The minimum-norm current estimates

This page describes the mathematical concepts and the computation of the minimum-norm estimates needed in order to obtain the linear inverse operator using mne.minimum_norm.make_inverse_operator(). Its usage is presented in the tutorial Source localization with MNE/dSPM/sLORETA/eLORETA.

Minimum-norm estimates

This section describes the mathematical details of the calculation of minimum-norm estimates. In Bayesian sense, the ensuing current distribution is the maximum a posteriori (MAP) estimate under the following assumptions:

  • The viable locations of the currents are constrained to the cortex. Optionally, the current orientations can be fixed to be normal to the cortical mantle.

  • The amplitudes of the currents have a Gaussian prior distribution with a known source covariance matrix.

  • The measured data contain additive noise with a Gaussian distribution with a known covariance matrix. The noise is not correlated over time.

The linear inverse operator

The measured data in the source estimation procedure consists of MEG and EEG data, recorded on a total of N channels. The task is to estimate a total of M strengths of sources located on the cortical mantle. If the number of source locations is P, M = P for fixed-orientation sources and M = 3P if the source orientations are unconstrained. The regularized linear inverse operator following from the Bayesian approach is given by the \(M \times N\) matrix

\[M = R' G^T (G R' G^T + C)^{-1}\ ,\]

where G is the gain matrix relating the source strengths to the measured MEG/EEG data, \(C\) is the data noise-covariance matrix and \(R'\) is the source covariance matrix. The dimensions of these matrices are \(N \times M\), \(N \times N\), and \(M \times M\), respectively. The \(M \times 1\) source-strength vector is obtained by multiplying the \(N \times 1\) data vector by \(M\).

The expected value of the current amplitudes at time t is then given by \(\hat{j}(t) = Mx(t)\), where \(x(t)\) is a vector containing the measured MEG and EEG data values at time t.

Note

For computational convenience, in MNE-Python the linear inverse operator is not computed explicitly. See Computation of the solution for mathematical details, and Calculating the inverse operator for a detailed example.

Regularization

The a priori variance of the currents is, in practise, unknown. We can express this by writing \(R' = R/ \lambda^2\), which yields the inverse operator

\[M = R G^T (G R G^T + \lambda^2 C)^{-1}\ ,\]

where the unknown current amplitude is now interpreted in terms of the regularization parameter \(\lambda^2\). Small \(\lambda^2\) corresponds to large current amplitudes and complex estimate current patterns while a large \(\lambda^2\) means the amplitude of the current is limited and a simpler, smooth, current estimate is obtained.

We can arrive in the regularized linear inverse operator also by minimizing the cost function

\[S = \tilde{e}^T \tilde{e} + \lambda^2 j^T R^{-1} j\ ,\]

where the first term consists of the difference between the whitened measured data (see Whitening and scaling) and those predicted by the model while the second term is a weighted-norm of the current estimate. It is seen that, with increasing \(\lambda^2\), the source term receive more weight and larger discrepancy between the measured and predicted data is tolerable.

Whitening and scaling

The MNE software employs data whitening so that a ‘whitened’ inverse operator assumes the form

\[\tilde{M} = R \tilde{G}^T (\tilde{G} R \tilde{G}^T + I)^{-1}\ ,\]

where \(\tilde{G} = C^{-^1/_2}G\) is the spatially whitened gain matrix. The expected current values are \(\hat{j} = Mx(t)\), where \(x(t) = C^{-^1/_2}x(t)\) is a the whitened measurement vector at t. The spatial whitening operator is obtained with the help of the eigenvalue decomposition \(C = U_C \Lambda_C^2 U_C^T\) as \(C^{-^1/_2} = \Lambda_C^{-1} U_C^T\). In the MNE software the noise-covariance matrix is stored as the one applying to raw data. To reflect the decrease of noise due to averaging, this matrix, \(C_0\), is scaled by the number of averages, \(L\), i.e., \(C = C_0 / L\).

As shown above, regularization of the inverse solution is equivalent to a change in the variance of the current amplitudes in the Bayesian a priori distribution.

Convenient choice for the source-covariance matrix \(R\) is such that \(\text{trace}(\tilde{G} R \tilde{G}^T) / \text{trace}(I) = 1\). With this choice we can approximate \(\lambda^2 \sim 1/SNR\), where SNR is the (power) signal-to-noise ratio of the whitened data.

Note

The definition of the signal to noise-ratio/ \(\lambda^2\) relationship given above works nicely for the whitened forward solution. In the un-whitened case scaling with the trace ratio \(\text{trace}(GRG^T) / \text{trace}(C)\) does not make sense, since the diagonal elements summed have, in general, different units of measure. For example, the MEG data are expressed in T or T/m whereas the unit of EEG is Volts.

See Computing a covariance matrix for example of noise covariance computation and whitening.

Regularization of the noise-covariance matrix

Since finite amount of data is usually available to compute an estimate of the noise-covariance matrix \(C\), the smallest eigenvalues of its estimate are usually inaccurate and smaller than the true eigenvalues. Depending on the seriousness of this problem, the following quantities can be affected:

  • The model data predicted by the current estimate,

  • Estimates of signal-to-noise ratios, which lead to estimates of the required regularization, see Regularization,

  • The estimated current values, and

  • The noise-normalized estimates, see Noise normalization.

