Mathematical Analysis of Neural Dynamics with Multiple Frequencies

神经系统在所有认知状态下都产生电活动，并且这种活动通常在多个频带中的任何给定时间显示出显著的功率。 该项目侧重于以前的大规模模拟所提出的数学问题，并寻求更深入地了解单细胞中固有电流和离子电流的定性特性如何塑造各种大型和小型网络模拟中所看到的复杂行为。 这项工作侧重于两个具体情况，其中提出了许多一般性问题。 第一个是海马体中伽马（40-90 Hz）和θ（4-12 Hz）节律的相互作用。 如先前的数值和实验工作所示，伽马和θ节律似乎是由海马神经元的不同子网络在体外产生的，其中一些组件是共同的;模拟表明，参数的变化可以切换对共同元件的控制，并改变不同频带中的功率。 该项目认为，全球分叉涉及开关的控制。 第二种情况涉及麻醉剂异丙酚存在时脑节律的变化，在生物物理水平上，异丙酚主要通过增加GABA_A介导的抑制的衰减时间和幅度起作用。 之前建模工作的一个中心问题是所谓的“β嗡嗡声”的起源，其中低剂量的异丙酚会使患者兴奋而不是镇静，β频段的功率增加（13-30赫兹）和较低和较高频段的减少。 模拟显示了具有多个组件的模型网络中的这种非直观行为，特别是通过将抑制性细胞“聚类”成反相放电的亚组，将低频转换为高频。 Kopell指导了许多研究生和博士后研究员;该项目将数学分析与其他专注于功能的工作联系起来，从而使学员能够看到数学如何用于从生物物理学到功能的桥梁。神经系统在所有认知状态下都会产生电活动，并且这种活动通常在任何给定时间在多个频带中显示出显着的功率。 该项目侧重于以前的大规模模拟所提出的数学问题，并寻求更深入地了解单细胞中固有电流和离子电流的定性特性如何塑造各种大型和小型网络模拟中所看到的复杂行为。 这项工作解决了大脑如何产生多种频率的一般问题，以及生物物理学的变化如何改变动态成分的混合。 在第一个分项目中，这与网络如何对具有反映信息编码的空间和时间结构的输入作出反应的中心问题有关。 在第二，这项工作有助于桥梁的麻醉剂的生物物理效应的知识，其功能特性（诱导意识丧失），通过绑定的生物物理变化的动态变化已知的认知状态。 所提议的分析工具已在简单得多的情况下使用。 这项工作将需要开发扩展功能，以便应用于更大和更复杂的网络。 扩展应该适用于广泛的神经应用。 Kopell是波士顿大学生物动力学中心和数学与计算神经科学项目的联合主任。 在这种情况下，她的作品和导师大量的研究生和博士后，其中包括许多妇女。 该项目将数学分析与其他专注于功能的工作联系起来，从而使这些和其他学员能够看到数学如何用于从生物物理学到功能的桥梁。

The nervous system produces electrical activity in all cognitive states, and this activity generally displays significant power at any given time in multiple frequency bands. This project focuses on the mathematical issues that have been raised by previous large-scale simulations, and seeks to get a deeper understanding of how the qualitative properties of intrinsic and ionic currents in single cells shape the complex behaviour that has been seen in a variety of simulations of large and small networks. The work focuses on two specific situations which present many of the general issues. The first is the interaction of the gamma (40-90 Hz) and theta (4-12 Hz) rhythms in the hippocampus. As shown in previous numerical and experimental work, the gamma and theta rhythms appear to be produced in vitro by different sub-networks of hippocampal neurons, with some components in common; simulations have shown that changes in parameters can switch control of the common elements and change the power in the different frequency bands. The project considers the global bifurcations involved in switches of control. The second situation concerns changes of brain rhythms in the presence of the anesthetic propofol which, at the biophysical level, acts mainly by increasing the decay time and amplitude of GABA_A mediated inhibition. A central question of the previous modeling work is the origin of the so-call "beta buzz", in which a low dose of propofol excites, rather than sedates, the patient, with an increase in the power in the beta frequency bands (13-30 Hz) and a decrease in lower and higher frequency bands. Simulations have shown this un-intuitive behaviour in model networks having multiple components, notably by the creation of "clustering" of inhibitory cells into subgroups firing in antiphase, transforming low frequencies into higher ones. Kopell mentors many graduate students and postdoctoral fellows; this project ties mathematical analysis to other work focusing on function, and thus allows trainees to see how mathematics can be used to bridge from biophysics to function.The nervous system produces electrical activity in all cognitive states, and this activity generally displays significant power at any given time in multiple frequency bands. This project focuses on the mathematical issues that have been raised by previous large-scale simulations, and seeks to get a deeper understanding of how the qualitative properties of intrinsic and ionic currents in single cells shape the complex behaviour that has been seen in a variety of simulations of large and small networks. The work addresses the general issues of how the brain produces its multiple frequencies, and how changes in biophysics can change the mixture of dynamic components. In the first sub-project, this is linked with the central question of how networks react to inputs with spatial and temporal structure reflecting the coding of information. In the second, the work helps to bridge the knowledge of biophysical effects of an anesthetic to its functional properties (inducing loss of consciousness) by tying the biophysical changes to changes in dynamics known to be related to cognitive state. The analytical tools proposed have been used in much simpler contexts. The work will require the development of extensions to allow application to larger and more complex networks. The extensions should be applicable to a wide range of neural applications. Kopell is co-Director of the Center for BioDynamics and the Program in Mathematical and Computational Neuroscience at Boston University. In this context, she works with and mentors a large number of graduate students and postdocs, including many women. This project ties mathematical analysis to other work focusing on function, and thus allows these and other trainees to see how mathematics can be used to bridge from biophysics to function.