An Analytical Description for Action Potential Thresholds Defined by Concavity Changes

A novel mathematical framework to define the threshold of action potentials in excitable cells is presented. Unlike previously applied methods that rely on approximations or specific fixed-point bifurcations, the approach focuses on the geometry of membrane potential trajectories. Specifically, the focus is on the concavity changes during the upstroke of an electrical pulse. These changes in concavity form a curve of inflection points that defines a region in phase space crossed by all the action potentials in the system, and containing no non-action potential trajectories. Such region is called the excitability region and its size can be measured, thus providing a measure for the excitability of a dynamical system, and a way to compare the excitability between systems representing different biological phenotypes and stimulus conditions. The work transforms the traditionally vague physiological concept of excitability into a rigorous analytical description applicable across continuous, single compartment models of electrical excitability.