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iCAPs_v2/packages/spm12/spm_DEM_diff.m
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spm_DEM_diff.m
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function [u,dg,df] = spm_DEM_diff(M,u)
% Evaluate an active model given innovations z{i} and w{i}
% FORMAT [u dgdv dgdx dfdv dfdx] = spm_DEM_diff(M,u);
%
% M - generative model
%
% u.v - causal states - updated
% u.x - hidden states - updated
% u.z - innovation (causal state)
% u.w - innovation (hidden states)
% u.a - [active states]
%
% dg.dv, ... components of the Jacobian in generalised coordinates
%
% The system is evaluated at the prior expectation of the parameters
%__________________________________________________________________________
% Copyright (C) 2008-2014 Wellcome Trust Centre for Neuroimaging
% Karl Friston
% $Id: spm_DEM_diff.m 6132 2014-08-06 19:59:46Z karl $
% Check for action (ADEM)
%==========================================================================
try
M.a;
ADEM = 1;
catch
u.a = u.v;
[M(:).a] = deal(sparse(0,1));
[M(:).k] = deal(0);
[u.a{:}] = deal(sparse(0,1));
ADEM = 0;
end
% number of levels and order parameters (n = 1 for static models)
%--------------------------------------------------------------------------
nl = size(M,2); % number of levels
n = M(1).E.n + 1; % order of embedding
% initialise arrays for hierarchical form
%--------------------------------------------------------------------------
dfdvi = cell(nl,nl);
dfdxi = cell(nl,nl);
dfdai = cell(nl,nl);
dgdvi = cell(nl,nl);
dgdxi = cell(nl,nl);
dgdai = cell(nl,nl);
for i = 1:nl
dgdvi{i,i} = sparse(M(i).l,M(i).l);
dgdxi{i,i} = sparse(M(i).l,M(i).n);
dgdai{i,i} = sparse(M(i).l,M(i).k);
dfdvi{i,i} = sparse(M(i).n,M(i).l);
dfdxi{i,i} = sparse(M(i).n,M(i).n);
dfdai{i,i} = sparse(M(i).n,M(i).k);
end
% partition states {x,v,z,w} into distinct vector arrays v{i}, ...
%--------------------------------------------------------------------------
vi = spm_unvec(u.v{1},{M.v});
xi = spm_unvec(u.x{1},{M.x});
ai = spm_unvec(u.a{1},{M.a});
zi = spm_unvec(u.z{1},{M.v});
% Derivatives for Jacobian
%==========================================================================
vi{nl} = zi{nl};
for i = (nl - 1):-1:1
% evaluate
%----------------------------------------------------------------------
if ADEM
[dgdx,g] = spm_diff(M(i).g,xi{i},vi{i + 1},ai{i + 1},M(i).pE,1);
[dfdx,f] = spm_diff(M(i).f,xi{i},vi{i + 1},ai{i + 1},M(i).pE,1);
dgdv = spm_diff(M(i).g,xi{i},vi{i + 1},ai{i + 1},M(i).pE,2);
dfdv = spm_diff(M(i).f,xi{i},vi{i + 1},ai{i + 1},M(i).pE,2);
dgda = spm_diff(M(i).g,xi{i},vi{i + 1},ai{i + 1},M(i).pE,3);
dfda = spm_diff(M(i).f,xi{i},vi{i + 1},ai{i + 1},M(i).pE,3);
else
[dgdx,g] = spm_diff(M(i).g,xi{i},vi{i + 1},M(i).pE,1);
[dfdx,f] = spm_diff(M(i).f,xi{i},vi{i + 1},M(i).pE,1);
dgdv = spm_diff(M(i).g,xi{i},vi{i + 1},M(i).pE,2);
dfdv = spm_diff(M(i).f,xi{i},vi{i + 1},M(i).pE,2);
dgda = [];
dfda = [];
end
% g(x,v) & f(x,v)
%----------------------------------------------------------------------
gi{i} = g;
fi{i} = f;
vi{i} = spm_vec(gi{i}) + spm_vec(zi{i});
% and partial derivatives
%----------------------------------------------------------------------
dgdxi{i, i} = dgdx;
dgdvi{i,i + 1} = dgdv;
dgdai{i,i + 1} = dgda;
dfdxi{i, i} = dfdx;
dfdvi{i,i + 1} = dfdv;
dfdai{i,i + 1} = dfda;
end
% concatenate hierarchical arrays
%--------------------------------------------------------------------------
dg.da = spm_cat(dgdai);
dg.dv = spm_cat(dgdvi);
dg.dx = spm_cat(dgdxi);
df.da = spm_cat(dfdai);
df.dv = spm_cat(dfdvi);
df.dx = spm_cat(dfdxi);
% update generalised coordinates
%--------------------------------------------------------------------------
u.v{1} = spm_vec(vi);
u.x{2} = spm_vec(fi) + u.w{1};
for i = 2:(n - 1)
u.v{i} = dg.dv*u.v{i} + dg.dx*u.x{i} + dg.da*u.a{i} + u.z{i};
u.x{i + 1} = df.dv*u.v{i} + df.dx*u.x{i} + df.da*u.a{i} + u.w{i};
end
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