SYDE 556/750 — Assignment 3
Student ID: 00000000
Note: Please include your numerical student ID only, do not include your name.
Note: Refer to the PDF for the full instructions (including some hints), this notebook contains abbreviated instructions only. Cells you need to fill out are marked with a “writing hand” symbol. Of course, you can add new cells in between the instructions, but please leave the instructions intact to facilitate marking.
# Import numpy and matplotlib -- you shouldn't need any other libraries
import numpy as np
import matplotlib.pyplot as plt
from numpy import fft
from uuid import uuid4
# Fix the numpy random seed for reproducible results
s = 18945
np.random.seed(s)
# Some formating options
%config InlineBackend.figure_formats = ['svg']
def rmse(x1, x2):
return np.sqrt(np.mean(np.power(x1 - x2, 2)))
def rms(x):
return np.sqrt(np.mean(np.power(x, 2)))
1. Decoding from a population
a) Tuning curves. Plot the tuning curves (firing rate of each neuron for different $x$ values between $-2$ and $2$).
tau_ref = 2 / 1000
tau_rc = 20 / 1000
def maxJ(tau_ref=tau_ref, tau_rc=tau_rc, max_rate=200):
return 1 / (1 - np.exp((tau_ref - 1 / max_rate) / tau_rc))
def gain(j_max=2, e=1, intercept=0):
return (j_max - 1) / (2 - e * intercept)
def bias(alpha=1, e=1, intercept=0):
return 1 - alpha * e * intercept
def lif_encode_2d(neuron, xy):
J = neuron.a * np.vdot(xy, neuron.circ) + neuron.j_bias
if J > 1:
return 1 / (tau_ref - tau_rc * np.log(1 - 1 / J))
return 0
def print_block(title, data):
print(title + " ----------")
print(data)
print("-----------------")
class Population:
def __init__(self, num_neurons=1, state=None, neuron_overrides=[]):
self.num_neurons = num_neurons
self.rates = []
self.spikes = []
if state == None:
self.default_neuron_states = {
"min_rate": 100,
"max_rate": 200,
"encoder": [-1, 1],
"tau_ref": 2 / 1000,
"tau_rc": 20 / 1000,
"min_x_int": -2,
"max_x_int": 2,
}
else:
self.default_neuron_states = state
self.neurons = []
if len(neuron_overrides) == 0:
for idx in range(self.num_neurons):
neuron = Neuron(self.default_neuron_states)
self.neurons.append(neuron)
else:
for neuron_override in neuron_overrides:
print("Override Neuron " + str(neuron_override.get_id()))
self.neurons.append(neuron_override)
""" Cleans out a population """
def nuke(self):
self.neurons = []
def clear_rates(self):
self.rates = []
def clear_spikes(self):
self.spikes = []
""" Applies a mutator to each neuron in the population """
def mutate(self, mutator):
if len(self.neurons) == 0:
return
else:
for neuron in self.neurons:
mutator(neuron)
def spike(self, X, dT, cap=None):
O = []
for neuron in self.neurons:
spikes = neuron.spikies(X, dT, cap)
O.append(spikes)
return O
def get_curves(self, input):
for neuron in self.neurons:
self.rates.append(neuron.rates(input))
return self.rates
def get_neurons(self):
return self.neurons
def get_neuron_by_index(self, idx):
return self.neurons[idx]
def get_neuron_by_id(self, target_id):
match = None
for neuron in self.neurons:
if target_id == neuron.get_id():
match = neuron
return match
def get_spikes(self):
spikes = []
for neuron in self.neurons:
spikes.append(neuron.get_spiketrend()[:, 1])
self.spikes = spikes
return spikes
def get_voltages(self):
voltages = []
for neuron in self.neurons:
voltages.append(neuron.get_spiketrend()[:, 0])
self.voltages = voltages
return voltages
def get_ordered_encoders(self):
encoders = []
for neuron in self.neurons:
encoders.append(neuron.get_encoder())
return encoders
def get_neuron_count(self):
return len(self.neurons)
def get_neurons_by_rate(self, rate_range=[20, 50], stim=0):
target_neurons = []
for neuron in self.neurons:
rate = neuron.encode(stim)
is_less_then_max = rate < rate_range[1]
is_more_then_min = rate > rate_range[0]
if is_less_then_max and is_more_then_min:
target_neurons.append(neuron)
return target_neurons
class Neuron(Population):
def __init__(self, state=None, override=None):
if override == None:
self.x_int = np.random.uniform(state["min_x_int"], state["max_x_int"])
self.max_rate = np.random.uniform(state["min_rate"], state["max_rate"])
self.e = np.random.choice(state["encoder"])
self.tau_ref = state["tau_ref"]
self.tau_rc = state["tau_rc"]
J_max = maxJ(
tau_ref=self.tau_ref, tau_rc=self.tau_rc, max_rate=self.max_rate
)
self.alpha = gain(J_max, self.e, self.x_int)
self.j_bias = bias(alpha=self.alpha, e=self.e, intercept=self.x_int)
self.id = uuid4()
self.spiketrend = []
self.firing_rates = []
else:
self.x_int = override["x_int"]
self.max_rate = override["max_rate"]
self.e = override["encoder"]
self.tau_ref = override["tau_ref"]
self.tau_rc = override["tau_rc"]
self.alpha = override["alpha"]
self.j_bias = override["j_bias"]
self.id = uuid4()
self.spiketrend = override["spiketrend"]
self.firing_rates = override["firing_rates"]
def whoisthis(self):
print(self.__dict__)
def encode(self, x):
J = self.alpha * x * self.e + self.j_bias
if J > 1:
return 1 / (self.tau_ref - self.tau_rc * np.log(1 - 1 / J))
return 0
def encodeJ(self, x):
return self.alpha * (x) * self.e + self.j_bias
def voltage(self, J, V, dT):
return V + (dT * (1 / self.tau_rc) * (J - V))
def howmanyspikes(self):
spike_points = self.spiketrend[:, 1]
num_spikes = int(spike_points.tolist().count(1))
return num_spikes
def get_intercept(self):
return self.x_int
def get_gain(self):
return self.alpha
def get_bias(self):
return self.j_bias
def get_max_rate(self):
return self.max_rate
def get_tau_rc(self):
return self.tau_rc
def get_tau_ref(self):
return self.tau_ref
def get_id(self):
return self.id
def set_encoder(self, encoder):
self.e = encoder
def get_encoder(self):
return self.e
def me(self):
return self
def dangerously_set_id(self, ied):
self.id = ied
def set_rate(self, rate):
self.max_rate = rate
def set_bias(self, bias):
