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    filename: DEM_demo_fMRI_PEB.m
    path of file: ./repos/spm/toolbox/DEM
    
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% Demonstration of PEB for multisession spectral DCM studies
%__________________________________________________________________________
% This demonstration routine illustrates the analysis of a multisession
% fMRI study using spectral DCM. Crucially, the between session effects are
% characterised using empirical Bayes and Bayesian model reduction. This
% means that the original session data are only inverted once (at the
% within session level). The resulting posterior estimates and then used to
% make inferences about between session effects (e.g., time or drug
% effects). The basic question addressed in this sort of analysis is where
% between session effects are expressed in terms of connectivity or
% parameters of neuronal fluctuations. These sorts of effects are specified

    
function DEM_demo_fMRI_PEB
% Demonstration of PEB for multisession spectral DCM studies
%__________________________________________________________________________
% This demonstration routine illustrates the analysis of a multisession
% fMRI study using spectral DCM. Crucially, the between session effects are
% characterised using empirical Bayes and Bayesian model reduction. This
% means that the original session data are only inverted once (at the
% within session level). The resulting posterior estimates and then used to
% make inferences about between session effects (e.g., time or drug
% effects). The basic question addressed in this sort of analysis is where
% between session effects are expressed in terms of connectivity or
% parameters of neuronal fluctuations. These sorts of effects are specified
% in a second level design matrix in the usual way and can be identified
% using Bayesian model reduction.
%
% in this example, we analyse three sessions with a monotonic change in the
% intrinsic (self) connectivity over three sessions. This involves
% decreases in diagonal A parameters at the first two levels of a simple
% three node hierarchy - and an increase at the highest (third) level.
% Physiologically, this corresponds to a decrease in self-inhibition (or
% increase in excitability) in the lower notes for regions, as time goes
% on.
%__________________________________________________________________________
 
% Karl Friston
% Copyright (C) 2008-2022 Wellcome Centre for Human Neuroimaging
 
 
 
% Simulate timeseries
%==========================================================================
rng('default')
 
% assume we have data from eight subjects with a TR of two seconds
% -------------------------------------------------------------------------
T  = 256*8;                           % number of observations (scans)
TR = 2;                               % repetition time or timing
t  = (1:T)*TR;                        % observation times
n  = 3;                               % number of regions or nodes
 
% setup model
% -------------------------------------------------------------------------
options.nonlinear  = 0;
options.two_state  = 0;
options.stochastic = 0;
options.centre     = 1;
options.induced    = 1;
 
A   = ones(n,n);
B   = zeros(n,n,0);
C   = zeros(n,n);
D   = zeros(n,n,0);
pP  = spm_dcm_fmri_priors(A,B,C,D,options);
 
 
% true parameters - a simple hierarchy of three nodes
% -------------------------------------------------------------------------
pP.A = [  0  -.2    0;
         .2    0  -.1
          0   .3    0];
pP.C = eye(n,n);
pP.transit = randn(n,1)/16;
 
% spectral density of neuronal fluctuations and observation noise (eg)
% -------------------------------------------------------------------------
[Gu,Gn,Hz,dt] = spm_csd_fmri_gu(pP,TR);
Gu            = Gu(:,1,1)*ones(1,n);
Gn            = Gn(:,1,1)*ones(1,n);
 
 
% session specific parameters: here, intrinsic connectivity
% -------------------------------------------------------------------------
X    = exp(-(0:2))';                    % between session decay
B    = [  1    0    0;                  % of intrinsic connections
          0    1    0
          0    0   -1];
X    = X - mean(X);                     % detrend