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Understanding Dynamics via Phase Plane Analysis

While an experimenter typically can measure membrane potential, it is usually impossible to monitor other dynamic variables, like ionic currents, during non-clamped activity. For a theoretical model, we must compute explicitly the time courses of all dependent variables. Thus we have more information at hand and so we can obtain additional insight by comparing the time courses and identifying the contributions and temporal relationships of the different dynamic variables. A valuable way to simultaneously view the response of multiple variables and their relationship to physiological functions is by phase-plane profiles, i.e. curves of one dependent variable against another. Moreover, such plots allow us also to represent and interpret geometrically aspects of the model (e.g., activation curves) along with the response trajectories. At a glance we can see if the model has one or multiple steady states, and which stimuli might invoke switching between states, and where these steady states lie in relation to activation and I-V characteristics. While the phase plane view provides a full description for two-variable models, judicious two-dimensional projections from phase spaces of higher order systems can yield some of these same insights.

Phase plane analysis was used effectively by FitzHugh [10,11,12] to understand various aspects of the HH equations and the two-variable FitzHugh-Nagumo model. His review [12] also defines some basic mathematical terminology of nonlinear dynamics and supplements our presentation. For additional mathematical introduction, we recommend the books [6,34].

Bard Ermentrout
Mon Jul 29 17:47:46 EDT 1996