Dynamic Functional Connectivity

The most common and easiest method used to compute thedFCis the pairwise sliding-window Pearson correlation which consists in computing the Pearson Correlation coefficient within a temporal window with lengthW (Figure1.3, top image). Then, the window is shifted by a time-stepT and the Pearson Correlation coefficient is computed again, repeating this process until the whole time-course has been used to computed the Pearson Correlation coefficient [2]. Considering the parcellation of the brain into N different ROIs, the output is a temporal sequence of N ×N symmetric dFC matrices (Figure1.3, bottom image).

However, this approach has some limitations, starting from the window lengthW:

Figure 1.3:Sliding-window Pearson Correlation method to extract dFC (adapted from [2]). On the top are rep- resented two time-courses from two different ROIs and the window where the Pearson Correlation coefficient is computed. On the center is the result of the correlation across all windows. In the bottom are represented the dFC matrices across time, which is the result of the correlation across all windows (center image) for all pairs of ROIs

• a too short window length may have too few samples to compute a reliable correlation and it may increase the risk of introducing too much noise in the estimation of thedFC[36].

• if it is too long then some temporal variations of interest may be missed

So there is a need of a trade-off between specificity (long enough window length to detect reliable fluctuations) and sensitivity (short enough window length not to miss temporal fluctuations of interest) [2].

As a rule of thumb, a lower limit can be set to the largest wavelength present in the pre-processed fMRI time-courses in order to avoid artifacts [36], however there is no best length. Furthermore, having a fixed window length does not allow the analysis ofdFCfluctuations with a higher frequency than the imposed by the window period. [2].

Despite the disadvantages presented, the simplicity of the method and its ability to capture useful features of thedFCmade the approach one of the most preferred in the analysis ofdFC[2]. Although the window length is a parameter that cannot be improved, the window shape can, by adopting a ta- pered window instead of a rectangular window. The disadvantage of the rectangular window is that is gives equal weight to all observations inside the window, so if a highly influential outlier is included/re-

moved from one window to another, it will cause a sudden change in thedFCtime-course that could be misunderstood as an important change in brain connectivity [37].

The use of a sliding-window is not restricted to the calculation of the Pearson Correlation coefficient.

Other methods to compute theFCcan be used in a sliding-window approach in order to obtain a dynamic FC. It has been done withReHo[38] and also by computing sliding-windowICA[39], whereBOLDtime- courses are decomposed through ICA and spatial components are obtained for each window, being possible to observe their evolution over time.

Another family of approaches, which does not use sliding-windows, is based on time-frequency anal- ysis. Yaesoubi and colleagues [40] used the Wavelet Transform Coherence to estimate time-varying patterns ofFCthat were associated with specific frequencies.

One other method used to measuredFCis based on the computation of phase synchronization, or PC, across brain regions. It is a quite well-established tool in Magnetoencephalography (MEG)/EEG studies, however infMRIit is not very explored. Glerean and colleagues [9] were one of the first groups to apply it onfMRIand more recently Cabral and colleagues [41] and Figueroa and colleagues [42] also used this method, calling itPC.

In this method, the first step is to compute the analytic representation of the real valued signalx(t) (theBOLDsignal), which is a complex signalxa(t), built using the Hilbert transform:

xa(t) =x(t) +jH[x(t)] (1.1)

WhereH[·]refers to the Hilbert transform andj is the imaginary unit. Now, ifx(t)is a narrowband signal, it can be modeled as the product of an amplitude-modulated low-pass signala(t)and a sinusoidal carrierθ(t):

x(t) =a(t)cos[θ(t)] (1.2)

Using the representation in (1.2), by the Bedrosian’s theorem [43] the Hilbert transform ofx(t)is:

H[x(t)] =a(t)sin[θ(t)] (1.3)

So, the analytic representation in (1.1) becomesxa(t) =a(t)cos[θ(t)] +j(a(t)sin[θ(t)]), which by the Euler’s formula is:

x_a(t) =a(t)e^jθ(t) (1.4)

Where a(t)is the instantaneous amplitude andθ(t) the instantaneous phase. Given the phases of theBOLDsignals from two regionsnandp, at timet,P C(n, p, t)is computed by:

P C(n, p, t) = cos(θ(n, t)−θ(p, t)) (1.5) Wherecos()refers to the cosine function. Ascos(0) = 1, when two areas n and p have synchronized BOLDsignals at time t, thendF C(n, p, t) = 1. Otherwise,dF C(n, p, t) = 0when theBOLDsignals are orthogonal. The result of computing thedFCfor all brain areas, in each time point, is aN×N matrix (N denotes the number of brain areas), symmetric across the diagonal, thus the upper (or lower) triangular parts of thedFCmatrix capture all the relevant values.

Another method that can be used to compute the instantaneous phase is by convolving with a com- plex Morlet Wavelet. The complex Morlet wavelets are defined at each time pointt and frequency f by: [44]

G(t, f) =A×e^−t2/2σ2t ×e^{2πif t} (1.6) where A =1/p

σ_t×√

π, σ_t =1/2πσ_f,σ_f =f /ω₀. ω₀ corresponds to the ”wavelet factor” and governs the time and frequency resolution. However, nofMRIstudies have used complex wavelet convolution to retrieve the instantaneous phase and then computingPC.

The big advantage ofPCis that it allows to have a maximum temporal resolution of thedFC, as each dFCmatrix is computed at each Repetition Time (TR), being possible to analyze faster fluctuations of theFCthan using a sliding-window.