Department of Applied Mathematics, Zhejiang University of Technology, Hangzhou 310023, China; Key Laboratory of Systems Biology, Center for Excellence in Molecular Cell Science, Institute of Biochemistry and Cell Biology, Shanghai Institute of Biological Sciences, Chinese Academy of Sciences, Shanghai 200031, China.
Department of Mathematics and Statistics, College of Science, Huazhong Agricultural University, Wuhan 430070, China.
J Theor Biol. 2019 Jun 7;470:1-16. doi: 10.1016/j.jtbi.2019.03.008. Epub 2019 Mar 8.
In this paper, we build a basal ganglia-cortex-thalamus model to study the oscillatory mechanisms and boundary conditions of the beta frequency band (13-30 Hz) that appears in the subthalamic nucleus. First, a theoretical oscillatory boundary formula is obtained in a simplified model by using the Laplace transform and linearization process of the system at fixed points. Second, we simulate the oscillatory boundary conditions through numerical calculations, which fit with our theoretical results very well, at least in the changing trend. We find that several critical coupling strengths in the model exert great effects on the oscillations, the mechanisms of which differ but can be explained in detail by our model and the oscillatory boundary formula. Specifically, we note that the relatively small or large sizes of the coupling strength from the fast-spiking interneurons to the medium spiny neurons and from the cortex to the fast-spiking interneurons both have obvious maintenance roles on the states. Similar phenomena have been reported in other neurological diseases, such as absence epilepsy. However, some of those interesting mutual regulation mechanisms in the model have rarely been considered in previous studies. In addition to the coupling weight in the pathway, in this work, we show that the delay is a key parameter that affects oscillations. On the one hand, the system needs a minimum delay to generate oscillations; on the other hand, in the appropriate range, a longer delay leads to a higher activation level of the subthalamic nucleus. In this paper, we study the oscillation activities that appear on the subthalamic nucleus. Moreover, all populations in the model show the dynamic behaviour of a synchronous resonance. Therefore, we infer that the mechanisms obtained can be expanded to explore the state of other populations, and that the model provides a unified framework for studying similar problems in the future. Moreover, the oscillatory boundary curves obtained are all critical conditions between the stable state and beta frequency oscillation. The method is also suitable for depicting other common frequency bands during brain oscillations, such as the alpha band (8-12 Hz), theta band (4-7 Hz) and delta band (1-3 Hz). Thus, the results of this work are expected to help us better understand the onset mechanism of parkinson's oscillations and can inspire related experimental research in this field.
本文构建了一个基底节-皮层-丘脑模型，旨在研究亚丘脑核中出现的β频带（13-30 Hz）的振荡机制和边界条件。首先，通过使用系统在固定点的拉普拉斯变换和线性化过程，在简化模型中获得了一个理论的振荡边界公式。其次，我们通过数值计算模拟了振荡边界条件，结果与我们的理论结果非常吻合，至少在变化趋势上是如此。我们发现模型中的几个临界耦合强度对振荡有很大的影响，其机制虽然不同，但可以通过我们的模型和振荡边界公式详细解释。具体来说，我们注意到，来自快放电中间神经元到中等棘突神经元以及来自皮层到快放电中间神经元的耦合强度的相对较小或较大尺寸对状态都有明显的维持作用。在其他神经疾病中，如失神性癫痫，也有类似的现象。然而，模型中一些有趣的相互调节机制在以前的研究中很少被考虑。除了通路中的耦合权重外，在这项工作中，我们还表明，延迟是影响振荡的关键参数。一方面，系统需要最小的延迟才能产生振荡；另一方面，在适当的范围内，较长的延迟会导致亚丘脑核的激活水平更高。本文研究了出现在亚丘脑核上的振荡活动。此外，模型中的所有群体都表现出同步共振的动态行为。因此，我们推断所获得的机制可以扩展到探索其他群体的状态，并且该模型为未来研究类似问题提供了一个统一的框架。此外，所得到的振荡边界曲线都是稳定状态和β频带振荡之间的临界条件。该方法也适用于描绘大脑振荡过程中的其他常见频带，如α频带（8-12 Hz）、θ频带（4-7 Hz）和δ频带（1-3 Hz）。因此，这项工作的结果有望帮助我们更好地理解帕金森氏症振荡的起始机制，并能激发该领域的相关实验研究。
