Analysis of neural clusters due to deep brain stimulation pulses
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ORIGINAL ARTICLE
Analysis of neural clusters due to deep brain stimulation pulses Daniel Kuelbs1 · Jacob Dunefsky2 · Bharat Monga3 · Jeff Moehlis4 Received: 4 July 2020 / Accepted: 20 November 2020 / Published online: 9 December 2020 © The Author(s), under exclusive licence to Springer-Verlag GmbH, DE part of Springer Nature 2020
Abstract Deep brain stimulation (DBS) is an established method for treating pathological conditions such as Parkinson’s disease, dystonia, Tourette syndrome, and essential tremor. While the precise mechanisms which underly the effectiveness of DBS are not fully understood, several theoretical studies of populations of neural oscillators stimulated by periodic pulses have suggested that this may be related to clustering, in which subpopulations of the neurons are synchronized, but the subpopulations are desynchronized with respect to each other. The details of the clustering behavior depend on the frequency and amplitude of the stimulation in a complicated way. In the present study, we investigate how the number of clusters and their stability properties, bifurcations, and basins of attraction can be understood in terms of one-dimensional maps defined on the circle. Moreover, we generalize this analysis to stimuli that consist of pulses with alternating properties, which provide additional degrees of freedom in the design of DBS stimuli. Our results illustrate how the complicated properties of clustering behavior for periodically forced neural oscillator populations can be understood in terms of a much simpler dynamical system. Keywords Neural oscillators · Clustering · Phase models · Deep Brain Stimulation
1 Introduction A primary motivation for this study is Parkinson’s disease, which can cause an involuntary shaking that typically affects the distal portion of the upper limbs, and difficulty initiating motion. For patients with advanced Parkinson’s disease who do not respond to drug therapy, electrical deep brain stimulation (DBS), an FDA-approved therapeutic procedure, Communicated by Jonathan Rubin.
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Jeff Moehlis [email protected] Daniel Kuelbs [email protected] Jacob Dunefsky [email protected] Bharat Monga [email protected]
1
Stanford University, Palo Alto, CA 94305, USA
2
Yale University, New Haven, CT 06520, USA
3
Department of Mechanical Engineering, University of California, Santa Barbara, CA 93106, USA
4
Department of Mechanical Engineering, Program in Dynamical Neuroscience, University of California, Santa Barbara, CA 93106, USA
may offer relief (Benabid et al. 1991). Here, a neurosurgeon guides a small electrode into the sub-thalamic nucleus or globus pallidus interna (GPi); the electrode is connected to a pacemaker implanted in the chest which sends periodic electrical pulses directly into the brain tissue. The efficacy of DBS for the treatment of Parkinson’s disease has been found to depend on the frequency of stimulation, with highfrequency stimulation (70 to 1000 Hz and beyond) being therapeutically effective (Benabid et al. 1991; Rizzone et al. 2001; Moro
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