Wavelet-Based Localization of Oscillatory Sources From ... - IEEE Xplore

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IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 61, NO. 8, AUGUST 2014

Wavelet-Based Localization of Oscillatory Sources From Magnetoencephalography Data J. M. Lina∗ , R. Chowdhury, E. Lemay, E. Kobayashi, and C. Grova

Abstract—Transient brain oscillatory activities recorded with Eelectroencephalography (EEG) or magnetoencephalography (MEG) are characteristic features in physiological and pathological processes. This study is aimed at describing, evaluating, and illustrating with clinical data a new method for localizing the sources of oscillatory cortical activity recorded by MEG. The method combines time–frequency representation and an entropic regularization technique in a common framework, assuming that brain activity is sparse in time and space. Spatial sparsity relies on the assumption that brain activity is organized among cortical parcels. Sparsity in time is achieved by transposing the inverse problem in the wavelet representation, for both data and sources. We propose an estimator of the wavelet coefficients of the sources based on the maximum entropy on the mean (MEM) principle. The full dynamics of the sources is obtained from the inverse wavelet transform, and principal component analysis of the reconstructed time courses is applied to extract oscillatory components. This methodology is evaluated using realistic simulations of single-trial signals, combining fast and sudden discharges (spike) along with bursts of oscillating activity. The method is finally illustrated with a clinical application using MEG data acquired on a patient with a right orbitofrontal epilepsy. Index Terms—Electrophysiological imaging, epilepsy, inverse problem, magnetoencephalography (MEG), maximum entropy on the mean (MEM), wavelet representation.

Manuscript received August 24, 2011; revised January 15, 2012; accepted February 11, 2012. Date of publication March 6, 2012; date of current version July 15, 2014. This work was supported in part by the Natural Sciences and Engineering Research Council of Canada (Discovery Grant Program), in part by the Canadian Institutes of Health Research, and in part by the Montreal Neurological Institute-CECR. The work of E. Kobayashi was supported by the AES Early Career Clinician-Scientist Award and the Fonds de Recherche en Santé du Québec (FRSQ). The work of C. Grova was supported by the Fonds de Recherche en Santé du Québec (FRSQ). The work of R. Chowdhury was supported by a Savoy Foundation scholarship. Asterisk indicates corresponding author. ∗ J. M. Lina is with the Department of Electrical Engineering, Ecole ´ de Technologie Supérieure, Montréal H3C 3K7, Canada, and also with the Centre de Recherches Mathématiques, Montréal 6128, Canada (e-mail: jmlina@ ele.etsmtl.ca). R. Chowdhury is with the Department of Biomedical Engineering, McGill University, Montréal H3A 0G4, Canada (e-mail: [email protected]). E. Lemay is with the Department of Electrical Engineering, Ecole de Technologie Supérieure, Montréal H3C 3K7, Canada (e-mail: etienne.lemay.1@ ens.etsmtl.ca). E. Kobayashi is with the Department of Neurology and Neurosurgery, McGill University, Montreal H3A 0G4, Canada, and also with the Montreal Neurological Institute, Montreal, QC H3A 2B4, Canada (e-mail: mni.kobayashi@ gmail.com). C. Grova is with the Department of Biomedical Engineering, McGill University, Montréal H3A 0G4, Canada, and also with the Montreal Neurological Institute, Montreal QC H3A 2B4, Canada (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TBME.2012.2189883

I. INTRODUCTION RAIN functional imaging from noninvasive electrophysiological measurements still benefits considerably methodological developments, as cognitive and clinical investigations call for accurate localization of the neuronal substrates of brain activity. With a high-frequency sampling rate, electroencephalography (EEG) recordings and more recently magnetoencephalography (MEG) recordings provide a direct measurement of the electromagnetic activity of the brain. However, to be considered as functional imaging techniques, EEG and MEG require careful numerical pre- and postprocessing in order to provide a reliable functional map from scalp recordings of brain activity [1]. Advanced models and numerical methods rely on trustworthy physiological assumptions, exploring data in their full spatial, temporal, and spectral ranges [2]. Indeed, the brain is a complex system that presents different properties at various spatial and temporal scales. Among these, the so-called cortical oscillations play a crucial role in information processing and encoding [3], [4]. Brain activity recruits one or few regions of the brain in which massive synchronous postsynaptic discharges generate local currents that correspond to local sources of electrophysiological signals measured outside the brain. Neurons can spontaneously (or through stimulation) produce oscillatory activity [5] and when a critical mass is firing synchronously, oscillations can be recorded either on the scalp (EEG) [6] [7] or through the magnetic flux measured by MEG sensors [8]. Typical brain oscillations are defined in the wellestablished spectral bands: delta (0.5–4 Hz), theta (4–8 Hz), alpha (8–20 Hz), beta (20–30 Hz), gamma (30–80 Hz), and ripple ranges (80–150 Hz) [7]. Several studies have investigated bursts of oscillatory MEG or EEG signals in cognition, suggesting that these processes may be interpreted as the synchronous “binding” of multiple brain areas involved during a specific task [4], [9]. From a clinical point of view, the occurrence of bursts of spikes and waves or oscillatory activities is also relevant in epilepsy [10], especially during the interictal period (i.e., between the seizures) [11]. Recent investigations on transient oscillatory interictal activities have revealed a close relationship between fast oscillations and the epileptogenic regions of the brain [12]–[14]. Whereas EEG and MEG interictal spikes are spontaneous transient events which are most frequently identified and sometimes localized during noninvasive investigation of the epileptogenic focus [15], [16], source localization of extended generators of oscillatory activity in specific frequency bands remains an open field. Indeed, interictal gamma discharges and even faster discharges which are not “visible” in the standard time scales used for browsing of EEG signals

B

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LINA et al.: WAVELET-BASED LOCALIZATION OF OSCILLATORY SOURCES FROM MAGNETOENCEPHALOGRAPHY DATA

have been shown to be important biomarkers in gold standard intracranial recordings [17], [18]. This study describes in detail a source localization methodology dedicated to image cortical sources associated with oscillatory transients from MEG signals. Such oscillations of interest can be undesirably removed when applying methods on averaged trials or coregistered events, as they are not necessarily mutually phase locked. Consequently, our methodology is aimed at addressing the issue of localizing single-trial events, which are usually associated with low signal-to-noise ratios (SNR). Assuming a distributed model of dipolar sources along the cortical surface [19], the proposed approach relies on a sparsity assumption in both the time and space dimensions. Sparsity in time amounts to representing the data in a discrete wavelet basis, while sparsity in space relies on a parcelization of the cortical surface in a few putative functional regions. Time and space are, thus, considered at various scales in this formalism, in which the final estimation of cortical activity is based on the maximum entropy on the mean (MEM) approach. This new methodology leads to a unified paradigm to handle spatiotemporal MEG/EEG time series. MEM is a probabilistic approach that has been introduced both in the probabilistic literature [20] and in the signal and information processing [21]. It was quickly applied in neuroscience [22]–[24], and further developed in the context of MEG [25]. We have applied MEM for source localization associated with epileptic discharges [26] as well as for comparison with hemodynamic signals [27]. The ability of the MEM to recover generators of epileptic activity with their spatial extent has been further demonstrated in [28]. A. Related Works Time–frequency (t–f) analysis of electrophysiological data is not new. Thirty years ago, anticipating the filter banks approaches of the wavelet analysis, de Weerd et al. [29], [30] proposed a framework for studying the frequency structure of EEG signals. More recent studies proposed to localize oscillating activity using equivalent current dipole (ECD) modeling in order to explain either band-passed filtered MEG data [31] or short epoch of oscillatory activity [32]. In this framework, brain activity is summarized with a very small number of focal dipolar sources. An imaging approach for localizing oscillatory activities based on the time–frequency representation of singletrial MEG data has been proposed in [2]. In this approach, the minimum norm estimate (MNE) maps the continuous wavelet representation of the data over the cortical surface. This made it possible to quantify the power, the induced power, and the synchronicity of the sources among trials in the frequency bands of interest. Despite its undeniable interest, this approach has some limitations, like for instance, the redundancy of the continuous wavelet representation that prevents any stable reconstruction of the dynamics of the sources. Source imaging on t–f atoms has been investigated in [33] and [34] in the framework of multivariate matching pursuit applied in the sensor space. This approach tends to localize the brain activity from isolated t–f atoms selected from a dictionary using a greedy algorithm. This suboptimal sparse representation of the data may generate unstable

