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# berg

## (Toolbox/berg.m in BrainStorm 2.0 (Alpha))

### Function Synopsis

    [mu_berg,lam_berg,f] = berg(R,sigma,nmax,J,mu_berg_init,lam_berg_init)

### Help Text

    BERG - PARAMETER CALCULATION FOR EEG MULTILAYER SPHERICAL FORWARD MODEL (berg.m)
     function [mu_berg,lam_berg,f] = berg(R,sigma,nmax,J,mu_berg_init,lam_berg_init)
          -or- [mu_berg,lam_berg] = berg(R,sigma,nmax,J)
          -or- [mu_berg,lam_berg] = berg(R,sigma,nmax)
          -or- [mu_berg,lam_berg] = berg(R,sigma)
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

     This function computes the Berg Eccentricity and Magnitude parameters associated with
     a series Approximiation of a Single Dipole in a Multilayer Sphere -by-
     multiple dipoles in a single shell. Zhang: Eq (1i,2i,1i',5i'' and 7i).

     This function has been generalized to compute Berg parameters for an approximation
     based on a total of J (user-specified) dipoles (J=3 is recommended)

     (Ref: Z. Zhang "A fast method to compute surface potentials generated by dipoles
     within multilayer anisotropic spheres" (Phys. Med. Biol. 40, pp335-349,1995)

     INPUTS (Required):
             R    : Radii(in meters) of sphere from
                    INNERMOST to OUTERMOST                                     NL x 1
             sigma: conductivity from INNERMOST to OUTERMOST                   NL x 1

     INPUTS (Optional):
             nmax : # of terms used in Legendre Expansion used to
                   "fit" Berg Parameters (Default: 100)                        scalar
               J  : Number of Berg Dipoles (Default: 3)                        scalar
      mu_berg_init: User specified initial value for Berg eccentricity
                    factors (default values shown below used otherwise)        J x 1
     lam_berg_init: User specified initial value for Berg magnitude
                    factors (default values shown below used otherwise)        J x 1

                    where: NL = # of sphere layers; J = # of Berg Dipoles

     OUTPUTS:
            mu_berg: Computed Value of Berg eccentricity factors                J x 1
           lam_berg: Computed Value of Berg magnitude factors                   J x 1
                  f: Legendre Expansion Weights used to fit Berg Parameters     nmax x 1

     External Functions and Files:
        zhang_fit.m: External Function used to fit Berg Parameters (Zhang Eq# 5i")

     By John Ermer 5/5/99
        6/21/99: Addded Legendre Expansion weights as optional output (JE)
        9/30/99: Fixed Logic to pass Initial Berg Parameters (JE)
        Nov 2001: SB - replaced fmins syntax (obsolete in Matlab R12) by fminsearch
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

### Cross-Reference Information

This function calls

    zhang_fit                          C:\BrainStorm_2001\Toolbox\zhang_fit.m

This function is called by

    bst_headmodeler                    C:\BrainStorm_2001\Toolbox\bst_headmodeler.m
    zhang_fit                          C:\BrainStorm_2001\Toolbox\zhang_fit.m

