notebook.community

Edit and run 





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%matplotlib inline
%reload_ext autoreload
%autoreload 2



    











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from IPython.display import HTML

HTML('''<script>
code_show=true; 
function code_toggle() {
 if (code_show){
 $('div.input').hide();
 $('div.output_stderr').hide();
 } else {
 $('div.input').show();
 $('div.output_stderr').show();
 }
 code_show = !code_show
} 
$( document ).ready(code_toggle);
</script>
<form action='javascript:code_toggle()'><input STYLE='color: #4286f4' type='submit' value='Click here to toggle on/off the raw code.'></form>''')



    


















    
Out[2]:

























In [3]:



    


import numpy as np
import matplotlib.pyplot as plt



    











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from spectral_connectivity import Multitaper
from spectral_connectivity import Connectivity

Test Power Spectrum

200 Hz signal





In [5]:



    


frequency_of_interest = 200
sampling_frequency = 1500
time_extent = (0, 50)
n_time_samples = ((time_extent[1] - time_extent[0]) * sampling_frequency) + 1
time = np.linspace(time_extent[0], time_extent[1], num=n_time_samples, endpoint=True)
signal = np.sin(2 * np.pi * time * frequency_of_interest)
noise = np.random.normal(0, 4, len(signal))

plt.figure(figsize=(15, 6))
plt.subplot(2, 2, 1)
plt.plot(time, signal)
plt.xlabel('Time')
plt.ylabel('Amplitude')
plt.title('Signal', fontweight='bold')
plt.xlim((0.95, 1.05))
plt.ylim((-10, 10))

plt.subplot(2, 2, 2)
plt.plot(time, signal + noise)
plt.xlabel('Time')
plt.ylabel('Amplitude')
plt.title('Signal + Noise', fontweight='bold')
plt.xlim((0.95, 1.05))
plt.ylim((-10, 10))

plt.subplot(2, 2, 3)
m = Multitaper(signal,
               sampling_frequency=sampling_frequency,
               time_halfbandwidth_product=3,
               start_time=time[0])
c = Connectivity(fourier_coefficients=m.fft(),
                 frequencies=m.frequencies)
plt.plot(c.frequencies, c.power().squeeze());


plt.subplot(2, 2, 4)
m = Multitaper(signal + noise,
               sampling_frequency=sampling_frequency,
               time_halfbandwidth_product=3,
               start_time=time[0])
c = Connectivity(fourier_coefficients=m.fft(),
                 frequencies=m.frequencies)

plt.plot(c.frequencies, c.power().squeeze());

30 Hz signal





In [6]:



    


frequency_of_interest = 30
sampling_frequency = 1500
time_extent = (0, 50)
n_time_samples = ((time_extent[1] - time_extent[0]) * sampling_frequency) + 1
time = np.linspace(time_extent[0], time_extent[1], num=n_time_samples, endpoint=True)
signal = np.sin(2 * np.pi * time * frequency_of_interest)
noise = np.random.normal(0, 4, len(signal))

plt.figure(figsize=(15, 6))
plt.subplot(2, 2, 1)
plt.plot(time, signal)
plt.xlabel('Time')
plt.ylabel('Amplitude')
plt.title('Signal', fontweight='bold')
plt.xlim((0.95, 1.05))
plt.ylim((-10, 10))

plt.subplot(2, 2, 2)
plt.plot(time, signal + noise)
plt.xlabel('Time')
plt.ylabel('Amplitude')
plt.title('Signal + Noise', fontweight='bold')
plt.xlim((0.95, 1.05))
plt.ylim((-10, 10))

plt.subplot(2, 2, 3)
m = Multitaper(signal,
               sampling_frequency=sampling_frequency,
               time_halfbandwidth_product=3,
               start_time=time[0])
c = Connectivity(fourier_coefficients=m.fft(),
                 frequencies=m.frequencies)
plt.plot(c.frequencies, c.power().squeeze());


plt.subplot(2, 2, 4)
m = Multitaper(signal + noise,
               sampling_frequency=sampling_frequency,
               time_halfbandwidth_product=3,
               start_time=time[0])
c = Connectivity(fourier_coefficients=m.fft(),
                 frequencies=m.frequencies)

plt.plot(c.frequencies, c.power().squeeze());

Spectrogram

No trials, 200 Hz signal with 50 Hz signal starting at 25 seconds





In [7]:



    


frequency_of_interest = [200, 50]
sampling_frequency = 1500
time_extent = (0, 50)
n_time_samples = ((time_extent[1] - time_extent[0]) * sampling_frequency) + 1
time = np.linspace(time_extent[0], time_extent[1], num=n_time_samples, endpoint=True)
signal = np.sin(2 * np.pi * time[:, np.newaxis] * frequency_of_interest)
signal[:n_time_samples // 2, 1] = 0 
signal = signal.sum(axis=1)
noise = np.random.normal(0, 4, signal.shape)

fig, axes = plt.subplots(nrows=3, ncols=2, figsize=(15, 9))
axes[0, 0].plot(time, signal)
axes[0, 0].set_xlabel('Time')
axes[0, 0].set_ylabel('Amplitude')
axes[0, 0].set_title('Signal', fontweight='bold')
axes[0, 0].set_xlim((24.90, 25.10))
axes[0, 0].set_ylim((-10, 10))

axes[0, 1].plot(time, signal + noise)
axes[0, 1].set_xlabel('Time')
axes[0, 1].set_ylabel('Amplitude')
axes[0, 1].set_title('Signal + Noise', fontweight='bold')
axes[0, 1].set_xlim((24.90, 25.10))
axes[0, 1].set_ylim((-10, 10))

m = Multitaper(signal,
               sampling_frequency=sampling_frequency,
               time_halfbandwidth_product=3,
               start_time=time[0])
c = Connectivity(fourier_coefficients=m.fft(),
                 frequencies=m.frequencies)
axes[1, 0].plot(c.frequencies, c.power().squeeze())

m = Multitaper(signal + noise,
               sampling_frequency=sampling_frequency,
               time_halfbandwidth_product=3,
               start_time=time[0])
c = Connectivity(fourier_coefficients=m.fft(),
                 frequencies=m.frequencies)
axes[1, 1].plot(c.frequencies, c.power().squeeze())


m = Multitaper(signal,
               sampling_frequency=sampling_frequency,
               time_halfbandwidth_product=3,
               time_window_duration=0.600,
               time_window_step=0.300,
               start_time=time[0])
c = Connectivity(fourier_coefficients=m.fft(),
                 frequencies=m.frequencies,
                 time=m.time)
mesh = axes[2, 0].pcolormesh(c.time, c.frequencies, c.power().squeeze().T,
                             vmin=0.0, vmax=0.03, cmap='viridis')
axes[2, 0].set_ylim((0, 300))
axes[2, 0].axvline(time[int(np.fix(n_time_samples / 2))], color='black')

m = Multitaper(signal + noise,
               sampling_frequency=sampling_frequency,
               time_halfbandwidth_product=3,
               time_window_duration=0.600,
               time_window_step=0.300,
               start_time=time[0])
c = Connectivity(fourier_coefficients=m.fft(),
                 frequencies=m.frequencies,
                 time=m.time)
mesh = axes[2, 1].pcolormesh(c.time, c.frequencies, c.power().squeeze().T,
                             vmin=0.0, vmax=0.03, cmap='viridis')
axes[2, 1].set_ylim((0, 300))
axes[2, 1].axvline(time[int(np.fix(n_time_samples / 2))], color='black')

plt.tight_layout()
cb = fig.colorbar(mesh, ax=axes.ravel().tolist(), orientation='horizontal',
                  shrink=.5, aspect=15, pad=0.1, label='Power')
cb.outline.set_linewidth(0)

With trial structure (time x trials)





In [8]:



    


time_halfbandwidth_product = 1

frequency_of_interest = [200, 50]
time_extent = (0, 0.600)
n_trials = 100
sampling_frequency = 1500
n_time_samples = int(((time_extent[1] - time_extent[0]) * sampling_frequency) + 1)
time = np.linspace(time_extent[0], time_extent[1], num=n_time_samples, endpoint=True)
signal = np.sin(2 * np.pi * time[:, np.newaxis] * frequency_of_interest)
signal[:n_time_samples // 2, 1] = 0 
signal = signal.sum(axis=1)[:, np.newaxis, np.newaxis]
noise = np.random.normal(0, 2, size=(n_time_samples, n_trials, 1))

fig, axes = plt.subplots(nrows=3, ncols=2, figsize=(15, 9))
axes[0, 0].plot(time, signal.squeeze())
axes[0, 0].set_xlim(time_extent)
axes[0, 0].set_xlabel('Time')
axes[0, 0].set_ylabel('Amplitude')
axes[0, 0].set_title('Signal', fontweight='bold')
axes[0, 0].set_ylim((-10, 10))

axes[0, 1].plot(time, signal[:, 0, 0] + noise[:, 0, 0])
axes[0, 1].set_xlabel('Time')
axes[0, 1].set_ylabel('Amplitude')
axes[0, 1].set_title('Signal + Noise', fontweight='bold')
axes[0, 1].set_xlim(time_extent)
axes[0, 1].set_ylim((-10, 10))

m = Multitaper(signal,
               sampling_frequency=sampling_frequency,
               time_halfbandwidth_product=3,
               start_time=time[0])
c = Connectivity(fourier_coefficients=m.fft(),
                 frequencies=m.frequencies)
axes[1, 0].plot(c.frequencies, c.power().squeeze())

m = Multitaper(signal + noise,
               sampling_frequency=sampling_frequency,
               time_halfbandwidth_product=3,
               start_time=time[0])
c = Connectivity(fourier_coefficients=m.fft(),
                 frequencies=m.frequencies)
axes[1, 1].plot(c.frequencies, c.power().squeeze())


m = Multitaper(signal,
               sampling_frequency=sampling_frequency,
               time_halfbandwidth_product=time_halfbandwidth_product,
               time_window_duration=0.060,
               time_window_step=0.060,
               start_time=time[0])
c = Connectivity(fourier_coefficients=m.fft(),
                 frequencies=m.frequencies,
                 time=m.time)
mesh = axes[2, 0].pcolormesh(c.time, c.frequencies, c.power().squeeze().T,
                             vmin=0.0, vmax=0.03, cmap='viridis')
axes[2, 0].set_ylim((0, 300))
axes[2, 0].axvline(time[int(np.fix(n_time_samples / 2))], color='black')

m = Multitaper(signal + noise,
               sampling_frequency=sampling_frequency,
               time_halfbandwidth_product=time_halfbandwidth_product,
               time_window_duration=0.060,
               time_window_step=0.060,
               start_time=time[0])
c = Connectivity(fourier_coefficients=m.fft(),
                 frequencies=m.frequencies,
                 time=m.time)
mesh = axes[2, 1].pcolormesh(c.time, c.frequencies, c.power().squeeze().T,
                             vmin=0.0, vmax=0.03, cmap='viridis')
axes[2, 1].set_ylim((0, 300))
axes[2, 1].axvline(time[int(np.fix(n_time_samples / 2))], color='black')

plt.tight_layout()
cb = fig.colorbar(mesh, ax=axes.ravel().tolist(), orientation='horizontal',
                  shrink=.5, aspect=15, pad=0.1, label='Power')
cb.outline.set_linewidth(0)
print('frequency resolution: {}'.format(m.frequency_resolution))