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localization of brain activity. In [35], the t–f representation is part of the model of the sources, and a Bayesian approach infers the source dynamics from the MEG/EEG time series. This kind of generative model, based on an ad hoc t–f dictionary, has been further studied in [36] where the sparsity constraints, both in time and space, regularize the mean least-squares optimization solver. Beamforming techniques have also been used in combination with time–frequency representation of the data [37], [38]. It is worth mentioning here the dynamic imaging of coherent source (DICS) approach proposed by Gross et al. [39], which is based on the beamforming techniques. Despite the net advantage for scanning the entire brain volume, such spatial filtering approaches assume decorrelated sources, which may be inappropriate for localizing coherent oscillatory activity over extended brain regions networks. They also require a reliable and accurate estimate of frequency-tagged cross-covariance matrices which may be difficult to invert in some circumstances [40]. B. Notations Bold m(t) will denote the MEG data vector of dimension equal to the number sensors, Nd . The sources q(t) are dipoles (nA.m units) distributed over a dense mesh that tessellates the cortical surface. In general, we consider Ns 4000 dipoles uniformly distributed over the gray/white matter interface. C. Assumptions, Contributions, and Organization of This Paper We propose a new imaging method to localize brain’s oscillatory activities measured using MEG and based on the assumption that this activity results from processes occurring at various time scales (i.e., frequency bands) located in some extended cortical areas. This study introduces a new approach that combines, in a unified framework, the temporal modeling using the discrete wavelet representation of both data and sources (see Section II) together with a spatial clustering of the brain activity in homogeneous parcels (see Section IV). Each parcel is characterized with a hidden state variable that controls whether or not the parcel is active, thus modeling the probability distribution of the wavelet coefficient of the dipolar sources inside the parcel (see Section III). In this model, we assume that recorded MEG signals result from a mixture of a physiological background and a signal of interest that we aim to localize. This is particularly true when dealing with localizing single trials, while averaging data over trials will generally reduce the influence of the physiological background. The MEM principle is then applied to infer the wavelet coefficients of each dipolar source in each parcel involved in the oscillatory process. This allows us to focus on a few specific time-scale components that correspond to the specific oscillations to be localized or to reconstruct the entire temporal dynamics of the sources from the estimated wavelet coefficients. Both approaches are evaluated (see Section V) and compared in the context of localizing epileptic spikes and rhythmic oscillations, using a realistic simulation framework. Application of the method is then illustrated in the context of epilepsy for localization of epileptic spikes and bursts of fast oscillating interictal discharges (see Section VI). A discussion and a conclusion will follow.

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II. TIME–FREQUENCY FORWARD MODEL The general forward model in MEG relies on a lead field matrix G. Based on the electromagnetic origin of the signals, this matrix expresses the most realistic relationship between the bioelectric sources and the set of sensors with which they are coregistered. In this study, a realistic head model was extracted from an individual anatomical MRI with dipolar sources distributed along the cortical surface and oriented perpendicularly to it. To estimate the lead field matrix, steady-state Maxwell equations were solved using a one-layer boundary element model (BEM) [41] with OpenMEEG software [42]. Then, the following model m(t) = Gq(t) + σ 1/2 (t)

(1)

relates the Ns -dimensional source vector q with the Nd dimensional sensor vector m, at any given time t. (t) is an i.i.d. noise of unit variance and σ is an Nd × Nd variance–covariance matrix describing the correlations in the data space. Given a discrete orthogonal wavelet basis {ψj,k , j ∈ Z, k ∈ Z}, we consider the wavelet transform of the temporal series for both the sources and the measurements (see Appendix A): wj,k ψj,k (t) (2) q(t) = m(t) =

j

k

j

k

dj,k ψj,k (t).

(3)

In these expansions, ψj,k (t) are translations (index k) of the mother wavelet appropriately dilated to scale 2j (the scale with j=0 being the sampling scale). In terms of the wavelet expansions, (1) can be written as dj,k = Gwj,k + wj,k

(4)

where dj,k , wj,k , and wj,k are the wavelet coefficients for the data, the sources, and the measurement noise, respectively. Writing the source wavelet coefficients as a sum of a deter∗ ) that we aim to estimate and a zeroministic process (wj,k mean random process (zj,k ), we define the random process such that ηj,k = Gzj,k + wj,k ∗ dj,k = Gwj,k + ηj,k .

(5)

Since ηj,k is a sum of a large number of random processes through the linear operator G, we may assume that ηj,k is a zeromean Gaussian multivariate random variable of dimension Nd , with a variance–covariance matrix Σd . It is worth mentioning that we can preprocess the data in order to increase the SNR using a standard shrinkage of the wavelet coefficients [43] d∗j,k = W −1 δτ (W dj,k ).

(6)

−1/2

In this expression, W = Σd is the whitening matrix that can be estimated from a baseline recording in which we can assume ∗ ≈ 0. δτ (w) is the the absence of any signal of interest, i.e., wj,k soft shrinkage of the wavelet coefficients [43], [44] defined by δτ (w) = (1 − τ /|w|)+ w with w+ = w if w > 0, and 0 otherwise. For each channel, the threshold τ is obtained from the variance of the high-

Fig. 1. MEG recording of oscillatory activities in epilepsy. (a) Unfiltered raw data of one channel. A 500-ms burst of rapid oscillation (box) between two spikes (and a third one following immediately) indicated with arrows. (b) Corresponding wavelet-based denoised signal. 70% of the wavelet coefficients have been set to 0 in the preprocessing that preserves most of the oscillatory activities of interest.

est frequency wavelet coefficients, estimated using √the median estimator (see [45, p. 565] for instance), τ = η˜ 2 ln N . η˜ = med(|d1,. |)/0.6745 is an estimator of the noise level and N is the number of temporal samples. A wavelet coefficient below τ will be considered as contributing to noise and set to 0. It is assumed that the nonlinear denoising preprocessing step (6), illustrated in Fig. 1, maintains the signal and the spatial covariance to be explained with the generative model, at least for the SNR level of the expected value of the source intensities. Then, the equation to be solved is of the form ∗ ∗ + ηj,k d∗j,k = Gwj,k

(7)

∗ is seen as the expected value Ep [wj,k ] for some where now wj,k ∗ is a residual noise. probability p to be found and ηj,k The main contribution of this study is to provide an estimate ∗ . Note that of the wavelet coefficients of the sources, i.e., wj,k wavelet coefficients of the data we want to localize d∗j,k can either be obtained from the aforementioned denoising preprocessing step or, more conventionally, by averaging data over multiple trials, assuming such an average will extract the activity of interest (e.g., evoked related fields).

