As an applied mathematician, my research has been done in collaboration with experimental neurobiologists. As part of my masters research project I have performed electrophysiology experiments myself. I intend to continue active collaborations with experimentalists and dynamicists, extending my work in the area of epilepsy and expanding into new areas of mathematical biology. My work involves computer modeling and simulation of reaction diffusion systems consisting of discrete sources and sinks. Predictions of network behaviors are made based on the details of the local phase plane dynamics of the network elements.

My doctoral research involved analysis of the dynamical properties of individual elements that lead to synchronization and propagation in a network of neurons. Seizure like activity related to epilepsy can be modeled as a reaction diffusion system. In the case of a population of neurons, the system may also be considered to be a network of discrete sources and sinks coupled weakly by diffusion. The essential experimental observations are electrophysiological recordings from hippocampal CA1 slices bathed in a low calcium solution. Under these conditions the very weakly coupled neuronal network spontaneously generates waves of synchronized activity characterized by an increase of extracellular potassium concentration and a decrease in extracellular field potential. Neuronal activity produces potassium transients that diffuse through the surrounding network. I developed a model to simulate the network behavior. Analysis of the model aims to establish the role of local dynamics in the onset and propagation of low calcium epileptiform activity in the hippocampal network. The model is a discretization of a reaction-diffusion system set up to investigate the capacity for wave generation and propagation based on the local properties of individual elements (neurons).

Typically, repetitively firing neurons are modeled as limit cycle oscillators, that is, a set of differential equations with a periodic solution. In my model, neurons are more simply represented as potassium source functions where the frequency of short duration events having a high rate of potassium release (action potentials) is determined by the potassium concentration at the neuron. In future work the model will be extended to consider stochastic behavior that more realistically models the dynamics of hippocampal neurons. In addition, there is a smaller, continuous release of potassium from a neuron that increases as the potassium concentration at the neuron increases (and the neuron is depolarized). Potassium is removed from the system by glial cells and sodium/potassium pumps, represented in the model as a potassium sink function, and by diffusion in the extracellular space. The properties of the model combine to produce local increases in potassium that, given sufficient size, propagate across the space of the model as a wave.

My dissertation presents a methodology for analyzing propagation of activity in three spatial dimensions for systems of discrete sources and sinks, such as the hippocampal network in low calcium conditions. The bifurcation structure of local dynamical elements is described with respect to the strength of the diffusive flux at a point. Propagation occurs when elements at a wavefront generate sufficient output such that the flux at nodes in the direction of propagation exceeds a critical value. Applying this method to my hippocampal network model suggests that potassium diffusion alone can account for experimental observations. I am currently in the process of compiling manuscripts based on this research to be submitted for publication.

My masters research involved the application of computer modeling and nonlinear dynamics to problems in neurobiology. Mathematical neuron models, computer simulation and electrophysiology experiments were combined in an integrated approach to the problem of abnormal synchronization found in epilepsy. Computer programming played a large role in my work, from the numerical methods involved in implementing and integrating systems of stiff differential equations to solution continuation methods involved in bifurcation analysis. I designed and implemented data acquisition hardware and software used to control electrophysiology experiments. I have both written my own applications and used packaged software such as Matlab, Axon’s Pclamp suite and the phase plane package XPPAUT. I have also written graphical demonstration software for Windows to illustrate dynamic properties of a neuron.

The main aim of my work was to understand and use the dynamics of a neural population to facilitate desynchronization. Abnormal synchronization, as seen in epileptic seizures, can be induced in rodent brain slices by perfusion with saline solutions containing elevated potassium concentrations. Electrophysiological recordings of this activity are used to investigate the characteristics and mechanisms of synchronous population bursting activity and its modulation by applied electric fields. One method used is phase resetting. Phase resetting theory applied to an oscillatory system can reveal information about the dynamics of that system. The existence of weak resetting and strong resetting parameter regions in this system implies the existence of a phase singularity, that is, a combination of stimulus parameters that will annihilate bursting activity. In parameter regions where bistability is present, annihilation of repetitive activity can be long term.

Publications:

Bikson M, Hahn PJ, Fox JE and Jefferys JGR (2003) “Depolarization block of neurons during maintenance of electrographic seizures.” J. Neurophysiol. 90:2402-2408.

Shuai J, Bikson M. , Hahn PJ, Lian J, and Durand DM (2003) “Ionic mechanisms underlying spontaneous CA1 neuronal firing in Ca2+-free solution.” Biophys J. 84:2099-2111.

Hahn PJ and Durand D (2001) “Bistable dynamics in simulations of neural activity in high extracellular potassium conditions.” J. Comp. Neuro., 11:5-18.

Shuai J, Lian J, Hahn PJ and Durand DM (2001) “Positive Lyapunov exponents calculated from time series of strange nonchaotic attractors.” Phys. Rev. E, 64:026220.

Bikson M, Lian J, Hahn PJ, Stacey WC, Sciortino C and Durand DM (2001) “Suppression of epileptiform activity by high frequency sinusoidal fields in rat hippocampal slices.” J. Physiol., 531:181-191.

Lian J, Shuai J, Hahn PJ, and Durand DM (2001) “Nonlinear dynamic properties of low calcium induced epileptiform activity.” Brain Res., 890:246-254.

Conference Abstracts and Presentations:

Fox JE, Bikson M, Hahn PJ and Jefferys JGR (2002) “Neuronal firing is not necessary for the maintenance of ictal epileptiform events.” FENS Abstract.

Durand DM, Nakagawa K, Bikson M, Hahn PJ and Lian J (2000) “Patterns of evoked activity in an intact rat hippocampal preparation.” Society for Neuroscience Abstract.

Bikson M, Durand DM and Hahn PJ (1999) “Modulation of non-synaptic epileptiform activity by osmolality.” Society for Neuroscience Abstract.

Hahn PJ and Durand D (1998) “Dynamics of neuronal activity in high potassium.” Invited talk. Annals of Biomed. Eng. 26:S-93.

Hahn PJ, Bikson M and Durand DM (1998) “A novel intact preparation for studying patterns of activity in the hippocampus.” Annals of Biomed. Eng. 26:S-105.

Hahn PJ and Durand D.M. (1997) “Dynamical analysis of bursting in hippocampal CA3 pyramidal cells.” Society for Neuroscience Abstract.