Georgi S. Medvedev - US grants

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.

Synchronization is an important mode of collective behavior in diverse physical, biological, and technological networks. In many applications, local dynamics is modeled by systems of differential equations and the interaction schemes are defined by weighted graphs. This research is aimed at advancing the mathematical theory of synchronization and pattern formation in coupled systems of differential equations on graphs. Networks with different types of local dynamics, such as those generated by limit cycles, chaotic attractors, or induced by noise, are considered under general assumptions on the network architecture. The principal investigator (PI) develops mathematically rigorous yet practically efficient conditions guaranteeing synchronization, studies robustness of synchrony to noise, and analyzes patterns of electrical activity in gap-junctionally coupled neuronal networks. The graph-theoretic interpretation of the analytical results is emphasized. The PI seeks systematic ways of quantifying the contribution of the network topology to the dynamics of coupled oscillators by integrating combinatorial techniques into dynamical systems analysis.

Synchronization is a universal phenomenon with abundant applications across science and technology. Power grid safety, effective communication in information networks, and coordination of unmanned vehicles are just three of many areas of technology where synchronization is crucial. Furthermore, synchronization plays a prominent role in the mechanisms of many vital physiological and cognitive processes such as respiration, sleep, and attention, as well as in the mechanisms of several severe neurodegenerative disorders such as Parkinson's Disease and epilepsy. The PI develops new mathematical tools and uses them to study synchronization in biophysical models including that of the Locus Coeruleus network, a group of neurons in the mammalian brainstem involved in the regulation of cognitive performance and behavior. This study enhances our understanding of how to achieve, control, or destroy synchrony in an important class of models. This investigation fosters research at the interface between theories of dynamical systems, stochastic processes, and algebraic graph theory. The results of this research will be integrated into graduate courses in dynamical systems and mathematical neuroscience that are taught by the PI at Drexel University. This grant supports one graduate student and sponsors summer research for two undergraduate students.

A number of very important problems in science and technology lead to the analysis of large networks of interacting dynamical systems. Our ability to predict epileptic seizures, to effectively control a power grid, or to coordinate a group of robots rely on our understanding of the principles underlying collective behavior in coupled dynamical systems. Many natural and man-made networks around us feature extraordinary richness and complexity of interconnections. Mathematical modeling of such networks poses new challenges for nonlinear science and requires new approaches incorporating combinatorial and probabilistic methods into dynamical analysis of complex systems. Graph Theory holds an extraordinary potential to inspire new powerful techniques for extended dynamical systems and applications to technological, social, economic, and biological networks. In this research, the Principal Investigator (PI) combines state-of-the-art techniques of Graph Theory with analytical methods for Dynamical Systems to develop an effective set of tools for studying coupled dynamical systems and their applications in neuroscience. The results of this research will be integrated into graduate courses in Dynamical Systems and Mathematical Neuroscience.

In this project, the PI develops a unified approach for studying dynamical networks as evolution equations on three types of spatial domains: Caley graphs, quasirandom graphs, and graph limits. For problems in each class, analytical and algebraic techniques, which mesh well with the underlying spatial structures, are identified. These techniques are used to study stability of spatial patterns in systems of coupled phase oscillators on Caley graphs, synchronization in systems of coupled chaotic maps, and differential equations on different random graphs including small-world and graphs that exhibit power law behavior, as well as those that exhibit temporally structured stable patterns in a neural network model of learning. The PI seeks systematic ways for describing the role of network connectivity in shaping the dynamics in coupled systems. This work is aimed toward development of a theory for interacting dynamical systems on regular and random graphs.

A large number of important structures and phenomena in nature, society, and technology can be modeled by networks of interacting dynamical systems. Examples include power and communication networks in technology, neuronal and genetic networks in biology, as well as social and economic networks, to name a few. Understanding behavior of these systems requires advanced mathematical techniques for studying dynamics in complex networks. This project will use mathematical modeling, analysis, and numerical simulations to elucidate the link between the structure and dynamics in complex networks. In particular, the investigator aims to find new ways for describing network organization in dynamical models and study critical phenomena, such as the onset of synchronization and emergence and bifurcations of spatial patterns in networks of coupled oscillators. The results of this research and the tools that will be developed in its course, will enhance our ability to understand, predict, and control the behavior of real world networks.

The mean field approximation is one of the most effective analytical tools available for studying large ensembles of interacting dynamical systems. Originally developed for problems in statistical physics, this method has been extremely successful for studying collective dynamics in coupled oscillator models of various physical, chemical, biological, and technological systems. The analysis of synchronization in the Kuramoto model of coupled phase oscillators and the bifurcation analysis of chimera states rely on the mean field equation formally derived in the limit, as the number of oscillators goes to infinity. Despite its spectacular success in applications, the mathematical basis of the mean field approximation of coupled dynamical systems on networks is not well understood.  In this research, the aim is to derive and rigorously justify the mean field description of dynamics of coupled systems on convergent families of graphs. This advances the mean field theory for coupled systems in two ways: first, by extending the main results of this theory to a large class of random networks, including those with small-world and scale free connectivity, and, second, by relaxing the key regularity assumptions, blocking the application of this theory to real world networks. The new theoretical results are used to study the onset of synchronization in systems of coupled phase oscillators with spatially structured interactions, stable dynamical regimes in the Kuramoto model on power law graphs, and pattern formation in neural fields in random media.

We live in the age of networks. Our everyday lives depend on robust and predictable performance of different technological networks around us (such as power grids and communication networks). Human physiology relies on coordinated activity of hundreds of cellular networks (for example, networks of neural cells in the brain, cells in the heart and pancreas, to name a few). The patterns of connections in real world networks can be complex and exhibit nontrivial statistical properties. Understanding how the structural organization of a network affects its dynamics is the principal challenge in the theory of interacting dynamical systems that sets it apart from the theories for classical spatially extended dynamical systems, such as partial differential equations or lattice dynamical systems. The Principal Investigator (PI) seeks to develop a systematic mathematical approach to the analysis of dynamical networks. Through the development of new mathematical techniques and analyzing representative mathematical models, the PI aims to elucidate the relation between the structure and dynamics in complex networks. The PI is committed to teaching and training students. A six-month long Research Co-op for two undergraduate students will be organized in the course of this research. The PI will continue to organize minisymposia on dynamical networks at conferences on differential equations and dynamical systems.

This research advances the theory for interacting dynamical systems through the development of new theoretical results and analyzing selected models. The PI and colleagues identify new dynamical phenomena and study them using the combination of tools from graph theory, probability, and analysis. The emphasis will be on the effects of random spatial organization and noise on dynamics of large networks. The following problems are addressed: the Large Deviation Principle for interacting dynamical systems on random graphs, metastability in the continuum Kuramoto model forced by noise, and synchronization and pattern formation in the Kuramoto model with inertia. The Kuramoto model of coupled phase oscillators plays a central role in the theory of synchronization with many important applications in science and technology. In particular, the analysis of synchronization and stability of clusters developed in this research elucidates the dynamics emerging in high-voltage power grids.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.