III. TIME–FREQUENCY MEM ESTIMATION OF BRAIN ACTIVITY The MEM principle solves the generative model of the data (7), with respect to the joint probability distribution over the intensities of the distributed sources. This approach is usually applied to solve the inverse problem in the time domain [25]. It can also be applied in the wavelet representation, where the time index is changed for time–frequency indices, as in (7). A. wMEM: MEM in the Wavelet Domain Let us denote as p(w) the joint probability of the wavelet coefficients of all sources, at a particular discrete time (k) and scale (j). In this section, the indices k and j have been removed for clarity. The MEM estimation infers the expectation


Ep [w] assuming a reference probability μ(w) from which the entropy deviation is minimized under the goodness-of-fit data constraint. More explicitly, starting with the entropy of any μdensity p(w) = f (w)μ(w) defined by f (w) ln f (w) μ(w) dw (8) Sμ (f ) = −

In this expansion, the null parcel “p = 0” plays a special role, which will be described later. Defining the log-partition function for each parcel, the optimization problem will now involve the following functional:

w ∗

where d are given by (6). From the log-partition function associated with the reference density μ(w) t ln Z(λ) = Fμ∗ (Gt λ) with Fμ∗ (ξ) = ln eξ w μ(w) dw (10) ∗t the solution of (9) is of the form p∗ (w) = Z(λ∗ )−1 eλ G w μ(w) where λ∗ is the unique solution of

λ∗ = argmax D(λ) λ

(11)

∗

D(λ) = λ d −

Fμ∗ (Gt λ)

1 − λt η ∗ λ 2

(12)

where η ∗ is the covariance matrix of the residual noise in (7). Then, the a posteriori mean estimate of the wavelet coefficients of the sources is the expectation of the coefficient with respect to the optimal μ-density f ∗ , given by the following expression: d ∗ Fμ (ξ) w∗ = wf ∗ (w)μ(w) dw = . (13) ∗ dξ ξ =G t λ This is what we propose to compute for each discrete point (j, k) of the time-scale plane. Then, to address the source localization problem, we consider either the spatial cortical map of the wavelet coefficients of the sources at a particular time and scale (i.e., frequency band), or the time courses of the sources using the inverse wavelet transform. Let us observe in (8) that the reference law μ(w) entering in the MEM formalism is the zero-entropy source configuration and will be defined in terms of the hypothesis and some information extracted from the data. This probability density is discussed in the next sections. B. MEM Formalism for a Parcelized Cortex We first assume that the cortical surface can be segmented into spatially homogeneous areas of a given size. The next section will describe a data-driven parcelization approach [46]. Given such a collection of P + 1 parcels of dipoles and assuming that they are spatially independent, we factorize the joint probability law with respect to it μ(w) =

P p=0

μp (wp ).

(14)

P p=0

1 Fμ∗p (Gtp λ) − λt η ∗ λ 2

(15)

where Gp is the Nd × np dimensional subleadfield matrix corresponding to the parcel “p.” Each parcel gathers dipoles in a common state Sp that controls the activation status of the parcel. The collection of those P hidden state variables {S1 , S2 , . . . , SP } was introduced in [25] and leads to the following mixture law: μp (w) = (1 − αp )μp,0 (w) + αp μp,1 (w), 0 ≤ αp ≤ 1 (16) where αp represents the probability that parcel p will be in the active state Sp = 1. Using normal distributions to describe the wavelet coefficients of the dipoles in parcel “p,” at a specific time and scale, either in the silent state, where Sp = 0, or in the active state, where Sp = 1, Sp = 0 : N (0, σ0 I) and Sp = 1 : N (ωp , Σp )

(17)

we obtain

∗ ∗ Fμ∗p (ξ) = ln (1 − αp ) eF p , 0 (ξ ) + αp eF p , 1 (ξ )

with the functional [21] t

∗

D(λ) = λ d − t

w

the optimal solution will be given by w∗ = Ep ∗ [w], where p∗ (w) = f ∗ (w) μ(w) and ⎧ ∗ ⎪ ⎨ f = argmaxf Sμ (f ) (9) ⎪ ⎩ with G wf (w)μ(w) dw = d∗

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(18)

∗ ∗ in (15), with Fp,0 (ξ) = σ20 ξ t ξ, Fp,1 (ξ) = 12 ξ t Σp ξ + ξ t ωp . The parameters (σ0 , ωp , Σp ) will be defined in Section IV. Solving (11) for λ∗ , the solution is finally found to be of the following form:

wp∗ = αp∗ ωp + (1 − αp∗ )σ0 Gp t λ∗ + αp∗ Σp Gp t λ∗

(19)

where αp∗ updates the probability that parcel “p” will be active: αp∗ =

αp ∗ αp + (1 − αp )e−Δ F p

(20)

with 1 ∗t λ Gp (Σp − σ0 I)Gp t λ∗ + λ∗ t Gp ωp . (21) 2 To summarize, from the wavelet coefficients of the MEG data d∗j,k injected into functional (15) and solving the optimization problem (11), (19) and (20) estimate the wavelet coefficients wj,k for all sources of parcel “p,” at specific values of discrete time k and scale j. The first term on the r.h.s of (19) is nothing but the shrinkage of the mean value of the initial wavelet intensity in the parcel “p.” The other two terms further correct this wavelet coefficient in order to describe the sources inside the parcel. It is worth mentioning that the MEM estimate is not linear and thus does not commute with the wavelet transform. ΔFp∗ =

IV. DEFINITION AND INITIALIZATION OF THE REFERENCE MODEL The previous section emphasizes the definition of the reference μ that somehow contains the prior information that regularizes the inversion. This zero-entropy solution represents source activity related to some physiological background. Again, this definition is expressed for the wavelet coefficients of the sources,

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in any specific box in the time-scale representation. Our first objective is to define such a reference distribution μ(w) from which the MEM estimation will be obtained. Two main features must be specified: the set of P parcels and the parameters (αp , ωp , Σp , σ0 )p=1,...,P of the reference distribution inside each parcel. A. Parcelization of the Cortex The parcelization will be designed such that parcels can nicely represent the activity that we are looking for, if any. As in [46] and [26], we use a growing-region technique from selected sources (seed points), constrained to be convex regions along the geodesic cortical surface. Parcel size is the single ad hoc parameter, and a greedy selection of seed sources will rely on scores computed for each source. Those scores a(i), between 0 and 1, are computed from the data using the multivariate source prelocalization (MSP) approach [47] described in Appendix B. The higher the MSP score, the more probable the contribution of the corresponding sources to the source localization solution. Consequently, parcels are constructed mostly from sources with high a(i) scores. Defining the entire set of sources as the initial null parcel, each parcel is constructed iteratively from selected seed points using a growing region algorithm along the cortical surface, up to a prespecified scale s. At any iteration p = 1, 2, . . ., sources of the parcel p are removed from the null parcel “p = 0.” This pruning algorithm stops when the score of the next seed point is below some threshold a∗ . All sources remaining in the null parcel will be involved in describing only the physiological background or, as modeled in the previous state model, they will be characterized by α0 = 0 in (16). The threshold a∗ can eventually be set to 0, in which case the null parcel would be empty and all the sources would be involved in the localization solution, as evaluated in [26]. However, if a reliable physiological background recording is available, the threshold can be obtained using the false detection rate (FDR) [48] approach described in the next section. B. Null Parcel The null parcel is the set of sources that remains unselected in the previous parcelization algorithm. This set depends on the threshold a∗ that defines the lowest score of the seed of an a priori active parcel. Assuming that some baseline data recording describes the physiological background activity, our objective is to extract significant activity of interest that can differentiate from the background. To do so, our strategy is to ensure that selected sources with high MSP scores minimize the risk of constructing a false-positive parcel. Let us define as the null hypothesis the distribution of the MSP scores corresponding to some baseline data, thus modeling physiological background (see Fig. 2). As described in Appendix B, we use the baseline data in order to compute Pb (a) = Prob(score > a|baseline). Then, given data with putative sources of interest to be localized, it is possible with the FDR criterion to control the expected proportion of false positives relative to the number of selected sources [48]. At each time sample, a centered window of a given size (corresponding to the temporal scale) defines a piece of data

