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    Metadata details:
    filename: FEP_fluctuations.m
    path of file: ./repos/bspm/thirdparty/spm12/toolbox/DEM
    
    the code of the file until where you have to start completion: function FEP_fluctuations
% This demonstration  uses an ensemble of particles with intrinsic (Lorentz
% attractor) dynamics and (Newtonian) short-range coupling.  The focus of
% this routine is to unpack the Bayesian perspective. We first simulate
% dynamics  to nonequilibrium steady-state, identify the Markov blanket and
% then examine the encoding of external states by internal states; in terms
% of their expected values.
%
% The crucial aspect of this implicit inference (and the basis of the free
% energy principle) is the existence of a conditional synchronisation

    
function FEP_fluctuations
% This demonstration  uses an ensemble of particles with intrinsic (Lorentz
% attractor) dynamics and (Newtonian) short-range coupling.  The focus of
% this routine is to unpack the Bayesian perspective. We first simulate
% dynamics  to nonequilibrium steady-state, identify the Markov blanket and
% then examine the encoding of external states by internal states; in terms
% of their expected values.
%
% The crucial aspect of this implicit inference (and the basis of the free
% energy principle) is the existence of a conditional synchronisation
% manifold, when conditioning internal and external states on the Markov
% blanket. This provides the basis for a mapping between internal and
% external states that can be interpreted in terms of a probabilistic
% representation or inference.
%
% This Bayesian perspective is illustrated in terms of a mapping between
% the canonical modes of internal and external states (as approximated
% with a polynomial expansion). The canonical modes her are evaluated
% using an estimate of the conditional expectations  based upon the
% Euclidean proximity of Markov blanket states. The ensuing posterior over
% external states is than illustrated, in relation to the actual external
% states. We also  simulate event related potentials by identifying
% several points in time when the Markov blankets revisit the same
% neighbourhood. Finally, to illustrate the underlying dynamics, the
% Jacobians or coupling among internal and external states are
% presented; using different orders of coupling (i.e., degrees of
% separation)
%
%__________________________________________________________________________
% Copyright (C) 2008 Wellcome Trust Centre for Neuroimaging
 
% Karl Friston
% $Id: FEP_fluctuations.m 7163 2017-09-04 09:12:50Z karl $
 
 
% default settings (GRAPHICS sets movies)
%--------------------------------------------------------------------------
rng('default')
GRAPHICS = 1;
 
% Demo of synchronization manifold using coupled Lorenz attractors
%==========================================================================
N    = 128;                         % number of (Lorenz) oscillators
T    = 2048;                        % number of time bins
dt   = 1/32;                        % time interval
 
% parameters
%--------------------------------------------------------------------------
P.k  = 1 - exp(-rand(1,N)*4);       % variations in temporal scale
P.d  = 1/8;                         % amplitude of random fluctuations
 
% states
%--------------------------------------------------------------------------
x.p  = randn(2,N)*4;                % microstates (position)
x.v  = zeros(2,N);                  % microstates (velocity)
x.q  = randn(3,N)/32;               % microstates (states)
u    = zeros(1,T);                  % exogenous fluctuations
 
 
% generate an dynamics from initial conditions
%==========================================================================
spm_figure('GetWin','Markov blanket');clf
if GRAPHICS
    subplot(2,2,1)
else
    subplot(2,1,1)
end