David Terman - US grants

Reaction-diffusion equations arise as models in many areas of Mathematical Biology and Chemistry. A feature common to many of these systems is the existence of traveling wave solutions. These correspond to solutions which appear to be traveling with constant shape and velocity. The objective of this project is to develop new techniques for proving the existence and stability of traveling wave solutions. There are two basic steps in the program outlined in this project. The first consists of developing general methods for proving the existence and stability of traveling wave solutions. A basic ingredient for these methods is the Conley index, a relatively recent tool developed by the late Professor Charles Conley in 1977-1978. The second part consists of applications of the basic theory. Specific applications to be considered are the existence and stability of traveling wave solutions arising from ecology and combustion, and the existence of radially symmetric solutions of semilinear elliptic systems. This last problem is closely related to the question of the existence of traveling wave solutions. The project presented by Professor Terman contains several sub- projects related to various potential connections of this mathematical theory to other scientific problems. Professor Terman is a young talented mathematician who has already established his leading position in this area of research.

Reaction diffusion equations may give rise to a variety of pattern like solutions. These include traveling wave solutions, stable steady solutions, cellular patterns, oscillatory solutions, and even solutions which behave in a chaotic manner. All of the systems which the principle investigator will consider arise as a mathematical model of a biological or chemical system. This project will consider models for combusion, chemical reactions on catalytic surfaces, phase transitions, and bursting in excitable membranes. The primary goals of this research project are to investigate through which mathematical mechanisms these patterns may arise, analyse the stability properties of each pattern, and understand what physical significance each pattern has for the underlying physical problem. The problems to be addressed in this project raise a number of fundamental mathematical issues. Of particular interest are patterns arising in more than one space dimension. Much of the analysis will be concerned with how higher dimensional patterns bifurcate from basically one dimensional objects such as planar wave solutions and radially symmetric solutions. Simple examples demonstrate that the structure of the bifurcating patterns may be incredibly complex.

The investigator studies several mathematical models for biological and chemical systems. These include models for electrically excitable cells such as pancreatic beta-cells, and models for pattern formation on isothermal catalyst surfaces. Numerous models have been developed to test different hypothesis for the biophysical mechanisms that underlie the electrical activity in these cells. New analytic methods will be developed to predict and classify the variety of different behaviors of solutions to these recent models. Systems that model coupled beta-cells are also considered. Very complex behaviors including standing waves, rotating spirals, and chaotic dynamics have been observed experimentally on catalyst surfaces. Several models are studied in order to understand the underlying mathematical mechanisms responsible for this rich structure of pattern formation. Pancreatic beta-cells are responsible for the secretion of insulin. This analysis will be used to determine the relative contributions of different ion currents to the beta-cell electrical activity. It will help in understanding which of these currents are responsible for bursting activity in the cells, and which currents govern the duration of each burst. Moreover, this analysis will be used to identify glucose-sensing mechanisms and determine the role of the electrical activity in stimulating insulin release. This work contributes to fundamental understanding of pancreatic functions. Topics on the formation of surface patterns arise in connection with studies of the behavior of catalysts.

Terman The investigator studies large networks of relaxation-type oscillators with coupling that mimics chemical synapses. These lead to large systems of differential equations that include numerous physical parameters as well as multiple time scales. The models typically display an incredibly rich structure of dynamic behavior; this depends in a highly nontrivial way on the various parameters. The investigator systematically studies what sorts of collective behavior are possible in a given network, determines the mathematical mechanisms responsible for this rich oscillatory behavior, classifies the sorts of bifurcations that can take place as parameters in the models are varied, and determines the physical consequences of the underlying mathematical structures. The types of networks considered in this project arise in numerous applications. These include several areas of the nervous system, respiration, movement, secretion, and models for scene segmentation. The project examines models for bursting oscillations in pancreatic beta cells. These cells are responsible for the secretion of insulin. It also analyzes models for oscillations in various parts of the nervous system, including the hippocampus and the thalamus. The investigator also continues a study of scene segmentation and other aspects of visual processing. This work contributes to fundamental understanding of pancreatic functions and a variety of neurophysiological questions.

