this code allows to compute the spectral exponent of the resting EEG, based on the Power Spectral Density (PSD), over a given scaling region.
The spectral exponent describes the decay of the PSD. it is computed as the slope of an OLS line, fit on log-freq vs log-PSD, excluding oscillatory peaks (and their base).
(Note: while the Y axis label reads as mV^2/Hz, it should really read µ^2/Hz. Thus, the proper version should be microVolt (and not milliVolt) squared over Hz.
you should have in your workspace:
sRate: sampling Rate, e.g. 1450 for Nexstim amplifier
signal: one EEG channel, a vector of datapoints.
% use one eeg channel after preprocessing and filtering (but be aware of how filters impact the psd)
% if high-pass filter (butter order 5) is set at 0.5 Hz, then begin fitting at 1 Hz
% if low-pass filter (butter order 5) is set at 60 Hz, then stop fitting at approximately 45 Hz.
% Better even, do NOT use any low-pass
Or, for the sake of quickly testing this code, you can generate 1 minute of an EEG-looking signal
%%%%%%%%%%%%%%%%%%%%%%%%%%% FAKE SIGNAL GENERATION %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% generate signal data (a sum of sinusoids with random phase) with a given PSD exponent (-1,5)
sRate= 1000;
freqs= linspace( .5, sRate/2, 10000);
time= (1 : 60*sRate )./sRate; %1: sRate*5*60 +2*sRate;
exp= -1.5;
signal= zeros(size(time));
for ii = 1: length(freqs)
pwr = freqs(ii)^exp;
signal= signal + sqrt(pwr) *sin(2*pi*freqs(ii)*time + rand(1)*2*pi ) ; % + randn(size(time)).*sqrt(pwr)
end
%%% add to the signal alpha and beta oscillations , leading to two bell-shaped peaks in the PSD
pks= [8 13; 16 24];
for fb= 1 : size(pks,1)
ampFac= hamming(200);
ffs= linspace(pks(fb,1) , pks(fb,2) ,200);
for ii= 1 : 200;
amp = 60/( ffs(ii)^ 1.75)* ampFac(ii); % 60 1.75 % 90 2
signal= signal+ sin( 2*pi* ffs(ii)*time + rand(1)*2*pi )* amp;
end
end
% figure, plot(signal)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%now that you have generated a fake signal (or imported a real signal), you can compute the PSD and estimate the spectral exponent in a given frequency range (1-40 Hz)
% first compute the PSD
epLen= 3* sRate; epShift= 1*sRate; numFFT=[];
[myPSD,frex]= pwelch( signal , epLen, epShift,numFFT, sRate);
frBand=[1 40];
frBins= dsearchn( frex, frBand(1) ): dsearchn( frex, frBand(2));
XX= frex(frBins);
YY= myPSD(frBins);
robRegMeth= 'ols'; % method to perform linear regression. see >> help robustfit
doPlot= 1; figure;
thisCol= [0 0 1];
[intSlo, stat, Pows, Deviants, stat0, intSlo0] = fitPowerLaw3steps(XX,YY, robRegMeth, doPlot, thisCol)
spectralExponent= intSlo(2);
% repeat for every scalp electrode, and compute the average spectral exponent across the scalp
%% fitPowerLaw3steps %%
%%%%%%%%% INPUT
% XX ---> are the frequency bins of interest (hz)
% YY ---> are the PSD relative to the bins of interest (mV^2 /Hz)
% robRegMeth --->is the method for robust regression fit (default: ols)
% type help robustfit for more details
% doPlot ---> 0 (default) means no plot ;
% 1 is plot XX and YY on log-log scale;
% 2 is plot log(XX) and log(YY) on linear scale;
% thisCol ---> if plotting, is the rgb triplet
%%%%%%%%% OUTPUT
% intSlo(1) intercept of 2nd (final) powerLaw Fit
% intSlo(2) slope of 2nd (final) powerLaw Fit. i.e. the spectral exponent
% stat is a structure with informations on the 2nd fit, (see 2nd argument of robustfit)
% Pows is a structure with
% pred: 1*length(frBins) doubles, predicted PSD by the power-law fit
% obs: 1*length(frBins) doubles, observed PSD = YY
% res: 1*length(frBins) doubles, residuals (pred-obs) PSD
% frex: 1*length(frBins) doubles, observed frex = XX
% meanPred: 1 double, mean of the predicted PSD
% meanObs: 1 double, mean of the observed PSD
% meanRes: 1 double, mean of the residuals of the PSD
% meanResPos: 1 double, mean of the residuals of the PSD (exclude negative values)
% cenFrObs: 1 double, central Frequency of the observed PSD
% cenFrRes: 1 double, central Frequency of the residual PSD
% Deviants is a structure with
% res2: 1*length(frBins) doubles,
% res: 1*length(frBins) doubles,
% rej: length(frBins)*1 logicals, 1 if bins are rejected, 0 if kept for fitting the 2nd power-law
% fr: min and max freqs considered
% thre: threshold used for bins adjacent to the peaks
% clusts: structure with information on the contiguous freq. bins
% stat0 is a structure with informations on the 1st fit, (see 2nd argument of robustfit)
% intSlo(1) intercept of 2nd (final) powerLaw Fit
% intSlo(2) slope of 2nd (final) powerLaw FitFirst, generate 1 minute of an EEG-looking signal ( or use one channel of your own clean EEG signal)