    


















    






frequency resolution: 16.666666666666668

Decrease frequency resolution by decreasing time_halfbandwidth





In [9]:



    


time_halfbandwidth_product = 3

frequency_of_interest = [200, 50]
time_extent = (0, 0.600)
n_trials = 100
sampling_frequency = 1500
n_time_samples = int(((time_extent[1] - time_extent[0]) * sampling_frequency) + 1)
time = np.linspace(time_extent[0], time_extent[1], num=n_time_samples, endpoint=True)
signal = np.sin(2 * np.pi * time[:, np.newaxis] * frequency_of_interest)
signal[:n_time_samples // 2, 1] = 0 
signal = signal.sum(axis=1)[:, np.newaxis, np.newaxis]
noise = np.random.normal(0, 2, size=(n_time_samples, n_trials, 1))

fig, axes = plt.subplots(nrows=3, ncols=2, figsize=(15, 9))
axes[0, 0].plot(time, signal.squeeze())
axes[0, 0].set_xlim(time_extent)
axes[0, 0].set_xlabel('Time')
axes[0, 0].set_ylabel('Amplitude')
axes[0, 0].set_title('Signal', fontweight='bold')
axes[0, 0].set_ylim((-10, 10))

axes[0, 1].plot(time, signal[:, 0, 0] + noise[:, 0, 0])
axes[0, 1].set_xlabel('Time')
axes[0, 1].set_ylabel('Amplitude')
axes[0, 1].set_title('Signal + Noise', fontweight='bold')
axes[0, 1].set_xlim(time_extent)
axes[0, 1].set_ylim((-10, 10))

m = Multitaper(signal,
               sampling_frequency=sampling_frequency,
               time_halfbandwidth_product=time_halfbandwidth_product,
               start_time=time[0])
c = Connectivity(fourier_coefficients=m.fft(),
                 frequencies=m.frequencies)
axes[1, 0].plot(c.frequencies, c.power().squeeze())

m = Multitaper(signal + noise,
               sampling_frequency=sampling_frequency,
               time_halfbandwidth_product=time_halfbandwidth_product,
               start_time=time[0])
c = Connectivity(fourier_coefficients=m.fft(),
                 frequencies=m.frequencies)
axes[1, 1].plot(c.frequencies, c.power().squeeze())


m = Multitaper(signal,
               sampling_frequency=sampling_frequency,
               time_halfbandwidth_product=time_halfbandwidth_product,
               time_window_duration=0.060,
               time_window_step=0.060,
               start_time=time[0])
c = Connectivity(fourier_coefficients=m.fft(),
                 frequencies=m.frequencies,
                 time=m.time)
mesh = axes[2, 0].pcolormesh(c.time, c.frequencies, c.power().squeeze().T,
                             vmin=0.0, vmax=0.03, cmap='viridis')
axes[2, 0].set_ylim((0, 300))
axes[2, 0].axvline(time[int(np.fix(n_time_samples / 2))], color='black')

m = Multitaper(signal + noise,
               sampling_frequency=sampling_frequency,
               time_halfbandwidth_product=time_halfbandwidth_product,
               time_window_duration=0.060,
               time_window_step=0.060,
               start_time=time[0])
c = Connectivity(fourier_coefficients=m.fft(),
                 frequencies=m.frequencies,
                 time=m.time)
mesh = axes[2, 1].pcolormesh(c.time, c.frequencies, c.power().squeeze().T,
                             vmin=0.0, vmax=0.03, cmap='viridis')
axes[2, 1].set_ylim((0, 300))
axes[2, 1].axvline(time[int(np.fix(n_time_samples / 2))], color='black')

plt.tight_layout()
cb = fig.colorbar(mesh, ax=axes.ravel().tolist(), orientation='horizontal',
                  shrink=.5, aspect=15, pad=0.1, label='Power')
cb.outline.set_linewidth(0)
print('frequency resolution: {}'.format(m.frequency_resolution))



    


















    






frequency resolution: 50.0

Coherence

No trials, 200 Hz, $\pi / 2$ phase offset





In [10]:



    


time_halfbandwidth_product = 5
frequency_of_interest = 200
sampling_frequency = 1500
time_extent = (0, 50)
n_signals = 2
n_time_samples = ((time_extent[1] - time_extent[0]) * sampling_frequency) + 1
time = np.linspace(time_extent[0], time_extent[1], num=n_time_samples, endpoint=True)
signal = np.zeros((n_time_samples, n_signals))
signal[:, 0] = np.sin(2 * np.pi * time * frequency_of_interest)
phase_offset = np.pi / 2
signal[:, 1] = np.sin((2 * np.pi * time * frequency_of_interest) + phase_offset)
noise = np.random.normal(0, 4, signal.shape)

plt.figure(figsize=(15, 6))
plt.subplot(2, 2, 1)
plt.title('Signal', fontweight='bold')
plt.plot(time, signal[:, 0], label='Signal1')
plt.plot(time, signal[:, 1], label='Signal2')
plt.xlabel('Time')
plt.ylabel('Amplitude')
plt.xlim((0.95, 1.05))
plt.ylim((-10, 10))
plt.legend()

plt.subplot(2, 2, 2)
plt.title('Signal + Noise', fontweight='bold')
plt.plot(time, signal + noise)
plt.xlabel('Time')
plt.ylabel('Amplitude')
plt.xlim((0.95, 1.05))
plt.ylim((-10, 10))
plt.legend()

m = Multitaper(signal,
               sampling_frequency=sampling_frequency,
               time_halfbandwidth_product=time_halfbandwidth_product,
               start_time=time[0])
c = Connectivity(fourier_coefficients=m.fft(),
                 frequencies=m.frequencies)
plt.subplot(2, 2, 3)
plt.plot(c.frequencies, c.coherence_magnitude()[0, :, 0, 1])


m = Multitaper(signal + noise,
               sampling_frequency=sampling_frequency,
               time_halfbandwidth_product=time_halfbandwidth_product,
               start_time=time[0])
c = Connectivity(fourier_coefficients=m.fft(),
                 frequencies=m.frequencies)
plt.subplot(2, 2, 4)
plt.plot(c.frequencies, c.coherence_magnitude()[0, :, 0, 1]);



    


















    






No handles with labels found to put in legend.

With trial structure (time x trials), 200 Hz, $\pi / 2$ phase offset





In [11]:



    


time_halfbandwidth_product = 5
frequency_of_interest = 200
sampling_frequency = 1500
time_extent = (0, 0.600)
n_trials = 100
n_signals = 2
n_time_samples = int(((time_extent[1] - time_extent[0]) * sampling_frequency) + 1)
time = np.linspace(time_extent[0], time_extent[1], num=n_time_samples, endpoint=True)[:, np.newaxis]
signal = np.zeros((n_time_samples, n_trials, n_signals))
signal[:, :, 0] = np.sin(2 * np.pi * time * frequency_of_interest)
phase_offset = np.pi / 2
signal[:, :, 1] = np.sin((2 * np.pi * time * frequency_of_interest) + phase_offset)
noise = np.random.normal(0, 4, signal.shape)

plt.figure(figsize=(15, 6))
plt.subplot(2, 2, 1)
plt.title('Signal', fontweight='bold')
plt.plot(time, signal[:, 0, 0], label='Signal1')
plt.plot(time, signal[:, 0, 1], label='Signal2')
plt.xlabel('Time')
plt.ylabel('Amplitude')
plt.xlim(time_extent)
plt.ylim((-2, 2))
plt.legend()

plt.subplot(2, 2, 2)
plt.title('Signal + Noise', fontweight='bold')
plt.plot(time, signal[:, 0, :] + noise[:, 0, :])
plt.xlabel('Time')
plt.ylabel('Amplitude')
plt.xlim(time_extent)
plt.ylim((-10, 10))
plt.legend()

m = Multitaper(signal,
               sampling_frequency=sampling_frequency,
               time_halfbandwidth_product=time_halfbandwidth_product,
               start_time=time[0])
c = Connectivity(fourier_coefficients=m.fft(),
                 frequencies=m.frequencies)
plt.subplot(2, 2, 3)
plt.plot(c.frequencies, c.coherence_magnitude()[0, :, 0, 1])


m = Multitaper(signal + noise,
               sampling_frequency=sampling_frequency,
               time_halfbandwidth_product=time_halfbandwidth_product,
               start_time=time[0])
c = Connectivity(fourier_coefficients=m.fft(),
                 frequencies=m.frequencies)
plt.subplot(2, 2, 4)
plt.plot(c.frequencies, c.coherence_magnitude()[0, :, 0, 1]);



    


















    






No handles with labels found to put in legend.

Cohereograms





In [12]:



    


time_halfbandwidth_product = 2
frequency_of_interest = 200
sampling_frequency = 1500
time_extent = (0, 2.400)
n_trials = 100
n_signals = 2
n_time_samples = int(((time_extent[1] - time_extent[0]) * sampling_frequency) + 1)
time = np.linspace(time_extent[0], time_extent[1], num=n_time_samples, endpoint=True)[:, np.newaxis]
signal = np.zeros((n_time_samples, n_trials, n_signals))
signal[:, :, 0] = np.sin(2 * np.pi * time * frequency_of_interest)
phase_offset = np.random.uniform(-np.pi, np.pi, size=(n_time_samples, 1))
phase_offset[np.where(time > 1.5), :] = np.pi / 2
signal[:, :, 1] = np.sin((2 * np.pi * time * frequency_of_interest) + phase_offset)
noise = np.random.normal(0, 1, signal.shape)

fig, axes = plt.subplots(nrows=3, ncols=2, figsize=(15, 9))
axes[0, 0].set_title('Signal', fontweight='bold')
axes[0, 0].plot(time, signal[:, 0, 0], label='Signal1')
axes[0, 0].plot(time, signal[:, 0, 1], label='Signal2')
axes[0, 0].set_xlabel('Time')
axes[0, 0].set_ylabel('Amplitude')
axes[0, 0].set_xlim(time_extent)
axes[0, 0].set_ylim((-2, 2))
plt.legend()

axes[0, 1].set_title('Signal + Noise', fontweight='bold')
axes[0, 1].plot(time, signal[:, 0, :] + noise[:, 0, :])
axes[0, 1].set_xlabel('Time')
axes[0, 1].set_ylabel('Amplitude')
axes[0, 1].set_xlim(time_extent)
axes[0, 1].set_ylim((-10, 10))
plt.legend()

m = Multitaper(signal,
               sampling_frequency=sampling_frequency,
               time_halfbandwidth_product=time_halfbandwidth_product,
               start_time=time[0])
c = Connectivity(fourier_coefficients=m.fft(),
                 frequencies=m.frequencies)
axes[1, 0].plot(c.frequencies, c.coherence_magnitude()[..., 0, 1].squeeze())
axes[1, 0].set_xlim((0, m.nyquist_frequency))

