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A tomography of neural sources could be constructed from EEG/MEG recordings once the neuroelectromagnetic inverse problem (NIP) is solved. Unfortunately the NIP lacks a unique solution and therefore additional constraints are needed to achieve uniqueness. Researchers are then confronted with the dilemma of choosing one solution on the basis of the advantages publicized by their authors. This study aims to help researchers to better guide their choices by clarifying what is hidden behind inverse solutions oversold by their apparently optimal properties to localize single sources. Here, we introduce an inverse solution (ANA) attaining perfect localization of single sources to illustrate how spurious sources emerge and destroy the reconstruction of simultaneously active sources. Although ANA is probably the simplest and robust alternative for data generated by a single dominant source plus noise, the main contribution of this manuscript is to show that zero localization error of single sources is a trivial and largely uninformative property unable to predict the performance of an inverse solution in presence of simultaneously active sources. We recommend as the most logical strategy for solving the NIP the incorporation of sound additional a priori information about neural generators that supplements the information contained in the data.

Determining the neural origin and strength of sources producing scalp maps of electric or magnetic fields requires the solution of an inverse problem. This so-called neuroelectromagnetic inverse problem (NIP) lacks a unique solution. In spite of this serious difficulty, there is an active past and ongoing research on this field (see [

Several linear and nonlinear solutions based on a diversity of approaches have been proposed. However, independently of the approach used, we need to evaluate the reliability of the estimates provided by the inverse procedure selected. While there is interesting ongoing research on this topic [

Some authors center their attention on the columns of the MRM, also called point spread functions (PSFs), that allow inferring how the solutions behave for single punctual sources. These authors consider the PSF as an adequate measure of the “goodness” of a linear inverse [

The bias in Dipole localization (BDL) defined in terms of the accuracy in estimating the location of each Cartesian component of the dipole. As such, it is a linear measure fully compliant with the linearity involved in the definition of the Model Resolution Matrix and can be directly estimated from the PSF.

The Dipole Localization Error (DLE) defined as the error attained in localizing the modulus of the current density vector. This definition conceptually disagrees with the use of MRM and PSF since the modulus is a nonlinear transformation of the individual dipole components not directly reflected by the PSF. Besides, linking the dipole localization error with the superposition principle is a blatant error since the basis of superposition is linearity. Although it certainly holds that the PSF of two simultaneously active dipoles is the sum of their individual PSFs, this is not the case for the DLEs. The widespread use of the dipole localization error concept obeys to historical and practical reasons since the modulus is the magnitude currently displayed in brain imaging.

All along this paper we will use the term single source to denote each of the three orthonormal (i.e., orthogonal with unitary norm) dipoles associated with a solution point. This is in agreement with the structure of the model resolution matrix where each solution point is represented by three columns. Consequently, each column corresponds to one and only one of three Cartesian components of a dipole. As typically used in this field, the term perfect localization will be used whenever the DLE or the BDL of a single source is zero independently of the off-diagonal elements of that column.

Two linear inverse solutions have been reported in the NIP literature to explicitly optimize the localization of single sources. The EPIFOCUS solution [

Authors advocating the use of PSF employ the appealing argument of the superposition principle [

In this paper we introduce a “trivial” and easy-to-compute linear inverse solution coined Adjoint Normalized Approximation (ANA) that transforms the original inverse problem into a space in which the model resolution matrix shows optimal properties for single source localization. We demonstrate that in the transformed space, ANA inverse solution is able to correctly localize single sources in full extent, that is, with zero dipole localization bias and perfectly accurate strength. These properties are shown to be satisfied for arbitrary lead field models independently of the amount of scalp sensors. ANA solution is used to build a simple didactical example illustrating that perfect localization of single sources in position and strength

The neuroelectromagnetic inverse problem (NIP), that is, the reconstruction of the current density vector inside the brain responsible for the electric and magnetic fields measured near/over the scalp, can be represented by a (first kind) Fredholm linear integral equation, denoting the relationship between the data measured at the external point,

The (vector) lead field function

Under experimental conditions, neither the measurements nor the lead field function is known for arbitrary surface/brain locations. However, assuming that the integral equation can be approximated by a discrete sum, (

Vectors

All linear solutions of (

Substitution of the measured data, as described in (

Here,

For the noisy case where

For the particular example discussed here, the unknown current density vector contains the three Cartesian components at each solution point. Correspondingly, each solution point will be represented by 3 columns and 3 rows of the MRM. The rows of

It is evident that for every invertible matrix

To obtain a unique solution to (

According to (

From this, it follows that the resolution matrix of ANA inverse solution is the product of the transposed normalized lead field times the normalized lead field. Therefore the resolution matrix is symmetric.

