# Closed-loop firing rate regulation of two interacting excitatory and inhibitory neural populations of the basal ganglia

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## Abstract

This paper develops a new closed-loop firing rate regulation strategy for a population of neurons in the subthalamic nucleus, derived using a model-based analysis of the basal ganglia. The system is described using a firing rate model, in order to analyse the generation of beta-band oscillations. On this system, a proportional regulation of the firing rate reduces the gain of the subthalamo-pallidal loop in the parkinsonian case, thus impeding pathological oscillation generation. A filter with a well-chosen frequency is added to this proportional scheme, in order to avoid a potential instability of the feedback loop due to actuation and measurement delays. Our main result is a set of conditions on the parameters of the stimulation strategy that guarantee both its stability and a prescribed delay margin. A discussion on the applicability of the proposed method and a complete set of mathematical proofs is included.

## Keywords

Neural oscillations Firing rate models Time-delay systems Basal ganglia Parkinson’s disease Deep brain stimulation Closed-loop stimulation## Notes

### Acknowledgments

This work was financially supported by the European Commission through the FP7 NoE HYCON2 and by the region Ile-de-France through the Neurosynch project (RTRA Digiteo). The work of the fourth author was supported by the Government of the Russian Federation (grant 074-U01).

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