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.

深部脑刺激 (DBS) 是治疗帕金森病、肌张力障碍、图雷特综合征和特发性震颤等病理状况的既定方法。虽然 DBS 有效性背后的精确机制尚未完全了解，但对周期性脉冲刺激的神经振荡器群体的几项理论研究表明，这可能与聚类有关，其中神经元的亚群是同步的，但亚群是彼此不同步。聚类行为的细节以复杂的方式取决于刺激的频率和幅度。在本研究中，我们研究了簇的数量及其稳定性特性、分岔、吸引力盆地可以通过定义在圆圈上的一维地图来理解。此外，我们将此分析推广到由具有交替特性的脉冲组成的刺激，这为 DBS 刺激的设计提供了额外的自由度。我们的结果说明了如何从更简单的动态系统的角度理解周期性强制神经振荡器群体的聚类行为的复杂特性。