FitzHugh–Nagumo model

FitzHugh–Nagumo model is a simplified version of the Hodgkin–Huxley model used to describe the electrical activity of a single neuron. It was proposed independently by Richard FitzHugh in 1961 and by J. Nagumo and his colleagues in 1962 as a two-dimensional model to explain how action potentials in neurons are initiated and propagated. The model captures the essential features of excitability that can be found in nerve cell membranes and is widely used in neuroscience and biophysics for theoretical studies of neuronal dynamics.

Overview

The FitzHugh–Nagumo model simplifies the complex four-variable Hodgkin–Huxley model into a two-variable system that describes the interaction between a neuron's membrane potential and a recovery variable. The model equations are given by:

\[ \begin{align} \frac{dv}{dt} &= v - \frac{v^3}{3} - w + I \\ \frac{dw}{dt} &= \epsilon(v + a - bw) \end{align} \]

where \(v\) represents the membrane potential, \(w\) is the recovery variable, \(I\) is the external current, and \(\epsilon\), \(a\), and \(b\) are parameters that can be adjusted to mimic different types of neuronal behavior. The term \(v - \frac{v^3}{3}\) represents a cubic nonlinearity that is crucial for the generation of action potentials, while the recovery variable \(w\) provides feedback that controls the timing of these events.

Applications

The FitzHugh–Nagumo model is used in various fields of neuroscience and biophysics to study the qualitative behavior of excitable systems. It serves as a fundamental model for understanding the mechanisms of action potential initiation and propagation in neurons. Additionally, it has been applied in the study of cardiac tissue to model the electrical activity of heart cells and in the analysis of pattern formation in spatially extended systems.

Mathematical Analysis

The FitzHugh–Nagumo model is an example of a dynamical system that can exhibit a wide range of behaviors, including fixed points, limit cycles, and bifurcations, depending on the values of its parameters. The model can undergo a Hopf bifurcation, leading to oscillatory behavior that mimics the repetitive firing of neurons. Mathematical analysis of the model helps in understanding how the properties of the system change as parameters are varied, providing insights into the mechanisms of neuronal excitability and the conditions under which abnormal rhythms such as arrhythmias may arise.

Limitations

While the FitzHugh–Nagumo model captures the essential features of neuronal excitability, it is a simplification of the more detailed Hodgkin–Huxley model and thus may not accurately represent all aspects of neuronal dynamics. The model's simplicity, however, makes it a valuable tool for theoretical studies and educational purposes, offering insights into the complex behavior of excitable cells with a more tractable mathematical framework.

See Also

• Hodgkin–Huxley model
• Neuron
• Action potential
• Dynamical systems
• Bifurcation theory

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  FitzHugh–Nagumo model

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  Time domain graph of the FitzHugh–Nagumo model

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  FitzHugh–Nagumo model with b = 0.8

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  FitzHugh–Nagumo model with b = 1.25

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  FitzHugh–Nagumo model with b = 2.0

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  FitzHugh–Nagumo model with b = 2.0, separatrix

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  FitzHugh–Nagumo model with b = 2.0, I_ext = 5.37, with stable and unstable manifolds marked

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  FitzHugh–Nagumo model with b = 2.0, with stable and unstable manifolds marked