Further properties of the resolution matrix

The point-spread functions (columns of

Because the diagonal of the resolution matrix is one (due to normalization), the intensity of the estimated source is exactly the intensity activity of the original source.

Since

The ideal properties of ANA’s resolution matrix described in the previous section are independent from the lead field model. This implies that they will hold even for arbitrarily small sensor configurations and very large solution spaces provided that there are no collinear columns in the lead field. We have exploited this fact to construct a simple numerical example that might help to shed light on several aspects influencing the behavior of linear inverse solutions in the presence of multiple active sources. The computational simplicity of ANA will facilitate the task to readers interested in further simulating its behavior with simultaneous sources.

The example given here considers the case of two EEG sensors and four solution points as depicted in Figure

In the case of this simple example, the current density vector is a 12 component vector of the form

This vector is formed by the three Cartesian components of the dipoles (subscripts

Table

1 | 0.48 | 0.94 | 0.48 | −0.84 | −0.75 | −0.67 | −0.86 | −0.93 | −0.10 | −0.93 | 0.16 |

0.48 | 1 | 0.74 | 0.99 | −0.87 | −0.94 | −0.97 | −0.85 | −0.13 | 0.81 | −0.14 | 0.94 |

0.94 | 0.74 | 1 | 0.75 | −0.97 | −0.92 | −0.88 | −0.98 | −0.75 | 0.23 | −0.76 | 0.48 |

0.48 | 0.99 | 0.75 | 1 | −0.87 | −0.94 | −0.97 | −0.85 | −0.14 | 0.81 | −0.15 | 0.94 |

−0.84 | −0.87 | −0.97 | −0.87 | 1 | 0.98 | 0.96 | 0.99 | 0.60 | −0.43 | 0.61 | −0.66 |

−0.75 | −0.94 | −0.92 | −0.94 | 0.98 | 1 | 0.99 | 0.98 | 0.46 | −0.57 | 0.47 | −0.77 |

−0.67 | −0.97 | −0.88 | −0.97 | 0.96 | 0.99 | 1 | 0.95 | 0.36 | −0.66 | 0.37 | −0.83 |

−0.86 | −0.85 | −0.98 | −0.85 | 0.99 | 0.98 | 0.95 | 1 | 0.62 | −0.40 | 0.63 | −0.63 |

−0.93 | −0.13 | −0.75 | −0.14 | 0.60 | 0.46 | 0.36 | 0.62 | 1 | 0.45 | 0.99 | 0.20 |

−0.10 | 0.81 | 0.23 | 0.81 | −0.43 | −0.57 | −0.66 | −0.40 | 0.45 | 1 | 0.44 | 0.96 |

−0.93 | −0.14 | −0.76 | −0.15 | 0.61 | 0.47 | 0.37 | 0.63 | 0.99 | 0.44 | 1 | 0.19 |

0.16 | 0.94 | 0.48 | 0.94 | −0.66 | −0.77 | −0.83 | −0.63 | 0.20 | 0.96 | 0.19 | 1 |

The theoretical properties derived in the previous section obviously hold for the problem presented. The main diagonal is filled by ones that are the dominant elements within their respective rows (and columns since the matrix is symmetric). A first aspect to note is that while the recovery of each Cartesian component of the dipole (if alone) is perfect, the recovery of the modulus is not. Perfect recovery of the modulus can be obtained with ANA inverse by stating the original problem for the modulus rather than for the individual dipolar components. This can be done by determining a priori the orientation as in SAM beamformer [

The following two simple examples illustrate how the model resolution matrix is used to derive the inverse solution estimates for a single active source and for two simultaneously active sources.