self.j_bias = bias
def set_gain(self, gain):
self.alpha = gain
def get_spiketrend(self):
return self.spiketrend
def get_rates(self):
return self.firing_rates
def rates(self, x):
self.firing_rates = []
for point in x:
self.firing_rates.append(self.encode(point))
return self.firing_rates
def clear_rates(self):
self.firing_rates = []
def spikies(self, X, dT, cap=None):
N = np.floor(self.tau_ref / dT)
V_th = 1
V_rest = 0
spikes = np.array([np.zeros(len(X)), np.zeros(len(X))]).T
V = V_rest
V_prev = V_rest
ref_period = False
for idx, x in enumerate(X):
if ref_period == True:
V = V_rest
V_prev = V_rest
# voltage is 0
spikes[idx][0] = 0
# no spike so set it to 0
spikes[idx][1] = 0
# we have completed one ref cycle
ref_period = False
else:
J = self.encodeJ(x)
V = self.voltage(J, V_prev, dT)
if V >= V_th:
# we have a spike so assign second column to 1 to indicate a spike
spikes[idx][1] = int(1)
# start the ref period
ref_period = True
# assign the first collumn to the current voltage
# assign a constant spiking voltage to make identification easier
if cap != None:
spikes[idx][0] = cap
else:
spikes[idx][0] = V
# reset the voltage to 0
V = V_rest
V_prev = V_rest
else:
if V < V_rest:
V = V_rest
# no spikes to assign second column to 0
spikes[idx][1] = int(0)
# still capture the voltage
spikes[idx][0] = V
# assign the previous voltage to the current voltage for next iteration
V_prev = V
self.spiketrend = spikes
return spikes
# we want 20 neurons
num_neurons = 20
# with this default state
state = {
"min_rate": 100,
"max_rate": 200,
"encoder": [-1, 1],
"tau_ref": 2 / 1000,
"tau_rc": 20 / 1000,
"min_x_int": -2,
"max_x_int": 2,
}
# create a population of 20 neurons with the default states
ensemble1 = Population(
num_neurons,
state=state,
)
S = 81 # samples
X = np.linspace(-2, 2, S)
curves = np.array(ensemble1.get_curves(X))
plt.figure()
plt.suptitle(
"LIF Neuron Turning curves from population of 20 randomly generated neurons"
)
for curve in curves:
plt.plot(X, curve)
plt.xlabel("$x$ stimuli")
plt.ylabel("$\\alpha$ Hz")
plt.xlim([-2, 2])
plt.ylim([0, 200])
plt.show()
b) Decoder and error computation. Compute the decoders and plot $(x-\hat{x})$. When computing decoders, take into account noise ($\sigma=0.1 \cdot 200\,\mathrm{Hz}$). When computing $\hat{x}$, add random Gaussian noise with $\sigma=0.1 \cdot 200\,\mathrm{Hz}$ to the activity. Report the Root Mean-Squared Error (RMSE).
A = curves
X = X
noise_stdev = 0.1 * 200
w_noise = np.random.normal(scale=noise_stdev, size=np.shape(A))
A_NOISE = A + w_noise
N = len(X)
n = num_neurons
# find decoders via least squares solution
D = np.linalg.lstsq(
A @ A.T + 0.5 * N * np.square(noise_stdev) * np.eye(n), A @ X.T, rcond=None
)[0]
print_block("Decoders with noise", D)
X_hat = np.dot(D, A_NOISE)
X = np.array(X)
E = X - X_hat
plt.figure()
plt.suptitle("$x-\hat{x}$ with Noisey Decoders and Noise in the activities matrix $A$")
plt.plot(X, E)
plt.xlabel("stimuli $x$")
plt.ylabel("$x-\hat{x}$")
plt.xlim([-2, 2])
plt.show()
plt.figure()
plt.suptitle("$x$ and $\hat{x}$ relative to the stimuli $x$")
x1 = plt.plot(X, X_hat, label="$x$")
x2 = plt.plot(X, X, label="$\hat{x}$")
plt.xlim([-2, 2])
plt.legend(handles=[x1, x2], labels=[])
plt.xlabel("stimuli $x$")
plt.show()
print_block("RMSE", rmse(X, X_hat))
Decoders with noise ----------
[-0.00068122 -0.00101302 -0.00127595 0.0001646 -0.00072472 -0.00063046
0.0018315 0.0016065 -0.00056462 0.00216541 0.00076354 -0.00112071
0.00170102 0.00196816 -0.00276315 -0.00117113 0.00196086 -0.00127341
-0.00057206 0.00146749]
-----------------
RMSE ----------
0.12919565696552784
-----------------
2. Decoding from two spiking neurons
a) Synaptic filter. Plot the post-synaptic current \(h(t)= \begin{cases} 0 & \text{if } t < 0 \,, \\ \frac{e^{-t/\tau}}{\int_0^\infty e^{-t'/\tau} \mathrm{d}t'} & \text{otherwise} \,. \end{cases}\)
def signal_rms(signal):
return np.sqrt(np.mean(np.power(signal, 2)))
def im_normal_rand():
return np.random.normal() + np.random.normal() * 1j
def symmetry_exists(f, F):
neg = -f
return neg in F and f != 0, np.where(F == neg)
def locations(index):
return int(index[0])
def rescale(signal, ideal_rms):
cur_rms = signal_rms(signal)
rescaled_signal = [p * ideal_rms / cur_rms for p in signal]
return rescaled_signal
def zippify(F, Z):
return (list(tt) for tt in zip(*sorted(zip(F, Z))))
def generate_signal(T, dt, rms, limit, seed):
if seed != 0:
np.random.seed(int(seed))
timescale = np.arange(0, T, dt)
# get the number of points so that we can create a signal in the frequency domain
num_pts = len(timescale)
# convert to frequency domain
F = fft.fftfreq(num_pts, dt)
# create a frequenct signal of zeros
length_F = len(F)
# create zeros for the frequency domain
zeros = np.zeros(length_F)
Z = zeros.tolist()
for idx, f in enumerate(F):
if Z[idx] == 0:
magnitude_f = abs(f)
if magnitude_f <= limit:
im = im_normal_rand()
Z[idx] = im
# ensure that we account for the negative symmetric value
exists, index = symmetry_exists(f, F)
if exists:
location = locations(index)
# assig it to the complex conjugate
Z[location] = np.conj(im)
else:
continue
# perform inverse fft
z = fft.ifft(Z)
# select the real components
z = z.real
# rescale based on the current and ideal rmse
z = rescale(z, rms)
# convert back to frequency domain
Z = fft.fft(z)
# touple Z so that it aligns with our intial number of samples
F, Z = zippify(F, Z)
return z, Z
def plot_signal(
signal, domain="time", T=1, dt=1 / 1000, show_rmse=True, bandwidth=False
):
t = np.arange(0, T, dt)
if domain == "time":
plt.figure()
plt.plot(t, signal["x"])
if bandwidth:
plt.suptitle(
"$x(t)$ signal with " + str(signal["freq"]) + " Hz bandwidth",
)
else:
plt.suptitle(
"$x(t)$ signal with " + str(signal["freq"]) + " Hz limit",
)
plt.xlabel("$t$ sec.")