Hastings 9801227 The investigator and his colleagues organize an interdisciplinary conference to bring together biologists and mathematicians to discuss wave phenomena from their varying perspectives. The conference aims to give both biologists and mathematicians insights into the types of models that can be used for wave phenomena and the parameter ranges where such behavior can be expected. To this end, the conference includes both general lectures and more technical talks where particular techniques are explored more fully. Of special interest in techniques are continuation methods in models of long range interaction, where integral equation models are involved. One-dimensional traveling waves have long been of interest to biologists, particularly in neurobiology where they describe the propopagation of electrical signals down a nerve axon or as a plane wave across a two-dimensional collection of electrically active cells. Related phenomena include spiral and other patterns, such as those thought to be responsible for some pathogenic behavior in cardiac tissue. Similar patterns in the brain are of current interest as well. Such behavior is not limited to neurobiology, and appears in a wide variety of chemical and biological systems, such as the Belousov-Zhabotinsky reaction, slime molds, and many others. On the other hand, mathematicians have studied basic questions about waves for a variety of models, including biological and chemical settings. One focus of mathematical work has been to prove the existence and stability of traveling waves. In this regard continuation methods have become particularly interesting in models of long range interaction, where integral equations are involved. The conference brings together biologists and mathematicians to discuss wave phenomena from their different perspectives. The meeting fosters interactions between the two areas that should lead to greater understanding of a variety of phenomena important in biology.

Terman 9802339 The investigator and his colleagues develop mathematical tools for analyzing the population rhythms of biophysical models for neuronal networks. These models exhibit a rich structure of oscillatory behavior. The dynamics of even a single cell can be quite complicated; it may, for example, fire either periodic spikes or bursts of action potentials that are followed by a silent phase of near quiescent behavior. Examples of population rhythms include synchronous behavior, in which every cell in the network fires at the same time, and clustering, in which the entire population of cells breaks up into groups; cells within a single group fire synchronously and different groups are desynchronized from each other. Activity may propagate through the network in a wave-like manner. A network's population rhythm results from interactions between three separate components: the intrinsic properties of neurons, the synaptic properties of coupling between neurons, and the architecture of coupling. Each of these components may include numerous parameters and multiple time scales. The mathematical techniques that are developed by the investigator can help determine the role each of these components plays in shaping the emergent network behavior. This may lead to a classification of the possible rhythms that can emerge from a given network and help determine how complicated a model must be in order to display some observed dynamics. The types of rhythms that the investigator studies arise throughout the central nervous system. Consider, for example, thalamic networks: these have been implicated in the generation of sleep rhythms, certain forms of epilepsy, and Parkinson tremor. The investigator studies how the same set of neurons can exhibit the very different rhythms that take place during different stages of sleep and what changes must occur in the networks during the transition from one stage of sleep to another. Recent experiments have demons trated that neurons in the basal ganglia exhibit quite different population rhythms in normal and parkinsonian animals. The investigator develops and analyzes biophysical models for these networks in order to determine factors responsible for the generation of these rhythms.