Fig. 2. Typical probability distribution of the MSP scores. The black crosses are obtained from a baseline data recording (used to model the physiological background activity in Section VI, which is free of any epileptic discharge). The black line is a fit with a Beta law (α = 20.1, β = 31.4). The red line is obtained from the data related to the time–frequency box (14 Hz, onset of spike) for one of the epileptic spikes illustrated in Fig. 12. The blue dashed line indicates the FDR threshold (p < 0.05): all sources with a score on the r.h.s of this line will be kept for localizing the time–frequency box previously mentioned.

with which MSP scores are computed. From the distribution of these scores (red line in Fig. 2), the FDR threshold is found to be the highest value a∗ such that Pb (a)
a∗ . iv) initialize the reference probability μp of each parcel: ωp (23), αp (24), and Σp (26). v) solve (11) for λ∗ . vi) compute wj∗k as in (19). 3) The inverse wavelet transform of the sources ∗ wj,k ψj,k (t). q∗ (t) = j

Fig. 3. The principle of the MEM approach. The gray ellipse represents the set C of all the probability densities p = f μ that satisfy the data-goodness-offit (9). H 0 is the initial reference from which a first solution in C is found. We then consider the parcelization of the sources and initialize the parcel state parameters. This lead to the reference μ from which the MEM optimal solution p ∗ is finally obtained.

(Aij = 1 if sources i and j are distinct and linked, 0 otherwise), we first compute the discrete Green function as defined in [50], [51] Γ(ρ) =

L n =0

ρn

Añ n!

(25)

where A˜ij = Aij and A˜ii = − j Aij mimic a Laplacian operator of order 1. L represents the maximum geodesic length of local spatial correlations, and ρ scales the Laplacian operator in this expansion. Denoting by Γ(ρ)p the np columns of Γ(ρ) associated with the dipoles of the parcel “p,” the covariance matrix related to it is defined by [52] Σp = σ1 ωp2 Γ(ρ)tp Γ(ρ)p .

(26)

Here, σ1 scales the local variance in the parcel and calibrates the precision of our estimation. Here, we use σ1 = 0.05 and L = 8. Two cases will be considered: 1) the case ρ = 0 corresponds to Σp = I, i.e., no spatial correlation within the parcel; and 2) the case ρ = 0 (we will use 0.3) introduces a nontrivial spatial covariance, i.e., a local spatial smoothing within the parcel. D. Numerical Implementation Let us first restate the steps of the implemented algorithm illustrated in Fig. 3. 1) The ‘null distribution’ Pb (a) of the MSP scores. From the baseline MEG data, we model the “H0 ” distribution of the MSP scores (see Appendix B). 2) The discrete wavelet transform of the MEG data of interest. We use the real Daubechies filter banks with four vanishing moments in the following steps. a) Denoising: Wavelet shrinkage (6), dj k → d∗j k . b) For each multivariate wavelet coefficient d∗j k = 0, we i) compute the MSP scores a(i), i = 1, . . . Ns , from the time series associated with the selected t–f box. ii) compute the FDR threshold a∗ [with (22)].

k

The wavelet processing uses the WaveLab library [53], whereas the MEM is implemented within the MATLAB environment. It is worth mentioning that optimization (7) is solved in a space of dimensions equal to the number of sensors, in general more than ten times smaller than the number of sources. Consequently, convergence is generally well achieved in a reasonable computational time (between 5 and 20 s for each wavelet coefficient, with a standard PC equipped with a 1.8 GHz processor and 3 GB of RAM). Note that the original version of the MEM and the wavelet-based MEM have been made available in the Brainstorm library [54]. V. LOCALIZATION OF TRANSIENT OSCILLATIONS FROM MEG DATA: REALISTIC SIMULATIONS Following the validation framework we previously proposed in [26], we generated a set of realistic MEG data simulations in order to quantify the accuracy of the proposed wavelet-based MEM method, denoted wMEM. Our objective was to assess the accuracy of wMEM in localizing generators of two types of brain activity: one epileptic spike followed by a burst of fast oscillatory activity. We simulated some temporal overlap between these generators, and the main peak of the spike was set before the maximum of the oscillation. Receiver operating characteristic (ROC) curves were used as a validation metric to assess the detection accuracy of the wMEM method. A. Realistic Simulations of MEG Epileptic Spikes and Oscillations For realistic simulations over normal background MEG signals, we used segments free of epileptic discharges from a patient with mesial temporal lobe epilepsy, in whom we could clearly identify normal activity. Informed consent was obtained for the study, as approved by the Research Ethics Committee of the Montreal Neurological Institute (MNI). MEG acquisition was performed on a VSM-CTF system (275 MEG sensors, 600 Hz sampling rate) of the MEG Centre (Université de Montréal). Preprocessing of the physiological background consisted in a third-order gradient correction for magnetic external background removal, a band-pass filter 0.3–70 Hz, no prewhitening of the data but the denoising was performed using the wavelet shrinkage. Anatomical MRI data were acquired at the MNI on a Siemens Trio 3T scanner. Coregistration between the MEG sensors position and the MRI was ensured by locating fiducial points (left ear, right ear, and nasion) using a Polhemus digitalizating system.

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Fig. 4. Time-scale (with frequency correspondence in hertz) representations of the two simulated signals: spike and oscillatory burst. The two white crosses at j = 5 (i.e., ≈14 Hz) and j = 3 (i.e., ≈56 Hz) indicate the time-scale coefficients selected in order to localize the two signals in the time–frequency approach.