plt.ylabel("$x(t)$")
plt.xlim([0, T])
plt.show()
if show_rmse:
print("time-domain RMSE " + str(np.round(signal_rms(signal["x"]), 3)))
if domain == "frequency":
plt.figure()
plt.plot(t, signal["X"])
if bandwidth:
plt.suptitle(
"$x(\omega)$ signal with " + str(signal["freq"]) + " Hz bandwidth",
)
else:
plt.suptitle(
"$x(\omega)$ signal with " + str(signal["freq"]) + " Hz limit",
)
plt.xlabel("$w$ Hz.")
plt.ylabel("$x(\omega)$")
plt.show()
if show_rmse:
print("frequency-domain RMSE " + str(np.round(signal_rms(signal["X"]), 3)))
def synaptic_filter(tau=5 / 1000, dt=1 / 1000, ms=1000):
t_h = np.arange(ms) * dt - 0.5
h = (1 / tau) * np.exp(-t_h / tau)
h[np.where(t_h < 0)] = 0
h = h / np.linalg.norm(h, 1)
return h, t_h
def filter(x, h, t):
return np.convolve(x, h, mode="same")[: len(t)]
import copy
neuron_candidates = ensemble1.get_neurons_by_rate(rate_range=[20, 50], stim=0)
candidate = np.random.choice(neuron_candidates)
override = {
"x_int": candidate.get_intercept(),
"max_rate": candidate.get_max_rate(),
"encoder": candidate.get_encoder(),
"tau_ref": candidate.get_tau_ref(),
"tau_rc": candidate.get_tau_rc(),
"alpha": candidate.get_gain(),
"j_bias": candidate.get_bias(),
"spiketrend": candidate.get_spiketrend(),
"firing_rates": candidate.get_rates(),
}
neuron_1 = Neuron(state=None, override=override)
neuron_2 = Neuron(state=None, override=override)
neuron_1.set_encoder(1)
neuron_1.dangerously_set_id(uuid4())
neuron_2.set_encoder(-1)
neuron_1.set_encoder(1)
# create an ensemble wiht our neuron
ensemble2 = Population(state=None, neuron_overrides=[neuron_1, neuron_2])
# generate input signal
T = 1
dt = 1 / 1000
rms = 1
limit = 5
t = np.arange(0, T, dt)
x, X = generate_signal(T, dt, rms, limit, s)
# spike the neurons by giving them an input signal
ensemble2.spike(x, dt)
# generate synapic filter
h, t_h = synaptic_filter(tau=5 / 1000, dt=dt, ms=T * 1000)
plt.figure()
plt.suptitle("Post-Synaptic Current Filter $h(t)$ with $\\tau=0.005$")
plt.xlabel("$t$ (seconds)")
plt.ylabel("$h(t)$")
x1 = plt.plot(t_h, h, label="$h(t)$")
plt.legend(handles=[x1], labels=[])
plt.xlim([-0.4, 0.4])
plt.show()
Override Neuron ab990bf8-f759-458e-9ee7-1dafcc992cc5
Override Neuron 37165e31-1971-4bc5-96e9-69dd4117b7a8
b) Decoding using a synaptic filter. Plot the original signal $x(t)$, the spikes, and the decoded $\hat{x}(t)$ all on the same graph.
# from part a)
neuron_pos_encoder = ensemble2.get_neuron_by_index(0)
neuron_neg_encoder = ensemble2.get_neuron_by_index(1)
# make sure we have the correct signed encoder
assert neuron_pos_encoder.get_encoder() == 1
assert neuron_neg_encoder.get_encoder() == -1
# get the first colum of the outputs which is the voltages
v_out_pos = neuron_pos_encoder.get_spiketrend()[:, 0]
v_out_neg = neuron_neg_encoder.get_spiketrend()[:, 0]
# checking that they are reflections of themselves
assert np.array(v_out_neg).all() == -1 * np.array(v_out_pos).all()
# align spikes
spikes = np.array([v_out_pos, v_out_neg])
r = spikes[0] - spikes[1]
x_hat = filter(r, h, t)
plt.figure()
plt.suptitle(
"Spike train, input $x(t)$ and decoded output signal $\hat{x}(t)$ on 0 to 1 seconds"
)
a = plt.plot(t, r, label="Spikes $v(t)$", alpha=0.6)
b = plt.plot(t, x, label="input $x(t)$", alpha=0.8, color="#FF4D00")
c = plt.plot(t, x_hat, label="decoded output $\hat{x}(t)$", color="#003DDE")
plt.xlim([0, 1])
plt.legend(handles=[a, b, c], labels=[])
plt.xlabel("$t$")
plt.ylabel("Magnitude")
plt.show()
c) Error analysis. Compute the RMSE of the decoding.
print_block("RMSE",rmse(x,x_hat))
RMSE ----------
0.7451491125106602
-----------------
3. Decoding from many neurons
a) Exploring the error for an increasing neuron count. Plot the Root Mean-Squared Error as the number of neurons increases, on a log-log plot. Try $8$ neurons, $16$ neurons, $32$, $64$, $128$, up to $256$. For the RMSE for a particular number of neurons, average over at least $5$ randomly generated groups of neurons. For each group of neurons, randomly generate the signal $x(t)$. Use the same parameters as in question 2.