Terman
0103822
The investigator and his colleague develop geometric,
dynamical systems tools for analyzing biophysical,
conductance-based models for a broad class of neuronal networks.
These systems arise in numerous applications including motor
activity, sensory processing and learning. They also develop
computational models for the generation of sleep rhythms,
epilepsy and parkinsonian tremor. The investigators also
construct and refine a mathematical and computational model for
electrical activity in the subthalamic nucleus and external
segment of the globus pallidus. These are two nuclei in the
basal ganglia, a part of the brain involved in motor activity.
Dysfunction of the basal ganglia is associated with movement
disorders such as Parkinson's disease and Huntington's disease.
The model is used to test hypotheses on the role of the basal
ganglia in both normal and pathological movement. Numerous
experiments have demonstrated that neurons within the basal
ganglia display a rich variety of dynamic behavior; moreover,
patterns of neuronal activity, both spatial and temporal, are
different between a normal and a pathological state. The
investigators characterize the possible patterns of neuronal
activity that arise in the model and determine how these patterns
change with respect to modulations of network parameters and
structure. A long term goal is to develop a model realistic
enough so that it can shed light upon the key parameters and
mechanisms responsible for the generation and modulation of
observed activity patterns.
A mathematical theory for the analysis of neuronal dynamics
helps illuminate the role played by various components of a model
in generating a particular population rhythm. These components
may correspond to some intrinsic property of individual cells, or
to some network property such as the strength and type of
synaptic coupling or the probability that two cells communicate
with each other. Clarification of the mechanisms underlying
different activity patterns may lead to a classification of all
possible rhythms that can emerge from a given network, and enable
us to determine how complicated a model should be in order to
display some observed behavior. It also helps predict
transitions of the network behavior as parameters in the model
are varied. The investigators develop models for neuronal
dynamics, and in particular of electrical activity in the
subthalamic nucleus and external segment of the globus pallidus.
These are two nuclei in the basal ganglia, a part of the brain
involved in motor activity. Dysfunction of the basal ganglia is
associated with movement disorders such as Parkinson's disease
and Huntington's disease. The model is used to test hypotheses
on the role of the basal ganglia in both normal and pathological
movement. Such models may help illuminate both fundamental
neuroscience questions and clinical issues about how the brain
and central nervous system work.

Terman

The basal ganglia are a group of nuclei that play an important role in
the generation of movement. Dysfunction of the basal ganglia is associated
with movement disorders such as Parkinson's disease and Huntington's
chorea. Structures within the basal ganglia have in fact been the target
of recent therapeutic surgical procedures including pallidotomy and deep
brain stimulation. Numerous experiments have demonstrated that neurons
within the basal ganglia display a variety of dynamic behaviors; moreover,
patterns of neuronal activity differ between normal and pathological
states. Neither the origins of these neural firing patterns nor the
mechanisms that underlie the patterns are understood. The primary goals of
the project are to develop detailed computational models, based on
experimental data, for basal ganglia activity and to develop mathematical
tools for analyzing these models. The investigator systematically studies
mechanisms underlying complex activity patterns found in a broad class of
excitatory-inhibitory neuronal networks. These include a recently
constructed model for neurons within the so-called indirect pathway of the
basal ganglia. He also uses geometric, dynamical systems methods to
explore how these networks transform incoming firing patterns.

The investigator develops a computational model for neuronal activity
in the basal ganglia. The model is used to formulate and test hypotheses
concerning the functional role of the basal ganglia in both healthy and
diseased states. As it evolves, the model serves as a tool that can be
used to test hypotheses about the function basal ganglia, the pathologies
that develop in motor disorders, and the mechanisms through which
therapeutic interventions such as deep brain stimulation (DBS) may prove
effective. In particular, a central aspect of this research is to test
hypotheses about how DBS reduces parkinsonian motor symptoms. This may
lead to new strategies by which DBS can be applied. Topics in this project
present tremendous opportunities for cross-disciplinary seminars, workshops
and collaborations. The principal investigator has been very active in the
training of undergraduate, graduate, and postdoctoral students. He has
organized numerous workshops at international conferences and has lectured
at several summer educational programs. A primary goal of these activities
is to help nurture a community of scholars from different disciplines
involved in the mathematical biosciences.

