Epilepsy is a chronic medical condition that involves abnormal brain activity causing patients to lose control of awareness or motor activity. As a result, detection of pre-ictal states, before the onset of a seizure, can be lifesaving. The problem is challenging because it is difficult to discern between electroencephalogram signals in pre-ictal states versus signals in normal inter-ictal states. There are three key challenges that have not been addressed previously: (1) the inconsistent performance of prediction models across patients, (2) the lack of perfect prediction to protect patients from any episode, and (3) the limited amount of pre-ictal labeled data for advancing machine learning methods. This article addresses these limitations through a novel approach that uses adversarial examples with optimized tuning of a combined convolutional neural network and gated recurrent unit. Compared to the state of the art, the results showed an improvement of 3x in model robustness as measured in reduced variations and superior accuracy of the area under the curve, with an average increase of 6.7%. The proposed method also exhibited superior performance with other advances in the field of machine learning and customized for epilepsy prediction including data augmentation with Gaussian noise and multitask learning.

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A Kempe swap in a properly colored graph recolors one component of the subgraph induced by two colors, interchanging them on that component. Two k-colorings are Kempe k-equivalent if we can transform one into the other by a sequence of Kempe swaps, such that each intermediate coloring uses at most k colors. Meyniel proved that if G is planar, then all 5-colorings of G are Kempe 5-equivalent; this proof relies heavily on the fact that planar graphs are 5-degenerate. To prove an analogous result for toroidal graphs would require handling 6-regular graphs. That is the focus of this paper. We show that if G is a 6-regular graph that has an embedding in the torus with every non-contractible cycle of length at least 7, then all 5-colorings of G are Kempe 5-equivalent. Bonamy, Bousquet, Feghali, and Johnson asked specifically about the case that G is a triangulated toroidal grid, which is formed from the Cartesian product Cm Cn by adding a diagonal inside each 4-face, with all diagonals parallel. By slightly modifying the proof of our main result, we answer their question affirmatively when m ≥ 6 and n ≥ 6. IntroductionGiven a k-coloring ϕ of a graph G and colors α, β ∈ {1, . . . , k}, a Kempe swap Kempe swap recolors a component of the subgraph induced by colors α and β, interchanging those colors on that component, which is called a Kempe component Kempe component . Two k-colorings, ϕ 1 and ϕ 2 of a graph G are (Kempe) k-equivalent kequivalent

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