# ISS SMALL GAIN THEOREM FOR SPATIOTEMPORAL DELAYED DYNAMICS WITH APPLICATION TO FEEDBACK ATTENUATION OF PATHOLOGICAL BRAIN OSCILLATIONS

Topic: Stabilization | All

Séance du jeudi 18 Février 2016, Salle L 224, 14h00.

Antoine CHAILLET, Université Paris Sud 11 - Supélec - L2S - EECI

Several disorders are related to pathological brain oscillations. In the case of Parkinson’s disease, sustained low-frequency oscillations (especially in the beta-band, 13–30Hz) correlate with motor symptoms. Among the hypotheses to explain the generation of such pathological oscillations, one is the possible pacemaker role played by two feedback-interconnected neuronal populations, one of which is excitatory whereas the other one is inhibitory. In this scenario, the abnormal increase of synaptic weights between these two populations, combined with the inherent transmission delays, gives rise to some instability which translates into sustained oscillations.

In this talk, we rely on a spatiotemporal model of the neuronal populations involved to show that a simple proportional feedback on the excitatory population is enough to attenuate pathological oscillations, provided that the internal synaptic weights within the inhibitory population are sufficiently low. The model used is a delayed nonlinear integro-differential equation know as delayed neural field. Delays are allowed to be position-dependent in order to model longer transmission delays between more distant neurons. To the best of our knowledge, this constitutes the first analysis of stabilizability of delayed neural fields.

Our proof relies on Krasovskii-Lyapunov arguments. We first provide conditions under which each population is input-to-state stable (ISS) with respect to the inputs coming from the other population. We then adapt the ISS small-gain theorem to delayed spatiotemporal dynamics in order to conclude on the stability of the overall loop. The use of ISS allows in turn to evaluate the robustness of the proposed feedback with respect to imperfect actuation and control delays.

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Antoine CHAILLET, Université Paris Sud 11 - Supélec - L2S - EECI

Several disorders are related to pathological brain oscillations. In the case of Parkinson’s disease, sustained low-frequency oscillations (especially in the beta-band, 13–30Hz) correlate with motor symptoms. Among the hypotheses to explain the generation of such pathological oscillations, one is the possible pacemaker role played by two feedback-interconnected neuronal populations, one of which is excitatory whereas the other one is inhibitory. In this scenario, the abnormal increase of synaptic weights between these two populations, combined with the inherent transmission delays, gives rise to some instability which translates into sustained oscillations.

In this talk, we rely on a spatiotemporal model of the neuronal populations involved to show that a simple proportional feedback on the excitatory population is enough to attenuate pathological oscillations, provided that the internal synaptic weights within the inhibitory population are sufficiently low. The model used is a delayed nonlinear integro-differential equation know as delayed neural field. Delays are allowed to be position-dependent in order to model longer transmission delays between more distant neurons. To the best of our knowledge, this constitutes the first analysis of stabilizability of delayed neural fields.

Our proof relies on Krasovskii-Lyapunov arguments. We first provide conditions under which each population is input-to-state stable (ISS) with respect to the inputs coming from the other population. We then adapt the ISS small-gain theorem to delayed spatiotemporal dynamics in order to conclude on the stability of the overall loop. The use of ISS allows in turn to evaluate the robustness of the proposed feedback with respect to imperfect actuation and control delays.

Download PDF