The forward solution

Overview

This page covers the definitions of different coordinate systems employed in MNE software and FreeSurfer, the details of the computation of the forward solutions, and the associated low-level utilities.

MEG/EEG and MRI coordinate systems

The coordinate systems used in MNE software (and FreeSurfer) and their relationships are depicted in MEG/EEG and MRI coordinate systems. Except for the Sensor coordinates, all of the coordinate systems are Cartesian and have the “RAS” (Right-Anterior-Superior) orientation, i.e., the \(x\) axis points to the right, the \(y\) axis to the front, and the \(z\) axis up.

MEG/EEG and MRI coordinate systems

MEG/EEG and MRI coordinate systems

The coordinate transforms present in the fif files in MNE and the FreeSurfer files as well as those set to fixed values are indicated with \(T_x\), where \(x\) identifies the transformation.

The coordinate systems related to MEG/EEG data are:

Head coordinates

This is a coordinate system defined with help of the fiducial landmarks (nasion and the two auricular points). In fif files, EEG electrode locations are given in this coordinate system. In addition, the head digitization data acquired in the beginning of an MEG, MEG/EEG, or EEG acquisition are expressed in head coordinates. For details, see MEG/EEG and MRI coordinate systems.

Device coordinates

This is a coordinate system tied to the MEG device. The relationship of the Device and Head coordinates is determined during an MEG measurement by feeding current to three to five head-position indicator (HPI) coils and by determining their locations with respect to the MEG sensor array from the magnetic fields they generate.

Sensor coordinates

Each MEG sensor has a local coordinate system defining the orientation and location of the sensor. With help of this coordinate system, the numerical integration data needed for the computation of the magnetic field can be expressed conveniently as discussed in Coil geometry information. The channel information data in the fif files contain the information to specify the coordinate transformation between the coordinates of each sensor and the MEG device coordinates.

The coordinate systems related to MRI data are:

Surface RAS coordinates

The FreeSurfer surface data are expressed in this coordinate system. The origin of this coordinate system is at the center of the conformed FreeSurfer MRI volumes (usually 256 x 256 x 256 isotropic 1-mm3 voxels) and the axes are oriented along the axes of this volume. The BEM surface and the locations of the sources in the source space are usually expressed in this coordinate system in the fif files. In this manual, the Surface RAS coordinates are usually referred to as MRI coordinates unless there is need to specifically discuss the different MRI-related coordinate systems.

RAS coordinates

This coordinate system has axes identical to the Surface RAS coordinates but the location of the origin is different and defined by the original MRI data, i.e. , the origin is in a scanner-dependent location. There is hardly any need to refer to this coordinate system explicitly in the analysis with the MNE software. However, since the Talairach coordinates, discussed below, are defined with respect to RAS coordinates rather than the Surface RAS coordinates, the RAS coordinate system is implicitly involved in the transformation between Surface RAS coordinates and the two Talairach coordinate systems.

MNI Talairach coordinates

The definition of this coordinate system is discussed, e.g. , in http://imaging.mrc-cbu.cam.ac.uk/imaging/MniTalairach. This transformation is determined during the FreeSurfer reconstruction process.

FreeSurfer Talairach coordinates

The problem with the MNI Talairach coordinates is that the linear MNI Talairach transform does matched the brains completely to the Talairach brain. This is probably because the Talairach atlas brain is a rather odd shape, and as a result, it is difficult to match a standard brain to the atlas brain using an affine transform. As a result, the MNI brains are slightly larger (in particular higher, deeper and longer) than the Talairach brain. The differences are larger as you get further from the middle of the brain, towards the outside. The FreeSurfer Talairach coordinates mitigate this problem by additing a an additional transformation, defined separately for negatice and positive MNI Talairach \(z\) coordinates. These two transformations, denoted by \(T_-\) and \(T_+\) in MEG/EEG and MRI coordinate systems, are fixed as discussed in http://imaging.mrc-cbu.cam.ac.uk/imaging/MniTalairach (Approach 2).

The different coordinate systems are related by coordinate transformations depicted in MEG/EEG and MRI coordinate systems. The arrows and coordinate transformation symbols (\(T_x\)) indicate the transformations actually present in the FreeSurfer files. Generally,

\[\begin{split}\begin{bmatrix} x_2 \\ y_2 \\ z_2 \\ 1 \end{bmatrix} = T_{12} \begin{bmatrix} x_1 \\ y_1 \\ z_1 \\ 1 \end{bmatrix} = \begin{bmatrix} R_{11} & R_{12} & R_{13} & x_0 \\ R_{13} & R_{13} & R_{13} & y_0 \\ R_{13} & R_{13} & R_{13} & z_0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} x_1 \\ y_1 \\ z_1 \\ 1 \end{bmatrix}\ ,\end{split}\]