m = Multitaper(signal + noise,
               sampling_frequency=sampling_frequency,
               time_halfbandwidth_product=time_halfbandwidth_product,
               start_time=time[0])
c = Connectivity(fourier_coefficients=m.fft(),
                 frequencies=m.frequencies)
axes[1, 1].plot(c.frequencies,  c.coherence_magnitude()[..., 0, 1].squeeze())
axes[1, 1].set_xlim((0, m.nyquist_frequency))


m = Multitaper(signal,
               sampling_frequency=sampling_frequency,
               time_halfbandwidth_product=time_halfbandwidth_product,
               time_window_duration=0.080,
               time_window_step=0.080,
               start_time=time[0])
c = Connectivity(fourier_coefficients=m.fft(),
                 frequencies=m.frequencies,
                 time=m.time)
time_grid, freq_grid = np.meshgrid(
    np.append(c.time, time_extent[-1]),
    np.append(c.frequencies, m.nyquist_frequency))

mesh = axes[2, 0].pcolormesh(time_grid, freq_grid,  c.coherence_magnitude()[..., 0, 1].squeeze().T,
                             vmin=0.0, vmax=1.0, cmap='viridis')
axes[2, 0].set_ylim((0, 300))
axes[2, 0].set_xlim(time_extent)
axes[2, 0].axvline(1.5, color='black')

m = Multitaper(signal + noise,
               sampling_frequency=sampling_frequency,
               time_halfbandwidth_product=time_halfbandwidth_product,
               time_window_duration=0.080,
               time_window_step=0.080,
               start_time=time[0])
c = Connectivity(fourier_coefficients=m.fft(),
                 frequencies=m.frequencies,
                 time=m.time)

time_grid, freq_grid = np.meshgrid(
    np.append(c.time, time_extent[-1]),
    np.append(c.frequencies, m.nyquist_frequency))

mesh = axes[2, 1].pcolormesh(time_grid, freq_grid,  c.coherence_magnitude()[..., 0, 1].squeeze().T,
                             vmin=0.0, vmax=1.0, cmap='viridis')
axes[2, 1].set_ylim((0, 300))
axes[2, 1].set_xlim(time_extent)
axes[2, 1].axvline(1.5, color='black')

plt.tight_layout()
cb = fig.colorbar(mesh, ax=axes.ravel().tolist(), orientation='horizontal',
                  shrink=.5, aspect=15, pad=0.1, label='Coherence')
cb.outline.set_linewidth(0)
print('frequency resolution: {}'.format(m.frequency_resolution))



    


















    






No handles with labels found to put in legend.
No handles with labels found to put in legend.










    






frequency resolution: 25.0

Imaginary Coherence





In [13]:



    


time_halfbandwidth_product = 2
frequency_of_interest = 200
sampling_frequency = 1500
time_extent = (0, 2.400)
n_trials = 100
n_signals = 2
n_time_samples = int(((time_extent[1] - time_extent[0]) * sampling_frequency) + 1)
time = np.linspace(time_extent[0], time_extent[1], num=n_time_samples, endpoint=True)[:, np.newaxis]
signal = np.zeros((n_time_samples, n_trials, n_signals))
signal[:, :, 0] = np.sin(2 * np.pi * time * frequency_of_interest)
phase_offset = np.random.uniform(-np.pi, np.pi, size=(n_time_samples, 1))
phase_offset[np.where(time > 1.5), :] = np.pi / 2
signal[:, :, 1] = np.sin((2 * np.pi * time * frequency_of_interest) + phase_offset)
noise = np.random.normal(0, 1, signal.shape)

fig, axes = plt.subplots(nrows=3, ncols=2, figsize=(15, 9))
axes[0, 0].set_title('Signal', fontweight='bold')
axes[0, 0].plot(time, signal[:, 0, 0], label='Signal1')
axes[0, 0].plot(time, signal[:, 0, 1], label='Signal2')
axes[0, 0].set_xlabel('Time')
axes[0, 0].set_ylabel('Amplitude')
axes[0, 0].set_xlim(time_extent)
axes[0, 0].set_ylim((-2, 2))
plt.legend()

axes[0, 1].set_title('Signal + Noise', fontweight='bold')
axes[0, 1].plot(time, signal[:, 0, :] + noise[:, 0, :])
axes[0, 1].set_xlabel('Time')
axes[0, 1].set_ylabel('Amplitude')
axes[0, 1].set_xlim(time_extent)
axes[0, 1].set_ylim((-10, 10))
plt.legend()

m = Multitaper(signal,
               sampling_frequency=sampling_frequency,
               time_halfbandwidth_product=time_halfbandwidth_product,
               start_time=time[0])
c = Connectivity(fourier_coefficients=m.fft(),
                 frequencies=m.frequencies)
axes[1, 0].plot(c.frequencies, c.imaginary_coherence()[..., 0, 1].squeeze())
axes[1, 0].set_xlim((0, m.nyquist_frequency))

m = Multitaper(signal + noise,
               sampling_frequency=sampling_frequency,
               time_halfbandwidth_product=time_halfbandwidth_product,
               start_time=time[0])
c = Connectivity(fourier_coefficients=m.fft(),
                 frequencies=m.frequencies)
axes[1, 1].plot(c.frequencies,  c.imaginary_coherence()[..., 0, 1].squeeze())
axes[1, 1].set_xlim((0, m.nyquist_frequency))


m = Multitaper(signal,
               sampling_frequency=sampling_frequency,
               time_halfbandwidth_product=time_halfbandwidth_product,
               time_window_duration=0.080,
               time_window_step=0.080,
               start_time=time[0])
c = Connectivity(fourier_coefficients=m.fft(),
                 frequencies=m.frequencies,
                 time=m.time)
time_grid, freq_grid = np.meshgrid(
    np.append(c.time, time_extent[-1]),
    np.append(c.frequencies, m.nyquist_frequency))

mesh = axes[2, 0].pcolormesh(time_grid, freq_grid,  c.imaginary_coherence()[..., 0, 1].squeeze().T,
                             vmin=0.0, vmax=1.0, cmap='viridis')
axes[2, 0].set_ylim((0, 300))
axes[2, 0].set_xlim(time_extent)
axes[2, 0].axvline(1.5, color='black')

m = Multitaper(signal + noise,
               sampling_frequency=sampling_frequency,
               time_halfbandwidth_product=time_halfbandwidth_product,
               time_window_duration=0.080,
               time_window_step=0.080,
               start_time=time[0])
c = Connectivity(fourier_coefficients=m.fft(),
                 frequencies=m.frequencies,
                 time=m.time)

time_grid, freq_grid = np.meshgrid(
    np.append(c.time, time_extent[-1]),
    np.append(c.frequencies, m.nyquist_frequency))

mesh = axes[2, 1].pcolormesh(time_grid, freq_grid,  c.imaginary_coherence()[..., 0, 1].squeeze().T,
                             vmin=0.0, vmax=1.0, cmap='viridis')
axes[2, 1].set_ylim((0, 300))
axes[2, 1].set_xlim(time_extent)
axes[2, 1].axvline(1.5, color='black')

plt.tight_layout()
cb = fig.colorbar(mesh, ax=axes.ravel().tolist(), orientation='horizontal',
                  shrink=.5, aspect=15, pad=0.1, label='Imaginary Coherence')
cb.outline.set_linewidth(0)
print('frequency resolution: {}'.format(m.frequency_resolution))



    


















    






No handles with labels found to put in legend.
No handles with labels found to put in legend.










    






frequency resolution: 25.0

Phase Locking Value





In [14]:



    


time_halfbandwidth_product = 2
frequency_of_interest = 200
sampling_frequency = 1500
time_extent = (0, 2.400)
n_trials = 100
n_signals = 2
n_time_samples = int(((time_extent[1] - time_extent[0]) * sampling_frequency) + 1)
time = np.linspace(time_extent[0], time_extent[1], num=n_time_samples, endpoint=True)[:, np.newaxis]
signal = np.zeros((n_time_samples, n_trials, n_signals))
signal[:, :, 0] = np.sin(2 * np.pi * time * frequency_of_interest)
phase_offset = np.random.uniform(-np.pi, np.pi, size=(n_time_samples, 1))
phase_offset[np.where(time > 1.5), :] = np.pi / 2
signal[:, :, 1] = np.sin((2 * np.pi * time * frequency_of_interest) + phase_offset)
noise = np.random.normal(0, 1, signal.shape)

fig, axes = plt.subplots(nrows=3, ncols=2, figsize=(15, 9))
axes[0, 0].set_title('Signal', fontweight='bold')
axes[0, 0].plot(time, signal[:, 0, 0], label='Signal1')
axes[0, 0].plot(time, signal[:, 0, 1], label='Signal2')
axes[0, 0].set_xlabel('Time')
axes[0, 0].set_ylabel('Amplitude')
axes[0, 0].set_xlim(time_extent)
axes[0, 0].set_ylim((-2, 2))
plt.legend()

axes[0, 1].set_title('Signal + Noise', fontweight='bold')
axes[0, 1].plot(time, signal[:, 0, :] + noise[:, 0, :])
axes[0, 1].set_xlabel('Time')
axes[0, 1].set_ylabel('Amplitude')
axes[0, 1].set_xlim(time_extent)
axes[0, 1].set_ylim((-10, 10))
plt.legend()

m = Multitaper(signal,
               sampling_frequency=sampling_frequency,
               time_halfbandwidth_product=time_halfbandwidth_product,
               start_time=time[0])
c = Connectivity(fourier_coefficients=m.fft(),
                 frequencies=m.frequencies)
axes[1, 0].plot(c.frequencies, abs(c.phase_locking_value())[..., 0, 1].squeeze())
axes[1, 0].set_xlim((0, m.nyquist_frequency))

m = Multitaper(signal + noise,
               sampling_frequency=sampling_frequency,
               time_halfbandwidth_product=time_halfbandwidth_product,
               start_time=time[0])
c = Connectivity(fourier_coefficients=m.fft(),
                 frequencies=m.frequencies)
axes[1, 1].plot(c.frequencies,  abs(c.phase_locking_value())[..., 0, 1].squeeze())
axes[1, 1].set_xlim((0, m.nyquist_frequency))


m = Multitaper(signal,
               sampling_frequency=sampling_frequency,
               time_halfbandwidth_product=time_halfbandwidth_product,
               time_window_duration=0.080,
               time_window_step=0.080,
               start_time=time[0])
c = Connectivity(fourier_coefficients=m.fft(),
                 frequencies=m.frequencies,
                 time=m.time)
time_grid, freq_grid = np.meshgrid(
    np.append(c.time, time_extent[-1]),
    np.append(c.frequencies, m.nyquist_frequency))

mesh = axes[2, 0].pcolormesh(time_grid, freq_grid,  abs(c.phase_locking_value())[..., 0, 1].squeeze().T,
                             vmin=0.0, vmax=1.0, cmap='viridis')
axes[2, 0].set_ylim((0, 300))
axes[2, 0].set_xlim(time_extent)
axes[2, 0].axvline(1.5, color='black')

m = Multitaper(signal + noise,
               sampling_frequency=sampling_frequency,
               time_halfbandwidth_product=time_halfbandwidth_product,
               time_window_duration=0.080,
               time_window_step=0.080,
               start_time=time[0])
c = Connectivity(fourier_coefficients=m.fft(),
                 frequencies=m.frequencies,
                 time=m.time)

time_grid, freq_grid = np.meshgrid(
    np.append(c.time, time_extent[-1]),
    np.append(c.frequencies, m.nyquist_frequency))

mesh = axes[2, 1].pcolormesh(time_grid, freq_grid, abs(c.phase_locking_value())[..., 0, 1].squeeze().T,
                             vmin=0.0, vmax=1.0, cmap='viridis')
axes[2, 1].set_ylim((0, 300))
axes[2, 1].set_xlim(time_extent)
axes[2, 1].axvline(1.5, color='black')

plt.tight_layout()
cb = fig.colorbar(mesh, ax=axes.ravel().tolist(), orientation='horizontal',
                  shrink=.5, aspect=15, pad=0.1, label='Phase Locking Value')
cb.outline.set_linewidth(0)
print('frequency resolution: {}'.format(m.frequency_resolution))



    


















    






No handles with labels found to put in legend.
No handles with labels found to put in legend.