# create sets of neurons
neuron_sets = [8, 16, 32, 64, 128, 256]
num_sets = 5
# with this default state
state = {
"min_rate": 100,
"max_rate": 200,
"encoder": [-1, 1],
"tau_ref": 2 / 1000,
"tau_rc": 20 / 1000,
"min_x_int": -2,
"max_x_int": 2,
}
populations = []
# create populations
for amount in neuron_sets:
set = []
for count in range(num_sets):
ensemble = Population(num_neurons=amount, state=state)
set.append(ensemble)
populations.append(set)
T = 1
dt = 1 / 1000
rms = 1
limit = 5
t = np.arange(0, T, dt)
# generate synapic filter
h, t_h = synaptic_filter(tau=5 / 1000, dt=dt, ms=T * 1000)
# spike the neurons
population_errors_spike = []
population_errors_activities = []
for population_set in populations:
set_errors_spike = []
set_errors_activites = []
for idx, population in enumerate(population_set):
x, X = generate_signal(T, dt, rms, limit, s)
population.spike(x, dt)
curves = np.array(population.get_curves(x))
population_spikes = np.array(population.get_spikes())
ordered_encoders = np.array(population.get_ordered_encoders())
r = np.zeros(len(t))
for idx, population_spike in enumerate(population_spikes):
if ordered_encoders[idx] == 1:
r = r + population_spike
if ordered_encoders[idx] == -1:
r = r - population_spike
else:
r = r
x_hat = filter(r, h, t)
error_denom_spike = rmse(x, x_hat)
set_errors_spike.append(1 / error_denom_spike)
A = curves
noise_stdev = 0.1 * 200
w_noise = np.random.normal(scale=noise_stdev, size=np.shape(A))
A_NOISE = A + w_noise
N = len(x)
n = population.get_neuron_count()
x = np.array(x)
# find decoders via least squares solution
D = np.linalg.lstsq(
A @ A.T + 0.5 * N * np.square(noise_stdev) * np.eye(n), A @ x.T, rcond=None
)[0]
x_hat_activities = np.dot(D, A_NOISE)
x = np.array(x)
error_activities = rmse(x, x_hat_activities)
set_errors_activites.append(error_activities)
mean_error_spikes = np.mean(np.array(set_errors_spike))
mean_error_activites = np.mean(np.array(set_errors_activites))
population_errors_spike.append(mean_error_spikes)
population_errors_activities.append(mean_error_activites)
for idx, errors in enumerate(population_errors_spike):
print_block("RMSE for " + str(neuron_sets[idx]) + " neurons from spikes", errors)
print_block(
"RMSE for " + str(neuron_sets[idx]) + " neurons from activities",
population_errors_activities[idx],
)
n = [1 / N for N in neuron_sets]
fig = plt.figure()
ax = fig.add_subplot(1, 1, 1)
plt.title("Log Log trend of RMSE errors with respect to number of neurons")
x1 = ax.plot(neuron_sets, population_errors_spike, label="Spiking Errors")
x2 = ax.plot(neuron_sets, n, "--", label="$1/n$")
x3 = ax.plot(neuron_sets, population_errors_activities, "--", label="Activities Error")
ax.set_xscale("log")
ax.set_yscale("log")
ax.legend(handles=[x1, x2, x3], labels=[])
plt.xlabel("Neurons")
plt.ylabel("Squared Error")
plt.show()
fig = plt.figure()
ax = fig.add_subplot(1, 1, 1)
plt.title("No log trend of RMSE errors with respect to number of neurons")
x1 = ax.plot(neuron_sets, population_errors_spike, label="Spiking Errors")
x2 = ax.plot(neuron_sets, n, "--", label="$1/n$")
x3 = ax.plot(neuron_sets, population_errors_activities, "--", label="Activities Error")
ax.legend(handles=[x1, x2, x3], labels=[])
plt.xlabel("Neurons")
plt.ylabel("Squared Error")
plt.show()
RMSE for 8 neurons from spikes ----------
1.375865756122073
-----------------
RMSE for 8 neurons from activities ----------
0.2556622504392272
-----------------
RMSE for 16 neurons from spikes ----------
1.8411883465592354
-----------------
RMSE for 16 neurons from activities ----------
0.16655459817920598
-----------------
RMSE for 32 neurons from spikes ----------
1.8713342921109977
-----------------
RMSE for 32 neurons from activities ----------
0.10514449469879364
-----------------
RMSE for 64 neurons from spikes ----------
0.6888350588291419
-----------------
RMSE for 64 neurons from activities ----------
0.07087349168376288
-----------------
RMSE for 128 neurons from spikes ----------
0.24927442973092223
-----------------
RMSE for 128 neurons from activities ----------
0.04914791260802407
-----------------
RMSE for 256 neurons from spikes ----------
0.11778318037163224
-----------------
RMSE for 256 neurons from activities ----------
0.03514458778002315
-----------------
b) Discussion. Discuss your results. What is the systematic relationship between the neuron count and the error?
As we can see from the above graphs, there is a negative exponentialy decaying relationship beween the number of neurons and the error. As we increase the number of neurions we see a rapid decrease in the error until we reach about 200 neurons in the population at which point we begin to see a decay in the decrease in the error, relative to the increase in the number of neurons. What can be said is that generally speaking \(\begin{equation} Error \propto \frac{1}{N}, \Rightarrow \text{ as } N \to \infty, Error \to 0 \end{equation}\) where $N$ is the integer number of neurons encoding and decoding an input in a population
4. Connecting two groups of neurons
a) Computing a function. Show the behaviour of the system with an input of $x(t)=t-1$ for $1\,\mathrm{s}$ (a linear ramp from $-1$ to $0$). Plot the ideal $x(t)$ and $y(t)$ values, along with $\hat{y}(t)$.
def decoder(signal=[], A=[], num_neurons=200):
X = signal
noise_stdev = 0.1 * 200
w_noise = np.random.normal(scale=noise_stdev, size=np.shape(A))
A_NOISE = A + w_noise
N = len(X)
n = num_neurons
# find decoders via least squares solution
D = np.linalg.lstsq(
A @ A.T + N * np.square(noise_stdev) * np.eye(n),
A @ X.T,
rcond=None,
)[0].T
return D, A_NOISE
# we want 200 neurons
num_neurons = 200
# with this default state
state = {
"min_rate": 100,
"max_rate": 200,
"encoder": [-1, 1],
"tau_ref": 2 / 1000,
"tau_rc": 20 / 1000,
"min_x_int": -1,
"max_x_int": 1,
}
# create a population of 200 neurons with the default states
ensemble_x = Population(num_neurons=200, state=state)
ensemble_y = Population(num_neurons=200, state=state)
T = 1
dt = 1 / 1000
rms = 1
limit = 5
t = np.arange(-1, 1, dt)
h, t_h = synaptic_filter(tau=5 / 1000, dt=dt, ms=T * 1000)
# y(x) = x
x1 = np.linspace(-1, 0, len(t))
# y(x) = 2x+1
x2 = [2 * t + 1 for t in x1]
curves_x = np.array(ensemble_x.get_curves(x2))
curves_y = np.array(ensemble_y.get_curves(x1))
# create matrix of activities
A_X = curves_x
A_Y = curves_y
# generate decoders
D_X, A_X_NOISE = decoder(signal=np.array(x2), A=A_X, num_neurons=num_neurons)
D_Y, A_Y_NOISE = decoder(signal=np.array(x1), A=A_Y, num_neurons=num_neurons)
# make sure we have all our decoders
assert len(D_X) == num_neurons
assert len(D_Y) == num_neurons
# make test input
ls = np.linspace(-1, 0, len(t))
x = [t - 1 for t in ls]
ensemble_x.spike(x, dt)
spike_x = np.array(ensemble_x.get_spikes())
fspikes = []
for spike in spike_x:
fspike = np.convolve(spike, h, mode="same")
fspikes.append(fspike)
A = np.array(fspikes)
x_hat = np.dot(D_X, A / dt)
ensemble_y.spike(x_hat, dt)
spike_y = np.array(ensemble_y.get_spikes())
fspikes = []
for spike in spike_y:
fspike = np.convolve(spike, h, mode="same")
fspikes.append(fspike)
A = np.array(fspikes)
y_hat = np.dot(D_Y, A / dt)
plt.figure()
plt.suptitle("Decoded $\hat{y}$ and inputs")
a = plt.plot(ls, y_hat, label="$\hat{y}$")
plt.xlim([-1, 0])
b = plt.plot(ls, x, label="$x(t)=x-1$")
c = plt.plot(ls, x1, label="$f(y)=y$")
d = plt.plot(ls, x2, label="$y=2x+1$")
plt.legend(handles=[a, b, c], labels=[])
plt.show()
b) Step input. Repeat part (a) with an input that is ten randomly chosen values between -1 and 0, each one held for 0.1 seconds (a randomly varying step input)
values = []
num_values = 10
dt = 1 / 1000
for idx in range(num_values):
value = np.random.uniform(-1, 0)
values.append(value)
t = np.arange(-1, 0, dt)
x = []
for value in values:
x_n = np.ones(int(len(t) / len(values))) * value
x = x + x_n.tolist()
ensemble_x.spike(x, dt)
spike_x = np.array(ensemble_x.get_spikes())
fspikes = []
for spike in spike_x:
fspike = np.convolve(spike, h, mode="same")
fspikes.append(fspike)
A = np.array(fspikes)
x_hat = np.dot(D_X, A / dt)
ensemble_y.spike(x_hat, dt)
spike_y = np.array(ensemble_y.get_spikes())
fspikes = []
for spike in spike_y:
fspike = np.convolve(spike, h, mode="same")
fspikes.append(fspike)
A = np.array(fspikes)
y_hat = np.dot(D_Y, A / dt)
y = [2 * k + 1 for k in x]
plt.figure()
plt.suptitle("Decoded $\hat{y}$ and inputs")
a = plt.plot(t, y_hat, label="$\hat{y}$")
plt.xlim([-1, 0])
b = plt.plot(t, x, label="$x(t)=STEP$")
c = plt.plot(t, y, label="$y$")
plt.legend(
handles=[a, b, c],
labels=[],
)
plt.xlabel("$t$")
plt.show()
c) Sinusoidal input. Repeat part (a) with an input that is $x(t)=0.2\sin(6\pi t)$.
