#
# Fitzhugh-Nagumo toy model.
#
# This file is part of PINTS (https://github.com/pints-team/pints/) which is
# released under the BSD 3-clause license. See accompanying LICENSE.md for
# copyright notice and full license details.
#
#
import numpy as np
import pints
from . import ToyODEModel
[docs]class FitzhughNagumoModel(ToyODEModel, pints.ForwardModelS1):
r"""
Fitzhugh-Nagumo model of the action potential [1]_.
Has two states, and three phenomenological parameters: ``a`` , ``b``,
``c``. All states are visible
.. math::
\frac{d \mathbf{y}}{dt} = \mathbf{f}(\mathbf{y},\mathbf{p},t)
where
.. math::
\mathbf{y} &= (V,R)\\
\mathbf{p} &= (a,b,c)
The RHS, jacobian and change in RHS with the parameters are given by
.. math::
\begin{align}
\mathbf{f}(\mathbf{y},\mathbf{p},t) &=
\left[\begin{matrix}
c \left(R - V^{3}/3+V\right) \\
- \frac{1}{c} \left(R b + V - a\right)
\end{matrix}\right] \\
\frac{\partial \mathbf{f}}{\partial \mathbf{y}} &=
\left[\begin{matrix}
c \left(1- V^{2}\right) & c \\
- \frac{1}{c} & - \frac{b}{c}
\end{matrix}\right] \\
\frac{\partial \mathbf{f}}{\partial \mathbf{p}} &=
\left[\begin{matrix}
0 & 0 & R - V^{3}/3 + V\\
\frac{1}{c} & - \frac{R}{c} &
\frac{1}{c^{2}} \left(R b + V - a\right)
\end{matrix}\right]
\end{align}
Extends :class:`pints.ForwardModelS1`, `pints.toy.ToyODEModel`.
Parameters
----------
y0
The system's initial state. If not given, the default ``[-1, 1]`` is
used.
References
----------
.. [1] A kinetic model of the conductance changes in nerve membrane
Fitzhugh (1965) Journal of Cellular and Comparative Physiology.
https://doi.org/10.1002/jcp.1030660518
"""
def __init__(self, y0=None):
super(FitzhughNagumoModel, self).__init__()
# Check initial values
if y0 is None:
self._y0 = np.array([-1, 1], dtype=float)
else:
self._y0 = np.array(y0, dtype=float)
if len(self._y0) != 2:
raise ValueError('Initial value must have size 2.')
def _dfdp(self, y, t, p):
""" See :meth:`pints.ToyODEModel._dfdp()`. """
V, R = y
a, b, c = [float(param) for param in p]
ret = np.empty((2, 3))
ret[0, 0] = 0
ret[0, 1] = 0
ret[0, 2] = R - V**3 / 3 + V
ret[1, 0] = 1 / c
ret[1, 1] = -R / c
ret[1, 2] = (R * b + V - a) / c**2
return ret
[docs] def jacobian(self, y, t, p):
""" See :meth:`pints.ToyODEModel.jacobian()`. """
V, R = y
a, b, c = [float(param) for param in p]
ret = np.empty((2, 2))
ret[0, 0] = c * (1 - V**2)
ret[0, 1] = c
ret[1, 0] = -1 / c
ret[1, 1] = -b / c
return ret
[docs] def n_outputs(self):
""" See :meth:`pints.ForwardModel.n_outputs()`. """
return 2
[docs] def n_parameters(self):
""" See :meth:`pints.ForwardModel.n_parameters()`. """
return 3
def _rhs(self, y, t, p):
""" See :meth:`pints.ToyODEModel._rhs()`. """
V, R = y
a, b, c = [float(x) for x in p]
dV_dt = (V - V**3 / 3 + R) * c
dR_dt = (V - a + b * R) / -c
return dV_dt, dR_dt
[docs] def suggested_parameters(self):
""" See :meth:`pints.toy.ToyModel.suggested_parameters()`. """
return np.array([0.1, 0.5, 3])
[docs] def suggested_times(self):
""" See :meth:`pints.toy.ToyModel.suggested_times()`. """
return np.linspace(0, 20, 200)