# # HOW TO USE THIS CODE ON A GENERATED EEG SIGNAL
from scipy.signal import welch
def dsearchn(x, value):
dist_from_value = np.abs(x-value)
return np.where(dist_from_value == dist_from_value.min())[0][0]
################################### SIGNAL GENERATION ###################################
# generate noise data (a sum of sinusoids with random phase) with a given PSD exponent (-1.5)
fs = 1000 #sampling rate in Hz
time = np.arange(60, step=1/fs)
freqs = np.linspace(0.5,fs/2,10000)
exponent=-1.5
signal = np.zeros(time.shape)
for f in range(len(freqs)):
power = freqs[f]**(exponent)
signal += np.sqrt(power)*np.sin(2*np.pi*freqs[f]*time + np.random.uniform()*2*np.pi)
# add to the noise alpha and beta oscillations, leading to two bell-shaped peaks in the PSD
pks= np.array([[8, 13], [16, 24]])
for fb in range(len(pks)):
ampFac= np.hamming(200)
ffs= np.linspace(pks[fb,0] , pks[fb,1], 200)
for ii in range(len(ffs)):
amp = 60/(ffs[ii]**1.75)*ampFac[ii] #90 **2
signal += amp * np.sin(2*np.pi*ffs[ii]*time + np.random.uniform()*2*np.pi)
########## Compute the PSD and estimate the exponent in the range 1 to 40 Hz ########## now that you have generated a fake signal (or imported a real signal), you can compute the PSD and estimate the spectral exponent in a given frequency range (1-40 Hz)
#PSD
frex, psd_1chan = welch(signal, fs=fs, nperseg=fs*3, noverlap=fs*2, detrend='linear')
frband = [1, 40]
frBins = np.arange(dsearchn(frex, frband[0]), dsearchn(frex, frband[1])+1)
XX = frex[frBins]
# spectral exponent
YY = psd_1chan[frBins]
plt.figure(dpi=300, figsize=(4,4))
slope, inter, stats, vectors = compute_SpectralExponent(XX, YY, do_plot=True)
PSD and estimates of the spectral exponent (preliminary vs excluding peaks) of a generated EEG-looking signal
This generated EEG signal has a theoretical PSD spectral exponent of -1.5. As you can see, the first preliminary/naive fit (in the legend> slope fit0, grey dotted line) of the whole PSD is biased by the peaks in the alpha and beta frequency bands. The second and final fit (in the legend> slope 1-40 Hz, blue dotted line) , by excluding most of those peaks, well adheres to the aperiodic PSD background, and its slope is much closer to the theoretical value (not exact, each simulation gives slightly different values).
First transform PSD (YY) and frequencies (XX) each by log10, resample the values using logarithmically spaced bins (upsample 4x to obtain PSD bins evenly sampled in the loglog space)
A 1st order ordinary least square, or "best fitting straight line", is fit under the log-log axes
Find all the residual >0 (i.e. bins emerging from the naive power-law fit). Only large peaks (whose residuals are larger than 1*mad(residuals)) count as proper peaks. Find clusters of bins with residuals >0, and consider only those clusters of frequency bins where there is a peak, for subsequent rejection. i.e. reject large enough peaks (large residuals well above the naive power-law trend, where residuals > mad(residuals) ) along with the base of the peaks (adjacent bins till above the naive power-law trend, where residuals >0)
Perform the fit only on the remaining residuals (those closer to a power-law)
indexing the decay of the PSD over frequencies. statistics relative to the fit are also assessed
Obviously, temporal filters distort the shape of the PSD. Their effect is best estimated by directly comparing the PSD before and after filtering. A high-pass filter is often used to remove long drift in the signal. However, if the high-pass cutoff is high (>=1 Hz) or the filter-order is low (e.g. 3), this may induce a shoulder to the left of your PSD which extends into the standard fitting range (above 1 Hz). Further, if the low-pass cutoff is low (<=60 Hz) or the filter-order is low (e.g. 3), this may induce a shoulder to the right of your PSD which extends to the standard fitting range ( below 40 Hz). Recommended filter settings: • High-pass: Butter iir high-pass 0.5 Hz, 5th order. to avoid distortions above 1 Hz (a sharp roll-on) • Low-pass: no low-pass, or above 60 Hz to avoid distortions below 40 Hz (a smooth roll-off)
• EOG artifacts induce a big bump to the left of your PSD (most of the power below ~4 Hz), increasing the slope and intercept of the fit. • EMG artifacts induce a bump to the right of your PSD (most of the power above ~20 Hz), flattening the slope and decreasing the intercept of the fit. EOG and EMG artifacts must be throroughly cleaned before estimating the spectral exponent (for instance , using Independent Component Analysis).
