The major theme of my research is to understand dynamics of complex biological systems. To achieve this goal, my research approach applies mathematical analysis and computer simulation techniques to elucidate underlying mechanisms of biological systems, and to extract principles of biological function. Although my focus has been on the processing of sensory information in the nervous system, the mathematical methodologies that I use for neural systems could find broad application to many biological systems.

Modeling studies, such as those developed by my lab at OHSU, can help bridge the gap between the level of fundamental biological processes and the systems level. In addition, by developing new models of complex biological processes, we are forced to develop new mathematical methods and expand the domain of application for present methods. My background in theoretical physics has provided me with excellent training to develop new mathematical techniques in pursuit of answers to fundamental scientific questions.

I presently collaborate with biologists at OHSU and Washington State University, Vancouver who have expertise in exploring primary sensory processing. These scientists are presently involved in research programs in a variety of sensory modalities, such as auditory, vestibular, electrosensory, and visual. Our lab is involved in an expanding program of collaborations with other scientists in the Portland area to develop mathematical methods and introduce student with a quantitative background to biological research. These collaborations extend to the Systems Science Program at Portland State University, as well as the Nonlinear System Group and the Biological Signal Processing Lab, bringing expertise in artificial learning systems, hybrid and hierarchical systems, and observability issues in control.

* Auditory Processing*, (Collaborators: Christine Portfors, Claudio Mello, and James McNames)

A key challenge for the central auditory system is to filter predictable sounds so that novel sounds can be better processed and perceived. Auditory neurons achieve this filtering by altering their responses based on recent experience. This type of adaptive mechanism, or plasticity, has been observed in the mammalian dorsal cochlear nucleus (DCN); the first site in the central nervous system where acoustic cues are processed. Our long term goal is to understand the mechanisms of auditory processing in the DCN and determine how auditory processing is affected by adaptive mechanisms. We have adapted our models from the electrosensory system to predict the adaptive properties of processing in the DCN of the mammalian auditory system [31].

A fundamental function of the auditory system in humans is to process speech. When individuals suffer from age-related hearing loss, they have difficulty understanding basic speech sounds important in everyday life. The long term goal of this second research project on auditory processing is to understand how speech processing mechanisms are affected by age-related hearing loss. The objective of the current research is to understand how age-related hearing loss alters encoding of vocalizations in the inferior colliculus (IC). To achieve this objective, empirical studies of single neuron responses in the mouse IC will be combined with mathematical modeling to test our central hypothesis: age-related hearing loss disrupts the selectivity of neurons in IC to frequency combinations that are important for encoding vocalizations.

The long-term goal of a third project is to understand higher-order auditory processing in biological systems. Our research focuses on the auditory forebrain of songbirds, a system well-suited for the study of adaptive auditory processing. We use mathematical modeling and novel experimental paradigms to advance towards a better understanding of auditory processing and adaptation. One specific goal is to develop quantitative models of the auditory forebrain and simulate the adaptive responses of neurons to conspecific songs. The model predicts the adaptive responses of neurons in caudomedial nidopallium (NCM), and quantifies how properties of neural response properties change in response to selective changes of conspecific songs. The objective of this research is to develop and test a model of habituation to auditory patterns, a model that predicts the neural activity of auditory neurons to any spectral-temporal pattern [36].

* Dynamics of Neural Circuitry*, (Collaborators: M. Frerking, S. Prasad, and G. Lendaris)

Synaptic transmission is a major mechanism underlying the intercellular transfer and processing of signals in the nervous system. The output of synapses is highly dynamic, owing to the existence of several forms of activity-dependent synaptic plasticity that range in duration from milliseconds to hours. Although individual forms of synaptic plasticity have been well described, the simple stimulus patterns used to define each form of plasticity bear little resemblance to the activity seen in vivo, where most synapses are activated by temporally complex patterns of afferent firing. These complex patterns of activity are expected to engage several forms of synaptic plasticity simultaneously in a complex combination, which we will refer to as synaptic dynamics. Synaptic dynamics define how the synaptic output produced by each spike is influenced by the pattern of the preceding spike train. These dynamics are widely presumed to be an important component of signal processing during synaptic transmission, and may be affected by drugs or neurological diseases; however, synaptic dynamics during realistic patterns of afferent activity are poorly understood. We are actively developing mathematical methods to predict the changes in synaptic response to complex spike trains, and to predict how those responses change following activity dependent plasticity.

