model in (29).
For the set of parameters , the system will display a limit cycle Γi of period Ti.
Fig 9.
Isochrons and PRCs for the reduced epileptor.
For the sets of parameters and
in Table 1 we show: Limit cycle Γ+, Γ0 and Γ− and its isochrons
(left). The phase response curves for pulses in the v direction for different values of A (right). For the three cases we plot 16 equispaced isochrons. Consistently with our previous analysis, since the monotonicity of the slow vector field does not change, the slope of isochrons does not change sign. For numerical details about the computation of both the isochrons and the PRCs see Materials and methods section.
Fig 10.
Response of perturbations for the reduced epileptor (29).
We show the change in the seizure rate Δ for a random train of pulses whose mean inter impulse interval follows a normal distribution with a mean time Ts and standard deviation σ. Panels A, B, C correspond to the sets of parameters
and
in Table 1. Left Figures correspond to the periodic case
and right Figures to the random case
. Consistently with our theoretical analysis there is a direct correspondence between the mean distance between the lower branch and the slow nullcline and the minimal pulse amplitude A for which perturbations may lead to lock the system. Purple solid lines, bounding locking regimes, correspond to the bifurcations of the map (25). By drawing the same curve for the random case, we illustrate the resilience of locking states to noise. For the specific details about the numerical computation of this Figure, we refer the reader to Materials and Methods section.
Fig 11.
Slow vector field for the reduced epileptor and the phenomenor.
Each cycle is depicted in purple, the v-nullcline in black and the slow nullcline in green. Notice that the direction of the slow variable in both models is flipped, and thus is also the motion over the cycles and the sign of the derivative of the slow vector field in the fast direction .