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homogeneous_model.m
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homogeneous_model.m
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%% Whole-brain modelling in the homogeneous case
%This model based on Stuard-Landau oscillators, the local dynamics of single brain
%regions using the normal form of a Hopf bifurcation.
%The dynamics of the N brain regions were coupled through the connectivity matrix,
%which was given by the connectome of healthy subjects (C).
%coupling among areas is given by the SC-> by the g scaling parameter.
%For the homogeneous case, in which we seta=0 for all nodes.
%This choice was based on previous studies which suggest that the best fit
%to the empirical data arises at the brink of the Hopf bifurcation
%where a~1. So, here the only free parameter is the g.
%The fitting of the model is captures using the correlation of the FC and
%the Kolmogorov-Smirnov distance of the FCD.
%This version of the model is prepared for individualize subjects (the SC is a template).
%Ane López-González 14/07/2021
%% load data and basic variables
load('data_controls_all.mat') %Load the BOLD timeseries NsubjectsxNodesxTimepoints
load('SC_template.mat') %Structural connectivity matrix NodesxNodes
NSUB=36; %Number of subjects
for subj=1:NSUB
ldata=1;%In this version we run the model only for one subject
C=sc_healthy; %structural connectivity
Tmax=295;%Number of timepoints
wG=[0:0.1:4]; %Range of values of g (depends on the data)
ap=0.0005;%a values
idxSub=1;%
rng('shuffle');
nNodes = length(C);
nSubs = ldata;
si = 1:ldata;
nWeights = numel(wG);
%% Details of the model
Cfg.simulID = 'coma1';
Cfg.opt_a.nIters = 200;%Number of iterations for initial optimization
Cfg.opt_a.updateStrength = 0.1;%Update strength for a optimization
Cfg.opt_a.abortCrit = 0.1; %maximally 2.5% error
Cfg.opt_a.gref = 1;%This is reference g, for the initial optimization only. Ignore if not using newOpt
if ~isfield(Cfg, 'simulID'), Cfg.simulID = 'Unknown'; else end;
if ~isfield(Cfg, 'TRsec'), Cfg.TRsec = 2; else end;
if ~isfield(Cfg, 'opt_a'), Cfg.opt_a = []; else end;
if ~isfield(Cfg.opt_a, 'nIters'), Cfg.opt_a.nIters = 100; else end;
if ~isfield(Cfg.opt_a, 'updateStrength'), Cfg.opt_a.updateStrength = 0.1; else end
if ~isfield(Cfg.opt_a, 'abortCrit'), Cfg.opt_a.abortCrit = 0.1; else end
if ~isfield(Cfg, 'plots'), Cfg.plots = []; else end
%% Pre-modelling
%In this section, the calculation of the FC, power spectrum, omega and
%other variables that will be used in the model simulation.
fprintf(1, 'Fitting models for %d different weights\n', nWeights);
FC_simul = zeros(nNodes, nNodes, nWeights);
fitting = zeros(1, nWeights);
meta = zeros(1, nWeights);
ksP = zeros(1, nWeights);
Phases = zeros(nNodes, Tmax, nSubs, nWeights);
