The forward solution¶

Tessellation file format¶

The format of the text format surface files is the following:

<nvert>

<vertex 1>

<vertex 2>

…

<vertex nvert>

<ntri>

<triangle 1>

<triangle 2>

…

<triangle ntri> ,

where <nvert> and <ntri> are the number of vertices and number of triangles in the tessellation, respectively.

The format of a vertex entry is one of the following:

x y z

The x, y, and z coordinates of the vertex location are given in mm.

number x y z

A running number and the x, y, and z coordinates are given. The running number is not considered by mne_tri2fiff. The nodes must be thus listed in the correct consecutive order.

x y z nx ny nz

The x, y, and z coordinates as well as the approximate vertex normal direction cosines are given.

number x y z nx ny nz

A running number is given in addition to the vertex location and vertex normal.

Each triangle entry consists of the numbers of the vertices belonging to a triangle. The vertex numbering starts from one. The triangle list may also contain running numbers on each line describing a triangle.

Topology checks¶

If the --check option is specified, the following topology checks are performed:

• The completeness of each surface is confirmed by calculating the total solid angle subtended by all triangles from a point inside the triangulation. The result should be very close to \(4 \pi\). If the result is \(-4 \pi\) instead, it is conceivable that the ordering of the triangle vertices is incorrect and the --swap option should be specified.
• The correct ordering of the surfaces is verified by checking that the surfaces are inside each other as expected. This is accomplished by checking that the sum solid angles subtended by triangles of a surface \(S_k\) at all vertices of another surface \(S_p\) which is supposed to be inside it equals \(4 \pi\). Naturally, this check is applied only if the model has more than one surface. Since the surface relations are transitive, it is enough to check that the outer skull surface is inside the skin surface and that the inner skull surface is inside the outer skull one.
• The extent of each of the triangulated volumes is checked. If the extent is smaller than 50mm, an error is reported. This may indicate that the vertex coordinates have been specified in meters instead of millimeters.

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

where

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:

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 \(k\).

There are three accuracy levels for the numerical integration expressed above. The simple accuracy means the simplest description of the coil. This accuracy is not used in the MNE forward calculations. The normal or recommended accuracy typically uses two integration points for planar gradiometers, one in each half of the pickup coil and four evenly distributed integration points for magnetometers. This is the default accuracy used by MNE. If the --accurate option is specified, the forward calculation typically employs a total of eight integration points for planar gradiometers and sixteen for magnetometers. Detailed information about the integration points is given in the next section.

Implemented coil geometries¶

This section describes the coil geometries currently implemented in Neuromag software. The coil types fall in two general categories:

• Axial gradiometers and planar gradiometers and
• Planar magnetometers.

For axial sensors, the z axis of the local coordinate system is parallel to the field component detected, i.e., normal to the coil plane.For circular coils, the orientation of the x and y axes on the plane normal to the z axis is irrelevant. In the square coils employed in the Vectorview (TM) system the x axis is chosen to be parallel to one of the sides of the magnetometer coil. For planar sensors, the z axis is likewise normal to the coil plane and the x axis passes through the centerpoints of the two coil loops so that the detector gives a positive signal when the normal field component increases along the x axis.

Normal coil descriptions. lists the parameters of the normal coil geometry descriptions Accurate coil descriptions lists the accurate descriptions. For simple accuracy, please consult the coil definition file, see The coil definition file. The columns of the tables contain the following data:

• The number identifying the coil id. This number is used in the coil descriptions found in the FIF files.
• Description of the coil.
• Number of integration points used
• The locations of the integration points in sensor coordinates.
• Weights assigned to the field values at the integration points. Some formulas are listed instead of the numerical values to demonstrate the principle of the calculation. For example, in the normal coil descriptions of the planar gradiometers the weights are inverses of the baseline of the gradiometer to show that the output is in T/m.

The coil definition file¶

The coil geometry information is stored in the text file $MNE_ROOT/share/mne/coil_def.dat. In this file, any lines starting with the pound sign (#) are comments. A coil definition starts with a description line containing the following fields:

** <class>**

This is a number indicating class of this coil. Possible values are listed in Coil class values.