Fortunately, the latter two are least likely to be affected due to regularization of the estimates. However, in some cases especially the EEG part of the noise-covariance matrix estimate can be deficient, i.e., it may possess very small eigenvalues and thus regularization of the noise-covariance matrix is advisable.

Historically, the MNE software accomplishes the regularization by replacing a noise-covariance matrix estimate \(C\) with

\[C' = C + \sum_k {\varepsilon_k \bar{\sigma_k}^2 I^{(k)}}\ ,\]

where the index \(k\) goes across the different channel groups (MEG planar gradiometers, MEG axial gradiometers and magnetometers, and EEG), \(\varepsilon_k\) are the corresponding regularization factors, \(\bar{\sigma_k}\) are the average variances across the channel groups, and \(I^{(k)}\) are diagonal matrices containing ones at the positions corresponding to the channels contained in each channel group.

See How should I regularize the covariance matrix? for details.

Computation of the solution

The most straightforward approach to calculate the MNE is to employ expression for the original or whitened inverse operator directly. However, for computational convenience we prefer to take another route, which employs the singular-value decomposition (SVD) of the matrix

\[A = \tilde{G} R^{^1/_2} = U \Lambda V^T\]

where the superscript \(^1/_2\) indicates a square root of \(R\). For a diagonal matrix, one simply takes the square root of \(R\) while in the more general case one can use the Cholesky factorization \(R = R_C R_C^T\) and thus \(R^{^1/_2} = R_C\).

With the above SVD it is easy to show that

\[\tilde{M} = R^{^1/_2} V \Gamma U^T\]

where the elements of the diagonal matrix \(\Gamma\) are

\[\gamma_k = \frac{1}{\lambda_k} \frac{\lambda_k^2}{\lambda_k^2 + \lambda^2}\ .\]

With \(w(t) = U^T C^{-^1/_2} x(t)\) the expression for the expected current is

\[\hat{j}(t) = R^C V \Gamma w(t) = \sum_k {\bar{v_k} \gamma_k w_k(t)}\ ,\]

where \(\bar{v_k} = R^C v_k\), \(v_k\) being the \(k\) th column of \(V\). It is thus seen that the current estimate is a weighted sum of the ‘modified’ eigenleads \(v_k\).

It is easy to see that \(w(t) \propto \sqrt{L}\). To maintain the relation \((\tilde{G} R \tilde{G}^T) / \text{trace}(I) = 1\) when \(L\) changes we must have \(R \propto 1/L\). With this approach, \(\lambda_k\) is independent of \(L\) and, for fixed \(\lambda\), we see directly that \(j(t)\) is independent of \(L\).

Note

This is computed using mne.minimum_norm.make_inverse_operator() and its usage is illustrated in Calculating the inverse operator.

Noise normalization

The noise-normalized linear estimates introduced by Dale et al. require division of the expected current amplitude by its variance. Noise normalization serves three purposes:

  • It converts the expected current value into a dimensionless statistical test variable. Thus the resulting time and location dependent values are often referred to as dynamic statistical parameter maps (dSPM).

  • It reduces the location bias of the estimates. In particular, the tendency of the MNE to prefer superficial currents is eliminated.

  • The width of the point-spread function becomes less dependent on the source location on the cortical mantle. The point-spread is defined as the MNE resulting from the signals coming from a point current source (a current dipole) located at a certain point on the cortex.

In practice, noise normalization requires the computation of the diagonal elements of the matrix

\[M C M^T = \tilde{M} \tilde{M}^T\ .\]

With help of the singular-value decomposition approach we see directly that

\[\tilde{M} \tilde{M}^T\ = \bar{V} \Gamma^2 \bar{V}^T\ .\]

Under the conditions expressed at the end of Computation of the solution, it follows that the t-statistic values associated with fixed-orientation sources) are thus proportional to \(\sqrt{L}\) while the F-statistic employed with free-orientation sources is proportional to \(L\), correspondingly.

Note

A section discussing statistical considerations related to the noise normalization procedure will be added to this manual in one of the subsequent releases.

Note

The MNE software usually computes the square roots of the F-statistic to be displayed on the inflated cortical surfaces. These are also proportional to \(\sqrt{L}\).

Predicted data

Under noiseless conditions the SNR is infinite and thus leads to \(\lambda^2 = 0\) and the minimum-norm estimate explains the measured data perfectly. Under realistic conditions, however, \(\lambda^2 > 0\) and there is a misfit between measured data and those predicted by the MNE. Comparison of the predicted data, here denoted by \(x(t)\), and measured one can give valuable insight on the correctness of the regularization applied.

In the SVD approach we easily find

\[\hat{x}(t) = G \hat{j}(t) = C^{^1/_2} U \Pi w(t)\ ,\]

where the diagonal matrix \(\Pi\) has elements \(\pi_k = \lambda_k \gamma_k\) The predicted data is thus expressed as the weighted sum of the ‘recolored eigenfields’ in \(C^{^1/_2} U\).

