Irregular Firing in Dopaminergic Neurons and Related Problems

了解神经元中产生不同放电模式的机制以及它们之间的转换是理解神经系统如何处理信息的基础。在霍奇金和赫胥黎的一系列经典论文之后，非线性微分方程式成为模拟神经细胞中电活动的主要框架。今天，应用动力系统理论的语言和技术是理解计算生物学不可或缺的一部分。稳定性是动力系统理论中最重要的概念之一。从历史上看，DS理论的发展是由物理问题，特别是力学和电子学问题推动的。在这种情况下，研究稳定解(即那些在小扰动下持续存在的解)是很自然的，因为这样的解应该是物理上可以观察到的。在系统层面上，这导致了对结构稳定系统的研究，即其解在参数的微小变化下保持其定性性质的系统。结构失去稳定性的现象称为分叉。从物理角度来看，接近分叉的系统很少见。在对生物系统进行建模时，情况有所不同。生物模型的一个显著特点是，它们往往接近分叉。具体地说，许多已知的神经细胞模型位于分叉附近。接近分叉点在神经元模型中创造了一个可变性的来源，并对它们产生的放电模式产生了重大影响。在分叉附近，系统在生成形式和频率不同的动态模式方面获得了更大的灵活性。众所周知，某些细胞振荡频率的瞬时变化会影响神经递质释放和激素分泌的速度，以及其他重要的生理和认知过程。因此，了解不同放电模式的控制和可变性的机制对于确定神经细胞的功能至关重要。本研究的目的是研究有噪声和无噪声的Hodgkin-Huxley模型中分叉附近的含义。为此，PI使用了非线性微分方程组理论和随机过程理论的技术。在这项研究过程中将发展的理论将被应用于研究在具体的生物物理系统中产生激发模式的机制。后者包括(但不限于)哺乳动物中脑中的多巴胺能神经元、胰岛β细胞和无颗粒新皮质中的锥体细胞。这项研究的更广泛的科学影响有两个：第一，它通过使用先进的数学技术加强对复杂生物现象的理解；第二，它确定了由生物学驱动的新的数学问题。预计本项目的结果将引起从事非线性科学工作的广大研究人员的兴趣，并促进对非线性动力学的新研究。这项研究反映了PI所在机构的数学系与新的德雷克塞尔医学院建立更紧密联系的目标。PI将培训一名研究生，并让他/她参与与该项目相关的研究。部分基于这项研究的结果，PI将在德雷克塞尔大学开发和教授一门名为“计算神经科学”的课程。从这项研究中得出的适当问题将被整合到微分方程式的课程中，该课程是PI为德雷克塞尔大学的研究生和本科生教授的。

Understanding mechanisms for generating different firing patterns in neurons and transitions between them is fundamental for understanding how the nervous system processes information. After a classical series of papers by Hodgkin and Huxley, nonlinear differential equations became the main framework for modeling electrical activity in neural cells. Today the language and techniques of the applied dynamical systems theory are an indispensable part of understanding computational biology. One of the most important concepts of the dynamical systems theory is that of stability. Historically, the development of the theory of DS was motivated by physical problems, in particular, by problems in mechanics and electronics. In this context, it was natural to study stable solutions (i.e., those that persist under small perturbations), because such solutions are expected to be physically observable. On the system level, this led to study of structurally stable systems, i.e. systems whose solutions preserve their qualitative properties under small variations of parameters. A phenomenon of loss of structural stability is called a bifurcation. From a physical point of view, systems near a bifurcation are rare. The situation is different in modeling biological systems. A distinctive feature of biological models is that they are often close to a bifurcation. In particular, many known models of neural cells reside near a bifurcation. The proximity to a bifurcation creates a source of variability in neuronal models and has a significant impact on the firing patterns that they produce. Near a bifurcation systems acquire greater flexibility in generating dynamical patterns varying in form and frequency. Transient changes in the frequency of oscillations in certain cells are known to affect the rates of neurotransmitter release and hormone secretion, as well as other important physiological and cognitive processes. Therefore, understanding the mechanisms for control and variability of different modes of firing is essential for determining how neural cells function. The goal of the present research is to investigate the implications of the proximity to a bifurcation in the models of Hodgkin-Huxley type with and without noise. For this, the PI uses the techniques of the theory of nonlinear differential equations and the theory of random processes. The theory to be developed in the course of this research will be applied to study the mechanisms for generating firing patterns in concrete biophysical systems. The latter include (but are not limited to) dopaminergic neurons in the mammalian midbrain, pancreatic beta-cells, and pyramidal cells in agranular neocortex. The broader scientific impacts of this research are twofold: first, it enhances understanding of complex biological phenomena through the use of advanced mathematical techniques; second, it identifies new mathematical problems motivated by biology. The results of the present project are expected to generate interest in a broad community of researchers working in nonlinear science and to stimulate new research in nonlinear dynamics. This research reflects the goal of the Department of Mathematics in the PI's home institution to develop stronger links to the new Drexel College of Medicine. The PI will train a graduate student and engage him/her into research relevant to this project. Based in part on the results of this research, the PI will develop and teach a course 'Computational Neuroscience' at Drexel University. Appropriate problems drawn from this research will be integrated in the courses on differential equations, which the PI teaches for graduate and undergraduate students at Drexel University.