Detailed analysis of trajectories in the Morris water maze (Gehring et al. 2015)

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MATLAB code that can be used for detailed behavioural analyzes of the trajectories of animals be means of a semi-supervised clustering algorithm. The method is applied here to trajectories in the Morris Water Maze (see Gehring, T. V. et al., Scientific Reports, 2015) but the code can easily be adapted to other types experiments. For more information and the latest version of the code please refer to
1 . Gehring TV, Luksys G, Sandi C, Vasilaki E (2015) Detailed classification of swimming paths in the Morris Water Maze: multiple strategies within one trial. Sci Rep 5:14562 [PubMed]
Model Information (Click on a link to find other models with that property)
Model Type:
Brain Region(s)/Organism:
Cell Type(s):
Gap Junctions:
Simulation Environment: MATLAB;
Model Concept(s): Methods;
% distanceEllipsePoint      Computes the distances between an ellipse and
%                           an arbitrary number of points (in 3D)
%   [min_dist, f_min] = distanceEllipsePoint(XYZ, a,b,c,u,v)
% The input arguments are 
% =============================================================
%  name     description                           size
% =============================================================
%   XYZ     array of points                       (Nx3)
%   a       Ellipse's semi-major axis             (1x1)
%   b       Ellipse's semi-minor axis             (1x1)
%   c       Location of Ellipse's center          (1x3)
%   u       Direction of Ellipse's primary axis   (1x3)
%           (unit vector towards [a])             
%   v       Direction of Ellipse's secondary axis (1x3)
%           (unit vector towards [b])             
% The output arguments are
% =============================================================
%  name     description                           size
% =============================================================
% min_dist  distances between the points and the  (Nx1)
%           ellipse                 
% f_min     corresponding true anomalies on the   (Nx1)
%           ellipse (-pi <= f <= +pi)         
% Based on:
% Ik-Sung Kim: "An algorithm for finding the distance between two 
% ellipses". Commun. Korean Math. Soc. 21 (2006), No.3, pp.559-567
function [min_dist, f_min] = distanceEllipsePoints(XYZ, a,b,c,u,v)
% Please report bugs and inquiries to: 
% Name       : Rody P.S. Oldenhuis
% E-mail     :    (personal)
%      (professional)
% Affiliation: LuxSpace sàrl
% Licence    : GPL + anything implied by placing it on the FEX

% If you find this work useful, please consider a donation:

    % error traps
    assert( any(size(XYZ)==3),'distanceEllipsePoint:points_not_3D',...
            'At least one dimension of the array of points must be equal to 3.'); 
    assert( isscalar(a) && isscalar(b), 'distanceEllipsePoint:ab_not_scalar',...
            'Arguments [a] and [b] must be scalar.');  
    assert( isvector(c) && numel(c)==3, 'distanceEllipsePoint:c_not_3Dvector',...
            'Coordinates of the center [c] must be given as 3-D Cartesian coordinates.');    
    assert( isvector(u) && numel(u)==3, 'distanceEllipsePoint:u_not_3Dvector',...
            'Primary axis [u] must be given as 3-D Cartesian coordinates.');    
    assert( isvector(v) && numel(v)==3, 'distanceEllipsePoint:v_not_3Dvector',...
            'Secondary axis [v] must be given as 3-D Cartesian coordinates.');
    % make sure everything is correct shape & size
    c = c(:); u = u(:); v = v(:); XYZ = reshape(XYZ,[],3);    
    % make sure [u] and [v] are UNIT-vectors
    if (norm(u) ~= 1), u = u/norm(u); end
    if (norm(v) ~= 1), v = v/norm(v); end
    % initialize some variables to speed up computation    
    R = [u, v, cross(u,v)];   % rotation matrix to put ellipse in standard form    
    comp0 = [eye(3),[0;0;0]]; % part of a companion matrix for a quartic     
    % initialize output
    min_dist = zeros(size(XYZ,1),1);
    f_min    = min_dist;
    % loop through all points in [XYZ]
    for ii = 1:size(XYZ,1)
        % find optimal point on the ellipse
        s = R\(XYZ(ii,:).' - c); % transform current point
        A = a*s(1);              % The constants A,B and C follow from the
        B = b*s(2);              % condition dQ/dt = 0, with Q = Q(s,E,t) the
        C = b*b - a*a;           % XY-distance between point s and ellipse E

        % we have to find [t_hat], the true anomaly on the ellipse that minimizes
        % the distance between the associated point on the ellipse [E] and the
        % point [s]. The solution depends on the value of [C]. 
        % If C = 0, the solution is easy:
        if C == 0
            t_hat = atan2(B, A);

        % otherwise, we have to solve a quartic eqution in A,B,C, which is
        % done most quickly by using EIG() on its companion matrix:
            % associated companion matrix
            comp  = [-2*A/C, -(A*A+B*B-C*C)/C/C, +2*A/C, (A/C)^2; comp0];
            % solve this quartic (real values only)
            Roots = eig(comp);  
            Roots = Roots(imag(Roots)==0);        
            % extract optimal point
            sint1  = sqrt(1 - Roots.^2);    sint2 = -sint1;
            sints  = [sint1, sint2];        costs = [Roots,Roots];
            selld  = (s(1)-a*costs).^2 + (s(2)-b*sints).^2;
            [dummy, tind] = min(selld(:));%#ok
            sinth = sints(tind);            costh = costs(tind); 
            % t_hat
            t_hat = atan2(sinth, costh);


        % compute distance
        min_dist(ii) = sqrt( (s(1)-a*cos(t_hat))^2 + (s(2)-b*sin(t_hat))^2 + s(3)^2 );    
        % insert the optimal thetas
        f_min(ii) = t_hat;
end % function (Kim's method)

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