dt = 1 / 1000
t = np.arange(-1, 0, dt)
x = [0.2 * np.sin(6 * np.pi * pt) for pt in t]
ensemble_x.spike(x, dt)
spike_x = np.array(ensemble_x.get_spikes())
fspikes = []
for spike in spike_x:
fspike = np.convolve(spike, h, mode="same")
fspikes.append(fspike)
A = np.array(fspikes)
x_hat = np.dot(D_X, A / dt)
ensemble_y.spike(x_hat, dt)
spike_y = np.array(ensemble_y.get_spikes())
fspikes = []
for spike in spike_y:
fspike = np.convolve(spike, h, mode="same")
fspikes.append(fspike)
A = np.array(fspikes)
y_hat = np.dot(D_Y, A / dt)
y = [2 * k + 1 for k in x]
plt.figure()
plt.suptitle("Decoded $\hat{y}$ and inputs")
a = plt.plot(t, y_hat, label="$\hat{y}$")
plt.xlim([-1, 0])
plt.xlabel("$t$")
b = plt.plot(t, x, label="$x(t)=0.2\sin(6\pi t)$")
c = plt.plot(t, y, label="$y$")
plt.legend(
handles=[a, b, c],
labels=[],
)
plt.show()
d) Discussion. Briefly discuss the results for this question. Does the output match the ideal output? What kind of deviations do you see and why do those exist?
The output does not match the ideal input. while it seems to match the general “shape” of the ideal output, it is generally translated or scaled incorrectly, It does however, generally match the input $x(t)$ quite well. We was also observed is that feeding $\hat{x}$ into the second neuron population seemed to address some of the large amounts of noise in the signal, This makes sense because the signal would be undergoing a second filter with $h(t)$. Generally speaking, the neurons performed better than expected given that they were tuned on a completely different function than the one that was fed into the network.
5. Connecting three groups of neurons
a) Sinusoidal input. Plot $x(t)$, $y(t)$, the ideal $z(t)$, and the decoded $\hat{z}(t)$ for an input of $x(t)=\cos(3\pi t)$ and $y(t)=0.5 \sin (2 \pi t)$ (over $1\,\mathrm{s}$).
# we want 200 neurons
num_neurons = 200
# with this default state
state = {
"min_rate": 100,
"max_rate": 200,
"encoder": [-1, 1],
"tau_ref": 2 / 1000,
"tau_rc": 20 / 1000,
"min_x_int": -1,
"max_x_int": 1,
}
# create a population of 200 neurons with the default states
ensemble_x = Population(num_neurons=200, state=state)
ensemble_y = Population(num_neurons=200, state=state)
ensemble_z = Population(num_neurons=200, state=state)
T = 1
dt = 1 / 1000
rms = 1
limit = 5
t = np.arange(0, 1, dt)
h, t_h = synaptic_filter(tau=5 / 1000, dt=dt, ms=T * 1000)
# f(x)=0.5x
x = [0.5 * pt for pt in t]
# f(y)=2y
y = [2 * pt for pt in t]
# z(x,y) = 2y + 0.5x
tune_z = [pt[0] + pt[1] for pt in zip(x, y)]
curves_x = np.array(ensemble_x.get_curves(x))
curves_y = np.array(ensemble_y.get_curves(y))
curves_z = np.array(ensemble_z.get_curves(tune_z))
# create matrix of activities
A_X = curves_x
A_Y = curves_y
A_Z = curves_z
# generate decoders
D_X, A_X_NOISE = decoder(signal=np.array(x), A=A_X, num_neurons=num_neurons)
D_Y, A_Y_NOISE = decoder(signal=np.array(y), A=A_Y, num_neurons=num_neurons)
D_Z, A_Z_NOISE = decoder(signal=np.array(tune_z), A=A_Z, num_neurons=num_neurons)
# make sure we have all our decoders
assert len(D_X) == num_neurons
assert len(D_Y) == num_neurons
assert len(D_Z) == num_neurons
x_input = [np.cos(3 * np.pi * pt) for pt in t]
y_input = [0.5 * np.sin(2 * np.pi * pt) for pt in t]
z_ideal= [pt[0] + pt[1] for pt in zip(x_input, y_input)]
ensemble_x.spike(x_input, dt)
spike_x = np.array(ensemble_x.get_spikes())
fspikes = []
for spike in spike_x:
fspike = np.convolve(spike, h, mode="same")
fspikes.append(fspike)
Ax = np.array(fspikes)
x_hat = np.dot(D_X, Ax / dt)
ensemble_y.spike(y_input, dt)
spike_y = np.array(ensemble_y.get_spikes())
fspikes = []
for spike in spike_y:
fspike = np.convolve(spike, h, mode="same")
fspikes.append(fspike)
Ay = np.array(fspikes)
y_hat = np.dot(D_Y, Ay / dt)
z_input = [pt[0] + pt[1] for pt in zip(x_hat, y_hat)]
ensemble_z.spike(z_input, dt)
spike_z = np.array(ensemble_z.get_spikes())
fspikes = []
for spike in spike_z:
fspike = np.convolve(spike, h, mode="same")
fspikes.append(fspike)
Az = np.array(fspikes)
z_hat = np.dot(D_Z, Az / dt)
plt.figure()
plt.suptitle("Decoded $\hat{z}$ and inputs $x,y$ with ideal output $z$")
a = plt.plot(t, z_hat, label="$\hat{z}$")
plt.xlim([0, 1])
b = plt.plot(t, x_input, label="$x_{ideal}$")
c = plt.plot(t, y_input, label="$y_{ideal}$")
d = plt.plot(t, z_ideal, label="$z_{ideal}$")
plt.legend(handles=[a, b, c], labels=[])
plt.show()
b) Random input. Plot $x(t)$, $y(t)$, the ideal $z(t)$, and the decoded $\hat{z}(t)$ for a random input over $1\,\mathrm{s}$. For $x(t)$ use a random signal with a limit of $8\,\mathrm{Hz}$ and $\mathtt{rms}=1$. For $y(t)$ use a random signal with a limit of $5\,\mathrm{Hz}$ and $\mathtt{rms}=0.5$.
