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].

Present Research:
I am now investigating activity patterns of neural populations in vertebrates using mathematical methods drawn from statistical physics. These projects attempt to understand the dynamics of neural populations in response to sensory stimuli.
The storage of temporal patterns in cerebellum-like structures: I am presently the Principal Investigator of a research grant funded by the National Science Foundation that focuses on the mormyrid electric fish, a nocturnal fish that senses its environment by emitting a weak electric pulse and then detecting the distortions caused by external objects with electrosensory receptors in its skin. This research effort is in collaboration with Dr. Curtis Bell who will provide data from experiments performed in his lab. The site of initial electrosensory information processing is the electrosensory lateral line lobe (ELL). The responses of these neurons in the ELL are found to be highly adaptable to changing conditions that effect the electrosensory system. This adaptability leads to the ability of these neurons to "store" an image of the fish's expectation of its own electrical signal. However, due to the complexity of the ELL, it is unclear whether the rules of adaptive learning measured in certain experimental conditions can explain the collective neural activity observed under other experimental conditions. The difficulty of experimentally exploring the roles of various synaptic learning rules, sites of adaptive change, and intracellular connections make theoretical and modeling work a necessary adjunct to experimental study. I am using mathematical analyses and computer simulations to combine results from different experiments to predict changes in the responses neurons in the ELL during changing sensory conditions [14]. These predictions will then be used to test different mechanisms that may be responsible for the adaptive changes observed in the ELL.
Biological learning rules: Physiological experiments of long term changes in synaptic strength has revealed a precise sensitivity to the timing between pre- and postsynaptic spikes. For instance, in the rat neocortex, the synaptic efficacy increases if a postsynaptic spike follows a presynaptic spike by 10 milliseconds, but decreases if the postsynaptic spike precedes the presynaptic spike by the same amount. A research program is presently underway to analyze the neural dynamics that result from different biological learning rules [15]. For instance, using both mathematical analysis and computer simulations it was shown in [13] that the above learning rule results in synaptic change that is proportional to the rate-of-change of the postsynaptic neuron's average activity. This type of learning, referred to as differential Hebbian learning, has been previously associated with classical conditioning behavior. Since the timing relations of biological learning rules result from molecular events at the synapse, this line of research helps to link the implications of dynamics from the molecular level, through the network level, to the behavior of whole organisms.
Dynamics of neural activity in the cerebellum: The cerebellum has been implicated in the regulation of movement, the processing and interpretation of sensory information, and even involvement in cognition and language. The precise regularity of the neural anatomy in the cerebellar cortex makes this system tractable to analytical and computational investigations [10]. The uvula-nodulus is a region of the mammalian cerebellum that receives both visual and vestibular sensory input. Purkinje cells in this region modulate their activity in response to sensory input. It is unclear at present how much of the modulation is a result of synaptic plasticity and how much is due to the network dynamics. Dr. Neal Barmack will provide experimental data to test theoretical hypotheses regarding the mechanisms underlying neuronal activity patterns in the cerebellum. The model will predict Purkinje cell activity at multiple sites in the uvula-nodulus and be compared with multiple electrode recordings performed in Dr. Barmack's lab. This modeling effort will bridge the gap between cellular- and systems-level findings.