# [2D] Decision Making Model with SlowPointFinder

[2]:

import brainpy as bp
import brainpy.math as bm

bp.math.enable_x64()
bp.math.set_platform('cpu')

[3]:

import numpy as np
import matplotlib.pyplot as plt

[4]:

# parameters
gamma = 0.641  # Saturation factor for gating variable
tau = 0.06  # Synaptic time constant [sec]
a = 270.
b = 108.
d = 0.154

[5]:

JE = 0.3725  # self-coupling strength [nA]
JI = -0.1137  # cross-coupling strength [nA]
JAext = 0.00117  # Stimulus input strength [nA]

[6]:

mu = 20.  # Stimulus firing rate [spikes/sec]
coh = 0.5  # Stimulus coherence [%]
Ib1 = 0.3297
Ib2 = 0.3297

[7]:

@bp.odeint
def int_s1(s1, t, s2, coh=0.5, mu=20.):
I1 = JE * s1 + JI * s2 + Ib1 + JAext * mu * (1. + coh)
r1 = (a * I1 - b) / (1. - bm.exp(-d * (a * I1 - b)))
return - s1 / tau + (1. - s1) * gamma * r1

[8]:

@bp.odeint
def int_s2(s2, t, s1, coh=0.5, mu=20.):
I2 = JE * s2 + JI * s1 + Ib2 + JAext * mu * (1. - coh)
r2 = (a * I2 - b) / (1. - bm.exp(-d * (a * I2 - b)))
return - s2 / tau + (1. - s2) * gamma * r2

[9]:

def cell(s):
ds1 = int_s1.f(s[0], 0., s[1])
ds2 = int_s2.f(s[1], 0., s[0])
return bm.asarray([ds1.value, ds2.value])

[10]:

finder = bp.analysis.SlowPointFinder(f_cell=cell, )
finder.find_fps_with_gd_method(
candidates=bm.random.random((1000, 2)),
tolerance=1e-5, num_batch=200,
)
finder.filter_loss(1e-5)
finder.keep_unique()

print('fixed_points: ', finder.fixed_points)
print('losses:', finder.losses)

Optimizing to find fixed points:
Batches 1-200 in 0.40 sec, Training loss 0.0515620188
Batches 201-400 in 0.53 sec, Training loss 0.0043030679
Batches 401-600 in 0.41 sec, Training loss 0.0006742308
Batches 601-800 in 0.41 sec, Training loss 0.0001451481
Batches 801-1000 in 0.40 sec, Training loss 0.0000341044
Batches 1001-1200 in 0.41 sec, Training loss 0.0000080870
Stop optimization as mean training loss 0.0000080870 is below tolerance 0.0000100000.
Excluding fixed points with squared speed above tolerance 0.00001:
Kept 948/1000 fixed points with tolerance under 1e-05.
Excluding non-unique fixed points:
Kept 3/948 unique fixed points with uniqueness tolerance 0.025.
fixed_points:  [[0.28276717 0.40635691]
[0.70045189 0.00486431]
[0.01394651 0.65738904]]
losses: [3.27336633e-10 2.57846002e-30 4.80622382e-30]

[11]:

if len(finder.fixed_points):
jac = finder.compute_jacobians(finder.fixed_points)
for i in range(len(finder.fixed_points)):
eigval, eigvec = np.linalg.eig(np.asarray(jac[i]))
plt.figure()
plt.scatter(np.real(eigval), np.imag(eigval))
plt.plot([0, 0], [-1, 1], '--')
plt.xlabel('Real')
plt.ylabel('Imaginary')
plt.title(f'FP {i}')
plt.show()