Research Interests

My interests include dynamics of complex networks with applications in biological and social sciences as well as other math biology problems.

My Ph.D. thesis work focuses on a sequence of idealized models of insect olfaction. Modeling insect olfaction is motivated by the fact that olfactory systems appear structurally and functionally similar across phyla ranging from insects to mammals, such as humans. One particularly well-studied insect is the locust, whose olfaction dynamics exhibit strong similarities with those of mammals.

When a locust detects an odor, the stimulus triggers a specific sequence of network dynamics of the neurons in its antennal lobe. The odor response begins with a series of synchronous oscillations, followed by a short quiescent period, with a transition to slow patterning of the neuronal firing rates, before the system finally returns to a background level of activity.

We begin modeling this behavior using an integrate-and-fire neuronal network, composed of excitatory and inhibitory neurons, each of which has fast-excitatory, and fast- and slow-inhibitory conductance responses. We further derive a coarse- grained, firing-rate model for each (excitatory and inhibitory) neuronal population, which allows for more detailed analysis of and insight into the plausible olfaction mechanisms seen in experiments, prior models, and our numerical model. We conclude that the transition of the network dynamics through fast oscillations, a pause in network activity, and the slow modulation of firing rates can be described by system which has a limit cycle of the fast variables, slowly passes through a saddle-node-on- a-circle bifurcation eliminating the oscillations, and, eventually, slowly passes again through the bifurcation point, producing a new limit cycle with a slower period.

Experimental work and prior models suggest changes in ambient temperaturecan affect sleep patterns in humans. We have designed a mathematical model describing numerous features of the human sleep/wake cycle and aspects of REM/non-REM dynamics. The model simulates temperature changes detected by neurons in the POAH that, in turn, affect the REM/non-REM cycles during sleep through a state-dependent homeostatic process. Many connections in our brains are not yet fully determined and experimental and modeling work may shed new light on the communication between neuron populations in our brains that are responsible for typical and atypical sleeping patterns. 

[pdf] J. Best, S. Bañuelos, G. Huguet, A. Prieto Langarica, P. B. Pyzza, M. H. Schmidt, S. Wilson. “Effects of Thermoregulation on Human Sleep Patterns: A Mathematical Model of Sleep–Wake Cycles with REM–NREM Subcircuit” in Applications of Dynamical Systems in Biology and Medicine, vol. 158, T. Jackson, A. Radunskaya, Eds. New York: Springer, 2015, pp. 123–147.

Many real-world networks have high clustering among vertices: vertices that share neighbors are often also directly connected to each other. A network’s clustering can be a useful indicator of its connectedness and community structure. Algorithms for generating networks with high clustering have been developed, but typically rely on adding or removing edges and nodes, sometimes from a completely empty network. We introduce algorithms that create a highly clustered network by starting with an existing network and rearranging edges, without adding or removing them. These algorithms can preserve other network properties even as the clustering increases. These algorithms rely on local rewiring rules, in which a single edge changes one of its vertices in a way that is guaranteed to increase clustering. This greedy algorithm can be applied iteratively to transform a random network into a form with much higher clustering. Additionally, these algorithms grow the network’s clustering faster than they increase its path length, meaning that network enters a regime of comparatively high clustering and low path length: a small world. Thus, these algorithms may be a basis for how real-world networks rearrange themselves organically to achieve or maintain high clustering and small-world structure.

[arXiv] J. Alstott, C. Klymko, P. B. Pyzza, M. Radcliffe. “Local Rewiring Algorithms to Increase Clustering and Grow a Small World” Under review.

Recent work also includes research on creating a model for a dynamic sexual network, where the relationships in the network are changing over time, on which we can spread an epidemic (such as HPV). Then, we can analyze various vaccination strategies against this epidemic to determine effective means of lowering the incidence rate for the virus.

Professional Memberships

Society for Industrial and Applied Mathematics (SIAM)
Association for Women in Mathematics (AWM)
American Mathematical Society (AMS)
Society for Neuroscience (SfN)
Faculty for Undergraduate Neuroscience (FUN)