### Listing of function C:\BrainStorm_2001\Toolbox\berg.m

    function [mu_berg,lam_berg,f] = berg(R,sigma,nmax,J,mu_berg_init,lam_berg_init)
    %BERG - PARAMETER CALCULATION FOR EEG MULTILAYER SPHERICAL FORWARD MODEL (berg.m)
    % function [mu_berg,lam_berg,f] = berg(R,sigma,nmax,J,mu_berg_init,lam_berg_init)
    %      -or- [mu_berg,lam_berg] = berg(R,sigma,nmax,J)
    %      -or- [mu_berg,lam_berg] = berg(R,sigma,nmax)
    %      -or- [mu_berg,lam_berg] = berg(R,sigma)
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    %
    % This function computes the Berg Eccentricity and Magnitude parameters associated with
    % a series Approximiation of a Single Dipole in a Multilayer Sphere -by-
    % multiple dipoles in a single shell. Zhang: Eq (1i,2i,1i',5i'' and 7i).
    %
    % This function has been generalized to compute Berg parameters for an approximation
    % based on a total of J (user-specified) dipoles (J=3 is recommended)
    %
    % (Ref: Z. Zhang "A fast method to compute surface potentials generated by dipoles
    % within multilayer anisotropic spheres" (Phys. Med. Biol. 40, pp335-349,1995)
    %
    % INPUTS (Required):
    %         R    : Radii(in meters) of sphere from
    %                INNERMOST to OUTERMOST                                     NL x 1
    %         sigma: conductivity from INNERMOST to OUTERMOST                   NL x 1
    %
    % INPUTS (Optional):
    %         nmax : # of terms used in Legendre Expansion used to
    %               "fit" Berg Parameters (Default: 100)                        scalar
    %           J  : Number of Berg Dipoles (Default: 3)                        scalar
    %  mu_berg_init: User specified initial value for Berg eccentricity
    %                factors (default values shown below used otherwise)        J x 1
    % lam_berg_init: User specified initial value for Berg magnitude
    %                factors (default values shown below used otherwise)        J x 1
    %
    %                where: NL = # of sphere layers; J = # of Berg Dipoles
    %
    % OUTPUTS:
    %        mu_berg: Computed Value of Berg eccentricity factors                J x 1
    %       lam_berg: Computed Value of Berg magnitude factors                   J x 1
    %              f: Legendre Expansion Weights used to fit Berg Parameters     nmax x 1
    %
    % External Functions and Files:
    %    zhang_fit.m: External Function used to fit Berg Parameters (Zhang Eq# 5i")
    %
    % By John Ermer 5/5/99
    %    6/21/99: Addded Legendre Expansion weights as optional output (JE)
    %    9/30/99: Fixed Logic to pass Initial Berg Parameters (JE)
    %    Nov 2001: SB - replaced fmins syntax (obsolete in Matlab R12) by fminsearch
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

    %<autobegin> ---------------------- 14-Jun-2004 17:09:48 -----------------------
    % --------- Automatically Generated Comments Block Using AUTO_COMMENTS ---------
    %
    % CATEGORY: Forward Modeling
    %
    % At Check-in: $Author: Mosher $  $Revision: 14 $  $Date: 6/14/04 3:37p $
    %
    % This software is part of BrainStorm Toolbox Version 2.0 (Alpha) 14-Jun-2004
    %
    % Principal Investigators and Developers:
    % ** Richard M. Leahy, PhD, Signal & Image Processing Institute,
    %    University of Southern California, Los Angeles, CA
    % ** John C. Mosher, PhD, Biophysics Group,
    %    Los Alamos National Laboratory, Los Alamos, NM
    % ** Sylvain Baillet, PhD, Cognitive Neuroscience & Brain Imaging Laboratory,
    %    CNRS, Hopital de la Salpetriere, Paris, France
    %
    % See BrainStorm website at http://neuroimage.usc.edu for further information.
    %
    % Copyright (c) 2004 BrainStorm by the University of Southern California
    % This software distributed  under the terms of the GNU General Public License
    % as published by the Free Software Foundation. Further details on the GPL
    % license can be found at http://www.gnu.org/copyleft/gpl.html .
    %
    % FOR RESEARCH PURPOSES ONLY. THE SOFTWARE IS PROVIDED "AS IS," AND THE
    % UNIVERSITY OF SOUTHERN CALIFORNIA AND ITS COLLABORATORS DO NOT MAKE ANY
    % WARRANTY, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO WARRANTIES OF
    % MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE, NOR DO THEY ASSUME ANY
    % LIABILITY OR RESPONSIBILITY FOR THE USE OF THIS SOFTWARE.
    %<autoend> ------------------------ 14-Jun-2004 17:09:48 -----------------------

