Multiscale analysis

To study the neuronal dynamics at a multiscale level we opted to apply Empirical Mode Decomposition (EMD)⁸⁶, a method that separates a signal x(t) in a set of n so-called intrinsic mode functions (IMFs)—c_i, which represent the dynamics of the signal at different time scales:

where c_i(t) represents the n IMFs and r_n(t) is the residue (a constant or monotonic trend). It is noteworthy that the method does not result in a separation of the signal in predetermined frequency bands but rather, in a signal-dependent time-variant filtering⁸⁷, fully adaptive and, therefore, suitable for the nonstationary and nonlinearity of the electroencephalogram. Notwithstanding, the IMFs obtained have a defined bandwidth that can be related to the classic frequency bands used in clinical practice or neuroscientific research. Each mode has typically a power spectrum that peaks around a limited range of frequencies⁸⁷, and a characteristic frequency given by f_s/2ⁿ⁺¹, where f_s represents the sampling frequency and n the number of the mode (n = 1, 2, 3 …). Thus, to match this filtering method with the frequency ranges of the classical bands, the first six (IMF1–IMF6) correspond to a spectral energy with peaks within roughly the range 8–250 Hz and the last three have activity around 1–4 Hz (Fig. 1 and Supplementary Fig. ^S1). Following the aforementioned relationship between mode number and its main frequencies, we labelled IMF1 as high-gamma (γ_h), IMF2 as mid-gamma (γ_m), IMF3 as low-gamma (γ_l), IMF4 as β, IMF5 as α, IMF6 as θ, and finally, the sum of modes IMF7, 8 and 9 equivalent to δ. The method uses a sifting process which starts by identifying the extrema in the raw time series x(t). Next, two cubic splines are fitted, connecting the local maxima and the local minima. The average—m(t)—of these envelopes is performed and subtracted from x(t), this difference constitutes the first mode (IMF1). The residue signal (r₁ = x(t) − c₁) is treated as the new signal and the sifting process iterates until further modes are extracted. The optimal number of sifting is undetermined; we opted to use 10 as this choice preserves a dyadic filtering ratio across the signals⁸⁸. The code used is available online at http://perso.ens-lyon.fr/patrick.flandrin/emd.html.