where \(x_k\),:math:y_k,and \(z_k\) are the location coordinates in two coordinate systems, \(T_{12}\) is the coordinate transformation from coordinate system “1” to “2”, \(x_0\), \(y_0\),and \(z_0\) is the location of the origin of coordinate system “1” in coordinate system “2”, and \(R_{jk}\) are the elements of the rotation matrix relating the two coordinate systems. The coordinate transformations are present in different files produced by FreeSurfer and MNE as summarized in Coordinate transformations in FreeSurfer and MNE software packages.. The fixed transformations \(T_-\) and \(T_+\) are:

\[\begin{split}T_{-} = \begin{bmatrix} 0.99 & 0 & 0 & 0 \\ 0 & 0.9688 & 0.042 & 0 \\ 0 & -0.0485 & 0.839 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}\end{split}\]

and

\[\begin{split}T_{+} = \begin{bmatrix} 0.99 & 0 & 0 & 0 \\ 0 & 0.9688 & 0.046 & 0 \\ 0 & -0.0485 & 0.9189 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}\end{split}\]

Note

This section does not discuss the transformation between the MRI voxel indices and the different MRI coordinates. However, it is important to note that in FreeSurfer, MNE, as well as in Neuromag software an integer voxel coordinate corresponds to the location of the center of a voxel. Detailed information on the FreeSurfer MRI systems can be found at https://surfer.nmr.mgh.harvard.edu/fswiki/CoordinateSystems.

Coordinate transformations in FreeSurfer and MNE software packages.

Transformation

FreeSurfer

MNE

\(T_1\)

Not present

Measurement data files
Forward solution files (*fwd.fif)
Inverse operator files (*inv.fif)

\(T_{s_1}\dots T_{s_n}\)

Not present

Channel information in files containing \(T_1\).

\(T_2\)

Not present

MRI description files Separate
coordinate transformation files
saved from mne_analyze
Forward solution files
Inverse operator files

\(T_3\)

mri/*mgz files

MRI description files saved with mne_make_cor_set if the input is in mgz or mgh format.

\(T_4\)

mri/transforms/talairach.xfm

MRI description files saved with mne_make_cor_set if the input is in mgz or mgh format.

\(T_-\)

Hardcoded in software

MRI description files saved with mne_make_cor_set if the input is in mgz or mgh format.

\(T_+\)

Hardcoded in software

MRI description files saved with mne_make_cor_set if the input is in mgz or mgh format.

Note

The symbols \(T_x\) are defined in MEG/EEG and MRI coordinate systems. mne_make_cor_set /mne_setup_mri prior to release 2.6 did not include transformations \(T_3\), \(T_4\), \(T_-\), and \(T_+\) in the fif files produced.

The head and device coordinate systems

Head coordinate system

The head coordinate system

The MEG/EEG head coordinate system employed in the MNE software is a right-handed Cartesian coordinate system. The direction of \(x\) axis is from left to right, that of \(y\) axis to the front, and the \(z\) axis thus points up.

The \(x\) axis of the head coordinate system passes through the two periauricular or preauricular points digitized before acquiring the data with positive direction to the right. The \(y\) axis passes through the nasion and is normal to the \(x\) axis. The \(z\) axis points up according to the right-hand rule and is normal to the \(xy\) plane.

The origin of the MEG device coordinate system is device dependent. Its origin is located approximately at the center of a sphere which fits the occipital section of the MEG helmet best with \(x\) axis axis going from left to right and \(y\) axis pointing front. The \(z\) axis is, again, normal to the \(xy\) plane with positive direction up.

Note

The above definition is identical to that of the Neuromag MEG/EEG (head) coordinate system. However, in 4-D Neuroimaging and CTF MEG systems the head coordinate frame definition is different. The origin of the coordinate system is at the midpoint of the left and right auricular points. The \(x\) axis passes through the nasion and the origin with positive direction to the front. The \(y\) axis is perpendicular to the \(x\) axis on the and lies in the plane defined by the three fiducial landmarks, positive direction from right to left. The \(z\) axis is normal to the plane of the landmarks, pointing up. Note that in this convention the auricular points are not necessarily located on \(y\) coordinate axis. The file conversion utilities take care of these idiosyncrasies and convert all coordinate information to the MNE software head coordinate frame.

Coil geometry information

This Section explains the presentation of MEG detection coil geometry information the approximations used for different detection coils in MNE software. Two pieces of information are needed to characterize the detectors:

  • The location and orientation a local coordinate system for each detector.

  • A unique identifier, which has an one-to-one correspondence to the geometrical description of the coil.

Note

MNE ships with several coil geometry configurations. They can be found in mne/data. See Plotting sensor layouts of MEG systems for a comparison between different coil geometries, and Implemented coil geometries for detailed information regarding the files describing Neuromag coil geometries.