[unreadable] DESCRIPTION (provided by applicant): Sensory transduction by primary olfactory sensory cells in a diverse array of organisms sets up spatial neural codes for odors that evolve over time. A subject of debate is whether the temporal aspect of the sensory response in the vertebrate olfactory bulb or the insect antennal lobe contributes to detection and differentiation of odors. If the temporal aspect of odor representation is important, then a reduction in stimulus duration, stimulus intensity and/or the number of stimulus presentations will impair odor discrimination. This prediction will be tested using a combination of mathematical modeling, and behavioral and electrophysiological experiments with honeybees. The honeybee is an excellent model for studying olfactory processing because honeybees can be conditioned to respond to specific olfactory stimuli and also because it is possible to record simultaneously from multiple neurons in the honeybee antennal lobe. In the proposed behavioral experiments, it is expected that honeybees have more difficulty learning and discriminating among odors of decreased duration and intensity. In parallel, the responses of neurons in the antennal lobe are expected to correlate with the behavioral findings, such that the separation of spatiotemporal response patterns to odors decreases in parallel with impaired discriminability. Finally, the data will be integrated into a computational model of the antennal lobe. The model will be used to test and formulate hypotheses for mechanisms underlying the experimental data. [unreadable] [unreadable] The significant anatomical and functional similarities between the vertebrate olfactory bulb and insect antennal lobe, such and that of the honeybee, indicates that these different groups of animals have evolved the same type of neural solution to olfactory coding. Therefore, this work stands to reveal mechanisms of olfactory coding that are fundamental to most, if not indeed all, animals, including humans. [unreadable] [unreadable]

A primary goal of the proposed research is to develop mathematical tools for analyzing activity patterns in biophysical, conductance-based models for a broad class of neuronal systems. Specific issues are motivated by models for activity patterns in three systems. The first model arises in the study of the basal ganglia, a part of the brain that plays an important role in the generation of movements. Dysfunction of the basal ganglia is associated with movement disorders such as Parkinson's disease and Huntington's chorea. Experiments have demonstrated that neurons within the basal ganglia display a variety of dynamic behaviors; moreover, patterns of activity differ between normal and pathological states. The principal investigator is developing mathematical methods for analyzing the origins and mechanisms underlying these firing patterns. The principal investigator will also consider models of the insect antennal lobe. These neurons exhibit complex oscillatory dynamics in response to odors; however, the mechanisms underlying both single-cell and population rhythms are not fully understood. The principal investigator will develop mathematical models to explore mechanisms underlying odor representations in the insect antennal lobe and their modification with learning. Finally, the principal investigator is considering models of respiratory rhythm generation. Breathing movements in mammals are generated by networks of neurons in the lower brain stem that produce rhythmic oscillations of neural activity. One interesting feature of this network is the apparent high degree of heterogeneity among cells. The principal investigator will develop analytic tools to understand how a network of heterogeneous cells can exhibit synchronized activity and how synchronization is lost as parameters in the model are varied.

Oscillations and other patterns of neuronal activity arise throughout the central nervous system. These oscillations have been implicated in the generation of sleep rhythms, epilepsy, parkinsonian tremor, sensory processing, and learning. Oscillatory behavior also arises in such physiological processes as respiration, movement, and secretion. Models for the relevant neuronal networks often exhibit a rich structure of dynamic behavior. Examples of population rhythms include synchronized oscillations, propagating waves and chaotic dynamics. Computational models and mathematical analysis can be extremely useful in understanding the mechanisms underlying this complex dynamics and predicting how the dynamics may change with respect to parameters. Specific neuronal systems to be studied in this project include the basal ganglia, a part of the brain implicated in the generation of Parkinsonian rhythms, the insect antennal lobe and the pre-Botzinger complex, a brain nuclei believed to be the origin of respiratory rhythm generation.

