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.

[1]:
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 \(I_{ext} = 0.8\).

[2]:
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.
../_images/tutorials_dynamics_analysis_8_1.png

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.

[3]:
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)
../_images/tutorials_dynamics_analysis_10_0.png

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.

[4]:
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)
../_images/tutorials_dynamics_analysis_15_0.png

Codimension 2 bifurcation analysis

We simulaneously change \(I_{ext}\) and parameter \(a\).

[5]:
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)
../_images/tutorials_dynamics_analysis_18_0.png