Math Union Colloquium

This talk focuses on the mathematical modeling of neurons through the lens of dynamical systems theory. The 1952 Hodgkin-Huxley model revolutionized neuroscience by providing a quantitative description of the action potential, yet its complexity (four coupled, non-linear differential equations) often obscures the underlying dynamical principles. This talk bridges the gap between biological mechanism and mathematical intuition by exploring the transition from the full Hodgkin-Huxley system to its reduced geometric representations. We begin by establishing the biophysical foundations of the model as an electrical circuit before applying biologically-justified simplifications to achieve the two-dimensional 1981 Morris-Lecar model. We then distinguish between Type I and Type II excitability through the analysis of phase planes via bifurcations. From channels to geometry, this approach shifts the focus from biological detail to geometric structures that dictate the firing thresholds and frequency-current relationships of the neuron. No prior knowledge of biology and dynamical systems is assumed.