Separation of phase-locked sources in pseudo-real MEG data

2.1 Hilbert transform: phase of a real-valued signal

Usually, the signals under study are real-valued discrete signals. To obtain the phase of a real signal, one can use a complex Morlet (or Gabor) wavelet transform, which can be seen as a bank of bandpass filters[17]. Alternatively, one can use the Hilbert transform, which should be applied to a locally narrowband signal or be preceded by appropriate filtering[18] for the meaning of the phase extracted by the Hilbert transform to be clear. The two transforms have been shown to be equivalent for the study of brain signals[19], but they may differ for other kinds of signals. In this article, we chose to use the Hilbert transform. To ensure that this transform yields meaningful results, we will precede its use by band-pass filtering the pseudo-real MEG sources used in this article (see Section 4.1). Note that this is a very common preprocessing step in the analysis of real MEG signals (cf.,[20–22]).

Note that the Hilbert transform is a linear operator. The Hilbert filter h(t) is not causal and has infinite duration, which makes direct implementation of the above formula impossible. In practice, the Hilbert transform is usually computed in the frequency domain, where the above convolution becomes a product of the discrete Fourier transforms of x(t) and h(t). A more thorough mathematical explanation of this transform is given in[18, 23]. We used the Hilbert transform as implemented by MATLAB.

The analytic signal of x(t), denoted by$\tilde{x}\left(t\right)$, is given by$\tilde{x}\left(t\right)\equiv x\left(t\right)+\text{i}{x}_h\left(t\right)$, where$\text{i}=\sqrt{-1}$ is the imaginary unit. The phase of x(t) is defined as the angle of its analytic signal. In the remainder of the article, we drop the tilde notation; it should be clear from the context whether the signals under consideration are the real signals or the corresponding analytic signals.

2.2 Phase-locked sources

where a _j (t) are the amplitudes of the sources, which are by definition non-negative and real-valued. α _j is the constant dephasing (or phase lag) between the sources (it does not depend on the time t), ϕ(t) represents an oscillation common to all the sources (it does not depend on the source j), and δ _j (t) is the phase jitter, which represents the deviation of the j th source from its nominal phase α _j + ϕ(t). Throughout this article, we will assume that the phase jitter is Gaussian with zero mean and a standard deviation σ.

where ϕ _j (t) is the phase of oscillator j (it is unrelated to ϕ(t) in Equation (1)), ω _j (t) is its natural frequency, and κ _jk measures the strength of the interaction between oscillators j and k. If the κ _jk coefficients are large enough and ω _j (t) = ω _k (t) for all j,k, then the solutions of the Kuramoto model are of the form (1) with small δ _j (t).

2.3 PLF

where 〈·〉 is the time average operator. The PLF satisfies 0 ≤ ϱ _jk  ≤ 1. The value ϱ _jk  = 1 corresponds to two oscillators that are fully synchronized (i.e., their phase lag is constant). In terms of Equation (1), a PLF of 1 is obtained only if the phase jitter δ _j (t) is zero. The value ϱ _jk  = 0 is attained, for example, if the phase difference ϕ _j (t) − ϕ _k (t)modulo 2Π is uniformly distributed in [−Π,Π[. Values between 0 and 1 represent partial synchrony; in general, higher values of the standard deviation of the phase jitter δ _j (t) yield lower PLF values.

Note that a PLF of 1 is obtained if and only if ϕ _j (t) − ϕ _k (t) is constant^c. Thus, studying the separation of sources with constant phase lags can equivalently become the study of separation of sources with pairwise PLFs of 1.

Throughout this article, phase synchrony is measured using the PLF; two signals are perfectly synchronous if and only if they a PLF of 1. Other approaches exist, e.g., for chaotic systems or specific types of oscillators[27]. Studying separation algorithms based on such other definitions is outside of the scope of this article. The definition used here has the advantages of being tractable from an algorithmic point of view, and of being applicable to any situation where ϕ _j (t) − ϕ _k (t) is constant^d, regardless of the type of oscillator.

