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There are 5 distinct parametrisations of the FitzHugh-Nagumo model implemented in this package. They are explained in turn below
:regular
This is the most commonly encountered parametrisation of the FitzHugh-Nagumo model. The target, two dimensional process (Y,X) solves the following stochastic differential equation:
The proposal is taken to be a guided proposal with auxiliary law induced by the linear diffusion obtained by linearising FitzHugh-Nagumo diffusion at an end-point:
The auxiliary law is now induced by a pair: (I,B), where B is a scaled Brownian motion and I is an integrated B:
$$\begin{align*}
d I_t &= B_tdt,\\\
dB_t &= \frac{\sigma}{\epsilon}dW_t.
\end{align*}$$
:complexAlter
The stochastic differential equation solved by the target process is the same as in :simpleAlter. However, the auxliary law is induced by a two-dimensional diffusion, where the second coordinate is a linear diffusion obtained from linearising at an end-point and the first coordinate is an integrated second coordinate. If only the first coordinate is observed the proposal takes a form:
$$\begin{align*}
d I_t &= B_tdt,\\\
dB_t &= \sigma dW_t.
\end{align*}$$
:complexConjug
It is defined analogously to :complexAlter, the only difference being that an additional step is taken of redefining the parameters (just as it was done in :simpleConjug above). Consequently the target law is as given above, in the section on :simpleConjug parametrisation, whereas proposal law is given by: