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Perturbations both trigger and delay seizures due to generic properties of slow-fast relaxation oscillators

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 t_h. In this case, to approach Γ_ϵ, has to cross layers of x whose values are smaller than the ones surrounding Γ_ϵ. For this reason travels slower than z. Since and z have to meet after a time t_h at the same point on Γ_ϵ, 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.

doi: https://doi.org/10.1371/journal.pcbi.1008521.g004