2.4 Effect of linear mixing on synchrony

Assume that we have N sources which have PLFs of 1 with each other. Let s(t), for t = 1,…,T, denote the vector of sources and x(t) = A s(t) denote the mixed signals, where A is the mixing matrix, which is assumed to be square and non-singular^e. Our goal is to find a square unmixing matrix W such that the estimated sources y(t) = W ^T x(t) = W ^T As(t) are as close to the true sources as possible, up to permutation, scaling, and sign change.

The effect of linear mixing on the PLF matrix is illustrated in Figure1 for a set of simulated sources. This set has three sources, with PLFs of 1 with each other. These sources are of the form (1) with negligible phase jitter, and the phase lags α _j are 0,$\frac{\Pi }{6}$, and$\frac{\Pi }{3}$ radians, respectively. The common oscillation is a time-dependent sinusoid. The amplitudes are generated by adding a small constant baseline to a random number of “bursts” with Gaussian shape. Each “burst” has a random center and a random width, and each source amplitude has 1 to 5 such “bursts”.

The first row of Figure1 shows on the left the original sources and on the right their PLF matrix. The second row depicts the mixed signals x(t) on the left and their PLFs on the right; the mixing matrix has random entries uniformly distributed between −1 and 1. It is clear that the mixed signals have lower pairwise PLFs than the sources, although signals 2 and 3 still exhibit a rather high mutual PLF. This example suggests that linear mixing of synchronous sources reduces their synchrony, a fact that will be proved in Section 3.3, ahead; this fact will be used to extract the sources from the mixtures by trying to maximize the PLF of the estimated sources.

3.1 Preprocessing

3.2 Separation of phase-locked sources

The goal of the IPA algorithm is to separate a set of N fully phase-locked sources which have linearly been mixed. Since these sources have a maximal PLF with each other and the mixture components do not (as motivated in Section 2.4 above and proved in Section 3.3 below), we can unmix them by searching for projections that maximize the resulting PLFs. Specifically, this corresponds to finding a N × N matrix W such that the estimated sources, y(t) = W ^T x(t) = W ^T A s(t), have the highest possible PLFs.

where w _j is the j th column of W. In the first term, we sum the squared PLFs between all pairs of sources. The second term penalizes unmixing matrices that are close to singular, and λ is a parameter controlling the relative weights of the two terms. This second term serves the purpose of preventing the algorithm from finding, e.g., solutions where two columns j and k of W are colinear, which trivially yields ϱ _jk  = 1 (a similar term is used in some ICA algorithms[7]). Each column of W is constrained to have unit norm to prevent trivial decreases of that term.

The optimization problem in Equation (6) is highly non-convex: the objective function is a sum of two terms, each of which is non-convex in the variable W. Furthermore, the unit norm constraint is also non-convex. Despite this, as we show below in Section 3.3, it is possible to characterize all the global maxima of this problem for the case λ=0 and to devise an optimization strategy taking advantage of that result.

The above optimization problem can be tackled through various maximization algorithms. Our choice was to use a gradient ascent algorithm with momentum and adaptive step sizes; after this gradient algorithm has run for 200 iterations, we use the BFGS algorithm implemented in MATLAB to improve the solution. The result of this optimization for the sources shown in Figure1 is shown in Figure2 for λ = 0.1, illustrating that IPA successfully recovers the original sources for this dataset.

3.3 Unicity of solution

3.4 Comparison to ICA

The above result is simple, but some relevant remarks should be made. If the optimum is found using λ = 0 and the second assumption is not violated (or equivalently, det(C) = det(W)det(A) ≠ 0, which is equivalent to det(W) ≠ 0 if A is non-singular), then we can be certain that the correct solution has been found. However, if the optimization is made using λ = 0, there is a possibility that the algorithm will estimate a bad solution where, for example, some of the estimated sources are all equal to one another (in which case the PLFs between those estimated sources is trivially equal to 1). On the other hand, if we use λ ≠ 0 to guarantee that W is non-singular, the unicity result stated above cannot be applied to the complete objective function. We call “non-singular solutions” and “singular solutions” those in which det(W) ≠ 0 and det(W) = 0, respectively. The result expressed in Section 3.3 is thus equivalent to stating that “all non-singular global optima of Equation (6) with λ = 0 correspond to correct solutions”.