* Electrosensory Processing*, (Collaborators: Curtis C. Bell, Todd Leen, and Gerardo Lafferriere)

Primary sensory information is processed at the earliest stages by complex neural circuitry. Understanding how the central nervous system stores information about sensory signals is a prominent challenge facing neuroscience today. At present, little is known about what features are extracted, or how sensory processing is modulated by adaptation and recurrence from higher stages. Our understanding is particularly poor concerning the connection between synaptic plasticity at the cellular level and the storage of actual sensory patterns as examined in systems-level studies. New theoretical methods are needed to help in the design of experimental protocols and the analysis of data.

The first goal of this project has been to understand how the nervous system accurately stores the temporal flow of sensory information, and how past stimuli effect future sensory processing. Our studies focused on the cerebellum-like electrosensory lateral line lobe (ELL) in mormyrid electric fish in collaboration with NSI scientist Dr. C. Bell. In vitro experiments on these structures have revealed precise learning rules of spike-timing dependent synaptic plasticity. However, due to the complexity of cerebellum-like structures, it was unclear how the rules of adaptive learning measured in vitro explain the collective neural activity observed in vivo. The difficulty of experimentally exploring the roles of various learning rules, sites of adaptive change, and intracellular connections make theoretical and modeling work a necessary adjunct to experimental study. We developed mathematical methods to estimate the average effects of precise synaptic learning rules to correctly predict the system level adaptation [13, 14, 16, 17, 20].

The second goal of this project has been to identify the mechanisms of central control that are used in sensory processing and adaptation. Recent studies of primary sensory processing structures are revealing that they are far more than simple relays that transfer sensory information from the receptors to “higher centers." Not only does some significant sensory processing take place at these primary structures, but feedback from targets of these structures also influences the processing that takes place. However, it is presently unknown what “higher-level" processing takes place in these primary structures, or how that processing is influenced by experienced. We have studied mechanisms of recurrent control by constructing detailed mathematical, conductance-based models of neurons involved in the initial processing of sensory information and by determining how the interactions between peripheral and central inputs to cerebellum-like structures affect information transfer and adaptation. In addition, progress towards our goals necessitated the development of analytical methods to determine the stability of recurrent control, and to analyze the biological evidence for the presence of control mechanisms [15, 18, 22, 23, 25, 32].

The third goal of this project has been to understand how the nervous system accurately stores the temporal flow of sensory information in the presence of noise. This research investigates how noise effects the storage and retrieval of temporally ordered responses to sensory stimuli in the electrosensory system of mormyrid electric fish. The long-term objective is to understand how noise affects multimodal sensory processing. The types of noise that we investigate arise from the sensory receptors and the variability of synaptic contacts within the sensory processing circuit. We developed mathematical methods adapted from statistical physics to quantify how noise affects learning and memory [24, 26, 27].

* Vestibular Processing in the Cerebellum,* (Collaborators: Neal Barmack and David Rossi)

Our long-term goal is to understand how the cerebellum processes information and stores a representation of that information that is be used to modulate balance and orientation. Our immediate objective is to formalize the spike activity of parallel fibers in the cerebellum as dynamical equations in order to understand the spike responses of Purkinje cells to time-dependent input stimuli. Our research is expected to contribute to our knowledge of how the spike activity of Purkinje cells is dependent on circuitry, adaptive mechanisms, and properties of other neurons in the circuitry. By developing accurate mathematical predictions of how changes in spike activity of specific neurons affect the collective spike activity of the neural population, we will learn more about how brain circuitry works. Such predictions will ultimately provide a means to forecast how particular pharmaceutical interventions will affect the function of the brain.

