# FitzHugh-Nagumo analysis

The FitzHugh-Nagumo model is given by:

$\frac {dV} {dt} = V(1 - \frac {V^2} 3) - w + I_{ext}$
$\tau \frac {dw} {dt} = V + a - b w$
[1]:

import brainpy as bp
import brainpy.math as bm
bm.set_dt(dt=0.02)

[2]:

class FitzHughNagumo(bp.NeuGroup):

def __init__(self, size, a=0.7, b=0.8, tau=12.5, Vth=1.9, **kwargs):
super(FitzHughNagumo, self).__init__(size=size, **kwargs)

self.a = a
self.b = b
self.tau = tau
self.Vth = Vth

self.V = bm.Variable(bm.zeros(size))
self.w = bm.Variable(bm.zeros(size))
self.spike = bm.Variable(bm.zeros(size))
self.input = bm.Variable(bm.zeros(size))

@staticmethod
@bp.odeint(method='rk4')
def integral(V, w, t, Iext, a, b, tau):
dw = (V + a - b * w) / tau
dV = V - V * V * V / 3 - w + Iext
return dV, dw

def update(self, _t, _dt):
V, self.w[:] = self.integral(self.V, self.w, _t, self.input, self.a, self.b, self.tau)
self.spike[:] = (V >= self.Vth) * (self.V < self.Vth)
self.V[:] = V
self.input[:] = 0.


## Simulation

[3]:

FNs = FitzHughNagumo(2, monitors=['V'])

FNs.run(duration=300., inputs=('input', 1.), report=True)
bp.visualize.line_plot(FNs.mon.ts, FNs.mon.V, show=True)

FNs.run(duration=(300., 600.), inputs=('input', 0.6), report=True)
bp.visualize.line_plot(FNs.mon.ts, FNs.mon.V, show=True)

Compilation used 0.0000 s.
Start running ...
Run 100.0% used 3.087 s.
Simulation is done in 3.087 s.


Compilation used 0.0000 s.
Start running ...
Run 100.0% used 3.096 s.
Simulation is done in 3.096 s.



## Phase plane analysis

[4]:

phase = bp.symbolic.PhasePlane(FNs,
target_vars={'V': [-3, 2], 'w': [-2, 2]},
fixed_vars=None,
pars_update={'Iext': 1., "a": 0.7, 'b': 0.8, 'tau': 12.5})
phase.plot_nullcline()
phase.plot_fixed_point()
# phase.plot_trajectory(initials={'V': -1, 'w': 1}, duration=100.)
phase.plot_limit_cycle_by_sim(initials={'V': -1, 'w': 1}, duration=100.)
phase.plot_vector_field(show=True)


## Codimension 1 bifurcation analysis

[5]:

bifurcation = bp.symbolic.Bifurcation(FNs,
target_pars={'Iext': [-1, 1]},
target_vars={'V': [-3, 2], 'w': [-2, 2]},
fixed_vars=None,
pars_update={'a': 0.7, 'b': 0.8, 'tau': 12.5},
numerical_resolution=0.01)
bifurcation.plot_bifurcation(show=True)

[5]:

{'saddle node': {'p': [], 'V': [], 'w': []},
'center': {'p': [], 'V': [], 'w': []},
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'unstable star': {'p': [], 'V': [], 'w': []},
'unstable line': {'p': [], 'V': [], 'w': []},
'stable degenerate': {'p': [], 'V': [], 'w': []},
'unstable degenerate': {'p': [], 'V': [], 'w': []}}


## Codimension 2 bifurcation analysis

[6]:

bifurcation = bp.symbolic.Bifurcation(FNs,
target_pars=dict(a=[0.5, 1.], Iext=[0., 1.]),
target_vars=dict(V=[-3, 3], w=[-3., 3.]),
fixed_vars=None,
pars_update={'b': 0.8, 'tau': 12.5},
numerical_resolution=0.01)
bifurcation.plot_bifurcation(show=True)

[6]:

{'saddle node': {'p0': [], 'p1': [], 'V': [], 'w': []},
'center': {'p0': [], 'p1': [], 'V': [], 'w': []},
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'unstable star': {'p0': [], 'p1': [], 'V': [], 'w': []},
'unstable line': {'p0': [], 'p1': [], 'V': [], 'w': []},
'stable degenerate': {'p0': [], 'p1': [], 'V': [], 'w': []},
'unstable degenerate': {'p0': [], 'p1': [], 'V': [], 'w': []}}