A realistic head model was obtained by segmenting the surface of the brain from the subject’s MRI. The cortical surface was obtained from a segmentation of the gray/white matter interface from the MRI of the subject, using Brainvisa software [55], [56]. The forward matrix G was estimated using the BEM proposed in [41] and [42], using a one-layer model (i.e., the inner skull surface only). Monte Carlo simulations were generated using 100 configurations involving two spatially extended sources. The first one was associated with the time course of an epileptic spike and the second with the time course of a burst of oscillatory activity. For each simulation, the position of each source was selected by randomly choosing a seed point on the cortical surface mesh. The spatial extent of an active parcel was obtained by region growing around the seed, using a spatial neighborhood of order 3 along the geodesic surface (∼6 cm2 ). The amplitude of each vertex of the simulated source was set to 9.5 nA.m, generating an overall maximum signal of 1.5 pT for MEG when all the cortical sources were set as active. In order to generate realistic MEG simulations, we assessed the effect of source amplitude rather than the SNR of our simulated data. Thus, for a typical source amplitude, deep sources will result in lower SNR data than superficial sources. The amplitude of 9.5 nA.m was used as our reference and corresponds to an SNR of 0 dB for most superficial sources when MEG physiological background data are added. B. Simulated Spikes and Oscillations The time course of the first simulated source was the epileptic spike modeled with three Gamma functions as in [26]. The time course of the burst of oscillatory activity was modeled as a 40Hz sine wave damped using a Gaussian window (70 ms standard deviation), peaking 150 ms after the first peak of the spike (see Fig. 4). MEG signals were obtained by multiplying the simulated source intensities Jtheo by the forward model G, and further corrupted by adding real MEG background. Realistic physiological noise was extracted from a 3-min segment of MEG background activity, ensuring that no epileptic discharges occurred in this segment. Periods with motion and eye blinks

Fig. 5. Example of three simulated cases. The global field power over the MEG sensors is displayed for R = 1, 2, and 6. In case R = 1, the red and blue segments isolate the portion of the signal related to the spike and the burst, respectively. In cases R = 2 and R = 6, the blue segment indicates the full oscillatory activity (spike and oscillation).

were also excluded, resulting in the selection of 128 trials of 700 ms of MEG background activity. To mimic realistic data, each simulation consisted of one trial, using a background segment randomly chosen among the 128 segments. The amplitude of all 128 segments was scaled to ensure an SNR of 0 dB for most superficial sources when using a reference amplitude of 9.5 nA.m along a patch of ∼6 cm2 . An extra 2-s segment of MEG data, independent of the previous segments, was selected in order to model the null hypothesis (see Appendix B). Moreover, we chose to simulate single-trial data since bursts of oscillatory activity are rarely phase locked and will have a tendency to cancel out through averaging. The effect of the source amplitude was quantified using a signal-to-background ratio R = 0.5, 1, 1.5, 2, 4, and 6, resulting in source amplitudes ranging from 4.75 to 56 nA.m, respectively, whereas the amplitude of the MEG background segment used to corrupt the data was the same under all conditions. In this context, for each specific amplitude level (each R), simulation of deep sources actually results in simulated signals with lower amplitude than that of superficial sources. Fig. 5 illustrates the global field power (sum of the normalized energy over all the MEG sensors), for R = 1, 2, and 6. Colored segments indicate the spiking and oscillatory part of these particular single-trial simulated data. The effect of source depth was quantified by estimating the eccentricity of each simulated source. The eccentricity e was defined as the mean distance between each dipole of the simulated patch and the center of the head, defined as the mean point between the nasion and the left and right periauricular locations. For the anatomical data used for this simulation, e < 40 mm corresponds to subcortical structures, an eccentricity of 40 mm < e < 60 mm corresponds to generators located in the mesial and polar parts of the temporal lobes, and e > 60 mm corresponds to neocortical generators. C. ROC Analysis To assess the detection ability of wMEM to localize these simulated data, we used the area under the ROC curve (AUC), as a detection accuracy index. Denoting Jtheo as the theoretical


time course of the simulated sources, Jˆ is the corresponding ˆ being the corresource estimate obtained using wMEM (E sponding quadratic energy of the estimated source intensities). AUC was estimated at one particular box of the time–frequency plane (see Fig. 4) as well as at one particular time point t0 , once the dynamics of the source had been reconstructed through inverse wavelet transform. We estimated an AUC value for each of the two sources. The reference map Etheo1 (resp. Etheo2 ) was a binary map, for which only the dipoles belonging to the first simulated source (resp. to the second simulated source) ˆ were set to 1. The energy of each estimated source intensity E was first normalized between 0 and 1 to be compared with the corresponding reference Etheo1 or Etheo2 . By varying a threshold β between 0 and 1 and considering a dipole j as active ˆ t0 ) > β, the specificity and sensitivity of the localizaif E(j, tion method can be quantified. Note that while evaluating the AUC of the first source, dipoles corresponding to the second source were not taken into account (and vice versa). The ROC curves were obtained by plotting sensitivity(β) against (1 − specificity(β)) and detection accuracy was assessed by the area under the curves (AUC). However, the estimation of AUC is biased by the fact that among the Ns = 4000 dipoles of the model, only a few dipoles (say N ∗ ) were actually active compared to the large number of inactive dipoles (Ns − N ∗ ). A more accurate estimation of AUC was obtained by using as many inactive sources as active sources during the evaluation. This was done by randomly selecting N ∗ inactive or fictive sources among the (Ns − N ∗ ) available. To obtain an index that is more sensitive to the false positive rate, these N ∗ hypothetical sources were chosen among the local ˆ 0 ) (see [26] for details). An AUC score of 0.8 maxima of E(t or higher was considered as a good detection accuracy (80% of good detection accuracy). AUC scores were first evaluated for the estimated sources obtained for one specific box of the time–frequency plane, i.e., the box showing the maximum energy of the simulated spike and oscillation (see the cross in Fig. 4). This first evaluation was designed to quantify the ability of wMEM to accurately recover source distribution when one specific box of the discrete wavelet decomposition is used. This also summarizes the ability of the method to perform source localization in a specific frequency band. Once the full dynamics of the sources was reconstructed through an inverse wavelet transform, we evaluated the accuracy of the reconstructed activity at the first peak of the spike and at the central peak of the oscillation (see Fig. 4). In each case, one AUC value was estimated to quantify the accuracy of the reconstruction of the spike generator (discarding the influence of the dipoles belonging to the oscillation generator), and another AUC value was estimated to quantify the accuracy of the reconstruction of the oscillation generator (discarding the influence of the dipoles belonging to the spike generator). Two versions of the wMEM algorithm were evaluated: one uses local spatial smoothing within the parcels [i.e., ρ = 0.3 in (26)], while the other excludes local spatial smoothing (ρ = 0). In order to assess the influence of source depth, the AUC values were plotted as a function of the eccentricity (e) of the source. In order to assess the ability of the method to detect two

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Fig. 6. AUC scores for spike localization. (a) Localization with respect to the time–frequency box of maximum power for the simulated spike. Distribution of the AUC scores from 100 simulations shown for R ranging from 0.5 to 6, with local spatial smoothing (red) and without local spatial smoothing (blue). (b) From simulations with R ranging from 1 to 4, AUC scores for the time– frequency box with respect to the eccentricity of the simulated sources associated with the spike. The blue line is a moving average of the scores without spatial smoothing (×), and the red line corresponds to the localizations with a spatial smoothing in the parcels (+). (c) With all values of R ranging from 1 to 4, AUC scores for the time–frequency box with respect to the minimum Euclidian distance between the two simulated sources. Only configurations where both simulated sources had an eccentricity e > 60 mm were kept [same conventions as in (b)]. (d) Localization with respect to the peak of the simulated spike [same conventions as in (a)].

