FitzHugh-Nagumo

anypinn create my-project --template fitzhugh-nagumo

Two-field nonlinear neuron model. Recovers timescale \(\varepsilon\) and threshold parameter \(a\) from a partially observed voltage trace.

Background

The FitzHugh-Nagumo model, simplified by FitzHugh (1961) and Nagumo et al. (1962) from the four-variable Hodgkin-Huxley equations, captures the essential dynamics of neural spike generation with just two variables: a fast voltage-like variable \(v\) and a slow recovery variable \(w\). The model exhibits excitable dynamics: small perturbations decay, but a sufficiently large stimulus drives the system through a full action potential (spike) before returning to rest. The timescale separation parameter \(\varepsilon \ll 1\) governs how fast the recovery variable \(w\) responds relative to the voltage \(v\).

Governing Equations

\[ \begin{cases} \dfrac{dv}{dt} = v - \dfrac{v^3}{3} - w + I_{\text{ext}} \\[8pt] \dfrac{dw}{dt} = \varepsilon\,(v + a - b\,w) \end{cases} \]

where:

• \(v(t)\): membrane voltage (fast variable, observed)
• \(w(t)\): recovery current (slow variable, latent)
• \(\varepsilon\): timescale separation (to recover)
• \(a\): excitability threshold (to recover)
• \(b\): recovery sensitivity (known)
• \(I_{\text{ext}}\): external stimulus current (known)

Default Configuration

The generated template uses the following values.

Parameters to recover:

Symbol	Code constant	True value
\(\varepsilon\)	TRUE_EPSILON	\(0.08\)
\(a\)	TRUE_A	\(0.7\)

Known constants:

Symbol	Code constant	Value
\(b\)	B	\(0.8\)
\(I_{\text{ext}}\)	I_EXT	\(0.5\)

Initial conditions: \(v(0) = -1.0, \quad w(0) = 1.0\)

Domain: \(t \in [0, 50]\)

Observability: only \(v\) (voltage) is observed; \(w\) is a latent state reconstructed by the network.

Features Demonstrated

• Partial observability (only voltage is observed)
• Multi-parameter Parameter recovery (\(\varepsilon\) and \(a\))
• Neural excitation dynamics

Results