# Controlled oscillator¶

The controlled oscillator is an oscillator with an extra input that controls the frequency of the oscillation.

To implement a basic oscillator, we would use a neural ensemble of two dimensions that has the following dynamics:

$\begin{split}\dot{x} = \begin{bmatrix} 0 && - \omega \\\ \omega && 0 \end{bmatrix} x\end{split}$

where the frequency of oscillation is $$\omega \over {2 \pi}$$ Hz.

We need the neurons to represent three variables, $$x_0$$, $$x_1$$, and $$\omega$$. According the the dynamics principle of the NEF, in order to implement some particular dynamics, we need to convert this dynamics equation into a feedback function:

\begin{split} \begin{align*} \dot{x} &= f(x) \\\ &\implies f_{feedback}(x) = x + \tau f(x) \end{align*}\end{split}

where $$\tau$$ is the post-synaptic time constant of the feedback connection.

In this case, the feedback function to be computed is

\begin{split}\begin{align*} f_{feedback}(x) &= x + \tau \begin{bmatrix} 0 && - \omega \\\ \omega && 0 \end{bmatrix} x \\\ &= \begin{bmatrix} x_0 - \tau \cdot \omega \cdot x_1 \\\ x_1 + \tau \cdot \omega \cdot x_0\end{bmatrix} \end{align*}\end{split}

Since the neural ensemble represents all three variables but the dynamics only affects the first two ($$x_0$$, $$x_1$$), we need the feedback function to not affect that last variable. We do this by adding a zero to the feedback function.

$\begin{split}f_{feedback}(x) = \begin{bmatrix} x_0 - \tau \cdot \omega \cdot x_1 \\\ x_1 + \tau \cdot \omega \cdot x_0 \\\ 0 \end{bmatrix}\end{split}$

We also generally want to keep the ranges of variables represented within an ensemble to be approximately the same. In this case, if $$x_0$$ and $$x_1$$ are between -1 and 1, $$\omega$$ will also be between -1 and 1, giving a frequency range of $$-1 \over {2 \pi}$$ to $$1 \over {2 \pi}$$. To increase this range, we introduce a scaling factor to $$\omega$$ called $$\omega_{max}$$.

$\begin{split}f_{feedback}(x) = \begin{bmatrix} x_0 - \tau \cdot \omega \cdot \omega_{max} \cdot x_1 \\\ x_1 + \tau \cdot \omega \cdot \omega_{max} \cdot x_0 \\\ 0 \end{bmatrix}\end{split}$
In [1]:

import matplotlib.pyplot as plt
%matplotlib inline

import nengo


## Step 1: Create the network¶

In [2]:

tau = 0.1  # Post-synaptic time constant for feedback
w_max = 10  # Maximum frequency is w_max/(2*pi)

model = nengo.Network(label='Controlled Oscillator')
with model:
# The ensemble for the oscillator

# The feedback connection
def feedback(x):
x0, x1, w = x  # These are the three variables stored in the ensemble
return x0 + w * w_max * tau * x1, x1 - w * w_max * tau * x0, 0

nengo.Connection(oscillator, oscillator, function=feedback, synapse=tau)

# The ensemble for controlling the speed of oscillation
frequency = nengo.Ensemble(100, dimensions=1)

nengo.Connection(frequency, oscillator[2])


## Step 2: Create the input¶

In [3]:

from nengo.processes import Piecewise
with model:
# We need a quick input at the beginning to start the oscillator
initial = nengo.Node(Piecewise({0: [1, 0, 0], 0.15: [0, 0, 0]}))
nengo.Connection(initial, oscillator)

# Vary the speed over time
input_frequency = nengo.Node(
Piecewise({
0: 1,
1: 0.5,
2: 0,
3: -0.5,
4: -1
}))

nengo.Connection(input_frequency, frequency)


In [4]:

with model:
# Indicate which values to record
oscillator_probe = nengo.Probe(oscillator, synapse=0.03)


## Step 4: Run the Model¶

In [5]:

with nengo.Simulator(model) as sim:
sim.run(5)


## Step 5: Plot the Results¶

In [6]:

plt.figure()
plt.plot(sim.trange(), sim.data[oscillator_probe])
plt.xlabel('Time (s)')
plt.legend(['$x_0$', '$x_1$', r'$\omega$']);