x_input, X_input = generate_signal(T=1, dt=1 / 1000, rms=1, limit=8, seed=S)
y_input, Y_input = generate_signal(T=1, dt=1 / 1000, rms=0.5, limit=5, seed=S)
z_ideal = [pt[0] + pt[1] for pt in zip(x_input, y_input)]
ensemble_x.spike(x_input, dt)
spike_x = np.array(ensemble_x.get_spikes())
fspikes = []
for spike in spike_x:
fspike = np.convolve(spike, h, mode="same")
fspikes.append(fspike)
Ax = np.array(fspikes)
x_hat = np.dot(D_X, Ax / dt)
ensemble_y.spike(y_input, dt)
spike_y = np.array(ensemble_y.get_spikes())
fspikes = []
for spike in spike_y:
fspike = np.convolve(spike, h, mode="same")
fspikes.append(fspike)
Ay = np.array(fspikes)
y_hat = np.dot(D_Y, Ay / dt)
z_input = [pt[0] + pt[1] for pt in zip(x_hat, y_hat)]
ensemble_z.spike(z_input, dt)
spike_z = np.array(ensemble_z.get_spikes())
fspikes = []
for spike in spike_z:
fspike = np.convolve(spike, h, mode="same")
fspikes.append(fspike)
Az = np.array(fspikes)
z_hat = np.dot(D_Z, Az / dt)
plt.figure()
plt.suptitle("Decoded $\hat{z}$ and inputs $x,y$ with ideal output $z$")
a = plt.plot(t, z_hat, label="$\hat{z}$")
plt.xlim([0, 1])
b = plt.plot(t, x_input, label="$x_{ideal}$")
c = plt.plot(t, y_input, label="$y_{ideal}$")
d = plt.plot(t, z_ideal, label="$z_{ideal}$")
plt.legend(handles=[a, b, c], labels=[])
plt.show()
6. Computing with vectors
a) Constant inputs. Plot the decoded output $\hat{w}(t)$ and the ideal $w$ for \(x =(0.5,1), \quad y = (0.1,0.3), \quad z =(0.2,0.1), \quad q = (0.4,-0.2) \,.\)
# Create our 2D LIF Neurons and 2D LIF Neuron Populations
tau_ref = 2 / 1000
tau_rc = 20 / 1000
def maxJ2D(tau_ref=tau_ref, tau_rc=tau_rc, max_rate=200):
return 1 / (1 - np.exp((tau_ref - 1 / max_rate) / tau_rc))
def gain2D(j_max=2, e=1, intercept=0):
return (j_max - 1) / (2 - np.vdot(e, intercept))
def bias2D(alpha=1, e=1, intercept=0):
return 1 - alpha * np.vdot(e, intercept)
class Population2D:
def __init__(self, num_neurons=1, state=None, neuron_overrides=[]):
self.num_neurons = num_neurons
self.rates = []
self.spikes = []
if state == None:
self.default_neuron_states = {
"min_rate": 100,
"max_rate": 200,
"encoder": [0, 2 * np.pi],
"tau_ref": 2 / 1000,
"tau_rc": 20 / 1000,
"min_angle": 0,
"max_angle": 2 * np.pi,
"max_radius": 2,
}
else:
self.default_neuron_states = state
self.neurons = []
if len(neuron_overrides) == 0:
for idx in range(self.num_neurons):
neuron = Neuron2D(self.default_neuron_states)
self.neurons.append(neuron)
else:
for neuron_override in neuron_overrides:
print("Override Neuron " + str(neuron_override.get_id()))
self.neurons.append(neuron_override)
""" Cleans out a population """
def nuke(self):
self.neurons = []
def clear_rates(self):
self.rates = []
def clear_spikes(self):
self.spikes = []
""" Applies a mutator to each neuron in the population """
def mutate(self, mutator):
if len(self.neurons) == 0:
return
else:
for neuron in self.neurons:
mutator(neuron)
def spike(self, X, dT, cap=None):
O = []
for neuron in self.neurons:
spikes = neuron.spikies(X, dT, cap)
O.append(spikes)
return O
def get_curves(self, input):
for neuron in self.neurons:
self.rates.append(neuron.rates(input))
return self.rates
def get_neurons(self):
return self.neurons
def get_neuron_by_index(self, idx):
return self.neurons[idx]
def get_neuron_by_id(self, target_id):
match = None
for neuron in self.neurons:
if target_id == neuron.get_id():
match = neuron
return match
def get_spikes(self):
spikes = []
for neuron in self.neurons:
spikes.append(neuron.get_spiketrend()[:, 1])
self.spikes = spikes
return spikes
def get_voltages(self):
voltages = []
for neuron in self.neurons:
voltages.append(neuron.get_spiketrend()[:, 0])
self.voltages = voltages
return voltages
def get_ordered_encoders(self):
encoders = []
for neuron in self.neurons:
encoders.append(neuron.get_encoder())
return encoders
def get_neuron_count(self):
return len(self.neurons)
def get_neurons_by_rate(self, rate_range=[20, 50], stim=0):
target_neurons = []
for neuron in self.neurons:
rate = neuron.encode(stim)
is_less_then_max = rate < rate_range[1]
is_more_then_min = rate > rate_range[0]
if is_less_then_max and is_more_then_min:
target_neurons.append(neuron)
return target_neurons
class Neuron2D(Population2D):
def __init__(self, state=None, override=None):
if override == None:
self.angle = np.random.uniform(state["min_angle"], state["max_angle"])
self.r = np.random.uniform(0, state["max_radius"])
self.x_int = [self.r * np.cos(self.angle), self.r * np.sin(self.angle)]
self.max_rate = np.random.uniform(state["min_rate"], state["max_rate"])
self.encoder_angle = np.random.uniform(
state["min_angle"], state["max_angle"]
)
self.e = [np.cos(self.encoder_angle), np.sin(self.encoder_angle)]
self.tau_ref = state["tau_ref"]
self.tau_rc = state["tau_rc"]
J_max = maxJ2D(
tau_ref=self.tau_ref, tau_rc=self.tau_rc, max_rate=self.max_rate
)
self.alpha = gain2D(J_max, self.e, self.x_int)
self.j_bias = bias2D(alpha=self.alpha, e=self.e, intercept=self.x_int)
self.id = uuid4()
self.spiketrend = []
self.firing_rates = []
else:
self.x_int = override["x_int"]
self.max_rate = override["max_rate"]
self.e = override["encoder"]
self.tau_ref = override["tau_ref"]
