Applied Mathematics Colloquium: Dr Gregory Handy

Title: Extending mathematical frameworks to investigate stochastic fluctuations in diverse brain cell types  

Abstract: Stochastic fluctuations drive biological processes from particle diffusion to neuronal spike times. Today, we will use a variety of mathematical frameworks to understand such fluctuations and derive insight into the corresponding applications. We start by considering a novel stochastic process motivated by astrocytes. These glial cells ensheath neuronal synapses, positioning them to remove signaling molecules from the synaptic cleft. We generalize this setup to considering n diffusing particles that may leave the domain by either ‘escaping’ through an absorbing boundary (i.e., astrocyte) or being ‘captured’ by traps (i.e., neurotransmitter receptor) that must recharge

between captures. We prove that the number of captured particles grows the order of (log n) because of this recharge time, suggesting that many neurotransmitters interact with these neighboring astrocytes. We then generalize the framework further to investigate the celebrated formula of Berg and Purcell, which models the rate that cell surface receptors capture extracellular molecules, in the context of such recharging receptors. We end by exploring how the brain leverages interneuron diversity and noisy recurrent connections to assist with cortical computations. Specifically, we analyze a spatial model of the visual cortex with linear response theory and show how interneurons modulate the level of synchrony in visually induced gamma rhythms.