    






frequency resolution: 25.0

Phase Lag Index





In [15]:



    


time_halfbandwidth_product = 2
frequency_of_interest = 200
sampling_frequency = 1500
time_extent = (0, 2.400)
n_trials = 100
n_signals = 2
n_time_samples = int(((time_extent[1] - time_extent[0]) * sampling_frequency) + 1)
time = np.linspace(time_extent[0], time_extent[1], num=n_time_samples, endpoint=True)[:, np.newaxis]
signal = np.zeros((n_time_samples, n_trials, n_signals))
signal[:, :, 0] = np.sin(2 * np.pi * time * frequency_of_interest)
phase_offset = np.random.uniform(-np.pi, np.pi, size=(n_time_samples, 1))
phase_offset[np.where(time > 1.5), :] = np.pi / 2
signal[:, :, 1] = np.sin((2 * np.pi * time * frequency_of_interest) + phase_offset)
noise = np.random.normal(0, 1, signal.shape)

fig, axes = plt.subplots(nrows=3, ncols=2, figsize=(15, 9))
axes[0, 0].set_title('Signal', fontweight='bold')
axes[0, 0].plot(time, signal[:, 0, 0], label='Signal1')
axes[0, 0].plot(time, signal[:, 0, 1], label='Signal2')
axes[0, 0].set_xlabel('Time')
axes[0, 0].set_ylabel('Amplitude')
axes[0, 0].set_xlim(time_extent)
axes[0, 0].set_ylim((-2, 2))
plt.legend()

axes[0, 1].set_title('Signal + Noise', fontweight='bold')
axes[0, 1].plot(time, signal[:, 0, :] + noise[:, 0, :])
axes[0, 1].set_xlabel('Time')
axes[0, 1].set_ylabel('Amplitude')
axes[0, 1].set_xlim(time_extent)
axes[0, 1].set_ylim((-10, 10))
plt.legend()

m = Multitaper(signal,
               sampling_frequency=sampling_frequency,
               time_halfbandwidth_product=time_halfbandwidth_product,
               start_time=time[0])
c = Connectivity(fourier_coefficients=m.fft(),
                 frequencies=m.frequencies)
axes[1, 0].plot(c.frequencies, c.phase_lag_index()[..., 0, 1].squeeze())
axes[1, 0].set_xlim((0, m.nyquist_frequency))

m = Multitaper(signal + noise,
               sampling_frequency=sampling_frequency,
               time_halfbandwidth_product=time_halfbandwidth_product,
               start_time=time[0])
c = Connectivity(fourier_coefficients=m.fft(),
                 frequencies=m.frequencies)
axes[1, 1].plot(c.frequencies,  c.phase_lag_index()[..., 0, 1].squeeze())
axes[1, 1].set_xlim((0, m.nyquist_frequency))


m = Multitaper(signal,
               sampling_frequency=sampling_frequency,
               time_halfbandwidth_product=time_halfbandwidth_product,
               time_window_duration=0.080,
               time_window_step=0.080,
               start_time=time[0])
c = Connectivity(fourier_coefficients=m.fft(),
                 frequencies=m.frequencies,
                 time=m.time)
time_grid, freq_grid = np.meshgrid(
    np.append(c.time, time_extent[-1]),
    np.append(c.frequencies, m.nyquist_frequency))

mesh = axes[2, 0].pcolormesh(time_grid, freq_grid,  c.phase_lag_index()[..., 0, 1].squeeze().T,
                             vmin=-1.0, vmax=1.0, cmap='RdBu_r')
axes[2, 0].set_ylim((0, 300))
axes[2, 0].set_xlim(time_extent)
axes[2, 0].axvline(1.5, color='black')

m = Multitaper(signal + noise,
               sampling_frequency=sampling_frequency,
               time_halfbandwidth_product=time_halfbandwidth_product,
               time_window_duration=0.080,
               time_window_step=0.080,
               start_time=time[0])
c = Connectivity(fourier_coefficients=m.fft(),
                 frequencies=m.frequencies,
                 time=m.time)

time_grid, freq_grid = np.meshgrid(
    np.append(c.time, time_extent[-1]),
    np.append(c.frequencies, m.nyquist_frequency))

mesh = axes[2, 1].pcolormesh(time_grid, freq_grid,  c.phase_lag_index()[..., 0, 1].squeeze().T,
                             vmin=-1.0, vmax=1.0, cmap='RdBu_r')
axes[2, 1].set_ylim((0, 300))
axes[2, 1].set_xlim(time_extent)
axes[2, 1].axvline(1.5, color='black')

plt.tight_layout()
cb = fig.colorbar(mesh, ax=axes.ravel().tolist(), orientation='horizontal',
                  shrink=.5, aspect=15, pad=0.1, label='Phase Lag Index')
cb.outline.set_linewidth(0)
print('frequency resolution: {}'.format(m.frequency_resolution))



    


















    






No handles with labels found to put in legend.
No handles with labels found to put in legend.










    






frequency resolution: 25.0

Weighted Phase Lag Index





In [16]:



    


time_halfbandwidth_product = 2
frequency_of_interest = 200
sampling_frequency = 1500
time_extent = (0, 2.400)
n_trials = 100
n_signals = 2
n_time_samples = int(((time_extent[1] - time_extent[0]) * sampling_frequency) + 1)
time = np.linspace(time_extent[0], time_extent[1], num=n_time_samples, endpoint=True)[:, np.newaxis]
signal = np.zeros((n_time_samples, n_trials, n_signals))
signal[:, :, 0] = np.sin(2 * np.pi * time * frequency_of_interest)
phase_offset = np.random.uniform(-np.pi, np.pi, size=(n_time_samples, 1))
phase_offset[np.where(time > 1.5), :] = np.pi / 2
signal[:, :, 1] = np.sin((2 * np.pi * time * frequency_of_interest) + phase_offset)
noise = np.random.normal(0, 1, signal.shape)

fig, axes = plt.subplots(nrows=3, ncols=2, figsize=(15, 9))
axes[0, 0].set_title('Signal', fontweight='bold')
axes[0, 0].plot(time, signal[:, 0, 0], label='Signal1')
axes[0, 0].plot(time, signal[:, 0, 1], label='Signal2')
axes[0, 0].set_xlabel('Time')
axes[0, 0].set_ylabel('Amplitude')
axes[0, 0].set_xlim(time_extent)
axes[0, 0].set_ylim((-2, 2))
plt.legend()

axes[0, 1].set_title('Signal + Noise', fontweight='bold')
axes[0, 1].plot(time, signal[:, 0, :] + noise[:, 0, :])
axes[0, 1].set_xlabel('Time')
axes[0, 1].set_ylabel('Amplitude')
axes[0, 1].set_xlim(time_extent)
axes[0, 1].set_ylim((-10, 10))
plt.legend()

m = Multitaper(signal,
               sampling_frequency=sampling_frequency,
               time_halfbandwidth_product=time_halfbandwidth_product,
               start_time=time[0])
c = Connectivity(fourier_coefficients=m.fft(),
                 frequencies=m.frequencies)
axes[1, 0].plot(c.frequencies, c.weighted_phase_lag_index()[..., 0, 1].squeeze())
axes[1, 0].set_xlim((0, m.nyquist_frequency))

m = Multitaper(signal + noise,
               sampling_frequency=sampling_frequency,
               time_halfbandwidth_product=time_halfbandwidth_product,
               start_time=time[0])
c = Connectivity(fourier_coefficients=m.fft(),
                 frequencies=m.frequencies)
axes[1, 1].plot(c.frequencies,  c.weighted_phase_lag_index()[..., 0, 1].squeeze())
axes[1, 1].set_xlim((0, m.nyquist_frequency))


m = Multitaper(signal,
               sampling_frequency=sampling_frequency,
               time_halfbandwidth_product=time_halfbandwidth_product,
               time_window_duration=0.080,
               time_window_step=0.080,
               start_time=time[0])
c = Connectivity(fourier_coefficients=m.fft(),
                 frequencies=m.frequencies,
                 time=m.time)
time_grid, freq_grid = np.meshgrid(
    np.append(c.time, time_extent[-1]),
    np.append(c.frequencies, m.nyquist_frequency))

mesh = axes[2, 0].pcolormesh(time_grid, freq_grid,  c.weighted_phase_lag_index()[..., 0, 1].squeeze().T,
                             vmin=-1.0, vmax=1.0, cmap='RdBu_r')
axes[2, 0].set_ylim((0, 300))
axes[2, 0].set_xlim(time_extent)
axes[2, 0].axvline(1.5, color='black')

m = Multitaper(signal + noise,
               sampling_frequency=sampling_frequency,
               time_halfbandwidth_product=time_halfbandwidth_product,
               time_window_duration=0.080,
               time_window_step=0.080,
               start_time=time[0])
c = Connectivity(fourier_coefficients=m.fft(),
                 frequencies=m.frequencies,
                 time=m.time)

time_grid, freq_grid = np.meshgrid(
    np.append(c.time, time_extent[-1]),
    np.append(c.frequencies, m.nyquist_frequency))

mesh = axes[2, 1].pcolormesh(time_grid, freq_grid,  c.weighted_phase_lag_index()[..., 0, 1].squeeze().T,
                             vmin=-1.0, vmax=1.0, cmap='RdBu_r')
axes[2, 1].set_ylim((0, 300))
axes[2, 1].set_xlim(time_extent)
axes[2, 1].axvline(1.5, color='black')

plt.tight_layout()
cb = fig.colorbar(mesh, ax=axes.ravel().tolist(), orientation='horizontal',
                  shrink=.5, aspect=15, pad=0.1, label='Weighted Phase Lag Index')
cb.outline.set_linewidth(0)
print('frequency resolution: {}'.format(m.frequency_resolution))



    


















    






No handles with labels found to put in legend.
No handles with labels found to put in legend.










