I have done research at Rensselaer Polytechnic Institute, Bryn Mawr College, and the ECRI Insititute. The picture you see in the background comes from my research in Computational Neuroscience. It is a snapshot of firing neurons forming a spiral pattern.
Rensselaer Polytechnic Institute - Neuronal Networks as an Excitable Medium:
The research area I am investigating in my doctoral dissertation is Computational Neuroscience, which utilizes computational models to describe the dynamics of biological neurons or neuronal systems. Such computational models are used to verify hypotheses in current neuro-physiological experiments and reveal mechanisms behind the observed phenomena.
My particular interest in Computational Neuroscience is to understand pattern formation in neuronal networks with relatively simple architecture. Such patterns are ubiquitous in excitable media in physics, chemistry, and biology. They are usually described by Ginzburg-Landau-type partial differential equations, but have also been observed in simple neuronal models. I am studying spiral- and target-like patterns in networks of conductance-based, integrate-and-fire, point model neurons synaptically connected to their close neighbors.
In the integrate-and-fire model, a neuron's membrane potential follows a first-order, linear current-voltage relation until it crosses the firing threshold. At that moment, the potential returns to its reset value, and the neuron sends an electro-chemical signal, ``spike," to every neighboring neuron to which it is connected. Each neuron is modeled as a point on a rectangular grid; the spatial structure of its axons and dendrites is not modeled explicitly but through the imposed connectivity architecture. The effect of each synaptic ``spike" on a target neuron is to temporarily elevate its conductance (i.e., inverse resistance). In the differential equation describing the membrane potential, the conductance does not multiply the potential alone, but rather the difference between the membrane potential and the respective ``reversal" potential. This latter potential is above the firing threshold if the incoming spike is generated by an excitatory neuron, and below the reset if the incoming spike is generated by an inhibitory neuron. In addition to spikes arriving from neighboring neurons, each model neuron receives drive in the form of an excitatory spike train, with the spike times distributed according to the Poisson law.
The geometry I choose for the model is a lattice of about 60,000 excitable neurons on a torus, coupled to nearest and next-nearest neighbors. The external spike train is chosen so that, with the neuron-to-neuron synaptic connections turned off, the average voltage of each neuron is very close to the firing threshold, which is the excitable medium regime. Due to the fluctuations in the incoming spike trains, a neuron is occasionally driven to firing, which provides a seed for the formation of firing patterns when neighboring neurons are coupled. In my simulations, I have found both spiral and target patterns. I am currently in the process of exploring the parameter space to determine the regimes in which patterns emerge and the possible bifurcations they undergo. I am computing the pattern wavelengths and the velocities of their propagation. In addition, I am computing diagnostics such as voltage and conductance averages and correlation functions, as well as temporal power spectra. I am also computing the statistical distribution of the incoming spike times of any given neuron in the network.
Coarse graining in neuroscience is similar to that in fluid mechanics. Just as in fluid mechanics individual molecules are replaced by a continuous fluid --- in neuroscience individual neurons are replaced by a continuous neuronal tissue. In this way, large sets of ordinary differential equations are replaced by partial or integro-differential equations. Different types of coarse grained neuronal models are available. Mean-driven models give the firing rate in terms of the membrane potential alone. Kinetic models give the statistics of the membrane potential and conductances, and can be used to describe the situations in which the mean membrane potential is well below the firing threshold. The neuronal firing in this case is triggered by the membrane potential fluctuations. I am currently in the process of deriving such coarse-grained models for the network whose dynamics I am simulating.
Bryn Mawr College Undergraduate Thesis - Cellular Lines and Ant Algorithms:
Many digital display companies are researching how to maximize the number of pixels on a screen to create a clearer picture. My thesis focused on the converse --- given a fixed resolution, what is best way to create a clearer picture. The most simplistic case is creating a line. Most digital displays use Bresenham's algorithm to create a line which is based on an imaginary Euclidean line and sub-grids on the cellular array. Due to the lack of a precise geometry on the array, Bresenham's algorithm runs into contradictions and ambiguities as to which cell to highlight in order to create a line. My task was to create a definition of a line on a cellular array that was independent of Euclidean geometry then present an algorithm to construct cellular lines.
I first defined the properties of the planar surface or cellular array. Then using Euclidean geometry as an example, I came up with properties helped me define a line. For example, I defined uniqueness of a line to parallel uniqueness in Euclidean geometry --- there is only one line that goes through two distinct points. I also developed the a metric in order to implement an algorithm to create cellular lines.
The algorithm I chose to create cellular lines was the food forging ant algorithm. I was intrigued by its use in optimization problems such as the Traveling Salesman Problem and the emergence of straight lines in nature. I was able to utilize both the optimization and straightness characteristics of the ant algorithm to build an algorithm that could be used to generate cellular lines.
ECRI Institute - Biostatistical Comparison of Chemonucleolysis and Laminectomy:
Herniated disks are a common ailment where the cushion or spinal disc between two spinal vertebra slips out of position or ruptures. This leads to a pinched nerve or compressed spinal cord. Often, physical therapy and over the counter medicines are enough to correct or lessen the symptoms. However, in extreme cases and often for elderly patients, it is necessary to remove the slipped spinal disc. Currently the most popular method is a surgical procedure, laminectomy. A few years ago, a non-surgical procedure, chemonucleolysis, gained a following which is an enzyme injection that dissolves the herniated disk. The United States Department of Social Security asked ECRI Institute to research the two procedures to see which procedure has a higher success rate.
To tackle this issue, I first searched research databases such as Nexis-Lexis for medical papers and experimental data comparing the two procedures. After collecting papers, I carefully studied each paper to see which were appropriate for the study. Some of the authors used the same experiment and data set for multiple papers. I also had to note the test conditions and control set. I was delighted to see that the papers I had found and selected, matched exactly to the set the hired team chose.
Since the data came from multiple papers with different experimental conditions, it was inappropriate to use basic statistical methods. My mentor showed me how to use meta-analysis and SAS, a statistics package, to analyze and combine the data from the papers. We found that laminectomy was more successful and our findings led to the US Department of Social Security support for laminectomy over chemonucleolysis.