bifpar = zeros(nWeights, nNodes);
%--------------------------------------------------------------------------
%CALCULATE FUNCTIONAL CONNECTIVITY MATRIX
%--------------------------------------------------------------------------
r = zeros(nNodes, nNodes, nSubs);
ts = zeros(nNodes, Tmax, nSubs);
i=1;
ts(:,:,1) = squeeze(ts_all(subj,:,:));
r(:,:,1) = corrcoef(ts(:,:,1)');
FC_emp=mean(r,3);
C=C/max(max(C))*0.2;%
%--------------------------------------------------------------------------
%COMPUTE POWER SPECTRA FOR
%NARROWLY FILTERED DATA WITH LOW BANDPASS (0.04 to 0.07 Hz)
%WIDELY FILTERED DATA (0.04 Hz to justBelowNyquistFrequency)
%[justBelowNyquistFrequency depends on TR,
%for a TR of 2s this is 0.249 Hz]
%--------------------------------------------------------------------------
TT=Tmax;
Ts = TT*Cfg.TRsec;
freq = (0:TT/2-1)/Ts;
[~, idxMinFreq] = min(abs(freq-0.04));
[~, idxMaxFreq] = min(abs(freq-0.07));
nFreqs = length(freq);
delt = 2; % sampling interval (TR)
fnq = 1/(2*delt); % Nyquist frequency
k = 2; % 2nd order butterworth filter
%WIDE BANDPASS
flp = .04; % lowpass frequency of filter
fhi = fnq-0.001; % highpass needs to be limited by Nyquist frequency, which in turn depends on TR
Wn = [flp/fnq fhi/fnq]; % butterworth bandpass non-dimensional frequency
[bfilt_wide, afilt_wide] = butter(k,Wn); % construct the filter
%NARROW LOW BANDPASS
flp = .04; % lowpass frequency of filter
fhi = .07; % highpass
Wn=[flp/fnq fhi/fnq]; % butterworth bandpass non-dimensional frequency
[bfilt_narrow,afilt_narrow] = butter(k,Wn); % construct the filter
PowSpect_filt_narrow = zeros(nFreqs, nNodes, nSubs);
PowSpect_filt_wide = zeros(nFreqs, nNodes, nSubs);
for seed=1:nNodes
%idxSub=1;
signaldata = squeeze(ts_all(subj,:,:));
x=detrend(demean(signaldata(seed,:)));
ts_filt_narrow =zscore(filtfilt(bfilt_narrow,afilt_narrow,x));
pw_filt_narrow = abs(fft(ts_filt_narrow));
PowSpect_filt_narrow(:,seed,1) = pw_filt_narrow(1:floor(TT/2)).^2/(TT/2);
ts_filt_wide =zscore(filtfilt(bfilt_wide,afilt_wide,x));
pw_filt_wide = abs(fft(ts_filt_wide));
PowSpect_filt_wide(:,seed,1) = pw_filt_wide(1:floor(TT/2)).^2/(TT/2);
end
Power_Areas_filt_narrow_unsmoothed = mean(PowSpect_filt_narrow,3);
Power_Areas_filt_wide_unsmoothed = mean(PowSpect_filt_wide,3);
Power_Areas_filt_narrow_smoothed = zeros(nFreqs, nNodes);
Power_Areas_filt_wide_smoothed = zeros(nFreqs, nNodes);
vsig = zeros(1, nNodes);
for seed=1:nNodes
Power_Areas_filt_narrow_smoothed(:,seed)=gaussfilt(freq,Power_Areas_filt_narrow_unsmoothed(:,seed)',0.01);
Power_Areas_filt_wide_smoothed(:,seed)=gaussfilt(freq,Power_Areas_filt_wide_unsmoothed(:,seed)',0.01);
%relative power in frequencies of interest (.04 - .07 Hz) with respect
%to entire power of bandpass-filtered data (.04 - just_below_nyquist)
vsig(seed) =...