    






frequency resolution: 25.0










    
























In [17]:



    


time_halfbandwidth_product = 2
frequency_of_interest = 200
sampling_frequency = 1500
time_extent = (0, 2.400)
n_trials = 100
n_signals = 2
n_time_samples = int(((time_extent[1] - time_extent[0]) * sampling_frequency) + 1)
time = np.linspace(time_extent[0], time_extent[1], num=n_time_samples, endpoint=True)[:, np.newaxis]
signal = np.zeros((n_time_samples, n_trials, n_signals))
signal[:, :, 0] = np.sin(2 * np.pi * time * frequency_of_interest)
phase_offset = np.random.uniform(-np.pi, np.pi, size=(n_time_samples, 1))
phase_offset[np.where(time > 1.5), :] = np.pi / 2
signal[:, :, 1] = np.sin((2 * np.pi * time * frequency_of_interest) + phase_offset)
noise = np.random.normal(0, 1, signal.shape)

fig, axes = plt.subplots(nrows=3, ncols=2, figsize=(15, 9))
axes[0, 0].set_title('Signal', fontweight='bold')
axes[0, 0].plot(time, signal[:, 0, 0], label='Signal1')
axes[0, 0].plot(time, signal[:, 0, 1], label='Signal2')
axes[0, 0].set_xlabel('Time')
axes[0, 0].set_ylabel('Amplitude')
axes[0, 0].set_xlim(time_extent)
axes[0, 0].set_ylim((-2, 2))
plt.legend()

axes[0, 1].set_title('Signal + Noise', fontweight='bold')
axes[0, 1].plot(time, signal[:, 0, :] + noise[:, 0, :])
axes[0, 1].set_xlabel('Time')
axes[0, 1].set_ylabel('Amplitude')
axes[0, 1].set_xlim(time_extent)
axes[0, 1].set_ylim((-10, 10))
plt.legend()

m = Multitaper(signal,
               sampling_frequency=sampling_frequency,
               time_halfbandwidth_product=time_halfbandwidth_product,
               start_time=time[0])
c = Connectivity(fourier_coefficients=m.fft(),
                 frequencies=m.frequencies)
axes[1, 0].plot(c.frequencies, c.debiased_squared_phase_lag_index()[..., 0, 1].squeeze())
axes[1, 0].set_xlim((0, m.nyquist_frequency))

m = Multitaper(signal + noise,
               sampling_frequency=sampling_frequency,
               time_halfbandwidth_product=time_halfbandwidth_product,
               start_time=time[0])
c = Connectivity(fourier_coefficients=m.fft(),
                 frequencies=m.frequencies)
axes[1, 1].plot(c.frequencies,  c.debiased_squared_phase_lag_index()[..., 0, 1].squeeze())
axes[1, 1].set_xlim((0, m.nyquist_frequency))


m = Multitaper(signal,
               sampling_frequency=sampling_frequency,
               time_halfbandwidth_product=time_halfbandwidth_product,
               time_window_duration=0.080,
               time_window_step=0.080,
               start_time=time[0])
c = Connectivity(fourier_coefficients=m.fft(),
                 frequencies=m.frequencies,
                 time=m.time)
time_grid, freq_grid = np.meshgrid(
    np.append(c.time, time_extent[-1]),
    np.append(c.frequencies, m.nyquist_frequency))

mesh = axes[2, 0].pcolormesh(time_grid, freq_grid,  c.debiased_squared_phase_lag_index()[..., 0, 1].squeeze().T,
                             vmin=0.0, vmax=1.0, cmap='viridis')
axes[2, 0].set_ylim((0, 300))
axes[2, 0].set_xlim(time_extent)
axes[2, 0].axvline(1.5, color='black')

m = Multitaper(signal + noise,
               sampling_frequency=sampling_frequency,
               time_halfbandwidth_product=time_halfbandwidth_product,
               time_window_duration=0.080,
               time_window_step=0.080,
               start_time=time[0])
c = Connectivity(fourier_coefficients=m.fft(),
                 frequencies=m.frequencies,
                 time=m.time)

time_grid, freq_grid = np.meshgrid(
    np.append(c.time, time_extent[-1]),
    np.append(c.frequencies, m.nyquist_frequency))

mesh = axes[2, 1].pcolormesh(time_grid, freq_grid,  c.debiased_squared_phase_lag_index()[..., 0, 1].squeeze().T,
                             vmin=0.0, vmax=1.0, cmap='viridis')
axes[2, 1].set_ylim((0, 300))
axes[2, 1].set_xlim(time_extent)
axes[2, 1].axvline(1.5, color='black')

plt.tight_layout()
cb = fig.colorbar(mesh, ax=axes.ravel().tolist(), orientation='horizontal',
                  shrink=.5, aspect=15, pad=0.1, label='Debiased Squared Phase Lag Index')
cb.outline.set_linewidth(0)
print('frequency resolution: {}'.format(m.frequency_resolution))



    


















    






No handles with labels found to put in legend.
No handles with labels found to put in legend.










    






frequency resolution: 25.0

Debiased Squared Weighted Phase Lag Index





In [18]:



    


time_halfbandwidth_product = 2
frequency_of_interest = 200
sampling_frequency = 1500
time_extent = (0, 2.400)
n_trials = 100
n_signals = 2
n_time_samples = int(((time_extent[1] - time_extent[0]) * sampling_frequency) + 1)
time = np.linspace(time_extent[0], time_extent[1], num=n_time_samples, endpoint=True)[:, np.newaxis]
signal = np.zeros((n_time_samples, n_trials, n_signals))
signal[:, :, 0] = np.sin(2 * np.pi * time * frequency_of_interest)
phase_offset = np.random.uniform(-np.pi, np.pi, size=(n_time_samples, 1))
phase_offset[np.where(time > 1.5), :] = np.pi / 2
signal[:, :, 1] = np.sin((2 * np.pi * time * frequency_of_interest) + phase_offset)
noise = np.random.normal(0, 1, signal.shape)

fig, axes = plt.subplots(nrows=3, ncols=2, figsize=(15, 9))
axes[0, 0].set_title('Signal', fontweight='bold')
axes[0, 0].plot(time, signal[:, 0, 0], label='Signal1')
axes[0, 0].plot(time, signal[:, 0, 1], label='Signal2')
axes[0, 0].set_xlabel('Time')
axes[0, 0].set_ylabel('Amplitude')
axes[0, 0].set_xlim(time_extent)
axes[0, 0].set_ylim((-2, 2))
plt.legend()

axes[0, 1].set_title('Signal + Noise', fontweight='bold')
axes[0, 1].plot(time, signal[:, 0, :] + noise[:, 0, :])
axes[0, 1].set_xlabel('Time')
axes[0, 1].set_ylabel('Amplitude')
axes[0, 1].set_xlim(time_extent)
axes[0, 1].set_ylim((-10, 10))
plt.legend()

m = Multitaper(signal,
               sampling_frequency=sampling_frequency,
               time_halfbandwidth_product=time_halfbandwidth_product,
               start_time=time[0])
c = Connectivity(fourier_coefficients=m.fft(),
                 frequencies=m.frequencies)
axes[1, 0].plot(c.frequencies, c.debiased_squared_weighted_phase_lag_index()[..., 0, 1].squeeze())
axes[1, 0].set_xlim((0, m.nyquist_frequency))

m = Multitaper(signal + noise,
               sampling_frequency=sampling_frequency,
               time_halfbandwidth_product=time_halfbandwidth_product,
               start_time=time[0])
c = Connectivity(fourier_coefficients=m.fft(),
                 frequencies=m.frequencies)
axes[1, 1].plot(c.frequencies,  c.debiased_squared_weighted_phase_lag_index()[..., 0, 1].squeeze())
axes[1, 1].set_xlim((0, m.nyquist_frequency))


m = Multitaper(signal,
               sampling_frequency=sampling_frequency,
               time_halfbandwidth_product=time_halfbandwidth_product,
               time_window_duration=0.080,
               time_window_step=0.080,
               start_time=time[0])
c = Connectivity(fourier_coefficients=m.fft(),
                 frequencies=m.frequencies,
                 time=m.time)
time_grid, freq_grid = np.meshgrid(
    np.append(c.time, time_extent[-1]),
    np.append(c.frequencies, m.nyquist_frequency))

mesh = axes[2, 0].pcolormesh(time_grid, freq_grid,  c.debiased_squared_weighted_phase_lag_index()[..., 0, 1].squeeze().T,
                             vmin=0.0, vmax=1.0, cmap='viridis')
axes[2, 0].set_ylim((0, 300))
axes[2, 0].set_xlim(time_extent)
axes[2, 0].axvline(1.5, color='black')

m = Multitaper(signal + noise,
               sampling_frequency=sampling_frequency,
               time_halfbandwidth_product=time_halfbandwidth_product,
               time_window_duration=0.080,
               time_window_step=0.080,
               start_time=time[0])
c = Connectivity(fourier_coefficients=m.fft(),
                 frequencies=m.frequencies,
                 time=m.time)

time_grid, freq_grid = np.meshgrid(
    np.append(c.time, time_extent[-1]),
    np.append(c.frequencies, m.nyquist_frequency))

mesh = axes[2, 1].pcolormesh(time_grid, freq_grid,  c.debiased_squared_weighted_phase_lag_index()[..., 0, 1].squeeze().T,
                             vmin=0.0, vmax=1.0, cmap='viridis')
axes[2, 1].set_ylim((0, 300))
axes[2, 1].set_xlim(time_extent)
axes[2, 1].axvline(1.5, color='black')

plt.tight_layout()
cb = fig.colorbar(mesh, ax=axes.ravel().tolist(), orientation='horizontal',
                  shrink=.5, aspect=15, pad=0.1, label='Debiased Weighted Squared Phase Lag Index')
cb.outline.set_linewidth(0)
print('frequency resolution: {}'.format(m.frequency_resolution))



    


















    






No handles with labels found to put in legend.
No handles with labels found to put in legend.










