Mathematics Colloquia and Seminars

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Talk 1: Bifurcation Structure of Traveling Wave Solutions in Electrophysiological Models of Cardiac Tissue In the heart, repeated waves of electrochemical activity propagate through the tissue, triggering the coordinated contractions that allow the heart to pump blood. The propagated waves can be modeled as traveling wave solutions in spatially-distributed excitable medium in which local dynamics have been represented by highly idealized models (such as the two-variable FitzHugh-Nagumo equations) and high-dimensional biophysically-detailed models (such as the ten-variable ten Tusscher et al. model). In general, traveling waves in models of cardiac tissue cannot be computed analytically, and therefore suitable numerical methods are a necessity. It is common to solve for traveling wave profiles with root finding methods, and then continuation methods can capture the dependence of the existence of waves and their properties, such as wave speed, on cardiac model parameters and the frequency of the waves. Current numerical methods for computing traveling wave solutions are adequate for idealized models but are lacking for the more computationally demanding detailed-biophysical models. We develop robust and efficient numerical methods for computing traveling wave solutions, dispersion curves, and bifurcation scenarios for the high-dimensional models. We use these numerical methods to uncover potential fundamental differences between the bifurcation structure of travelling waves in the highly idealized and in biophysically-detailed cardiac models.

Talk 2: The influence of electrical brain stimulation pulse sequence on dopamine release dynamics The dopamine system plays an important role in arousal, movement, memory, and reward-related behavior. Electrical brain stimulation (EBS) targeting dopaminergic (DA) neurons is an important tool for the investigation of dopamine circuits and the treatment of diseases such as Parkinson’s disease. However, our understanding of how the parameters of EBS, as well as the firing dynamics and synaptic plasticity of the DA neurons, regulate dopamine delivery at target neural structures remains limited. In this study, we investigate how brain stimulation parameters such as stimulation frequency, variability, and the brain region targeted impact the time course of dopamine release and dopamine's role in reward-guided learning by developing data-driven mathematical modeling to capture various experiments to measure variations in dopamine delivery in response to stimulation. Our model provides detailed information on how electrical brain stimulation (EBS) affects specific brain regions, thereby revealing the potential mechanisms underlying the therapeutic effects of EBS in conditions such as Parkinson’s disease and schizophrenia.

Also available on Zoom: https://ucdavis.zoom.us/j/98969645841