** <id>**

Coil id value. This value is listed in the first column of Tables Normal coil descriptions. and Accurate coil descriptions.

** <accuracy>**

The coil representation accuracy. Possible values and their meanings are listed in Coil representation accuracies..

** <np>**

Number of integration points in this representation.

** <size/m>**

The size of the coil. For circular coils this is the diameter of the coil and for square ones the side length of the square. This information is mainly included to facilitate drawing of the coil geometry. It should not be employed to infer a coil approximation for the forward calculations.

** <baseline/m>**

The baseline of a this kind of a coil. This will be zero for magnetometer coils. This information is mainly included to facilitate drawing of the coil geometry. It should not be employed to infer a coil approximation for the forward calculations.

** <description>**

Short description of this kind of a coil. If the description contains several words, it is enclosed in quotes.

Each coil description line is followed by one or more integration point lines, consisting of seven numbers:

** <weight>**

Gives the weight for this integration point (last column in Tables Normal coil descriptions. and Accurate coil descriptions).

** <x/m> <y/m> <z/m>**

Indicates the location of the integration point (fourth column in Tables Normal coil descriptions. and Accurate coil descriptions).

** <nx> <ny> <nz>**

Components of a unit vector indicating the field component to be selected. Note that listing a separate unit vector for each integration points allows the implementation of curved coils and coils with the gradiometer loops tilted with respect to each other.

Creating the coil definition file¶

The standard coil definition file $MNE_ROOT/share/mne/coil_def.dat is included with the MNE software package. The coil definition file can be recreated with the utility mne_list_coil_def as follows:

mne_list_coil_def –out $MNE_ROOT/share/mne/coil_def.dat

Purpose¶

Instead of using the convenience script mne_do_forward_solution it is also possible to invoke the forward solution computation program mne_forward_solution directly. In this approach, the convenience of the automatic file naming conventions present in mne_do_forward_solution are lost. However, there are some special-purpose options available in mne_forward_solution only. Please refer to Computing the forward solution for information on mne_do_forward_solution. See mne_forward_solution for command-line options.

Implementation of software gradient compensation¶

As described in Applying software gradient compensation the 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, mne_forward_solution takes them into account in the computations. If the data file specified with the --meas option has software gradient compensation activated, mne_forward_solution computes the field of at the reference sensors in addition to the main MEG sensor array and computes a compensated forward solution using the methods described in Applying software gradient compensation.

The EEG sphere model definition file¶

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 the --eegmodels option.

The EEG sphere model definition files may contain comment lines starting with a # and model definition lines in the following format:

<name>: <radius1>: <conductivity1>: <radius2>: <conductivity2>:…

When the file is loaded the layers are sorted so that the radiuses will be in ascending order and the radius of the outermost layer is scaled to 1.0. The scalp radius specified with the --eegrad option is then consulted to scale the model to the correct dimensions. Even if the model setup file is not present, a model called Default is always provided. This model has the structure given in Structure of the default EEG model

EEG forward solution in the sphere model¶

When the sphere model is employed, 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). mne_forward_solution approximates the solution with three dipoles in a homogeneous sphere whose locations and amplitudes are determined by minimizing the cost function:

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.

Field derivatives¶

If the --grad option is specified, mne_forward_solution includes the derivatives of the forward solution with respect to the dipole location coordinates to the output file. Let

be the \(N_{chan} \times 3\) matrix containing the signals produced by three orthogonal dipoles at location \(r_k\) making up \(N_{chan} \times 3N_{source}\) the gain matrix

With the --grad option, the output from mne_forward_solution also contains the \(N_{chan} \times 9N_{source}\) derivative matrix

where

where \(x_k\), \(y_k\), and \(z_k\) are the location coordinates of the \(k^{th}\) dipole. If the dipole orientations are to the cortical normal with the --fixed option, the dimensions of \(G\) and \(D\) are \(N_{chan} \times N_{source}\) and \(N_{chan} \times 3N_{source}\), respectively. Both \(G\) and \(D\) can be read with the mne_read_forward_solution Matlab function, see Table 10.1.

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 program averages both MEG and EEG forward solutions. Usually the EEG forward solution is identical across runs because the electrode locations do not change.