Paul C. Bressloff, Ph. D - US grants

This IGERT project will develop a graduate program of cross-disciplinary research and training in Mathematical Biology. The goal of the program is to give students a solid training in core mathematics and genuine expertise in an area of contemporary biology. Such training will bring to bear the power of mathematics on the exciting and challenging problems of modern biology. Students will be recruited from a broad spectrum of mathematical, scientific and cultural backgrounds. It is expected that the graduates of this program will receive Ph.D.'s in mathematics, but by virtue of their broad-based training will be able to contribute to collaborative research efforts in numerous academic and industrial settings. In the process, the program seeks to build many new bridges between mathematics and biology potentially reshaping research for a new generation of mathematical biologists. The research and training program will be organized around the four research themes of biofluids, ecology and evolutionary biology, neuroscience, and physiology. A unique feature of this research and training program will be the establishment of Special Interest Groups (SIG's). Each SIG will be led by one or more faculty members with activities that include discussion of research problems, discussion of recent seminars, formal and informal talks about recent papers, student presented talks on background literature, etc. The training of students will also include formal coursework in both Mathematics and Biology, laboratory rotations or field work in an area of the life sciences, mentoring by both mathematics and life science faculty, and journal clubs, laboratory group meetings, and workshops. In these ways, the training of students will put great emphasis on collaboration and interaction across traditional academic disciplines.

IGERT is an NSF-wide program intended to meet the challenges of educating U.S. Ph.D. scientists and engineers with the multidisciplinary backgrounds and the technical, professional, and personal skills needed for the career demands of the future. The program is intended to catalyze a cultural change in graduate education by establishing innovative new models for graduate education and training in a fertile environment for collaborative research that transcends traditional disciplinary boundaries. In the fifth year of the program, awards are being made to twenty-one institutions for programs that collectively span the areas of science and engineering supported by NSF.

One of the important challenges in theoretical neurobiology is understanding the relationship between spatially structured activity states in the brain and the underlying neural circuitry that supports them. This has led to considerable interest in analyzing reduced biological models of neuronal networks. Most analytical studies of these network models assume that the system is spatially homogeneous. Recently, however, the principal investigator has shown that the combined effect of a spatially localized inhomogeneous input and recurrent synaptic interactions between neurons can result in nontrivial forms of coherent oscillations and waves. This motivates the current research project, which will carry out a more detailed study of the cellular and network mechanisms underlying the generation of these oscillations and waves. The research program will be divided into three parts corresponding to three distinct neurobiological application areas: (I) epileptiform activity in a model of disinhibited neural tissue, (II) stimulus-induced coherent oscillations in a model of primary visual cortex, and (III) localized activity states in a two-layer thalamic network model of the head direction system. In each of these cases the existence and stability of coherent activity states will be analyzed, and their dependence on various biologically relevant parameters will be determined. The mathematical aspects of the work will also be applicable to other population-based biological systems, in which the basic elements at the molecular, cellular or organismal level interact nonlocally in space.

Analysis of the dynamical mechanisms underlying spatially structured activity states in neural tissue is crucially important for understanding a wide range of neurobiological phenomena, both naturally occurring and pathological. For example, neurological disorders such as epilepsy and migraine are characterized by waves propagating across the surface of the brain. Determining the various cellular and network properties underlying the onset of such disorders could ultimately help in developing clinical techniques for eliminating them. Spatially coherent activity states are also prevalent during the normal healthy functioning of the brain, encoding local properties of visual and auditory stimuli, encoding head direction and spatial location, and maintaining persistent activity states in short-term working memory.

In biochemical and gene networks, there is an important distinction between intrinsic and extrinsic noise; extrinsic noise refers to external sources of randomness associated with environmental factors, whereas intrinsic noise refers to random fluctuations arising from the discrete and probabilistic nature of chemical reactions at the molecular level, which are particularly significant when the number of reacting molecules is small. The main goal of this research project is to develop a mathematical theory of intrinsic and extrinsic noise in neuronal population dynamics, adapting analytical methods from the study of chemical master equations such as Langevin approximations, stochastic hybrid systems, and large deviation theory. It is assumed that intrinsic noise at the network level arises from fluctuations about an asynchronous state due to finite size effects, whereas extrinsic noise arises from fluctuating external inputs. The theory is applied to a variety of neurobiological phenomena where noise is thought to play a crucial role, including the stimulus-induced synchronization of neural oscillators during sensory processing, and the generation of oscillations and waves during binocular rivalry. The latter forms the basis for non-invasive studies of human vision.