The sensor coordinate system

The sensor coordinate system is completely characterized by the location of its origin and the direction cosines of three orthogonal unit vectors pointing to the directions of the x, y, and z axis. In fact, the unit vectors contain redundant information because the orientation can be uniquely defined with three angles. The measurement fif files list these data in MEG device coordinates. Transformation to the MEG head coordinate frame can be easily accomplished by applying the device-to-head coordinate transformation matrix available in the data files provided that the head-position indicator was used. Optionally, the MNE software forward calculation applies another coordinate transformation to the head-coordinate data to bring the coil locations and orientations to the MRI coordinate system.

If \(r_0\) is a row vector for the origin of the local sensor coordinate system and \(e_x\), \(e_y\), and \(e_z\) are the row vectors for the three orthogonal unit vectors, all given in device coordinates, a location of a point \(r_C\) in sensor coordinates is transformed to device coordinates (\(r_D\)) by

\[[r_D 1] = [r_C 1] T_{CD}\ ,\]

where

\[\begin{split}T = \begin{bmatrix} e_x & 0 \\ e_y & 0 \\ e_z & 0 \\ r_{0D} & 1 \end{bmatrix}\ .\end{split}\]

Calculation of the magnetic field

The forward calculation in the MNE software computes the signals detected by each MEG sensor for three orthogonal dipoles at each source space location. This requires specification of the conductor model, the location and orientation of the dipoles, and the location and orientation of each MEG sensor as well as its coil geometry.

The output of each SQUID sensor is a weighted sum of the magnetic fluxes threading the loops comprising the detection coil. Since the flux threading a coil loop is an integral of the magnetic field component normal to the coil plane, the output of the k th MEG channel, \(b_k\) can be approximated by:

\[b_k = \sum_{p = 1}^{N_k} {w_{kp} B(r_{kp}) \cdot n_{kp}}\]

where \(r_{kp}\) are a set of \(N_k\) integration points covering the pickup coil loops of the sensor, \(B(r_{kp})\) is the magnetic field due to the current sources calculated at \(r_{kp}\), \(n_{kp}\) are the coil normal directions at these points, and \(w_{kp}\) are the weights associated to the integration points. This formula essentially presents numerical integration of the magnetic field over the pickup loops of sensor \(k\).

Computing the forward solution

Examples on how to compute the forward solution using mne.make_forward_solution() can be found Compute forward solution and Computing the forward solution.

Note

Notice that systems such as CTF and 4D Neuroimaging data may have been subjected to noise cancellation employing the data from the reference sensor array. Even though these sensor are rather far away from the brain sources, this can be taken into account using mne.io.Raw.apply_gradient_compensation(). See Brainstorm CTF phantom dataset tutorial.

EEG forward solution in the sphere model

For the computation of the electric potential distribution on the surface of the head (EEG) it is necessary to define the conductivities (\(\sigma\)) and radiuses of the spherically symmetric layers. Different sphere models can be specified with through mne.make_sphere_model(). Here follows the default structure given when calling sphere = mne.make_sphere_model()

Structure of the default EEG model

Layer

Relative outer radius

\(\sigma\) (S/m)

Head

1.0

0.33

Skull

0.97

0.04

CSF

0.92

1.0

Brain

0.90

0.33

Although it is not BEM model per se the sphere structure describes the head geometry so it can be passed as bem parameter in functions such as mne.fit_dipole(), mne.viz.plot_alignment() or mne.make_forward_solution().

When the sphere model is employed to compute the forward model using mne.make_forward_solution(), the computation of the EEG solution can be substantially accelerated by using approximation methods described by Mosher, Zhang, and Berg, see Forward modeling (Mosher et al. and references therein). In such scenario, MNE approximates the solution with three dipoles in a homogeneous sphere whose locations and amplitudes are determined by minimizing the cost function:

\[S(r_1,\dotsc,r_m\ ,\ \mu_1,\dotsc,\mu_m) = \int_{scalp} {(V_{true} - V_{approx})}\,dS\]

where \(r_1,\dotsc,r_m\) and \(\mu_1,\dotsc,\mu_m\) are the locations and amplitudes of the approximating dipoles and \(V_{true}\) and \(V_{approx}\) are the potential distributions given by the true and approximative formulas, respectively. It can be shown that this integral can be expressed in closed form using an expansion of the potentials in spherical harmonics. The formula is evaluated for the most superficial dipoles, i.e., those lying just inside the inner skull surface.

Averaging forward solutions

Purpose

One possibility to make a grand average over several runs of a experiment is to average the data across runs and average the forward solutions accordingly. For this purpose, mne.average_forward_solutions() computes a weighted average of several forward solutions. The function averages both MEG and EEG forward solutions. Usually the EEG forward solution is identical across runs because the electrode locations do not change.