This contrasts strongly with ICA, where singular solutions are not an issue, because ICA algorithms attempt to find independent sources and one signal is never independent from itself[7]. In other words, singular solutions always yield poor values of the objective function of ICA algorithms. Here we are attempting to estimate phase-locked sources, and any signal is perfectly phase-locked with itself. Thus, one must always use λ≠0 in the objective function of Equation (6) when attempting to separate phase-locked sources.

We use a simple strategy to deal with this problem. We start by optimizing Equation (6) for a relatively large value of λ(λ = 0.4), and once convergence has been obtained, we use the result as the starting point for a new optimization, this time with λ = 0.2. The same process is repeated with the value of λ halved each time, until five such epochs have been run. The early optimization steps move the algorithm away from the singular solutions discussed above, whereas the final steps are done with a very low value of λ, where the above unicity conditions are approximately valid. As the following experimental results show, this strategy can successfully prevent singular solutions from being found, while making the influence of the second term of Equation (6) on the final result negligible.

4.1 Data generation

As mentioned earlier, the main goal of this study is to study the applicability of IPA to real-world electrophysiological data from human brain EEG and MEG. The choice of the data for this study was not trivial, since we need to know the true sources in order to quantitatively measure the quality of the results. On the one hand, to know the actual sources in the brain would require simultaneous data from outside the scalp (EEG or MEG, which would be the mixed signals) and from inside the scalp (intra-craneal recordings, corresponding to the sources). If intra-craneal recordings are not available, results cannot quantitatively be assessed; they can only qualitatively be assessed by experts who can tell whether the extracted sources are meaningful or not. On the other hand, due to their extreme simplicity, synthetic data such as those used so far to illustrate IPA, shown in Figure1, cannot be used to assess the usefulness of the method in real-world situations.

In an attempt to obtain “the best of both worlds”, we have generated a pseudo-real dataset from actual MEG recordings. By doing this, we know the true sources and the true mixing matrix, while still using sources that are of a nature similar to what one observes in real-world MEG. We begin by describing the process that we used to generate a perfectly phase-locked dataset; we then explain how we modified these data to analyze non-perfect cases as well. It is important to stress that the generation process described below has no relation to the one used to generate the data of Figure1, even though both processes generate sources with maximum PLF.

Our first step was to obtain a realistic mixing matrix. To do so, we used the well-known EEGIFT software package[29]. This package includes a real-world sample EEG dataset with 64 channels. Using all the default options of the software package, we extracted 20 independent components from the data of Subject 1 in that dataset. The results that was important for us, in this process, were not the independent components themselves (which were discarded), but rather the 64 × 20 mixing matrix. As discussed in Section 3.1, we have opted for using a square mixing matrix, with little loss of generality. Therefore, we selected N random rows and N random columns of that mixing matrix (without repetition), and formed an N × N mixing matrix from the corresponding values of the original 64 × 20 matrix. We will later show results for datasets ranging from N = 2 to N = 5 sources; in the following, assume, for the sake of concreteness, that N = 4.

Having generated a physiologically plausible mixing matrix, the next step was to generate a set of four sources. For this, we used the MEG dataset studied previously in[30]^h, which has 122 channels with 17,730 samples per channel. The sampling frequency is 297 Hz, and the data have already been subjected to low-pass filtering with cutoff at 90 Hz. Since band-pass filtering is a very common preprocessing step in the analysis of MEG data[20–22] and is useful for the use of the Hilbert transform, we performed a further band-pass filtering with no phase distortion, keeping only the 18–24 Hz bandⁱ. The resulting filtered data were used to generate a complex signal through the Hilbert transform; these data were whitened as described in Section 3.1, and from the whitened data we extracted the time-dependent amplitudes and phases.