Proposal: DMS-0635561
Principal Investigator: Friedman, Avner
Institution: Ohio State University
Proposal Title: Mathematical Biosciences Institute

ABSTRACT

This is a renewal proposal for the Mathematical Biosciences Institute (MBI) at Ohio State University (OSU). The mathematical biosciences include the biological, medical, and environmental sciences where the application of mathematical, statistical, and computational science may lead to significant progress in understanding key processes in the application area. The mission of the MBI is (i) to develop mathematical theories, statistical methods, and computational algorithms for the solution of fundamental problems in the biosciences; (ii) to involve mathematical scientists and bioscientists in the solution of these problems; and (iii) to nurture a community of scholars through education and support of students and researchers in the mathematical biosciences. The Institute organizes a yearly program consisting of 6-8 workshops and several tutorials around selected themes. Workshops emphasize discussions over formal talks and offer an environment that encourages developing collaborations. The themes selected for the next three years are Bioengineering (2007-'08), Developmental Biology (2008-'09), and From Genes to Cells: Networks, Scales, and Complexity (2009-'10). A unique postdoc program is designed to develop young researchers into truly interdisciplinary scientists. They are immersed in a broad range of topics and interact with workshop participants, visiting scientists, and local researchers, while being co-mentored by a mathematical scientist and a life scientist.

The Mathematical Biosciences Institute (MBI) at Ohio State University (OSU) is one of seven Mathematical Sciences Institutes (http://mathinstitutes.org/) supported by the NSF Division of Mathematical Sciences. The MBI supports programs that promote multidisciplinary research at the interface of the mathematical and life sciences. It hosts scientific programs on long (academic year) and short (one week or less) time scales. The programs are organized around selected themes, which range from Mathematical Neurosciences (2002-'03) to Systems Physiology (2006-'07) and From Genes to Cells: Networks, Scales, and Complexity (2009-'10). The MBI provides opportunities for researchers in the mathematical and life sciences to interact and form new collaborations. Most of the Institute's programs are conducted on the OSU campus, but the MBI also sponsors conferences and workshops at its academic Institute Partners. These Institute Partners and MBI's Corporate Partners contribute to the Institute's programming and planning through dedicated governance committees. The MBI website (http://www.mbi.ohio-state.edu/) describes upcoming and past programs, solicits ideas for future programs, and offers application forms for Institute activities.

Oscillations and other patterns of neuronal activity arise throughout the central nervous system. These activity patterns have been implicated in the generation of sleep rhythms, sensory processing, working memory and pathological rhythms that arise in diseased states such as schizophrenia and Parkinson's disease. While numerous mathematical models have been proposed for these systems, there has been very little mathematical analysis of these models. This is because the models are highly nonlinear, they exhibit an extremely complex structure of spatio-temporal dynamic behaviors and they depend in often unintuitive ways on the numerous parameters in the model. This research develops mathematical and computational tools for analyzing a general class of neuronal models. In particular, the research: (i) helps explain mechanisms underlying complex spatiotemporal patterns seen in recent experiments on early sensory processing in the insect antennal lobe; (ii) develops a novel model for working memory based on calcium dynamics and excitatory-inhibitory interactions within the prefrontal cortex; and (iii) develops new mathematical and computational tools for studying calcium and membrane potential wave propagation in spatial domains with complex geometries. The issues considered arise in many neuronal systems throughout the brain and the new mathematical tools will be very useful in the study of these other systems as well.

Everything that the brain does depends on the firing properties of neurons. This includes the control of movements, learning, memory, emotion and sensory processing. Changes in neuronal firing patterns are associated with memory and aging, and pathological patterns have been implicated in neurological diseases such as schizophrenia and Parkinson's disease. With the development of new sophisticated experimental techniques, neuroscientists are now beginning to better understand the functional roles of these firing patterns in normal brain processing, the biophysical mechanisms underlying these patterns and what is responsible for changes in these patterns during learning, aging and disease. However, it is becoming increasingly clear that mathematical modeling and computational methods are critically important in accounting for the massive amounts of new data, testing new hypotheses and understanding how complicated processes interact to generate complex brain rhythms. Novel mathematical tools are developed for analyzing detailed models that arise in numerous brain systems. In particular, models for working memory are constructed and analyzed; this corresponds to our ability to store and manipulate information for a short time in order to carry out complex tasks. Problems with working memory is associated with several neurological diseases including schizophrenia and the project explores how changes in neurotransmitters such as dopamine lead to pathological rhythms associated with these neurological diseases.

