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Delay-induced oscillations in Wilson and Cowan’s model: an analysis of the subthalamo-pallidal feedback loop in healthy and parkinsonian subjects. (English) Zbl 1267.92034

Summary: The model proposed by H.R. Wilson and J.D.Cowan [Biophys. J. 12, No. 1, 1–24) 1972)] describes the dynamics of two interacting subpopulations of excitatory and inhibitory neurons. It has been used to model neural structures like the olfactory bulb, whisker barrels, and the subthalamo-pallidal system. It is well-known that this system can exhibit an oscillatory behavior that is amplified by the presence of delays. In the absence of delays, the conditions for stability are well-known. The aim of our paper is to clarify these conditions when delays are included in the model. The first ingredient of our methods is a new necessary and sufficient condition for the existence of multiple equilibria. This condition is related to those for local asymptotic stability. In addition, a sufficient condition for global stability is also proposed. The second and main ingredient is a stability analysis of the system in the frequency-domain, based on the H. Nyquist criterion [Bell Syst. Technol. J. 11, No.1, 126–147 (1932)] that takes the four independent delays into account. The methods proposed in this paper can be applied to analyse the stability of the subthalamo-pallidal feedback loop, a deep brain structure involved in the Parkinson’s disease. Our stability conditions are easy to compute and characterize sharply the system’s parameters for which spontaneous oscillations appear.

MSC:

92C20 Neural biology
92C50 Medical applications (general)
92-08 Computational methods for problems pertaining to biology

Software:

DDE-BIFTOOL
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References:

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