spatially close generators, the AUC values were plotted against the minimum Euclidian distance d between the two simulated sources. In this case, only superficial sources (e > 60 mm) for both generators were considered. D. Evaluation Results Using Simulated Data Evaluation of wMEM for the localization of spikes occurring at the same time as a burst of oscillatory activity is presented in Fig. 6. Fig. 6(a) and Fig. 6(d) presents the distribution of the AUC values obtained for 100 simulations, at the t–f box showing the maximum energy of the spike and at the first peak of the spike in the time domain, respectively. Note that when evaluating AUC at the selected t–f box, we observed a percentage of AUC values exactly equal to 0 for low R values: 7% at R = 0.5, 1% at R = 1, and 1% at R = 1.5. These cases actually corresponded to situations in which the signal from the t–f box of interest was removed during the wavelet denoising preprocessing (too small wavelet coefficients). As no source localization was computed in these cases with AUC = 0, we excluded them from the evaluation. The same number of samples (here 93 samples) was further considered for each boxplot distribution in Fig. 6(a). Briefly, accuracy increases as a function of the amplitude of the simulated source (R), with a median AUC greater

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Fig. 7. AUC scores for oscillation localization. (a) Localization at the time– frequency box of maximum power for the simulated oscillation. Distribution of the AUC scores from 100 simulations shown for R ranging from 0.5 to 6, with spatial smoothing (red) and without spatial smoothing (blue). (b) From simulations with R ranging from 1 to 4, AUC scores for the time–frequency box with respect to the eccentricity of the simulated sources associated with the oscillation. The blue line is a moving average of the scores without spatial smoothing (×), and the red line corresponds to the localizations with a spatial smoothing in the parcels (+). (c) With all values of R ranging from 1 to 4, AUC scores for the time–frequency box with respect to the minimum Euclidian distance between the two simulated sources. Only configurations where both simulated sources had an eccentricity e > 60 mm were kept [same conventions as in (b)]. (d) Localization at the peak of the simulated oscillation, occurring 150 ms after the peak of the spike [same conventions as in (a)].

than 0.8 for R ≥ 2. There was also a clear improvement in detection accuracy when local spatial smoothing was imposed (ρ = 0). Similar AUC values were obtained when assessing the method in one box in the t–f domain, and at the peak of the spike in the temporal domain. The maximum energy of the spike in the time–frequency map may not correspond exactly to the first peak in the temporal domain (see Fig. 4), however. In each simulation configuration, all AUC showed a long tail, suggesting that many sources had probably been mislocalized, resulting in small AUC values. Fig. 6(b) shows that these outliers mainly correspond to deep sources. For e < 40 mm, most AUC values were below 0.8, as were a significant proportion of sources for 40 mm < e < 60 mm, corresponding probably to mesial temporal sources. Fig. 6(c) shows that when both sources were superficial (e > 60 mm), detection accuracy decreased slightly for close generators (d < 35 mm). Finally, we also observe an improvement through local smoothing (ρ = 0), which significantly increases the resolution in depth and the source separation power. Evaluation of wMEM to localize a burst of oscillatory activity is presented in Fig. 7. Fig. 7(a) and 7(d) displays the distribution of the AUC values obtained for 100 simulations, at the t–f box showing maximum energy of the oscillation and at the main peak in the time domain, respectively. When considering the

oscillations, we actually observed a higher percentage of AUC values exactly equal to 0 for low R values: 27% at R = 0.5, 7% at R = 1, and 5% at R = 1.5. These cases corresponded as well to t–f box removed during the wavelet denoising preprocessing. As no source localization was computed in these cases with AUC = 0, we excluded them from the evaluation [73 samples were then considered for each boxplot distribution in Fig. 7(a)]. The results presented in Fig. 7(d) and Fig. 6(d) are quite similar, showing that at the main peak of the generator in the time domain, AUC values increase with the simulated source amplitude R. Here again, we observed a clear improvement of detection accuracy when local spatial smoothing was imposed. An interesting finding was the excellent performance of wMEM in localizing oscillatory activity when assessed in the box of maximum energy in the t–f plane [see Fig. 7(a)], showing median AUC values greater than 0.8 for all simulated amplitudes R. As seen in Fig. 7(b), the lower AUC values observed in the tails of the boxplot representations are mainly due to deep sources (e < 40 mm). When compared to spike localization, fewer mislocalizations were observed. Fig. 7(c) shows that, for superficial sources, the distance between the two generators had no influence on the ability of wMEM to localize the oscillation. Overall it seems that when there is a competition between the two generators, wMEM tends to localize the oscillation more accurately than the spike. Two possible reasons explain this discrepancy. First, the discrete wavelet representation is a bit sparser for the oscillation than for the spike (see Fig. 4) and thus encapsulates more information with less wavelet coefficients. Such sparsity may explain the good performances of wMEM on the oscillations, especially when evaluated on a specific t–f box encoding most relevant information. Second, the background MEG signals considered to generate realistic simulations are characterized by a 1/f process in which the low frequencies dominate. Consequently, at a specific signal-to-background ratio R, the resulting SNR of the oscillation at 40 Hz was slightly better than the one of the spikes, since the main frequency band of the spike was around 14 Hz. Local smoothing in the model tends to improve the method’s performance significantly, either for the oscillation or for the spike localization. This improvement is probably related to the sources for which MEG is less sensitive (radial sources on the gyri and deep sources) and involved in the localization thanks to the smoothing constraint. E. Spikes and Oscillations: A PCA Approach If we know in advance the relevant t–f box in which source localization is of interest, then the wMEM approach will be appropriate and Fig. 8 illustrates the performance. Without such prior information, we must separate the generators of different types of oscillatory activity (i.e., spikes and burst) from the reconstructed sources dynamics. Principal component analysis (PCA) is a well-known unbiased approach for extracting the most relevant information from a high-dimensional dataset. In general, the PCA combines information across channels to reduce the dimensionality of the original data to a smaller set of meaningful components. In the


Fig. 8. Example of localization in specific time–frequency boxes (with R = 4). The parcels shown at left display (a) “spiking area” and (b) “oscillatory area” in the simulation. The right views display the corresponding areas found in time– frequency boxes associated with (a ) spike and (b ) oscillation. The boxes are indicated (white crosses) in Fig. 4. The spatial maps have been thresholded at Otsu’s level [60].

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Fig. 9. PCA of the sources dynamics (R = 1). (a) and (b) Simulated sources of spiking and oscillatory activities, respectively. Using a PCA on the source dynamics corresponding to the red window (see R = 1 in Fig. 9), the first component extracts the spatial map Q1 and temporal course T1 of the spike (a ). Using the blue temporal window (see R = 1 in Fig. 5), the first principal component extracts the 40 Hz oscillation and its corresponding spatial location (Q1 and T1 in b ). The spatial maps have been thresholded at Otsu’s level [60].

present case, the dataset consists in the temporal dynamics obtained for the distributed cortical sources. The PCA expends this high-dimensional spatiotemporal matrix in terms of tensor components that explain most of the variability, from the largest to the residual variance ones. If we denote this Ns × NT matrix by S, the PCA amounts considering the following expansion: S=

C

Qk Tk

(27)

k =1

where the sum involves C tensor products of spatial maps Qk (dimension Ns × 1) with a corresponding temporal course Tk (dimension 1 × NT ). C is the rank of the matrix S: in general, it is equal to NT since NT usually < Ns . This expansion is unique as PCA decomposition enforces orthonormality within the set of spatial maps. Then, from any vector Qk , the temporal dynamics is found to be equal to Tk = Qtk S. Our concern is to isolate sources associated with one of the two oscillating modes. We assume that PCA components, carefully defined on specific temporal windows, can do so. In our simulations, we assume that the time courses may constitute a prior knowledge in some circumstances, such that we can isolate the temporal components of the PCA in order to recover either the spiking activity or the oscillatory pattern. At the lowest value of R (R ≤ 1), PCA of the entire temporal window (spike and oscillation) cannot isolate the spike from the oscillation, and the latter dominates the time courses obtained for the sources. However, the first component found with a PCA over each of the two nonoverlapping temporal windows (the red and blue segments indicated for R = 1 in Fig. 5) isolates well the spiking area from the oscillating one. As illustrated in Fig. 9, each component (Q1 , T1 ) exhibits a temporal dynamic compatible with a spike and an oscillation.