According to (

In the same way, the reconstruction of each single active source of unitary strength is given by the PSF (column of MRM) linked to this source component. While the maximum always occurs at the right position and the source strength is correctly estimated, the reconstruction is rather noisy and contains spurious activity (ghost sources). This spurious activity appears at sites where the true source strength is zero and is a consequence of nonzero off-diagonal elements of the resolution matrix. To better understand the origin of nonzero off-diagonal elements in the MRM, we should remember that its

Not only will off-diagonal elements lead to noisy single source reconstruction but also, even worse, they will totally mislead multiple source reconstruction. To see how, let us return to our example of Table

1.16 | 1.42 | 1.42 | 1.43 | −1.50 | −1.52 | −1.51 | −1.50 | −0.72 | 0.86 | −0.74 | 1.16 |

The largest positive value of the reconstruction appears at source component number four and therefore at the second solution point. The largest absolute value appears at the source component number six which also belongs to the second solution point. The modulus of the vector, given in Table

2.32 | 2.57 | 2.25 | 1.62 |

As for a comparison, we depict on Table

0.00 | −0.02 | −0.02 | 0.00 | 0.00 | −0.01 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |

−0.02 | 0.5 | 0.45 | 0.00 | 0.15 | 0.02 | 0.00 | 0.09 | 0.01 | 0.00 | 0.09 | 0.11 |

−0.02 | 0.45 | 0.45 | −0.01 | −0.01 | 0.16 | 0.00 | 0.06 | 0.05 | 0.00 | 0.06 | 0.11 |

0.00 | 0.00 | −0.01 | 0.00 | 0.02 | −0.02 | 0.00 | 0.00 | −0.01 | 0.00 | 0.00 | 0.00 |

0.00 | 0.15 | −0.01 | 0.02 | 0.5 | −0.44 | 0.00 | 0.08 | −0.13 | 0.00 | 0.08 | −0.01 |

−0.01 | 0.02 | 0.16 | −0.02 | −0.44 | 0.45 | 0.00 | −0.05 | 0.13 | 0.00 | −0.06 | 0.04 |

0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |

0.00 | 0.09 | 0.06 | 0 | 0.08 | −0.05 | 0.00 | 0.02 | −0.02 | 0.00 | 0.02 | 0.01 |

0.00 | 0.01 | 0.05 | −0.01 | −0.13 | 0.13 | 0.00 | −0.02 | 0.04 | 0.00 | −0.02 | 0.01 |

0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |

0.00 | 0.09 | 0.06 | 0.00 | 0.08 | −0.06 | 0.00 | 0.02 | −0.02 | 0.00 | 0.02 | 0.01 |

0.00 | 0.11 | 0.11 | 0.00 | −0.01 | 0.04 | 0.00 | 0.01 | 0.01 | 0.00 | 0.01 | 0.03 |

We have shown so far that ANA solution is capable to provide perfect localization of single sources within the space of the transformed variable

In this section we present some simulation results to study how much the theoretical performance degrades with noise in the original source space

For reproducibility and compatibility with previous publications, we use in this section a lead field model corresponding to the sensor configuration and solution space described in ISBET NEWSLETTER number 6, December 1995, Grave and Gonzalez, 2000, Grave et al. 2001. Namely, a unit radius 3-shell spherical head model (Ary et al., 1981), with solution points confined to a maximum radius of 0.8. The sensor configuration comprises 148 electrodes. The solution space consists of 817 points on a regular grid with an intergrid distance of 0.133 cm, corresponding to 2451 focal sources. To simulate noisy data, we added to each electrode uncorrelated random noise in the range

the empirical Probability Distribution Function, defined as follows: Probability

the empirical density function defined for

Note that while the empirical density function describes the performance for each eccentricity range, the probability function provides a global assessment about how fast the maximum asymptotic value is attained.

Figure

Figure

The ANA inverse solution described in this paper is, to the best of our knowledge, the first linear solution to the NIP simultaneously fulfilling (in the transformed space) the three following properties: (1) symmetric resolution matrix; (2) perfect single source localization, and (3) perfect estimation of single source strength. Probably this is also the simplest (in the sense of numerical complexity) solution with these properties. Importantly, such properties stem from the theoretical resolution matrix and therefore hold for arbitrary (with noncollinear columns) lead field models.