Cortical patch statistics

use_cps parameter in mne.convert_forward_solution(), and mne.minimum_norm.make_inverse_operator() controls whether to use cortical patch statistics (CPS) to define normal orientations or not (see Cortical surface reconstruction with FreeSurfer). The CPS contain for each vertex of the cortical surface the information about the source space point closest to it as well as the distance from the vertex to this source space point. The vertices for which a given source space point is the nearest one define the cortical patch associated with with the source space point. Once these data are available, it is straightforward to compute the following for each source location \(d\):

  • The average over the normals of at the vertices in a patch, \(\bar{n_d}\),

  • The areas of the patches, \(A_d\), and

  • The average deviation of the vertex normals in a patch from their average, \(\sigma_d\), given in degrees.

The orientation constraints

The principal sources of MEG and EEG signals are generally believed to be postsynaptic currents in the cortical pyramidal neurons. Since the net primary current associated with these microscopic events is oriented normal to the cortical mantle, it is reasonable to use the cortical normal orientation as a constraint in source estimation. In addition to allowing completely free source orientations, the MNE software implements three orientation constraints based of the surface normal data:

  • Source orientation can be rigidly fixed by specifying fixed=True when calling mne.minimum_norm.make_inverse_operator(). In such case, the dipole orientations are fixed to be orthogonal to the surface of the cortex, pointing outwards. If cortical patch statistics are available the average normal over each patch, \(\bar{n_d}\), are used to define the source orientation. Otherwise, the vertex normal at the source space location is employed. See Fixed dipole orientations

  • A location independent or fixed loose orientation constraint (fLOC) can be employed by specifying fixed=False and loose=1.0 when calling mne.minimum_norm.make_inverse_operator(). In this approach, a source coordinate system based on the local surface orientation at the source location is employed. By default, the three columns of the gain matrix G, associated with a given source location, are the fields of unit dipoles pointing to the directions of the x, y, and z axis of the coordinate system employed in the forward calculation (usually the MEG head coordinate frame). For LOC the orientation is changed so that the first two source components lie in the plane normal to the surface normal at the source location and the third component is aligned with it. Thereafter, the variance of the source components tangential to the cortical surface are reduced by a factor defined by the loose parameter. See Loose dipole orientations

  • A variable loose orientation constraint (vLOC) can be employed by specifying fixed=False and loose parameters when calling mne.minimum_norm.make_inverse_operator(). This is similar to fLOC except that the value given with the loose parameter will be multiplied by \(\sigma_d\), defined above. See Limiting orientations, but not fixing them

Depth weighting

The minimum-norm estimates have a bias towards superficial currents. This tendency can be alleviated by adjusting the source covariance matrix \(R\) to favor deeper source locations. In the depth weighting scheme employed in MNE analyze, the elements of \(R\) corresponding to the \(p\) th source location are be scaled by a factor

\[f_p = (g_{1p}^T g_{1p} + g_{2p}^T g_{2p} + g_{3p}^T g_{3p})^{-\gamma}\ ,\]

where \(g_{1p}\), \(g_{2p}\), and \(g_{3p}\) are the three columns of \(G\) corresponding to source location \(p\) and \(\gamma\) is the order of the depth weighting. The maximal amount of depth weighting can be adjusted with depth parameter in mne.minimum_norm.make_inverse_operator().

Effective number of averages

It is often the case that the epoch to be analyzed is a linear combination over conditions rather than one of the original averages computed. As stated above, the noise-covariance matrix computed is originally one corresponding to raw data. Therefore, it has to be scaled correctly to correspond to the actual or effective number of epochs in the condition to be analyzed. In general, we have

\[C = C_0 / L_{eff}\]

where \(L_{eff}\) is the effective number of averages. To calculate \(L_{eff}\) for an arbitrary linear combination of conditions

\[y(t) = \sum_{i = 1}^n {w_i x_i(t)}\]

we make use of the the fact that the noise-covariance matrix

\[C_y = \sum_{i = 1}^n {w_i^2 C_{x_i}} = C_0 \sum_{i = 1}^n {w_i^2 / L_i}\]

which leads to

\[1 / L_{eff} = \sum_{i = 1}^n {w_i^2 / L_i}\]

An important special case of the above is a weighted average, where

\[w_i = L_i / \sum_{i = 1}^n {L_i}\]

and, therefore

\[L_{eff} = \sum_{i = 1}^n {L_i}\]

Instead of a weighted average, one often computes a weighted sum, a simplest case being a difference or sum of two categories. For a difference \(w_1 = 1\) and \(w_2 = -1\) and thus

\[1 / L_{eff} = 1 / L_1 + 1 / L_2\]

or

\[L_{eff} = \frac{L_1 L_2}{L_1 + L_2}\]

Interestingly, the same holds for a sum, where \(w_1 = w_2 = 1\). Generalizing, for any combination of sums and differences, where \(w_i = 1\) or \(w_i = -1\), \(i = 1 \dotso n\), we have

\[1 / L_{eff} = \sum_{i = 1}^n {1/{L_i}}\]