We then selected four random channels of these filtered MEG data. Since none of these MEG recordings were actually phase-locked (recall that they were themselves the result of a mixing process) and we wanted to study the performance on fully phase-locked sources (possibly corrupted by jitter, as explained below), we replaced the phase of the second of these channels with the phase of the first channel with a constant phase lag of$\frac{\Pi }{6}$ radians. The phase of the third channel was replaced with the phase of the first channel with a constant phase lag of$\frac{\Pi }{3}$ radians, and that of the fourth channel with the phase of the first channel with a lag of$\frac{\Pi }{2}$ radians. The amplitudes of the four sources were kept as the original amplitudes of the four random channels themselves. The process is illustrated in Figure3. The above process, including the choice of the 4 × 4 submatrix, was repeated 100 times, with different initializations of the random number generator. This way of constructing the data ensured that the sources were fully phase-locked.

We also constructed datasets in which the sources were not perfectly phase locked. For this, we used the same 100 sets of sources, but with those sources now corrupted by phase jitter: each sample t of each source j was multiplied by e ⁱ δ _j (t), where the phase jitter δ _j (t) was drawn from a random Gaussian distribution with zero mean and standard deviation σ. We tested IPA for σ from 0 to 20 degrees, in 5 degrees steps. One example with σ = 5 degrees is shown in Figure4, and one with σ = 20 degrees is shown in Figure5.

Finally, we studied the effect of N on the results of the proposed algorithm. We created 100 datasets similar to the jitterless datasets mentioned earlier, using N = 2,3 and 5. In all of these, and similarly to the data with N = 4, we used sources with phase lags multiple of$\frac{\Pi }{6}$.

4.2 Results

We measured the separation quality using two measures: the Amari performance index (API)[31] and the well-known signal-to-noise ratio (SNR). The API measures how far the gain matrix W ^T A is from a permutated diagonal matrix; the SNR measures how far the estimated sources are from the true sources. In summary, the API measures the quality of the estimation of the mixing matrix, while the SNR measures the quality of the estimation of the sources themselves.

Figure6 presents the means and standard deviations of these measures for the 100 runs mentioned in Section 4.1, for each of the jitter levels. The results indicate that IPA has very good performance on the jitterless case, in data of this kind, and that this level of performance is approximately maintained even in the presence of low levels of phase jitter, up to 5 degrees of standard deviation. Some deterioration in performance occurs from 5 to 10 degrees of phase jitter standard deviation, but with a SNR of 27 dB and an API below 0.1 the sources can still be considered to be well estimated.

The results for high jitter levels (sigma equal to 15 or 20 degrees) show that there is a limit to IPA’s robustness; this limit lies somewhere between 10 and 15 degrees. Equivalently, in terms of the PLF, the algorithm shows good robustness to PLF values smaller than 1 as long as they are above 0.95, but below that value its performance deteriorates progressively up to a PLF of approximately 0.9, at which point only partial separations are obtained.

Figure7 shows the effect of varying the number of sources N. The figure shows that IPA can handle values of N up to N = 5 with only a slight decrease in performance.

Figure7 also shows something rather peculiar: for N = 2, the results are mediocre (with an average API around 0.4)^j. This is not an effect of lowering the number of sources N, but rather an indirect effect of the phase lag between the sources. To verify this, we generated datasets of jitterless data with N = 2, using phase lags of$\frac{\Pi }{12}$,$\frac{2\Pi }{12}$ (the value used in Figure7),$\frac{3\Pi }{12}$, and$\frac{4\Pi }{12}$ (100 datasets for each of these values). Figure8 shows that a phase lag of$\frac{2\Pi }{12}$ yields poor API values, as we already knew, but$\frac{3\Pi }{12}$ yields very good values. Naively, one could conclude that when the sources have a phase lag of$\frac{2\Pi }{12}$, or less, the separation cannot be accurately performed.

The effect is, however, not so clear-cut. The results for N = 3,4,5 also involve sources with phase lags of$\frac{\Pi }{6}$, but the API values for those experiments are very good. We do not have a solid explanation for this fact; we conjecture that the presence of some pairs of sources with larger phase lags (e.g., for N = 4, the first and third sources have a phase lag of$\frac{\Pi }{3}$ and the first and fourth sources have a phase lag of$\frac{\Pi }{2}$) aids in the separation of all the sources.