Fig. 10. PCA of the sources dynamics (R = 2). Same convention as in Fig. 9 but the PCA is performed over the entire temporal source dynamics (see R = 2 in Fig. 5). This time, the first two principal components (Q 1 , T 1 ) and (Q 2 , T 2 ) isolate the spiking and oscillatory activities. The spatial maps have been thresholded at Otsu’s level [60].

For a higher signal-to-background ratio R, the first two principal components computed in the entire temporal window of the source dynamics exhibit either the spiking transient or the oscillation, with a consistent spatial localization. This is illustrated in Fig. 10 (R = 2). VI. LOCALIZATION OF THE EPILEPTIC ACTIVITY FROM CLINICAL MEG DATA A. Patient Selection and Data Acquisition In order to illustrate the performance of wMEM with clinical data, we applied our method to localizing epileptic activity in a patient (whom we denote as PA) who underwent a proper presurgical evaluation involving simultaneous EEG/MEG recordings as well as a “gold standard” intracranial EEG (iEEG) investigation. We, thus, know where the epileptogenic zone is, since iEEG determined a focal area generating interictal discharges and seizures in the right orbitofrontal region. Simultaneous

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Fig. 11. Epileptic spike discharge (left) and a burst of fast oscillations (right) on four MEG sensors. The order of the SNR is about 10 dB higher for the spike than for the burst. The vertical blue line indicates the burst onset as annotated by the expert epileptologist.

recordings of EEG–MEG data were obtained at the MEG Centre of Université of Montreal, while anatomical MRI data were acquired at the MNI. Informed consent was obtained as approved by the Research Ethics Committee of the MNI. MEG signals (1200 Hz sampling rate, band-pass filtered at 0.3–70 Hz) were browsed by an expert epileptologist and epileptic discharges were manually marked in a 6-min run (20 interictal spikes and 5 bursts of rhythmic activity). Preprocessing (MEG data, MEG/MRI coregistration, MRI segmentation) and forward model estimation followed what we have described in Section V. Around each spike marker, a segment starting 200 ms before the marker and ending 500 ms after the marker was extracted for source localization analysis. For rhythmic activities, windows of 2.5 s of unfiltered MEG signals were defined at the onset of the bursts, with 500 ms before the marker. Fig. 11 displays typical MEG recordings of epileptic discharges. The amplitude of MEG signals for physiological brain activity is expected to range from few femto-Teslas to pico-Teslas. As mentioned in [57], interictal spikes are spontaneous signals that can have relatively large amplitude (∼3 pT). This implies that such epileptic MEG signals are likely to arise from spatially extended regions (few cm2 ) of active cortex [57] [58]. Finally, a baseline segment of 2 s of physiological background (i.e., free from any notable epileptic activity) was visually selected. B. Sources of Interictal Spikes in MEG For each spike (denoted as “e”), we computed the entire source dynamics (Se ) and we considered a PCA over a temporal window (±100 ms) around the first peak. This window captures the main variability of the transient associated with the epileptic discharges. For each spike k = 1, . . . , Ne , we consid(e) ered the first ten spatial components Qi , i = 1, . . . , 10. Assuming a reproducibility of the spatial pattern between spikes, we computed the components that best maximize spatial cross correlations, i.e., (e) t

(e )

C(e, e ) = Qi∗ Qj ∗

with

(e) t (e ) (i∗ , j ∗ ) = argmaxi,j Qi Qj .

(28)

Fig. 12. Single-trial interictal spike localization for PA. The two views of the spatial map correspond to the average map Q ∗ obtained from localization of 20 spikes. Each single-trial dynamics T (e ) is shown (e = 1, 2, . . . , 20), with the averaged time course in black. The spatial maps have been thresholded at Otsu’s level [60].

In the present case, we obtained a very high consensus since i∗ = j ∗ = 1 for all pairs of spikes, with 0.51 ≤ C(e, e ) ≤ 0.87. (e) t

Fig. 12 shows the temporal component T (e) = Q1 Se for each single spike and the resulting average (black curve). Note (e) that the sign of the spatial components Q1 is arbitrary and we are free to normalize it so that the main peak of T (e) is (e) always positive. Then, the normalized Q1 can be averaged; the averaged map Q∗ is displayed in Fig. 12. C. Single-Trial Source Localization: Interictal Fast Rhythmic Discharge in MEG Source localization for a burst of rhythmic activity was performed around the onset (0.5 s before, 2 s after), at the temporal scale corresponding to the beta band. Since the sampling frequency is 1200 Hz, the beta band roughly corresponds to the scale j = 5, for which the spectral band is approximatively 19– 38 Hz. Doing so, we solved roughly 150 inverse problems rather than localizing at each of the 4800 temporal samples. Then, we reconstructed the temporal signal for all sources in all parcels, at scale j = 5, ∗ w5,k ψ5,k (t). (29) Sβ (t) = k

It is worth mentioning that our method does not filter out the beta band in a strict sense, but rather extracts fluctuating activity at a specific temporal scale (here j = 5, i.e., ≈25 ms), for any single burst. Considering the five bursts marked and the time window around each burst onset (−20 ms, +500 ms), Sβ matrices were analyzed with a PCA from which the first ten components were kept. Given five different bursts marked by (e) the neurologist, the spatial maps Qj ∗ showing large consensus (i.e., large correlations) among the bursts were computed from (e) Sβ (e = 1, . . . 5), and were used to compute the average Q∗ .


Fig. 13. Interictal beta-band activity for PA. The spatial map corresponds to (e ) Q ∗ for which the temporal course T β is shown for the particular burst (e = 1) displayed in Fig. 11. The vertical line indicates the burst onset as defined by the expert epileptologist. The spatial maps have been thresholded at Otsu’s level [60].

This resulting averaged spatial map could finally be considered as a spatial filter to estimate the time course associated with each burst: (e)

Tβ

= Q∗ t Sβ . (e)

(30)

Fig. 13 illustrates this procedure and shows that the lateral right orbitofrontal focus identified for the spiking activity also contributes to generating these bursts of fast oscillatory activity. Such a correspondence has been further investigated in a larger database in [59]. It is worth mentioning that spatial maps in Figs. 12 and 13 display the absolute value of the current density, normalized to its maximum activity and thresholded on the level of the background activity [60]. This nonstatistical threshold, which only tends to remove background activity, may explain the larger extent of the burst localization. VII. DISCUSSION AND CONCLUSION The electrophysiological sources of the brain oscillatory activity were the focuss of this study. The usual approaches estimate the source distribution instant by instant from the perspective of imaging brain activity over a continuous time period. In contrast, the wMEM formalism described here proposes a rhythmic imaging approach. Instead of localizing time-varying oscillatory activity in the Fourier domain, either windowed [61] or not [62], we consider a sparse representation of the brain activity provided by the discrete wavelet transform of the data. Besides potentially saving computational time provided by such a representation, the discrete wavelet representation also decorrelates the time courses leading to a robust and stable estimate of the continuous brain dynamics. This representation has also been used in the Bayesian framework developed in [35]. It is worth mentioning that other temporal representation of the data has been proposed in the past. In particular, Ou et al. [63] have proposed a temporal model based on the singular-value decom-