"Towards Mathematical Modeling of Neurological Disease from Cellular Perspective," is a series of workshops to be held at the Fields Institute in Toronto, Ontario, during May and June 2012. Oscillations and other complex dynamic behaviors have been implicated in several neurological diseases. Both animal models and humans with Parkinson?s disease, schizophrenia, epilepsy and sleep disorders exhibit changes in the firing properties of neurons within certain brain regions. These changes often take the form of increased correlation in spiking activity, changes in bursting and other rhythmic behavior and changes in the firing rates of neurons. Although the functional role of rhythmic activity in cognitive processing remains poorly understood, it is clear that these pathological rhythms depend on specific cell types and network properties. Recent technological advances such as optogenetics allow for the targeting of specific biological cells and the recording from multiple cells lying in separate brain regions. This provides novel opportunities for theoreticians to model the role of different cell types and network properties in the generation of neuronal activity. A series of five workshops will be conducted, each devoted to a separate neurological disease: Parkinson?s disease, schizophrenia, epilepsy, Alzheimer?s and anesthesiology/sleep disorders. Each workshop will bring together neuroscientists, mathematicians and clinicians, so that the most relevant issues and latest research associated with the workshop?s topic will be presented from several different viewpoints. In this way, the workshops will: (i) attract newcomers to the field; (ii) broaden and deepen perspectives of those in the field; and (iii) consider what mathematical tools and analyses would be most helpful, and where further developments are most needed, to move forth in tackling neurological disease.

Millions of people suffer from some form of neurological disease. Experiments have demonstrated that many brain disorders, including Parkinson?s disease, schizophrenia, sleep disorders and epilepsy, are associated with the onset of pathological neuronal firing patterns. For example, recordings from both monkeys and humans have shown that Parkinson?s disease is associated with increased synchronization among neurons within the basal ganglia. Changes in the synchronization properties of neurons may also play a critical role in epilepsy and pathological firing patterns within the prefrontal cortex, and other brain regions, have been implicated in patients suffering from schizophrenia. While there has been tremendous progress in developing experimental techniques for identifying cellular processes involved in the generation of neurological diseases, the mechanisms underlying these pathological firing patterns remain poorly understood. It has become increasingly evident that mathematical models, together with both computational and mathematical analysis of them, can play an important role in making sense of the data, testing hypotheses and generating new ones. A series of five workshops will be conducted, each devoted to a separate neurological disease: Parkinson?s disease, schizophrenia, epilepsy, Alzheimer?s and anesthesiology/sleep disorders. Each workshop will address the most recent and relevant research associated with the workshop's topic and will bring together scientists from a wide range of disciplines.

Cerebral ischemic stroke is the third leading cause of death in the United States, trailing only cancer and heart disease. Despite the urgent need, mechanisms underlying brain injuries during stroke remain largely unknown and current therapeutic strategies to combat stroke have been largely unsuccessful. This project will develop a detailed mathematical model, which will be used to identify those mechanisms and processes that play a decisive role in the generation of ischemic-like conditions. Calcium signaling and energy production play a critical role in many cellular processes and have been implicated in a wide range of neurological diseases, including Parkinson's, Alzheimer's, Huntington's and epilepsy. Insights that arise from this research will be relevant to the study of these and other diseases.

A detailed model for key processes involved in cerebral ischemic stroke will be developed. The model will be constrained by recent experimental studies and will be used to compare, contrast, and suggest mechanisms underlying observed behavior. Mathematical and computational methods will be used to: (i) identify key processes involved with ischemic stroke-like conditions; (ii) systematically understand mechanisms underlying the model's behavior; (iii) reduce the model's complexity. The model will also be used to better understand mechanisms underlying the neuroprotective role of stimulation of astrocyte ATP production.