    






frequency resolution: 25.0

Pairwise Phase Consistency





In [19]:



    


time_halfbandwidth_product = 2
frequency_of_interest = 200
sampling_frequency = 1500
time_extent = (0, 2.400)
n_trials = 100
n_signals = 2
n_time_samples = int(((time_extent[1] - time_extent[0]) * sampling_frequency) + 1)
time = np.linspace(time_extent[0], time_extent[1], num=n_time_samples, endpoint=True)[:, np.newaxis]
signal = np.zeros((n_time_samples, n_trials, n_signals))
signal[:, :, 0] = np.sin(2 * np.pi * time * frequency_of_interest)
phase_offset = np.random.uniform(-np.pi, np.pi, size=(n_time_samples, 1))
phase_offset[np.where(time > 1.5), :] = np.pi / 2
signal[:, :, 1] = np.sin((2 * np.pi * time * frequency_of_interest) + phase_offset)
noise = np.random.normal(0, 1, signal.shape)

fig, axes = plt.subplots(nrows=3, ncols=2, figsize=(15, 9))
axes[0, 0].set_title('Signal', fontweight='bold')
axes[0, 0].plot(time, signal[:, 0, 0], label='Signal1')
axes[0, 0].plot(time, signal[:, 0, 1], label='Signal2')
axes[0, 0].set_xlabel('Time')
axes[0, 0].set_ylabel('Amplitude')
axes[0, 0].set_xlim(time_extent)
axes[0, 0].set_ylim((-2, 2))
plt.legend()

axes[0, 1].set_title('Signal + Noise', fontweight='bold')
axes[0, 1].plot(time, signal[:, 0, :] + noise[:, 0, :])
axes[0, 1].set_xlabel('Time')
axes[0, 1].set_ylabel('Amplitude')
axes[0, 1].set_xlim(time_extent)
axes[0, 1].set_ylim((-10, 10))
plt.legend()

m = Multitaper(signal,
               sampling_frequency=sampling_frequency,
               time_halfbandwidth_product=time_halfbandwidth_product,
               start_time=time[0])
c = Connectivity(fourier_coefficients=m.fft(),
                 frequencies=m.frequencies)
axes[1, 0].plot(c.frequencies, c.pairwise_phase_consistency()[..., 0, 1].squeeze())
axes[1, 0].set_xlim((0, m.nyquist_frequency))

m = Multitaper(signal + noise,
               sampling_frequency=sampling_frequency,
               time_halfbandwidth_product=time_halfbandwidth_product,
               start_time=time[0])
c = Connectivity(fourier_coefficients=m.fft(),
                 frequencies=m.frequencies)
axes[1, 1].plot(c.frequencies,  c.pairwise_phase_consistency()[..., 0, 1].squeeze())
axes[1, 1].set_xlim((0, m.nyquist_frequency))


m = Multitaper(signal,
               sampling_frequency=sampling_frequency,
               time_halfbandwidth_product=time_halfbandwidth_product,
               time_window_duration=0.080,
               time_window_step=0.080,
               start_time=time[0])
c = Connectivity(fourier_coefficients=m.fft(),
                 frequencies=m.frequencies,
                 time=m.time)
time_grid, freq_grid = np.meshgrid(
    np.append(c.time, time_extent[-1]),
    np.append(c.frequencies, m.nyquist_frequency))

mesh = axes[2, 0].pcolormesh(time_grid, freq_grid,  c.pairwise_phase_consistency()[..., 0, 1].squeeze().T,
                             vmin=0.0, vmax=1.0, cmap='viridis')
axes[2, 0].set_ylim((0, 300))
axes[2, 0].set_xlim(time_extent)
axes[2, 0].axvline(1.5, color='black')

m = Multitaper(signal + noise,
               sampling_frequency=sampling_frequency,
               time_halfbandwidth_product=time_halfbandwidth_product,
               time_window_duration=0.080,
               time_window_step=0.080,
               start_time=time[0])
c = Connectivity(fourier_coefficients=m.fft(),
                 frequencies=m.frequencies,
                 time=m.time)

time_grid, freq_grid = np.meshgrid(
    np.append(c.time, time_extent[-1]),
    np.append(c.frequencies, m.nyquist_frequency))

mesh = axes[2, 1].pcolormesh(time_grid, freq_grid,  c.pairwise_phase_consistency()[..., 0, 1].squeeze().T,
                             vmin=0.0, vmax=1.0, cmap='viridis')
axes[2, 1].set_ylim((0, 300))
axes[2, 1].set_xlim(time_extent)
axes[2, 1].axvline(1.5, color='black')

plt.tight_layout()
cb = fig.colorbar(mesh, ax=axes.ravel().tolist(), orientation='horizontal',
                  shrink=.5, aspect=15, pad=0.1, label='Pairwise Phase Consistency')
cb.outline.set_linewidth(0)
print('frequency resolution: {}'.format(m.frequency_resolution))



    


















    






No handles with labels found to put in legend.
No handles with labels found to put in legend.










    






frequency resolution: 25.0

Group Delay

Signal #1 leads Signal #2





In [20]:



    


import scipy

sampling_frequency = 1000
time_extent = (0, 1)
n_trials = 500
time_halfbandwidth_product = 1

n_time_samples = ((time_extent[1] - time_extent[0]) * sampling_frequency) + 1
time = np.linspace(time_extent[0], time_extent[1], num=n_time_samples)

signal1 = (scipy.stats.norm.pdf(time, .40, .025) - scipy.stats.norm.pdf(time, .45, .025)) / 10
signal1 = signal1[:, np.newaxis] * np.ones((len(time), n_trials))
signal2 = (scipy.stats.norm.pdf(time, .43, .025) - scipy.stats.norm.pdf(time, .48, .025)) / 10
signal2 = signal2[:, np.newaxis] * np.ones((len(time), n_trials))

noise1 = np.random.normal(0, .2, size=(len(time), n_trials))
noise2 = np.random.normal(0, .1, size=(len(time), n_trials))
data1 = signal1 + noise1
data2 = signal2 + noise2

signals = np.stack((signal1, signal2), axis=-1)
data = np.stack((data1, data2), axis=-1)

fig, axis_handles = plt.subplots(5, 2, figsize=(12, 9))
axis_handles[0, 0].plot(time, signal1, color='blue');
axis_handles[0, 0].plot(time, signal2, color='green');
axis_handles[0, 0].set_xlabel('Time')

axis_handles[0, 1].plot(time, data1, color='blue');
axis_handles[0, 1].plot(time, data2, color='green');
axis_handles[0, 1].set_xlabel('Time')

m = Multitaper(signals,
               sampling_frequency=sampling_frequency,
               time_halfbandwidth_product=time_halfbandwidth_product,
               start_time=time[0])
c = Connectivity(fourier_coefficients=m.fft(),
                 frequencies=m.frequencies,
                 time=m.time)

axis_handles[1, 0].plot(c.frequencies, c.power()[..., 0].squeeze())
axis_handles[1, 0].plot(c.frequencies, c.power()[..., 1].squeeze())
axis_handles[2, 0].plot(c.frequencies, c.coherence_magnitude()[..., 0, 1].squeeze())
axis_handles[3, 0].plot(c.frequencies, c.coherence_phase()[..., 0, 1].squeeze(), linestyle='None', marker='8')

delay, slope, r_value = c.group_delay(frequencies_of_interest=[2, 10], frequency_resolution=m.frequency_resolution)
axis_handles[4, 0].bar([1, 2], [delay[..., 0, 1].squeeze(), delay[..., 1, 0].squeeze()], color=['b', 'g']);
axis_handles[4, 0].set_xlim((0.5, 2.5))
axis_handles[4, 0].axhline(0, color='black');
axis_handles[4, 0].set_xticks([1])
axis_handles[4, 0].set_xticklabels(['x1 → x2'])


m = Multitaper(data,
               sampling_frequency=sampling_frequency,
               time_halfbandwidth_product=time_halfbandwidth_product,
               start_time=time[0])
c = Connectivity(fourier_coefficients=m.fft(),
                 frequencies=m.frequencies,
                 time=m.time)

axis_handles[1, 1].plot(c.frequencies, c.power()[..., 0].squeeze())
axis_handles[1, 1].plot(c.frequencies, c.power()[..., 1].squeeze())
axis_handles[2, 1].plot(c.frequencies, c.coherence_magnitude()[..., 0, 1].squeeze())
axis_handles[3, 1].plot(c.frequencies, c.coherence_phase()[..., 0, 1].squeeze(), linestyle='None', marker='8')

delay, slope, r_value = c.group_delay(frequencies_of_interest=[2, 10], frequency_resolution=m.frequency_resolution)
axis_handles[4, 1].bar([1, 2], [delay[..., 0, 1].squeeze(), delay[..., 1, 0].squeeze()], color=['b', 'g']);
axis_handles[4, 1].set_xlim((0.5, 2.5))
axis_handles[4, 1].axhline(0, color='black');
axis_handles[4, 1].set_xticks([1])
axis_handles[4, 1].set_xticklabels(['x1 → x2']);

Signal #2 leads Signal #1





In [21]:



    


sampling_frequency = 1000
time_extent = (0, 1)
n_trials = 500
time_halfbandwidth_product = 1

n_time_samples = ((time_extent[1] - time_extent[0]) * sampling_frequency) + 1
time = np.linspace(time_extent[0], time_extent[1], num=n_time_samples)

signal1 = (scipy.stats.norm.pdf(time, .43, .025) - scipy.stats.norm.pdf(time, .48, .025)) / 10
signal1 = signal1[:, np.newaxis] * np.ones((len(time), n_trials))
signal2 = (scipy.stats.norm.pdf(time, .40, .025) - scipy.stats.norm.pdf(time, .45, .025)) / 10
signal2 = signal2[:, np.newaxis] * np.ones((len(time), n_trials))

noise1 = np.random.normal(0, .2, size=(len(time), n_trials))
noise2 = np.random.normal(0, .1, size=(len(time), n_trials))
data1 = signal1 + noise1
data2 = signal2 + noise2

signals = np.stack((signal1, signal2), axis=-1)
data = np.stack((data1, data2), axis=-1)

fig, axis_handles = plt.subplots(5, 2, figsize=(12, 9))
axis_handles[0, 0].plot(time, signal1, color='blue');
axis_handles[0, 0].plot(time, signal2, color='green');
axis_handles[0, 0].set_xlabel('Time')

axis_handles[0, 1].plot(time, data1, color='blue');
axis_handles[0, 1].plot(time, data2, color='green');
axis_handles[0, 1].set_xlabel('Time')

m = Multitaper(signals,
               sampling_frequency=sampling_frequency,
               time_halfbandwidth_product=time_halfbandwidth_product,
               start_time=time[0])
c = Connectivity(fourier_coefficients=m.fft(),
                 frequencies=m.frequencies,
                 time=m.time)

axis_handles[1, 0].plot(c.frequencies, c.power()[..., 0].squeeze())
axis_handles[1, 0].plot(c.frequencies, c.power()[..., 1].squeeze())
axis_handles[2, 0].plot(c.frequencies, c.coherence_magnitude()[..., 0, 1].squeeze())
axis_handles[3, 0].plot(c.frequencies, c.coherence_phase()[..., 0, 1].squeeze(), linestyle='None', marker='8')

delay, slope, r_value = c.group_delay(frequencies_of_interest=[2, 10], frequency_resolution=m.frequency_resolution)
axis_handles[4, 0].bar([1, 2], [delay[..., 0, 1].squeeze(), delay[..., 1, 0].squeeze()], color=['b', 'g']);
axis_handles[4, 0].set_xlim((0.5, 2.5))
axis_handles[4, 0].axhline(0, color='black');
axis_handles[4, 0].set_xticks([1])
axis_handles[4, 0].set_xticklabels(['x1 → x2']);


m = Multitaper(data,
               sampling_frequency=sampling_frequency,
               time_halfbandwidth_product=time_halfbandwidth_product,
               start_time=time[0])
c = Connectivity(fourier_coefficients=m.fft(),
                 frequencies=m.frequencies,
                 time=m.time)

axis_handles[1, 1].plot(c.frequencies, c.power()[..., 0].squeeze())
axis_handles[1, 1].plot(c.frequencies, c.power()[..., 1].squeeze())
axis_handles[2, 1].plot(c.frequencies, c.coherence_magnitude()[..., 0, 1].squeeze())
axis_handles[3, 1].plot(c.frequencies, c.coherence_phase()[..., 0, 1].squeeze(), linestyle='None', marker='8')

delay, slope, r_value = c.group_delay(frequencies_of_interest=[2, 10], frequency_resolution=m.frequency_resolution)
axis_handles[4, 1].bar([1, 2], [delay[..., 0, 1].squeeze(), delay[..., 1, 0].squeeze()], color=['b', 'g']);
axis_handles[4, 1].set_xlim((0.5, 2.5))
axis_handles[4, 1].axhline(0, color='black');
axis_handles[4, 1].set_xticks([1])
axis_handles[4, 1].set_xticklabels(['x1 → x2']);

