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.
