Lecture 12
The Fitzhugh Nagumo Model, Linear Stability Analysis, and the Trace-Determinant Plane
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Core Readings
- Linear Stability Analysis. Chapter 16 of Cellular Biophysics and Modeling: A primer on the systems biology of excitable cells. [DOI]
- Type II Excitability And Oscillations. Chapter 17 of Cellular Biophysics and Modeling: A primer on the systems biology of excitable cells. [DOI]
- Explore this MIT Mathlet on phase portraits of linear 2d systems of ODEs with Trace-Determinant Plane. [Mathlet]
Slides & Notes
Recordings & Tutorials
- The trace-determinant diagram
Practice
Models & Demonstrations
Further Reading & Resources
- Excitable Systems and the Fitzhugh-Nagumo Equations. Sections 11.1-11.4 (pp. 227-241) of A Primer on Mathematical Models in Biology by Leah Edelstein-Keshet and Lee Segel.
- Fitzhugh’s classic paper (1961).
The sign of beta in the quadratic formula
For a two-dimensional linear system with coefficient matrix A, let β = tr(A) and γ = det(A). The characteristic polynomial is λ² − βλ + γ = 0, so
λ = (β ± √(β² − 4γ)) / 2.
The trace appears with a positive sign in the numerator because its coefficient in the characteristic polynomial is negative. Use this relation when checking a linear-stability calculation.
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