# Dynamics Analysis¶

BrainPy provides fundamental methods for dynamics analysis of neuron models, including:

1. phase plane analysis for 1-dimensional and 2-dimensional systems;

1. codimension 1 and codimension 2 bifurcation analysis.

We take FitzHugh-Nagumo model as example to demonstrate how to perform dynamics analysis with BrainPy API.

The model is given by:

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

There are two variables $$V$$ and $$w$$, so this is a two-dimensional system with three parameters $$a, b$$ and $$\tau$$.

Let’s start by defining the model.

:

import brainpy as bp
import numpy as np

bp.profile.set(dt=0.02, numerical_method='rk4')

def get_FNmodel(a=0.7, b=0.8, tau=12.5, Vth=1.9):
state = bp.types.NeuState({'v': 0., 'w': 1., 'spike': 0., 'input': 0.})

@bp.integrate
def int_w(w, t, v):
return (v + a - b * w) / tau

@bp.integrate
def int_v(v, t, w, Iext):
return v - v * v * v / 3 - w + Iext

def update(ST, _t):
ST['w'] = int_w(ST['w'], _t, ST['v'])
v = int_v(ST['v'], _t, ST['w'], ST['input'])
ST['spike'] = np.logical_and(v >= Vth, ST['v'] < Vth)
ST['v'] = v
ST['input'] = 0.

return bp.NeuType(name='FitzHugh_Nagumo',
ST=state,
steps=update)


## Phase Plane Analysis¶

We provide brainpy.PhasePortraitAnalyzer to support phase plane analysis for 1-dimensional and 2-dimensional dynamical systems.

Two parameters should be specified to initialize a PhasePortraitAnalyzer:

• model: The neuron model to be analysis.

• target_vars: The variables to be analysis and its value range.

And two parameters are optional:

• fixed_vars: The slow variables to be fixed as stational variables (Optional for higher order system).

• pars_update: Parameters to update.

After defining a PhasePortraitAnalyzer, you can call the following functions:

• plot_nullcline(): Plot the nullclines.

• plot_vector_field(): Plot the vector field.

• plot_fixed_point(): Find and plot the fixed points, and perform stability analysis (print to the terminal).

• plot_trajectory(): Plot trajectories according to the settings (initial var1, initial var2, duration).

Here we perform a phase plane analysis with parameters $$a=0.7, b=0.8, \tau=12.5$$, and input $$I_{ext} = 0.8$$.

:

neuron = get_FNmodel(a=0.7, b=0.8, tau=12.5)

analyzer = bp.PhasePortraitAnalyzer(
model=neuron,
target_vars={'v': [-3, 3], 'w': [-3., 3.]},
fixed_vars={'Iext': 0.8})
analyzer.plot_nullcline()
analyzer.plot_vector_field()
analyzer.plot_fixed_point()
analyzer.plot_trajectory([(-2.8, -1.8, 100.)],
inputs=('ST.input', 0.8),
show=True)

Fixed point #1 at v=-0.2729009589972752, w=0.5338738012534059 is a unstable-node. We can see an unstable-node at the point (v=-0.27, w=0.53) inside a limit cycle. Then we can run a simulation with the same parameters and initial values to see the periodic activity that correspond to the limit cycle.

:

group = bp.NeuGroup(neuron, 1, monitors=['v', 'w'])
group.ST['v'] = -2.8
group.ST['w'] = -1.8
group.run(100., inputs=('ST.input', 0.8))
bp.visualize.line_plot(group.mon.ts, group.mon.v, legend='v', )
bp.visualize.line_plot(group.mon.ts, group.mon.w, legend='w', show=True) ## Bifurcation Analysis¶

We provide brainpy.BifurcationAnalyzer for users to perform bifucation analysis.

The PhasePortraitAnalyzer receives the following parameters:

• model: The neuron model to be analysis.

• target_pars: The parameters to be change and the ranges.

• dynamical_vars: The variables of the system and the change ranges.

• par_resolution: The numerical resolution of the bifurcation analysis.

### Codimension 1 bifurcation analysis¶

We will first see the codimension 1 bifurcation anlysis of the model. For example, we vary the input $$I_{ext}$$ between 0 to 1 and see how the system change it’s stability.

:

analyzer = bp.BifurcationAnalyzer(
model=neuron,
target_pars={'Iext': [0., 1.]},
dynamical_vars={'v': [-3, 3], 'w': [-3., 3.]},
par_resolution=0.001,
)
analyzer.plot_bifurcation(plot_vars=['v'], show=True) ### Codimension 2 bifurcation analysis¶

We simulaneously change $$I_{ext}$$ and parameter $$a$$.

:

analyzer = bp.BifurcationAnalyzer(
model=neuron,
target_pars={'a': [0.5, 1.], 'Iext': [0., 1.]},
dynamical_vars={'v': [-3, 3], 'w': [-3., 3.]},
par_resolution=0.01,
)
analyzer.plot_bifurcation(plot_vars=['v'], show=True) 