Geometric analysis of soft thresholds in action potential initiation and the consequences for understanding phase response curves and model tuning
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KeywordsFunction Choice Phase Response Curve Saddle Node Depolarization Block Neural Excitability
where the asymptotic steady state voltage V ∞ (m, n) is for constant h. Given that Open image in new window and Open image in new window in this neighborhood, the first inequality can be interpreted as saying that the fast sodium current must dominate the effect of the growing delayed rectifier potassium current, but not in such a way that Open image in new window becomes large too rapidly. We find that the curvature of the nullclines in this region is responsible for the truth of this condition (for standard parameter values used for Type I and II modes of H-H excitability).
We compare different parameter regimes for periodic and transient trajectories using our analysis. Under mild variation of parameters and initial conditions, and except in pathological circumstances that are related to the generation of sub-threshold oscillations and canards (e.g., see ref. ), we can predict the onset of AP initiation and its timing as the result of parameter changes or small voltage perturbations. This leads to insight about the origin of phase response curve shape in Type I vs. Type II neural excitability. Geometric features of the nullclines are measured during this analysis using PyDSTool , and the curvature conditions can be used to guide objective function choice for parameter tuning tasks where AP generation is found to be incomplete or imperfect (e.g., compare refs. [5, 6]). This has applications in situations where APs are pathologically changed due to genetic channel defects (especially in potassium channels), or where unwanted depolarization block occurs.
This research is supported in part by NSF EMT/BSSE award #0829742.
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