Noise has recently emerged as a key component of many biological systems including the brain. Randomness arises at multiple levels of brain function, ranging from molecular processes such as gene expression and the opening of ion channel proteins to complex networks of brain cells (neurons) that generate behavior. Indeed, the presence of noise has direct behavioral consequences, from setting perceptual and decision thresholds to influencing movement precision. Noise also contributes to the generation of spontaneous activity patterns during resting brain states, which are thought to play an important role in cognition. From one perspective, neuroscientists are interested in how, in spite of significant levels of noise, the brain appears to function reliably, consistent with the idea that it has evolved under the constraints that are imposed by noise. From another perspective, neuroscientists are interested in situations where the presence of noise can either be harmful to or, in certain cases, actually enhance brain function. The main goal of this project is to use mathematical and computational modeling to develop our understanding of how noise that is present at the molecular and cellular levels affects dynamics and information processing at the network level, both in healthy and diseased brains.

Numerous behaviors ranging from locomotion to cognitive tasks rely on oscillatory activity generated by networks of neurons in the brain. Despite the predominance and indispensability of brain oscillations, few theoretical tools are available for understanding how such oscillations are generated or controlled. A novel approach is used that combines biological experiments and mathematical analysis to break apart the complex interactions present in network components into simple building blocks. This allows core elements that are important in the generation of oscillations to be extracted and will clarify the role of other existing components in sculpting behavior using mathematical models. The models are tested through experiments that connect real time computer-simulated neurons to small oscillatory network in the crab central nervous systems. This project provides a framework for developing neural-based control systems with potential applications in robotics and bio-inspired computing.

This project is focused on developing improved insights into the functioning of the primary visual cortex of the brain. This is the first region of the cerebral cortex (the convoluted part of the brain responsible for higher cognitive function in primates) to receive and process visual information from the eyes. It can be viewed as a two-dimensional sheet of millions of brain cells (neurons) communicating with each other via electrical signals. The electrical activity patterns of these neurons encode information about a visual image, which is then processed by other regions of the brain, resulting in the visual perception of a dynamically changing three-dimensional world. Visual information is often represented by spatially structured or coherent activity patterns. Understanding the mechanisms that underpin the origin and maintenance of these dynamical patterns is not only important for understanding the normal functioning of the visual brain, but also the occurrence of pathological states during epileptic seizures and migraines. One of the major challenges in neuroscience is determining how the wiring of the visual brain contributes to the generation of cortical activity patterns. The investigator has developed mathematical models of the primary visual cortex based on models that describe the generation and spread of electrical activity across the two-dimensional cortical sheet. Recent experimental studies indicate, however, that the laminar or layered structure of the primary visual cortex plays a crucial role in the production of these activity patterns. This research project, which is part of a larger collaborative program with the Moran Eye Center at the University of Utah, aims to extend previous mathematical models in order to take into account the laminar structure and determine how it affects a range of spontaneous visual phenomena. The main focus of the collaboration is to use a combination of neurophysiology, anatomy, and computational modeling to understand the functional architecture of the primary visual cortex and its role in visual processing. The Moran group is currently developing the use of light to control genetically modified cells and virus labeling techniques in order to understand the fine-structure of the visual cortex, which will be used to refine the mathematical models. The underlying idea linking the two projects is that the neural circuits used in the mathematical models to understand spontaneous activity are the same as those used to explain observations of the normal response of the cortex to visual stimulations. This project promotes scientific progress in the interdisciplinary field of mathematical neuroscience and vision and contributes to the interdisciplinary training of graduate students and postdocs.

The modeling of the primary visual cortex involves the construction and analysis of continuum neural field models, in which the large-scale dynamics of spatially structured networks of neurons is described in terms of nonlinear, integro-differential equations. A major advantage of working with neural fields is that powerful methods from the mathematical theory of nonlinear partial differential equations can be adapted to analyze such models. Almost all previous studies of neural fields have ignored the fact that the cerebral cortex has a laminar structure, with neurons in distinct layers often having distinct stimulus response properties and participating in distinct circuits. There is also extensive coupling between layers via so-called vertical connections. In this project the laminar neural field models will be used study two important examples of spontaneous visual phenomena, namely, binocular rivalry waves and visual hallucinations. One possible mechanism for the latter is based on the idea that some chemical or physical disturbance can destabilize the visual part of the brain, inducing a spontaneous pattern of cortical activity. The geometry of the resulting hallucination thus reflects the intrinsic architecture and symmetry of the visual cortex. Analyzing such patterns can provide further insight in how the brain processes images in normal vision. Binocular rivalry is the phenomenon where perception switches back and forth between different images presented to the two eyes. The resulting fluctuations in perceptual dominance and suppression provide a basis for non-invasive studies of the human visual system and the identification of possible neural mechanisms underlying conscious visual awareness.