Signal #2 leads Signal #1 over time





In [22]:



    


sampling_frequency = 1000
time_extent = (0, 2)
n_trials = 500
time_halfbandwidth_product = 1

n_time_samples = ((time_extent[1] - time_extent[0]) * sampling_frequency) + 1
time = np.linspace(time_extent[0], time_extent[1], num=n_time_samples)

signal1 = (scipy.stats.norm.pdf(time, .43, .025) - scipy.stats.norm.pdf(time, .48, .025)) / 10
signal1 = signal1[:, np.newaxis] * np.ones((len(time), n_trials))
signal2 = (scipy.stats.norm.pdf(time, .40, .025) - scipy.stats.norm.pdf(time, .45, .025)) / 10
signal2 = signal2[:, np.newaxis] * np.ones((len(time), n_trials))

noise1 = np.random.normal(0, .2, size=(len(time), n_trials))
noise2 = np.random.normal(0, .1, size=(len(time), n_trials))
data1 = signal1 + noise1
data2 = signal2 + noise2

signals = np.stack((signal1, signal2), axis=-1)
data = np.stack((data1, data2), axis=-1)

fig, axis_handles = plt.subplots(2, 2, figsize=(12, 9))
axis_handles[0, 0].plot(time, signal1, color='blue');
axis_handles[0, 0].plot(time, signal2, color='green');
axis_handles[0, 0].set_xlabel('Time')
axis_handles[0, 0].set_xlim(time_extent);

axis_handles[0, 1].plot(time, data1, color='blue');
axis_handles[0, 1].plot(time, data2, color='green');
axis_handles[0, 1].set_xlabel('Time')
axis_handles[0, 1].set_xlim(time_extent);

m = Multitaper(signals,
               sampling_frequency=sampling_frequency,
               time_halfbandwidth_product=time_halfbandwidth_product,
                time_window_duration=0.500,
               time_window_step=0.100,
               start_time=time[0])
c = Connectivity(fourier_coefficients=m.fft(),
                 frequencies=m.frequencies,
                 time=m.time)

delay, slope, r_value = c.group_delay(frequencies_of_interest=[0, 15], frequency_resolution=m.frequency_resolution)
axis_handles[1, 0].plot(c.time + m.time_window_duration/2, delay[..., 0, 1]);
axis_handles[1, 0].set_xlim(time_extent)

m = Multitaper(data,
               sampling_frequency=sampling_frequency,
               time_halfbandwidth_product=time_halfbandwidth_product,
               time_window_duration=0.500,
               time_window_step=0.100,
               start_time=time[0])
c = Connectivity(fourier_coefficients=m.fft(),
                 frequencies=m.frequencies,
                 time=m.time)

delay, slope, r_value = c.group_delay(frequencies_of_interest=[0, 15], frequency_resolution=m.frequency_resolution)
axis_handles[1, 1].plot(c.time + m.time_window_duration/2, delay[..., 0, 1]);
axis_handles[1, 1].set_xlim(time_extent);

Phase Slope Index

Signal #1 leads Signal #2





In [23]:



    


sampling_frequency = 1000
time_extent = (0, 1)
n_trials = 500
time_halfbandwidth_product = 1

n_time_samples = ((time_extent[1] - time_extent[0]) * sampling_frequency) + 1
time = np.linspace(time_extent[0], time_extent[1], num=n_time_samples)

signal1 = (scipy.stats.norm.pdf(time, .40, .025) - scipy.stats.norm.pdf(time, .45, .025)) / 10
signal1 = signal1[:, np.newaxis] * np.ones((len(time), n_trials))
signal2 = (scipy.stats.norm.pdf(time, .43, .025) - scipy.stats.norm.pdf(time, .48, .025)) / 10
signal2 = signal2[:, np.newaxis] * np.ones((len(time), n_trials))

noise1 = np.random.normal(0, .2, size=(len(time), n_trials))
noise2 = np.random.normal(0, .1, size=(len(time), n_trials))
data1 = signal1 + noise1
data2 = signal2 + noise2

signals = np.stack((signal1, signal2), axis=-1)
data = np.stack((data1, data2), axis=-1)

fig, axis_handles = plt.subplots(5, 2, figsize=(12, 9))
axis_handles[0, 0].plot(time, signal1, color='blue');
axis_handles[0, 0].plot(time, signal2, color='green');
axis_handles[0, 0].set_xlabel('Time')

axis_handles[0, 1].plot(time, data1, color='blue');
axis_handles[0, 1].plot(time, data2, color='green');
axis_handles[0, 1].set_xlabel('Time')

m = Multitaper(signals,
               sampling_frequency=sampling_frequency,
               time_halfbandwidth_product=time_halfbandwidth_product,
               start_time=time[0])
c = Connectivity(fourier_coefficients=m.fft(),
                 frequencies=m.frequencies,
                 time=m.time)

axis_handles[1, 0].plot(c.frequencies, c.power()[..., 0].squeeze())
axis_handles[1, 0].plot(c.frequencies, c.power()[..., 1].squeeze())
axis_handles[2, 0].plot(c.frequencies, c.coherence_magnitude()[..., 0, 1].squeeze())
axis_handles[3, 0].plot(c.frequencies, c.coherence_phase()[..., 0, 1].squeeze(), linestyle='None', marker='8')

psi = c.phase_slope_index(frequencies_of_interest=[2, 10], frequency_resolution=m.frequency_resolution)
axis_handles[4, 0].bar([1, 2], [psi[..., 0, 1].squeeze(), psi[..., 1, 0].squeeze()], color=['b', 'g']);
axis_handles[4, 0].set_xlim((0.5, 2.5))
axis_handles[4, 0].axhline(0, color='black');

m = Multitaper(data,
               sampling_frequency=sampling_frequency,
               time_halfbandwidth_product=time_halfbandwidth_product,
               start_time=time[0])
c = Connectivity(fourier_coefficients=m.fft(),
                 frequencies=m.frequencies,
                 time=m.time)

axis_handles[1, 1].plot(c.frequencies, c.power()[..., 0].squeeze())
axis_handles[1, 1].plot(c.frequencies, c.power()[..., 1].squeeze())
axis_handles[2, 1].plot(c.frequencies, c.coherence_magnitude()[..., 0, 1].squeeze())
axis_handles[3, 1].plot(c.frequencies, c.coherence_phase()[..., 0, 1].squeeze(), linestyle='None', marker='8')

psi = c.phase_slope_index(frequencies_of_interest=[2, 10], frequency_resolution=m.frequency_resolution)
axis_handles[4, 1].bar([1, 2], [psi[..., 0, 1].squeeze(), psi[..., 1, 0].squeeze()], color=['b', 'g']);
axis_handles[4, 1].set_xlim((0.5, 2.5));
axis_handles[4, 1].axhline(0, color='black');

Signal #2 leads Signal #1





In [24]:



    


sampling_frequency = 1000
time_extent = (0, 1)
n_trials = 500
time_halfbandwidth_product = 1

n_time_samples = ((time_extent[1] - time_extent[0]) * sampling_frequency) + 1
time = np.linspace(time_extent[0], time_extent[1], num=n_time_samples)

signal1 = (scipy.stats.norm.pdf(time, .43, .025) - scipy.stats.norm.pdf(time, .48, .025)) / 10
signal1 = signal1[:, np.newaxis] * np.ones((len(time), n_trials))
signal2 = (scipy.stats.norm.pdf(time, .40, .025) - scipy.stats.norm.pdf(time, .45, .025)) / 10
signal2 = signal2[:, np.newaxis] * np.ones((len(time), n_trials))

noise1 = np.random.normal(0, .2, size=(len(time), n_trials))
noise2 = np.random.normal(0, .1, size=(len(time), n_trials))
data1 = signal1 + noise1
data2 = signal2 + noise2

signals = np.stack((signal1, signal2), axis=-1)
data = np.stack((data1, data2), axis=-1)

fig, axis_handles = plt.subplots(5, 2, figsize=(12, 9))
axis_handles[0, 0].plot(time, signal1, color='blue');
axis_handles[0, 0].plot(time, signal2, color='green');
axis_handles[0, 0].set_xlabel('Time')

axis_handles[0, 1].plot(time, data1, color='blue');
axis_handles[0, 1].plot(time, data2, color='green');
axis_handles[0, 1].set_xlabel('Time')

m = Multitaper(signals,
               sampling_frequency=sampling_frequency,
               time_halfbandwidth_product=time_halfbandwidth_product,
               start_time=time[0])
c = Connectivity(fourier_coefficients=m.fft(),
                 frequencies=m.frequencies,
                 time=m.time)

axis_handles[1, 0].plot(c.frequencies, c.power()[..., 0].squeeze())
axis_handles[1, 0].plot(c.frequencies, c.power()[..., 1].squeeze())
axis_handles[2, 0].plot(c.frequencies, c.coherence_magnitude()[..., 0, 1].squeeze())
axis_handles[3, 0].plot(c.frequencies, c.coherence_phase()[..., 0, 1].squeeze(), linestyle='None', marker='8')

psi = c.phase_slope_index(frequencies_of_interest=[2, 10], frequency_resolution=m.frequency_resolution)
axis_handles[4, 0].bar([1, 2], [psi[..., 0, 1].squeeze(), psi[..., 1, 0].squeeze()], color=['b', 'g']);
axis_handles[4, 0].set_xlim((0.5, 2.5))
axis_handles[4, 0].axhline(0, color='black');

m = Multitaper(data,
               sampling_frequency=sampling_frequency,
               time_halfbandwidth_product=time_halfbandwidth_product,
               start_time=time[0])
c = Connectivity(fourier_coefficients=m.fft(),
                 frequencies=m.frequencies,
                 time=m.time)

axis_handles[1, 1].plot(c.frequencies, c.power()[..., 0].squeeze())
axis_handles[1, 1].plot(c.frequencies, c.power()[..., 1].squeeze())
axis_handles[2, 1].plot(c.frequencies, c.coherence_magnitude()[..., 0, 1].squeeze())
axis_handles[3, 1].plot(c.frequencies, c.coherence_phase()[..., 0, 1].squeeze(), linestyle='None', marker='8')

psi = c.phase_slope_index(frequencies_of_interest=[2, 10], frequency_resolution=m.frequency_resolution)
axis_handles[4, 1].bar([1, 2], [psi[..., 0, 1].squeeze(), psi[..., 1, 0].squeeze()], color=['b', 'g']);
axis_handles[4, 1].set_xlim((0.5, 2.5));
axis_handles[4, 1].axhline(0, color='black');