Our methodological approach combines (1) formalized mathematical modeling with (2) empirical experimentation using in vitro patch clamp recordings in mammalian cerebellar slices and (3) analyses of in vivo recorded data from Purkinje cells in the cerebellar uvula-nodulus during vestibular stimulation. The modeling consists of: 1) numerical modeling of cerebellar granule cells and unipolar brush cells, and 2) analytical network modeling of cerebellar circuitry based on spiking neuron models of cerebellar neurons. Our analytic network models predict the spatial-temporal spike probability patterns of cerebellar neurons, and these results are be compared with numerical computer simulations of neural spike-activity. The in vitro recordings provide empirical data to test our predictions of synaptic plasticity in granule cells and characterize cellular dynamics of granule cells and UBCs. The in vivo recordings provide empirical data to test model predictions of spike-responses of cerebellular neurons to natural vestibular stimuli [16, 24, 26, 34].

**Postdoctoral research:**

In the spring of '93 I accepted a postdoctoral position in mathematical neuroscience. Following my
background in mathematical physics, I found an opportunity to pursue my long standing interest in
biological systems. In collaboration with Dr. Gin McCollum, I addressed some of the discrete
aspects of neural systems in motor control.

* Topological biomechanics:* In [6] we took advantage of global analysis used in the modern
approach to nonlinear dynamics, and applied these techniques to the problem of body movement.
After writing a set of coupled first order differential equations describing the possible movements
we applied constraints representing muscles activity using the methods of constrained Hamiltonian
mechanics. This reduced the relevant phase space to a plane, in which theorems from topological
dynamics elucidated the different strategies available to rise from sitting. The objective was to unite
continuous physical properties of a multijointed system with discrete functional properties found in
goal directed behavior.

* Conditional dynamics: *This mathematical formalism reveals the functional logic of the
system and organizes experimental observations. Application of the formalism determines a
mathematical structure that constrains observed behavior. In contrast to continuous modeling
methods, the formalism does not depend on numerous assumptions of system parameters, but
emphasizes functionally important aspects of the system and identifies gaps in experimental
knowledge. Conditional dynamics was applied to the movements of the foregut of decapod
crustaceans in [7] and provides predictions for possible behavioral modes.

** ***Rhythmic activity in small neural networks: *The study of rhythmic behavior of the
gastric mill in crustaceans led me to independently develop methods to study the neural
mechanisms underlying that behavior. In order to gain insight into the problem of predicting the
activity modes of multiple pattern generators such as the stomatogastric ganglion, the rhythm space
method was developed to classify the patterns of behavior to be expected given the synaptic
connectivity and cellular properties of a biological network. This method was applied to the
stomatogastric ganglion [9], the swim-reflex generator of* Tritonia* [11], cerebellar rhythmic
activity [8], and vestibular rhythms of the oblique nystagmus in [12].

**Graduate research:**

The majority of my graduate research work was the study of modular invariance
in string and conformal field theories (click for Dissertation
LaTex files). My tools of investigation were even, self-dual lattices to insure
modular invariance of the theory in question. This technique has proven to be
very powerful and I have applied it to the following three areas of study:

* (1) String phenomenology: *In [1] we made an orthodox application of the covariant lattice
approach to four dimensional string model building. We constructed some three generation
superstring models with a matter sector similar to the standard model. This study revealed the
advantages of the approach, as well as some of the drawbacks which stem from the large rank of
the gauge group.

*(2) Modular invariance of conformal field theories:* In [2] and [4] we found
that self dual lattices can be helpful in the classification of modular invariant
partition functions of conformal field theories (tex).
This has proven to be an efficient means of searching for new exceptional modular
invariants of higher rank Kac-Moody algebras.

*(3) Strings on non-compact group manifolds: *In [3] and [5] we constructed characters for
the SU(1,1) Kac-Moody algebra. We then used these characters to show that unitary, modular
invariant string models could propagate on a non-compact group manifold that represents curved
space-time.