    %%% THIS PART CHECKS INPUT PARAMETERS FOR THEIR DIMENSION AND VALIDITY %%%
    %
    NL = length(R);                          % # of concentric sphere layer
    %
    %%% THIS PART CHECKS FOR THE PRESENCE OF OPTIONAL PARAMETERS AND ASSIGNS %%%
    %%% DEFAULT VALUES IF NECESSARY %%%
    %
    if nargin < 3    % Check # terms to see if number of Legendre expansion terms is specified
        nmax = 200;  % Default # of Legendre Terms
    end
    %
    if nargin < 4  % Check # of Berg Paramters to be generated
        J=3;       % Default # of Berg Parameters
    end
    %
    if (nargin < 5)   % Check if user specified initial value for Berg Paramters
        mu_berg_init = (1/(J+2))*[1:1:J];  % Default Value for Initial Eccentricity Parameters
       lam_berg_init = 0.2*ones(1,J);      % Default Value for Initial Magnitude Parameters
    else
       if length(mu_berg_init)~=length(lam_berg_init)
          error('Initial Berg Eccen and Mag parameters are of Different Lengths!!!')
       elseif length(mu_berg_init) ~= J
          error('Initial Berg Param Size does not match number of dipoles specified!!!')
       end
    end
    %
    %%% THIS PART COMPUTES THE WEIGHTS fn ASSOCIATED WITH A LEGENDRE EXPANSION %%%%%%%%%%%%%%%
    %%% THE WEIGHTS fn DEPEND ONLY ON THE MULTISPHERE RADII AND CONDUCTIVITY %%%%%%%%%%%%%%%%%
    %%% (This portion for computing fn derived from gainp_sph.m by CCH,  Aug/20/1995)
    %
    Re_mag = R(NL);         % Radius of outermost layer (Sensor distance from origin)
    if NL==1
       f=ones(1,nmax);
    else
    %
    for k = 1:NL-1
       s(k) = sigma(k)/sigma(k+1);
    end
    a = Re_mag./R;
    ainv = R/Re_mag;
    sm1 = s-1;
    twonp1 = 2*[1:nmax]+1;
    twonp1 = twonp1(:);
    f = zeros(nmax,1);
    %
    for n = 1:nmax
     np1 = n+1;
     Mc = eye(2);
     for k = 2:NL-1
        Mc = Mc*[n+np1*s(k),  np1*sm1(k)*a(k)^twonp1(n);...
                 n*sm1(k)*ainv(k)^twonp1(n) , np1+n*s(k)];
     end
    Mc(2,:) = [n*sm1(1)*ainv(1)^twonp1(n) , np1+n*s(1)]*Mc; % Compute only components of interest
    Mc = Mc/(twonp1(n))^(NL-1);
    f(n) = n/(n*Mc(2,2)+np1*Mc(2,1));
    end
    end
    %
    %%%% End of Calculation of weights fn %%%%%%%
    %
    %%%% THIS PART COMPUTES THE BERG PARAMETERS ASSOCIATED WITH THE SPHERE RADII AND %%%%%%%%%%%%%
    %%%% AND CONDUCTIVITY BY MINIMIZING THE FUNCTION 5I" SPECIFIED IN ZHANG %%%%%%%%%%%%%%%%%%%%%%
    %
    % OPTIONS=zeros(1,18);                 % Reset all options flags
    % OPTIONS(1:3) = [0 1.e-9 1.e-9];      % Set options termination search criteria
    % OPTIONS(14) = J*1000;                  % Set max number of steps

    OPTIONS = optimset('MaxFunEvals',J*1000,'MaxIter',J*1000,'TolFun',[0 1.e-9 1.e-9]);

    [berg_out,fval,outputs]= fminsearch('zhang_fit',[mu_berg_init lam_berg_init(2:J)],OPTIONS,R,f);

    %num_steps = outputs.iterations;
    mu_berg = berg_out(1:J);
    lam_berg(2:J) = berg_out(J+1:2*J-1);
    lam_berg(1) = f(1) - sum(lam_berg(2:J));

    f = f';

    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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