Testing Critical Slowing Down as a Bifurcation Indicator in a Low-Dissipation Dynamical System

Epileptic-network-based prediction and control of seizures in humans

Epilepsy is now conceptualized as a network disease. The epileptic brain network comprises structurally and functionally connected cortical and subcortical brain regions - spanning lobes and hemispheres -, whose connections and dynamics evolve in time. With this concept, focal and generalized seizures as well as other related pathophysiological phenomena are thought to emerge from, spread via, and be terminated by network vertices and edges that also generate and sustain normal, physiological brain dynamics. Research over the last years has advanced concepts and techniques to identify and characterize the evolving epileptic brain network and its constituents on various spatial and temporal scales. Network-based approaches further our understanding of how seizures emerge from the evolving epileptic brain network, and they provide both novel insights into pre-seizure dynamics and important clues for success or failure of measures for network-based seizure control and prevention. In this review, we summarize the current state of knowledge and address several important challenges that would need to be addressed to move network-based prediction and control of seizures closer to clinical translation.

Analytical theory of gain-switched laser pulses: From Lotka-Volterra slow-fast dynamics to passive Q switching

Gain switching is a simple technique for generating short pulses through direct modulation of optical gain in lasers. Its mathematical description requires the connection between a slowly varying, low intensity solution and a short, high intensity solution. Previous studies constructed a complete pulse by patching these two partial solutions at an arbitrary point in the phase plane. Here, we develop an asymptotic theory in which slow and fast solutions are matched together through a third intermediate solution. The mathematical analysis of the laser problem benefits from a preliminary study of the Lotka-Volterra equations when the two competing populations exhibit different timescales. Since this particular limit has never been explored, we first analyze the Lotka-Volterra equations before applying the theory to the more complex laser equations. We show a significant effect of the transition layer on the pulse intensity. Last, we discuss the case of sustained laser pulses generated through the Q-switching technique and show how their description may benefit from our theory.

Towards a Data-Driven Estimation of Resilience in Networked Dynamical Systems: Designing a Versatile Testbed

Estimating resilience of adaptive, networked dynamical systems remains a challenge. Resilience refers to a system's capacity "to absorb exogenous and/or endogenous disturbances and to reorganize while undergoing change so as to still retain essentially the same functioning, structure, and feedbacks." The majority of approaches to estimate resilience requires exact knowledge of the underlying equations of motion; the few data-driven approaches so far either lack appropriate strategies to verify their suitability or remain subject of considerable debate. We develop a testbed that allows one to modify resilience of a multistable networked dynamical system in a controlled manner. The testbed also enables generation of multivariate time series of system observables to evaluate the suitability of data-driven estimators of resilience. We report first findings for such an estimator.

What Models and Tools can Contribute to a Better Understanding of Brain Activity?

Despite impressive scientific advances in understanding the structure and function of the human brain, big challenges remain. A deep understanding of healthy and aberrant brain activity at a wide range of temporal and spatial scales is needed. Here we discuss, from an interdisciplinary network perspective, the advancements in physical and mathematical modeling as well as in data analysis techniques that, in our opinion, have potential to further advance our understanding of brain structure and function.