Where are attractors used?
Hello!
Do you mean where are there examples of attractors in Nengo?
The following code you should be able to copy into a file and run with ‘python filename.py’
""" Implementing ddy = alpha * (beta * (y* - y) - dy) """
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):
tau = 0.1 # synaptic time constant
# 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, 0.0], [alpha*beta, 1.0]])
# discretize
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)
# config[nengo.Ensemble].neuron_type = nengo.Direct()
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=2, size_out=2)
# 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__':
time = 5 # number of seconds to run simulation
model = nengo.Network()
with model:
def goal_func(t):
return [float(int(t)) / time * 2 - 1, 0]
goal = nengo.Node(output=goal_func)
pa = generate(n_neurons=1000)
nengo.Connection(goal, pa.input, synapse=None)
probe_ans = nengo.Probe(goal)
probe = nengo.Probe(pa.output, synapse=.01)
sim = nengo.Simulator(model, dt=.001)
sim.run(time)
import matplotlib.pyplot as plt
plt.plot(sim.trange(), sim.data[probe])
plt.plot(sim.trange(), sim.data[probe_ans][:, 0], 'r--')
plt.legend(['continuous', 'discrete', 'desired'])
plt.show()
Does that help?
Yes, it helps. Thanks. There is a theory of chaotic attractors in the nose by Walter Freeman Jr. that would be nice to try out.