Bad segments (such as movements, chewing etc.) should be marked, and attention must be paid to avoid biasing PSD estimates (i.e.jumps in the signal are generated if bad-epochs are extruded from the data, by concatenating good epochs, leading to spurious extra power at several high- or mixed-freqeuncies). One may estimate the PSD using the Welch's method "explicity" (estimating the PSD over each single periodogram/ temporal window, with a given temporal overlap), and reject those windows where any datapoint belonging to a bad-segment is present, before averaging the PSD across periodograms (the pwelch method returns the periodograms-average PSD only, not the PSD of individual periodograms). In other words, each periodogram must always include only good data-points (no artifacts), and temporally adjacent time points (no time jumps). Obviously, bad epochs induce much larger PSD distortion than the concatenation of good epochs, but the latter strategy is best avoided.
The EEG typicaly displays an overall 1/f-like PSD decay in the 1-40 Hz range, above which the oscillatory activity induces various peaks in the theta-alpha-sigma or beta range. The EEG PSD shows two slightly different trends in the 1-20 Hz and in the 20-40 Hz range, with a bend near 20 Hz (difficult to pinpoint due to oscillatory activity in alpha and beta frequencies). Changing the fitting range should be done carefully, according to the signal properties, upon close inspection of the PSD in loglog axes. For instance, intracranial EEG may sometimes show a PSD where the decay begins at higher frequencies (e.g. 2 Hz), or the decay extends to a higher freqeuncy range. Sometimes, one may want to consider fitting even lower frequencies, if proper cleaning, filters and the PSD-shape allow.
Scientific References where we employed the present code to estimate the spectral exponent.
the first article of the series, the EEG spectral slope reflects the presence/absence of consciousness (according to delayed reports) across anesthetic agents all tailored to reach complete behavioural unresponsiveness, discriminating cases of unconsciousness (during propofol and xenon anesthesia) against baseline wakefulness and cases of sensorimotor disconnection from the environment (during ketamine)
Colombo, M. A., Napolitani, M., Boly, M., Gosseries, O., Casarotto, S., Rosanova, M., ... & Sarasso, S. (2019).
The spectral exponent of the resting EEG indexes the presence of consciousness
during unresponsiveness induced by propofol, xenon, and ketamine.
NeuroImage, 189, 631-644.
The EEG spectral exponent (dubbed spectral gradient) reflects the presence/absence of consciousness in severe brain-injury with Disorders of Cosnciousness, in conscious patients with brain injury, and even in cases of sensorimotor disconnection from the environment: those patients with unresponsive wakefulness syndrome (vegetative state) that nonetheless show capacity for consciousness, as shown by complex reactivity to direct cortical perturbation, (i.e. high values of Perturbational Complexity Index (PCI), assessed from TMS-EEG)
Colombo, M. A., Comanducci, A., Casarotto, S., Derchi, C. C., Annen, J., Viganò, A., ... & Rosanova, M. (2023).
Beyond alpha power: EEG spatial and spectral gradients robustly stratify disorders of consciousness.
Cerebral cortex, 33(11), 7193-7210.
link to Cerebral cortex article.
Wake and light-sleep EEG aperiodic activity diverge from deep sleep activity over age from infants to adolescents. EEG Spectral exponent flattens from infants to adolescent during wakefulness and light sleep, while the spread between wake and sleep values become progressively larger; further, the hot-spot where the PSD is the slowest migrates following a postero-anterior gradient over age.
Favaro, J., Colombo, M. A., Mikulan, E., Sartori, S., Nosadini, M., Pelizza, M. F., ... & Toldo, I. (2023).
The maturation of aperiodic EEG activity across development reveals a progressive differentiation of wakefulness from sleep.
NeuroImage, 277, 120264.
EEG Spectral exponent in patients with stroke (in a regimen of values compatible with consciousness) discriminate the lesioned from the contralateral hemisphere, and correlate with clinical indexes of stroke recovery
Lanzone, J., Colombo, M. A., Sarasso, S., Zappasodi, F., Rosanova, M., Massimini, M., ... & Assenza, G. (2022).
EEG spectral exponent as a synthetic index for the longitudinal assessment of stroke recovery.
Clinical Neurophysiology, 137, 92-101.
link to Clinical Neurophysiology article.
A good place explaining the basics; a brief description of the distinction between aperiodic and periodic activity
A youtube video where I present the findings of the EEG spectral exponent during various anesthetic protocols (Colombo et al., 2019)