self.tau_rc = override["tau_rc"]
self.alpha = override["alpha"]
self.j_bias = override["j_bias"]
self.id = uuid4()
self.spiketrend = override["spiketrend"]
self.firing_rates = override["firing_rates"]
def whoisthis(self):
print(self.__dict__)
def encode(self, x):
J = self.alpha * np.vdot(x, self.e) + self.j_bias
if J > 1:
return 1 / (self.tau_ref - self.tau_rc * np.log(1 - 1 / J))
return 0
def encodeJ(self, x):
return self.alpha * np.vdot(x, self.e) + self.j_bias
def voltage(self, J, V, dT):
return V + (dT * (1 / self.tau_rc) * (J - V))
def howmanyspikes(self):
spike_points = self.spiketrend[:, 1]
num_spikes = int(spike_points.tolist().count(1))
return num_spikes
def get_intercept(self):
return self.x_int
def get_gain(self):
return self.alpha
def get_bias(self):
return self.j_bias
def get_max_rate(self):
return self.max_rate
def get_tau_rc(self):
return self.tau_rc
def get_tau_ref(self):
return self.tau_ref
def get_id(self):
return self.id
def set_encoder(self, encoder):
self.e = encoder
def get_encoder(self):
return self.e
def me(self):
return self
def dangerously_set_id(self, ied):
self.id = ied
def set_rate(self, rate):
self.max_rate = rate
def set_bias(self, bias):
self.j_bias = bias
def set_gain(self, gain):
self.alpha = gain
def get_spiketrend(self):
return self.spiketrend
def get_rates(self):
return self.firing_rates
def rates(self, x):
self.firing_rates = []
for point in x:
self.firing_rates.append(self.encode(point))
return self.firing_rates
def clear_rates(self):
self.firing_rates = []
def spikies(self, X, dT, cap=None):
N = np.floor(self.tau_ref / dT)
V_th = 1
V_rest = 0
spikes = np.array([np.zeros(len(X)), np.zeros(len(X))]).T
V = V_rest
V_prev = V_rest
ref_period = False
for idx, x in enumerate(X):
if ref_period == True:
V = V_rest
V_prev = V_rest
# voltage is 0
spikes[idx][0] = 0
# no spike so set it to 0
spikes[idx][1] = 0
# we have completed one ref cycle
ref_period = False
else:
J = self.encodeJ(x)
V = self.voltage(J, V_prev, dT)
if V >= V_th:
# we have a spike so assign second column to 1 to indicate a spike
spikes[idx][1] = int(1)
# start the ref period
ref_period = True
# assign the first collumn to the current voltage
# assign a constant spiking voltage to make identification easier
if cap != None:
spikes[idx][0] = cap
else:
spikes[idx][0] = V
# reset the voltage to 0
V = V_rest
V_prev = V_rest
else:
if V < V_rest:
V = V_rest
# no spikes to assign second column to 0
spikes[idx][1] = int(0)
# still capture the voltage
spikes[idx][0] = V
# assign the previous voltage to the current voltage for next iteration
V_prev = V
self.spiketrend = spikes
return spikes
# we want 200 neurons
num_neurons = 200
# with this default state
state = {
"min_rate": 100,
"max_rate": 200,
"encoder": [0, 2 * np.pi],
"tau_ref": 2 / 1000,
"tau_rc": 20 / 1000,
"min_angle": 0,
"max_angle": 2 * np.pi,
"max_radius": 2,
}
# create 5 populations of 200 neurons with the default states
ensemble_x = Population2D(num_neurons=200, state=state)
ensemble_y = Population2D(num_neurons=200, state=state)
ensemble_z = Population2D(num_neurons=200, state=state)
ensemble_q = Population2D(num_neurons=200, state=state)
ensemble_w = Population2D(num_neurons=200, state=state)
DIM = 2
T = 1
dt = 1 / 1000
samples = int(T * (1 / dt))
t = np.array([np.ones(samples), np.ones(samples)])
tune_x = t
tune_y = -3 * t
tune_z = 2 * t
tune_q = -2 * t
tune_w = np.array([np.zeros(samples), np.zeros(samples)])
tune_w[0] = [
pt[0] + pt[1] + pt[2] + pt[3]
for pt in zip(tune_x[0], tune_y[0], tune_z[0], tune_q[0])
]
tune_w[1] = [
pt[0] + pt[1] + pt[2] + pt[3]
for pt in zip(tune_x[1], tune_y[1], tune_z[1], tune_q[1])
]
curves_x = np.array(ensemble_x.get_curves(tune_x.T))
curves_y = np.array(ensemble_y.get_curves(tune_y.T))
curves_z = np.array(ensemble_z.get_curves(tune_z.T))
curves_q = np.array(ensemble_q.get_curves(tune_q.T))
curves_w = np.array(ensemble_w.get_curves(tune_w.T))
# create matrix of activities
A_X = curves_x
A_Y = curves_y
A_Z = curves_z
A_Q = curves_q
A_W = curves_w
# generate decoders
D_X, A_X_NOISE = decoder(signal=np.array(tune_x), A=A_X, num_neurons=num_neurons)
D_Y, A_Y_NOISE = decoder(signal=np.array(tune_y), A=A_Y, num_neurons=num_neurons)
D_Z, A_Z_NOISE = decoder(signal=np.array(tune_z), A=A_Z, num_neurons=num_neurons)
D_Q, A_Q_NOISE = decoder(signal=np.array(tune_q), A=A_Q, num_neurons=num_neurons)
D_W, A_W_NOISE = decoder(signal=np.array(tune_w), A=A_W, num_neurons=num_neurons)
# X
x_input = np.array([np.ones(samples) * 0.5, np.ones(samples)])
ensemble_x.spike(x_input.T, dt)
spike_x = np.array(ensemble_x.get_spikes())
fspikes = []
for spike in spike_x:
fspike = np.convolve(spike, h, mode="same")
fspikes.append(fspike)
Ax = np.array(fspikes)
x_hat = np.dot(D_X, Ax / dt).T
# Y
y_input = np.array([np.ones(samples) * 0.1, np.ones(samples) * 0.3])
ensemble_y.spike(y_input.T, dt)
spike_y = np.array(ensemble_y.get_spikes())
fspikes = []
for spike in spike_y:
fspike = np.convolve(spike, h, mode="same")
fspikes.append(fspike)
Ay = np.array(fspikes)
y_hat = np.dot(D_Y, Ay / dt).T
# Z
z_input = np.array([np.ones(samples) * 0.2, np.ones(samples) * 0.1])