In case we accept that perfect single source localization, that is, correct estimation of the location and the source strength as in ANA or correct estimation of the location as in sLORETA, suffices to insure perfect multiple source reconstruction, we must conclude that ANA or sLORETA is the solution to the NIP. This statement is in flagrant contradiction to any rationale. The mistake resides in the assumption that perfect single source localization, defined as zero DLE or zero BDL, implies accurate multiple source localization. This implication is true only for the ideal resolution matrix with zero off-diagonal elements, which is impossible for an underdetermined problem. As demonstrated here, ANA solution is theoretically perfect for single source reconstruction but failed in the simplest case of two simultaneously active sources. As shown in the example, the reason for such failure is the existence of nonzero off-diagonal elements within the model resolution matrix that are ignored by the DLE or BDL. As we saw, nonzero off-diagonal elements appear as a consequence of the correlation between scalp potential (magnetic fields) patterns associated with different punctual sources. Such off-diagonal elements are inherent to the problem statement (the lead field model) and will appear for all linear inverse solutions (e.g., sLORETA, MPNL, EPIFOCUS, etc.), although to different extent. Note that while noiseless data imply the selection of a single MRM column, noisy data can be interpreted as an additional source (generating the noise) implying that multiple columns of the MRM should be added to get the final current density estimator. As shown before, off-diagonal elements might dominate such reconstruction even in the noiseless case. However, as long as the components of the additional source are lower than the correlation between patterns of neighboring dipoles, ANA (and the closely related EPIFOCUS) should yield low BDLs. Simulations suggest that this is not the case for sLORETA or MPNL with errors up to 6.5 grid units.

Importantly, it is widely accepted that localization accuracy will indefinitely improve by increasing the number of scalp recording sensors. While increasing the number of sensors augments the amount of information about the underlying sources, it does also enhance the correlation (redundancy) between the rows of the lead field matrix, that is, the way that one sensor sees all the sources. The increase in correlation between rows results in unstable (sensitive to noise) problems that need special regularization strategies to avoid noise amplification. This trade-off between the independent information conveyed by the new measurements and their redundancy will define a practical superior bound to the amount of electrodes to be used for source localization purposes.

We have seen that neither the perfect single source localization nor the unlimited increase in the amount of recording sensors will definitively solve the NIP. Obviously, the only remaining choice is to incorporate as much a priori information as possible about the generators into the problem. Such information should be independent of the information already contained in the measurements. A priori information can be incorporated within the discrete formalism by a right-side transformation of the lead field matrix, which in turn can be interpreted as a change of variable. Only this procedure, illustrated here for ANA solution (see (

The value of ANA solution is not only didactical. As shown by our simulation results, ANA can be applied to retrieve sources in the space of the original variable

It is worth mentioning that the limitations described here are not specific to linear inverse solutions, and they will certainly appear under a different mask for nonlinear inverse procedures. While these difficulties are easily analyzed within the linear framework because of the possibilities offered by the model resolution matrix formalism, they actually reflect the ill-posed nature of the original inverse problem. Therefore, unless useful a priori information is found that cannot be incorporated within linear inverses, we see no good reasons to replace the comfortable linear framework with its inherent computational and interpretational simplicity.

The evaluation and design of linear inverse solutions over last decade have been misguided by the idea that only solutions able to accurately localize a large proportion of single sources will succeed in the quest for constructing a tomography of neural generators [

Here we introduced a linear inverse solution coined ANA which fulfills several optimal properties for the localization of single sources. We demonstrated by means of the model resolution matrix formalism that ANA localizes correctly the location and the amplitudes of all single sources. These properties hold for arbitrary lead fields and for arbitrarily small sensor configurations. This fact was exploited to introduce simple examples that clarify how spurious sources are formed and their large relevance for simultaneous source reconstruction. We further showed that ANA solution is highly robust to noise, outperforming established methods for single source localization (sLORETA and EPIFOCUS). Its robustness to noise and computational simplicity make ANA a reasonable alternative for data generated by a single dominant source plus noise, as can be the case in epilepsy.

The most important contribution of this manuscript is to provide definitive evidence that the apparently reasonable (although naïve) idea of inferring the behavior of linear solutions from their single source localization properties proves false. It is thus concluded that zero localization error alone is a trivial and useless property unable to predict the performance of an inverse solution in presence of simultaneously active sources. We expect that these results will help researchers to guide their choices of inverse methods, in methods development as well as for clinical and neuroscientific applications. We also hope that it will stimulate further interest in finding neurophysiologically plausible constraints that can be used as a priori information in the NIP, which should be the ultimate goal in this endeavour.

The authors thank two anonymous reviewers for their constructive comments. This work was supported by the European Project FP6-IST-027140 (BACS) and the COST Action BM0601 “NeuroMath.” This paper only reflects the authors' views, and funding agencies are not liable for any use that may be made of the information contained herein. The COST Office is not responsible for the external websites referred to in this publication. We would like to acknowledge the financial support of the UK Medical Research Council for one of the authors (OH, U.1055.04.003.00001.01).