    


















    
























In [25]:



    


sampling_frequency = 1000
time_extent = (0, 2)
n_trials = 500
time_halfbandwidth_product = 1

n_time_samples = ((time_extent[1] - time_extent[0]) * sampling_frequency) + 1
time = np.linspace(time_extent[0], time_extent[1], num=n_time_samples)

signal1 = (scipy.stats.norm.pdf(time, .43, .025) - scipy.stats.norm.pdf(time, .48, .025)) / 10
signal1 = signal1[:, np.newaxis] * np.ones((len(time), n_trials))
signal2 = (scipy.stats.norm.pdf(time, .40, .025) - scipy.stats.norm.pdf(time, .45, .025)) / 10
signal2 = signal2[:, np.newaxis] * np.ones((len(time), n_trials))

noise1 = np.random.normal(0, .2, size=(len(time), n_trials))
noise2 = np.random.normal(0, .1, size=(len(time), n_trials))
data1 = signal1 + noise1
data2 = signal2 + noise2

signals = np.stack((signal1, signal2), axis=-1)
data = np.stack((data1, data2), axis=-1)

fig, axis_handles = plt.subplots(2, 2, figsize=(12, 9))
axis_handles[0, 0].plot(time, signal1, color='blue');
axis_handles[0, 0].plot(time, signal2, color='green');
axis_handles[0, 0].set_xlabel('Time')
axis_handles[0, 0].set_xlim(time_extent);

axis_handles[0, 1].plot(time, data1, color='blue');
axis_handles[0, 1].plot(time, data2, color='green');
axis_handles[0, 1].set_xlabel('Time')
axis_handles[0, 1].set_xlim(time_extent);

m = Multitaper(signals,
               sampling_frequency=sampling_frequency,
               time_halfbandwidth_product=time_halfbandwidth_product,
                time_window_duration=0.500,
               time_window_step=0.100,
               start_time=time[0])
c = Connectivity(fourier_coefficients=m.fft(),
                 frequencies=m.frequencies,
                 time=m.time)

psi = c.phase_slope_index(frequencies_of_interest=[0, 15], frequency_resolution=m.frequency_resolution)
axis_handles[1, 0].plot(c.time  + m.time_window_duration/2, psi[..., 0, 1], c.time  + m.time_window_duration/2, psi[..., 1, 0])
axis_handles[1, 0].set_xlim(time_extent)

m = Multitaper(data,
               sampling_frequency=sampling_frequency,
               time_halfbandwidth_product=time_halfbandwidth_product,
               time_window_duration=0.500,
               time_window_step=0.100,
               start_time=time[0])
c = Connectivity(fourier_coefficients=m.fft(),
                 frequencies=m.frequencies,
                 time=m.time)

psi = c.phase_slope_index(frequencies_of_interest=[0, 15], frequency_resolution=m.frequency_resolution)
axis_handles[1, 1].plot(c.time + m.time_window_duration/2, psi[..., 0, 1], c.time + m.time_window_duration/2, psi[..., 1, 0]);
axis_handles[1, 1].set_xlim(time_extent);

Canonical Coherence

The advantage of canonical coherence is that it can be more statistically powerful than coherence because it is combining coherence from groups.





In [26]:



    


from itertools import product

time_halfbandwidth_product = 2
frequency_of_interest = 200
sampling_frequency = 1500
time_extent = (0, 2.400)
n_trials = 100
n_signals = 4
n_time_samples = int(((time_extent[1] - time_extent[0]) * sampling_frequency) + 1)
time = np.linspace(time_extent[0], time_extent[1], num=n_time_samples, endpoint=True)

signal = np.zeros((n_time_samples, n_trials, n_signals))
signal[:, :, 0:2] = (
    np.sin(2 * np.pi * time * frequency_of_interest)[:, np.newaxis, np.newaxis] *
    np.ones((1, n_trials, 2)))

other_signals = (n_signals + 1) // 2
n_other_signals = n_signals - other_signals
phase_offset = np.random.uniform(-np.pi, np.pi, size=(n_time_samples, n_trials, n_other_signals))
phase_offset[np.where(time > 1.5), :] = np.pi / 2
signal[:, :, other_signals:] = np.sin(
    (2 * np.pi * time[:, np.newaxis, np.newaxis] * frequency_of_interest) + phase_offset)
noise = np.random.normal(0, 4, signal.shape)


m = Multitaper(signal + noise,
               sampling_frequency=sampling_frequency,
               time_halfbandwidth_product=time_halfbandwidth_product,
               time_window_duration=0.080,
               time_window_step=0.080,
               start_time=time[0])
c = Connectivity(fourier_coefficients=m.fft(),
                 frequencies=m.frequencies,
                 time=m.time)

time_grid, freq_grid = np.meshgrid(
    np.append(c.time, time_extent[-1]),
    np.append(c.frequencies, m.nyquist_frequency))

fig, axes = plt.subplots(nrows=n_signals, ncols=n_signals, figsize=(15, 9))
meshes = list()
for ind1, ind2 in product(range(n_signals), range(n_signals)):
    if ind1 == ind2:
        vmin, vmax = c.power().min(), c.power().max()
    else:
        vmin, vmax = 0, 0.5
    mesh = axes[ind1, ind2].pcolormesh(
        time_grid, freq_grid, c.coherence_magnitude()[..., ind1, ind2].squeeze().T,
        vmin=vmin, vmax=vmax, cmap='viridis')
    meshes.append(mesh)
    axes[ind1, ind2].set_ylim((0, 300))
    axes[ind1, ind2].set_xlim(time_extent)
    axes[ind1, ind2].axvline(1.5, color='black')

plt.suptitle('Coherence', y=1.02, fontsize=30)
plt.tight_layout()
cb = fig.colorbar(meshes[-2], ax=axes.ravel().tolist(), orientation='horizontal',
                  shrink=.5, aspect=15, pad=0.1, label='Coherence')
cb.outline.set_linewidth(0)
print('frequency resolution: {}'.format(m.frequency_resolution))


group_labels = (['a'] * (n_signals - n_other_signals)) + (['b'] * n_other_signals)
canonical_coherence, pair_labels = c.canonical_coherence(group_labels)
fig = plt.figure()
mesh = plt.pcolormesh(time_grid, freq_grid, canonical_coherence[..., 0, 1].squeeze().T,
               vmin=0, vmax=0.5, cmap='viridis')
plt.suptitle('Canonical Coherence', y=1.02, fontsize=30)
cb = fig.colorbar(mesh, ax=plt.gca(), orientation='horizontal',
                  shrink=.5, aspect=15, pad=0.1, label='Coherence')
cb.outline.set_linewidth(0);



    


















    






frequency resolution: 25.0

More signals, higher noise





In [27]:



    


from itertools import product

time_halfbandwidth_product = 2
frequency_of_interest = 200
sampling_frequency = 1500
time_extent = (0, 2.400)
n_trials = 100
n_signals = 6
n_time_samples = int(((time_extent[1] - time_extent[0]) * sampling_frequency) + 1)
time = np.linspace(time_extent[0], time_extent[1], num=n_time_samples, endpoint=True)

signal = np.zeros((n_time_samples, n_trials, n_signals))
signal[:, :, 0:2] = (
    np.sin(2 * np.pi * time * frequency_of_interest)[:, np.newaxis, np.newaxis] *
    np.ones((1, n_trials, 2)))

other_signals = (n_signals + 1) // 2
n_other_signals = n_signals - other_signals
phase_offset = np.random.uniform(-np.pi, np.pi, size=(n_time_samples, n_trials, n_other_signals))
phase_offset[np.where(time > 1.5), :] = np.pi / 2
signal[:, :, other_signals:] = np.sin(
    (2 * np.pi * time[:, np.newaxis, np.newaxis] * frequency_of_interest) + phase_offset)
noise = np.random.normal(10, 7, signal.shape)


m = Multitaper(signal + noise,
               sampling_frequency=sampling_frequency,
               time_halfbandwidth_product=time_halfbandwidth_product,
               time_window_duration=0.080,
               time_window_step=0.080,
               start_time=time[0])
c = Connectivity(fourier_coefficients=m.fft(),
                 frequencies=m.frequencies,
                 time=m.time)

time_grid, freq_grid = np.meshgrid(
    np.append(c.time, time_extent[-1]),
    np.append(c.frequencies, m.nyquist_frequency))

fig, axes = plt.subplots(nrows=n_signals, ncols=n_signals, figsize=(15, 9))
meshes = list()
for ind1, ind2 in product(range(n_signals), range(n_signals)):
    if ind1 == ind2:
        vmin, vmax = c.power().min(), c.power().max()
    else:
        vmin, vmax = 0, 0.5
    mesh = axes[ind1, ind2].pcolormesh(
        time_grid, freq_grid, c.coherence_magnitude()[..., ind1, ind2].squeeze().T,
        vmin=vmin, vmax=vmax, cmap='viridis')
    meshes.append(mesh)
    axes[ind1, ind2].set_ylim((0, 300))
    axes[ind1, ind2].set_xlim(time_extent)
    axes[ind1, ind2].axvline(1.5, color='black')

plt.suptitle('Coherence', y=1.02, fontsize=30)
plt.tight_layout()
cb = fig.colorbar(meshes[-2], ax=axes.ravel().tolist(), orientation='horizontal',
                  shrink=.5, aspect=15, pad=0.1, label='Coherence')
cb.outline.set_linewidth(0)
print('frequency resolution: {}'.format(m.frequency_resolution))


group_labels = (['a'] * (n_signals - n_other_signals)) + (['b'] * n_other_signals)
canonical_coherence, pair_labels = c.canonical_coherence(group_labels)
fig = plt.figure()
mesh = plt.pcolormesh(time_grid, freq_grid, canonical_coherence[..., 0, 1].squeeze().T,
               vmin=0, vmax=0.5, cmap='viridis')
plt.suptitle('Canonical Coherence', y=1.02, fontsize=30)
cb = fig.colorbar(mesh, ax=plt.gca(), orientation='horizontal',
                  shrink=.5, aspect=15, pad=0.1, label='Coherence')
cb.outline.set_linewidth(0);



    


















    






frequency resolution: 25.0










    

















    
























In [ ]:

Content source: Eden-Kramer-Lab/spectral_connectivity

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