ensemble_z.spike(z_input.T, dt)
spike_z = np.array(ensemble_z.get_spikes())
fspikes = []
for spike in spike_z:
fspike = np.convolve(spike, h, mode="same")
fspikes.append(fspike)
Az = np.array(fspikes)
z_hat = np.dot(D_Z, Az / dt).T
# Q
q_input = np.array([np.ones(samples) * 0.4, np.ones(samples) * -0.2])
ensemble_q.spike(q_input.T, dt)
spike_q = np.array(ensemble_q.get_spikes())
fspikes = []
for spike in spike_q:
fspike = np.convolve(spike, h, mode="same")
fspikes.append(fspike)
Aq = np.array(fspikes)
q_hat = np.dot(D_Q, Aq / dt).T
# FEED-FWD
w_input = np.array([np.zeros(samples), np.zeros(samples)]).T
for idx, point in enumerate(w_input):
point[0] = x_hat[idx][0] - 3 * y_hat[idx][0] + 2 * z_hat[idx][0] - 2 * q_hat[idx][0]
point[1] = x_hat[idx][1] - 3 * y_hat[idx][1] + 2 * z_hat[idx][1] - 2 * q_hat[idx][1]
ensemble_w.spike(w_input, dt)
spike_w = np.array(ensemble_w.get_spikes())
fspikes = []
for spike in spike_w:
fspike = np.convolve(spike, h, mode="same")
fspikes.append(fspike)
Aw = np.array(fspikes)
w_hat = np.dot(D_W, Aw / dt)
x_input = x_input.T
y_input = y_input.T
z_input = z_input.T
q_input = q_input.T
w_ideal = np.array([np.zeros(samples), np.zeros(samples)]).T
for idx, point in enumerate(w_ideal):
point[0] = (
x_input[idx][0]
- 3 * y_input[idx][0]
+ 2 * z_input[idx][0]
- 2 * q_input[idx][0]
)
point[1] = (
x_input[idx][1]
- 3 * y_input[idx][1]
+ 2 * z_input[idx][1]
- 2 * q_input[idx][1]
)
from mpl_toolkits import mplot3d
tt = np.arange(0, 1, dt)
w_ideal_transp = w_ideal.T
fig = plt.figure()
ax = plt.axes(projection="3d")
fig.suptitle("Decoded $\hat{w}$ and $w$")
a = ax.plot(tt, w_hat[0, :], zs=w_hat[1, :], label="$\hat{w}$")
b = ax.plot(tt, w_ideal_transp[0, :], zs=w_ideal_transp[1, :], label="$w$")
ax.legend(handles=[a, b], labels=[])
plt.xlabel("$t$")
plt.show()
b) Sinusoidal input. Produce the same plot for \(x =(0.5,1), \quad y = (\sin(4\pi t),0.3), \quad z =(0.2,0.1), \quad q = (\sin(4\pi t),-0.2) \,.\)
t = np.arange(0, T, dt)
assert len(t) == 1000
# X
x_input = np.array([np.ones(samples) * 0.5, np.ones(samples)])
ensemble_x.spike(x_input.T, dt)
spike_x = np.array(ensemble_x.get_spikes())
fspikes = []
for spike in spike_x:
fspike = np.convolve(spike, h, mode="same")
fspikes.append(fspike)
Ax = np.array(fspikes)
x_hat = np.dot(D_X, Ax / dt).T
# Y
y_input = np.array(
[np.array([np.sin(4 * np.pi * pt) for pt in t]), np.ones(samples) * 0.3]
)
ensemble_y.spike(y_input.T, dt)
spike_y = np.array(ensemble_y.get_spikes())
fspikes = []
for spike in spike_y:
fspike = np.convolve(spike, h, mode="same")
fspikes.append(fspike)
Ay = np.array(fspikes)
y_hat = np.dot(D_Y, Ay / dt).T
# Z
z_input = np.array([np.ones(samples) * 0.2, np.ones(samples) * 0.1])
ensemble_z.spike(z_input.T, dt)
spike_z = np.array(ensemble_z.get_spikes())
fspikes = []
for spike in spike_z:
fspike = np.convolve(spike, h, mode="same")
fspikes.append(fspike)
Az = np.array(fspikes)
z_hat = np.dot(D_Z, Az / dt).T
# Q
q_input = np.array(
[np.array([np.sin(4 * np.pi * pt) for pt in t]), np.ones(samples) * -0.2]
)
ensemble_q.spike(q_input.T, dt)
spike_q = np.array(ensemble_q.get_spikes())
fspikes = []
for spike in spike_q:
fspike = np.convolve(spike, h, mode="same")
fspikes.append(fspike)
Aq = np.array(fspikes)
q_hat = np.dot(D_Q, Aq / dt).T
# FEED-FWD
w_input = np.array([np.zeros(samples), np.zeros(samples)]).T
for idx, point in enumerate(w_input):
point[0] = x_hat[idx][0] - 3 * y_hat[idx][0] + 2 * z_hat[idx][0] - 2 * q_hat[idx][0]
point[1] = x_hat[idx][1] - 3 * y_hat[idx][1] + 2 * z_hat[idx][1] - 2 * q_hat[idx][1]
ensemble_w.spike(w_input, dt)
spike_w = np.array(ensemble_w.get_spikes())
fspikes = []
for spike in spike_w:
fspike = np.convolve(spike, h, mode="same")
fspikes.append(fspike)
Aw = np.array(fspikes)
w_hat = np.dot(D_W, Aw / dt)
x_input = x_input.T
y_input = y_input.T
z_input = z_input.T
q_input = q_input.T
w_ideal = np.array([np.zeros(samples), np.zeros(samples)]).T
for idx, point in enumerate(w_ideal):
point[0] = (
x_input[idx][0]
- 3 * y_input[idx][0]
+ 2 * z_input[idx][0]
- 2 * q_input[idx][0]
)
point[1] = (
x_input[idx][1]
- 3 * y_input[idx][1]
+ 2 * z_input[idx][1]
- 2 * q_input[idx][1]
)
w_ideal_transp = w_ideal.T
fig = plt.figure()
ax = plt.axes(projection="3d")
fig.suptitle("Decoded $\hat{w}$ and $w$")
a = ax.plot3D(t, w_hat[0, :],w_hat[1, :], label="$\hat{w}$")
b = ax.plot3D(t, w_ideal_transp[0, :], w_ideal_transp[1, :], label="$w$")
ax.legend(handles=[a, b], labels=[])
plt.xlabel("$t$")
plt.show()
c) Discussion. Describe your results and discuss why and how they stray from the expected answer.
The results show that the resulting vectors stray significantly from the expected result. We can see what appears to be a shift error over time of about $0.6$ in the $z$ direction. The results stray because we are adding significant noise to the system without changing our filtering technique or convolution strategy. One of the reasons why we may not be getting viable results has to do with the saturation level of the neurons which are capped at 1. This means that the populations will have trouble representing functions that extend beyond the saturation levels.