Action Potential Thresholds and Excitability from the Geometry of Membrane Potential

A groundbreaking mathematical framework has been developed to precisely define the threshold at which action potentials occur in excitable cells. Unlike previous methods that used approximations or bifurcation points, this new approach relies on the geometric properties of the membrane potential. By examining the changes in concavity during the upstroke of an action potential within a time series of voltage data, researchers can obtain direct information about the threshold.

The concavity criterion has been extended to models based on autonomous dynamical systems, allowing for analytical determination of the critical points in phase space where changes in concavity occur. These inflection points define a specific region known as the "inflection point manifold," which encompasses all the orbits that lead to action potentials. Any trajectory containing an action potential will also be within this region.

This analytical principle provides a solid foundation for defining excitability in dynamical systems, and it also offers a quantitative measure of excitability. This measure enables researchers to compare the excitability levels of different systems, such as neurons with varying electrophysiological characteristics and under diverse stimulus conditions.

By transforming the traditionally vague concept of electrical excitability into a rigorous analytical description, the criterion becomes applicable not only to smooth, single compartment models but also to higher-dimensional single compartment models and multicompartment models as well.