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position of the sensor data. Such a model reduces significantly the complexity of the source dynamics, assuming that the latter is well represented in a subspace of temporal eigenvectors. Based on such priors regarding the temporal dynamics of the sources, both [36] and [63] model spatial sparsity through an l1 -constrained minimum l2 -norm solver, which promotes focal sources together with smooth-varying time courses. In our approach, the “spatial sparsity” does not refer to a solution involving few focal sources, but it is rather a model based on a parcelized description of the cortical sources. This wellaccepted assumption of a sparse representation of the spatial brain activity (also known as segregation) is used as a regularization principle in the wMEM. Cortical parcels are constructed at each time–frequency point. They can be independently switched ON or OFF during the MEM inference, while still estimating a contrast of source intensities within an active parcel. Our ROC analysis shows that wMEM is sensitive to the spatial extent of the generators, even more when local spatial smoothness is imposed within the parcels. This confirms our previous evaluation of the standard MEM approach in the time domain on EEG [26] and MEG data [28]. Indeed, we previously demonstrated the ability of standard MEM to recover the spatial extent of sources ranging from 3 to 30 cm2 , more accurately and reliably than other standard solvers (MNE, LORETA). In [28], we notably showed that these performances were equivalent whatever was the scale of the parcelization, demonstrating the ability of MEM inference to adapt to many source configurations. Although this will require further investigation, we can assume similar behaviors for the wMEM method. It is worth mentioning that our sparsity principle is expressed through the representation of the time-space behavior of brain activity. The key element provided by wMEM is that our assumption of sparsity allows accurate localization of sources of oscillations with amplitude similar to that of the physiological background. With a slightly reduced performance, this approach also addresses the localization problem of single fluctuating transients, such as epileptic spikes. In addition, our new proposed method has demonstrated its ability to localize single-trial data. As opposed to averaged data, single-trial data or single spontaneous discharge are usually characterized by a low SNR within highly structured baseline data. The baseline consists of some physiological background activity of amplitude similar to the brain response of interest. In pathological events such as epileptic spikes, the discharge may be significantly larger than the background and visually detectable. Whereas the noise model incorporated in the MEM model was relatively simple (uncorrelated white noise), the FDR approach used for preselection of the parcels contributing to the solution took into account the structure of the underlying physiological background. In the previous application of MEM in the time domain [26], the so-called null parcel was empty. In the present approach, a new parcelization and estimation of the null parcel (depending on the FDR threshold) are obtained for each wavelet component of the time–frequency plane. The null parcel represents the sources that will not contribute to the solution, thus reducing the dimensionality of the source space. Reducing the source space when applying MEM in the time

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domain might be critical, as the whole localization will then depend on the accuracy of the FDR threshold. In contrast, by defining an independent parcelization and null parcel at each time and scale, wMEM is less sensitive to this constraint. The reconstruction of the continuous dynamics of the sources, using the inverse wavelet transform, will more likely involve all the dipolar sources of the model, with a reliable spatial extension. The wMEM methodology has been validated and illustrated in the context of epilepsy but it is a promising methodology that can be used to investigate any oscillatory patterns under healthy [4], [64] or pathological conditions.

of the sources, this method relies on the normalized magnetic ˜ and G ˜ the normalized data field topologies. We denote by M (column-wise) and forward matrices along the sensor dimen˜ = G NG ), ˜ = M NM (resp. G sion, respectively. We obtain M where NM (resp. NG ) is the diagonal matrix whose jth diagonal element is the inverse L2-norm of the jth data time sample vector (resp. jth forward field gj ). The MSP involves the following three steps. ˜ into a set of d 1) The singular-value decomposition of G mutually orthogonal eigenvectors ui : ˜= G

APPENDIX A

κ i u i vi t .

i

WAVELET PRIMER The usual wavelet-based time–frequency analysis of biomedical signals amounts to considering the continuous wavelet transform that maps a temporal signal s(t) toward bidimensional coefficients t−b 1 √ ψ d(a, b) = s, ψa,b with ψa,b (t) = (31) a a where a > 0 is a scale parameter and b is a time index. The wavelet ψ is any oscillating function ( ψ(t)dt = 0) with sufficiently fast spectral decay in frequency to warranty the global regularity of the wavelet. In this study, we consider the multiresolution framework for which a = 2j and b = k 2j with (j, k) ∈ Z 2 . The wavelet set {ψj,k (t) = 2−j /2 ψ(2−j t − k), (j, k) ∈ Z 2 } provides a nonredundant representation of the signals and can be endowed with fundamental properties: orthonormal basis, finite support, and vanishing moments (see [45]). Besides the performance of nonlinear thresholding estimators with orthogonal wavelet basis [44], the vanishing moments property makes it possible to remove slow polynomial drifts from the wavelet representation of the data. In practice, the discrete wavelet transform is performed using quadrature mirror filters (h, f ) associated with the wavelet of choice. They define two operators Hxk =

a(i) = Ps g˜i = g˜it Ps g˜i called the “MSP score.” This value which is between 0 and 1 quantifies the possible role of the source in the generation of the data. It happens that, for resting data, without specific activities but the background physiological fluctuations, the MSP scores follow a Beta distribution [47]:

h2k −n xn

n

F xk =

The ui s do form an orthogonal basis in the sensors space and ˜ , where U = [u1 , u2 , . . . ud ], are the orthogonal projecUt M tions of the normalized data over this new frame. 2) A selection Us = [u1 , u2 , . . . us ] corresponds to a subspace S that captures most of the projection (for instance 95% of the variability) and Qs = Us (Us t Us )−1 Us t = Us Us t is the projector onto this subspace. S and Qs characterize the “transfer function” of the imaging system (the MEG sensors array and the mesh of the brain sources). ˜ constitute a new 3) The projections of the data Ms = Qs M frame (not orthogonal) that represents the data that can be explained in S. This new frame and the associated projector Ps = Ms (Ms t Ms )−1 Ms t characterize the data with respect to the imaging system. How much one particular source can contribute to this data subspace? The answer is given by the norm of the projection of its related lead field

Pb (a) =

f2k −n xn .

n

1

p(a)da

with p(a) =

a

aα (1 − a)β . Γ(α)Γ(β)

(32)

dj. = F H j −1 m.

This probability density p(a) fits the real cases well, as illustrated with the black curve in Fig. 2. The empirical points are obtained by averaging the MSP scores obtained for each sources from a set of temporal windows of various sizes (e.g., between 7 and 31 samples), randomly chosen in the baseline data.

APPENDIX B

ACKNOWLEDGMENT

Then, given N samples of a time series m, the wavelet coefficients at any level j < log2 (N ) are given by

MSP TECHNIQUE The MSP [47] approach relies on the generative model (1) where the Nd observations (M ) are linearly linked to the underlying Ns sources (q) by M = Gq + . Because MSP is concerned with probable sources rather than the intensity estimation

The authors would like to thank M. Hämälainen (MGH, MA) and C.G. Benar (INSERM, France) for their constructive comments at the early stage of this work. They also acknowledge the review of the referees which has greatly improved the quality of the manuscript.


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