Abstract. In this paper, we provide a well-posedness theory for weak measure solutions of the Cauchy problem for a family of nonlocal interaction equations. These equations are continuum models for interacting particle systems with attractive/repulsive pairwise interaction potentials. The main phenomenon of interest is that, even with smooth initial data, the solutions can concentrate mass in finite time. We develop an existence theory that enables one to go beyond the blow-up time in classical norms and allows for solutions to form atomic parts of the measure in finite time. The weak measure solutions are shown to be unique and exist globally in time. Moreover, in the case of sufficiently attractive potentials, we show the finite time total collapse of the solution onto a single point, for compactly supported initial measures. Finally, we give conditions on compensation between the attraction at large distances and local repulsion of the potentials to have global-in-time confined systems for compactly supported initial data. Our approach is based on the theory of gradient flows in the space of probability measures endowed with the Wasserstein metric. In addition to classical tools, we exploit the stability of the flow with respect to the transportation distance to greatly simplify many problems by reducing them to questions about particle approximations.

In patients with idiopathic generalized epilepsies (IGE), bursts of generalized spike and wave discharges (GSWD) lasting ≥2 seconds are considered absence seizures. The location of the absence seizures generators in IGEs is thought to involve interplay between various components of thalamo-cortical circuits; we have recently postulated that medication resistance may, in part, be related to the location of the GSWD generators (1). In the present study we hypothesized that patients with medication-refractory IGE (R-IGE) and continued absence seizures may have location of the GSWD generators other than the thalamus, as typically seen in the IGE patients. Hence, the objective of this study was to determine the location of the GSWD generators in patients with R-IGE using EEG/fMRI. 83 patients with IGE received concurrent EEG/fMRI at 4T. Nine of them (ages 15-55) experienced absence seizures during EEG/fMRI and were included; all were diagnosed with R-IGE. Subjects participated in up to three 20-minute EEG/fMRI sessions (400 volumes; TR = 3 seconds) performed at 4T. After removing fMRI and ballistocardiographic artifacts, 36 absence seizures were identified. Statistical parametric maps were generated for each of these sessions correlating seizures to BOLD response. Timing differences between brain regions were tested using statistical parametric maps generated by modeling seizures with onset times shifted relative to the GSWD onsets. While thalamic BOLD responses peaked at approximately 6 seconds after the onset of absence seizures, other areas including the prefrontal and dorsolateral cortices showed brief and non-sustained peaks occurring ~2 seconds prior to the maximum of the thalamic peak. Temporal lobe peaks occurred at the same time as the thalamic peak with a cerebellar peak occurring ~1 second later. Confirmatory analysis averaging cross

Summary Purpose Idiopathic generalized epilepsy (IGE) resistant to treatment is common, but its neuronal correlates are not entirely understood. Thus, the aim of this study was to examine resting-state default mode network (DMN) functional connectivity in patients with treatment-resistant IGE. Methods Treatment-resistance was defined as continuing seizures despite an adequate dose of valproic acid (valproate, VPA). Data from 60 epilepsy patients and 38 healthy controls who underwent simultaneous EEG and resting-state fMRI were included (EEG/fMRI). Independent component analysis (ICA) and dual regression were used to quantify DMN connectivity. Confirmatory analysis using seed-based voxel correlation was performed. Key Findings There was a significant reduction of DMN connectivity in patients with treatment-resistant epilepsy when compared to patients who were treatment-responsive and healthy controls. Connectivity was negatively correlated with duration of epilepsy. Significance Our findings in this large sample of patients with IGE indicate the presence of reduced DMN connectivity in IGE and show that connectivity is further reduced in treatment-resistant epilepsy. DMN connectivity may be useful as a biomarker for treatment-resistance.