Irregular Firing in Dopaminergic Neurons and Related Problems

理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。在霍奇金和赫胥黎发表了一系列经典论文之后，非线性微分方程成为神经细胞电活动建模的主要框架。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。动力系统理论中最重要的概念之一是稳定性。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所推动的。在这种情况下，研究稳定解（即在小扰动下持续存在的解）是很自然的，因为这样的解在物理上是可以观察到的。在系统层面上，这导致了对结构稳定系统的研究，即在小的参数变化下其解保持其定性性质的系统。结构稳定性丧失的现象称为分岔。从物理角度来看，接近分岔的系统是罕见的。对生物系统进行建模的情况就不同了。生物学模型的一个显著特征是它们常常接近于一个分叉点。特别是，许多已知的神经细胞模型都位于分支附近。在神经元模型中，分叉的接近产生了变异性的来源，并对它们产生的放电模式产生了重大影响。在分岔附近，系统在产生形式和频率变化的动态模式方面具有更大的灵活性。已知某些细胞振荡频率的短暂变化会影响神经递质释放和激素分泌的速率，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机制和可变性对于确定神经细胞的功能至关重要。本研究的目的是探讨有噪声和无噪声霍奇金-赫胥黎型模型中接近分岔的含义。为此，PI使用了非线性微分方程理论和随机过程理论的技术。在本研究过程中发展的理论将应用于研究具体生物物理系统中产生放电模式的机制。后者包括（但不限于）哺乳动物中脑中的多巴胺能神经元、胰腺β细胞和颗粒状新皮层中的锥体细胞。这项研究的更广泛的科学影响是双重的：首先，它通过使用先进的数学技术增强了对复杂生物现象的理解；其次，它确定了由生物学引起的新的数学问题。本项目的结果有望引起广大从事非线性科学研究人员的兴趣，并激发非线性动力学方面的新研究。这项研究反映了PI所在机构数学系与新成立的德雷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。基于这项研究的部分结果，PI将在德雷塞尔大学开发并教授一门“计算神经科学”课程。从这项研究中得出的适当问题将整合到PI在德雷塞尔大学为研究生和本科生教授的微分方程课程中。

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.