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""" Implementing ddy = alpha * (beta * (y* - y) - dy)
NOTE: when connecting to the input, use synapse=None so that double
filtering of the input signal doesn't happen. """
import numpy as np
from scipy.linalg import expm
import nengo
def generate(net=None, n_neurons=200, alpha=1000.0, beta=1000.0/4.0,
dt=0.001, analog=False):
tau = 0.1 # synaptic time constant
synapse = nengo.Lowpass(tau)
# the A matrix for our point attractor
A = np.array([[0.0, 1.0],
[-alpha*beta, -alpha]])
# the B matrix for our point attractor
B = np.array([[0.0], [alpha*beta]])
# if you have the nengolib library you can do it this way
# from nengolib.synapses import ss2sim
# C = np.eye(2)
# D = np.zeros((2, 2))
# linsys = ss2sim((A, B, C, D), synapse=synapse, dt=None if analog else dt)
# A = linsys.A
# B = linsys.B
if analog:
# account for continuous lowpass filter
A = tau * A + np.eye(2)
B = tau * B
else:
# discretize state matrices
Ad = expm(A*dt)
Bd = np.dot(np.linalg.inv(A), np.dot((Ad - np.eye(2)), B))
# account for discrete lowpass filter
a = np.exp(-dt/tau)
A = 1.0 / (1.0 - a) * (Ad - a * np.eye(2))
B = 1.0 / (1.0 - a) * Bd
if net is None:
net = nengo.Network(label='Point Attractor')
config = nengo.Config(nengo.Connection, nengo.Ensemble)
config[nengo.Connection].synapse = nengo.Lowpass(tau)
with config, net:
net.ydy = nengo.Ensemble(n_neurons=n_neurons, dimensions=2,
# set it up so neurons are tuned to one dimensions only
encoders=nengo.dists.Choice([[1, 0], [-1, 0], [0, 1], [0, -1]]))
# set up Ax part of point attractor
nengo.Connection(net.ydy, net.ydy, transform=A)
# hook up input
net.input = nengo.Node(size_in=1, size_out=1)
# set up Bu part of point attractor
nengo.Connection(net.input, net.ydy, transform=B)
# hook up output
net.output = nengo.Node(size_in=1, size_out=1)
# add in forcing function
nengo.Connection(net.ydy[0], net.output, synapse=None)
return net
if __name__ == '__main__':
import matplotlib.pyplot as plt
plt.figure(figsize=(5, 12))
dt = 1e-3
time = 5 # number of seconds to run simulation
alphas = [10.0, 100.0, 1000.0]
for ii, alpha in enumerate(alphas):
probe_results = []
model = nengo.Network()
beta = alpha / 4.0
for option in [True, False]:
with model:
def goal_func(t):
return float(int(t)) / time * 2 - 1
goal = nengo.Node(output=goal_func)
pa = generate(n_neurons=1000, analog=option,
alpha=alpha, beta=beta, dt=dt)
nengo.Connection(goal, pa.input, synapse=None)
probe_ans = nengo.Probe(goal)
probe = nengo.Probe(pa.output, synapse=.01)
sim = nengo.Simulator(model, dt=dt)
sim.run(time)
probe_results.append(np.copy(sim.data[probe]))
plt.subplot(len(alphas), 1, ii+1)
lines_c = plt.plot(sim.trange(), probe_results[0], 'b')
lines_d = plt.plot(sim.trange(), probe_results[1], 'g')
line_ans = plt.plot(sim.trange(), sim.data[probe_ans][:, 0], 'r--')
plt.legend([lines_c[0], lines_d[0], line_ans],
['continuous', 'discrete', 'desired'])
plt.title('alpha=%.2f, beta=%.2f' % (alpha, beta))
plt.tight_layout()
plt.show()