sum(Power_Areas_filt_wide_smoothed(idxMinFreq:idxMaxFreq,seed))/sum(Power_Areas_filt_wide_smoothed(:,seed));
end
vmax=max(vsig); %consider computing this later where needed
vmin=min(vsig);%consider computing this later where needed
%a-minimization seems to only work if we use the indices for frequency of
%maximal power from the narrowband-smoothed data
[~, idxFreqOfMaxPwr]=max(Power_Areas_filt_narrow_smoothed);
f_diff = freq(idxFreqOfMaxPwr);
%FOR EACH AREA AND TIMEPOINT COMPUTE THE INSTANTANEOUS PHASE IN THE RANGE
%OF .04 TO .09 Hz
PhasesD = zeros(nNodes, Tmax, nSubs);
signaldata=squeeze(ts_all(subj,:,:));
for seed=1:nNodes
x = demean(detrend(signaldata(seed,:)));
xFilt = filtfilt(bfilt_narrow,afilt_narrow,x); % zero phase filter the data
Xanalytic = hilbert(demean(xFilt));
PhasesD(seed,:,1) = angle(Xanalytic);
end
%f_diff previously computed frequency with maximal power (of narrowly filtered data) by area
omega = repmat(2*pi*f_diff',1,2); %angular velocity
omega(:,1) = -omega(:,1);
%% FROM HERE ON SIMULATIONS AND FITTING
dt = 0.1;
sig = 0.04;
dsig = sqrt(dt)*sig;
a = repmat(-0.05*ones(nNodes,1),1,2);
a1 = a; %Starting values of a set to ~0
trackminm1 = zeros(Cfg.opt_a.nIters, nWeights); %for tracking the minimization (good for debugging)
for trial=1:10 %number of runs of the model
for idx_g = 1:nWeights
we = wG(idx_g);
fprintf(1, '-----------------------------------------\n');
fprintf(1, 'g(%d/%d) = %5.3f\n', idx_g, numel(wG), we);
fprintf(1, '-----------------------------------------\n');
xs = zeros(3000/2,nNodes);
wC = we*C; %structural connectivity matrix weighted with current global coupling parameter
sumC = repmat(sum(wC,2),1,2); % for sum Cij*xj
fprintf(1, 'SIMULATING HOMOGENEOUS MODEL.\n');
a=a1; %use those avalues that have been found to be optimal
bifpar(idx_g,:)=a(:,1)';%store them in an output variable
xs=zeros(Tmax*nSubs,nNodes);
z = 0.1*ones(nNodes,2); % --> x = z(:,1), y = z(:,2)
nn=0;
for t=1:dt:3000 %This part is to initialize the model and to warm the timeseries
suma = wC*z - sumC.*z; % sum(Cij*xi) - sum(Cij)*xj
zz = z(:,end:-1:1); % flipped z, because (x.*x + y.*y)
z = z + dt*(a.*z + zz.*omega - z.*(z.*z+zz.*zz) + suma) + dsig*randn(nNodes,2);
end
for t=1:dt:Tmax*Cfg.TRsec*nSubs %check for the specific data
suma = wC*z - sumC.*z; % sum(Cij*xi) - sum(Cij)*xj
zz = z(:,end:-1:1); % flipped z, because (x.*x + y.*y)
z = z + dt*(a.*z + zz.*omega - z.*(z.*z+zz.*zz) + suma) + dsig*randn(nNodes,2);
if mod(t,2)==0
nn=nn+1;
xs(nn,:)=z(:,1)';
end
end
fprintf(1, 'COMPUTING MODEL FIT.\n');
FC_simul(:, :, idx_g) = corrcoef(xs(1:nn,:));
cc=corrcoef(squareform(tril(FC_emp,-1)),squareform(tril(FC_simul(:, :, idx_g),-1)));
fitting(idx_g)=cc(2); %Fitting of the FC
%%%%%%%%%%%%%%%%%%
tini=(1-1)*Tmax;
for seed=1:nNodes
ts_simul = detrend(demean(xs(tini+1:tini+Tmax,seed)'));
ts_simul_filt_narrow = filtfilt(bfilt_narrow,afilt_narrow,ts_simul);
Xanalytic = hilbert(ts_simul_filt_narrow);
Phases(seed,:,1, idx_g) = angle(Xanalytic);
end
%This part has to be uncomment in case the fitting is also compared with
%the metastability.
% metastability22 = zeros(1, nSubs);
% T=1:Tmax;
% sync = zeros(1, numel(T));
% for t=T
% ku=sum(complex(cos(Phases(:,t,idxSub, idx_g)),sin(Phases(:,t,idxSub, idx_g))))/nNodes;
% sync(t, idx_g)=abs(ku);
% end
%
% metastability22(1)=std(sync(:, idx_g));
% meta(idx_g)=mean(metastability22);
pcD(:,1)=patternCons30(PhasesD(:,:),nNodes,Tmax); %empirical FCD
pcS(:,1)=patternCons30(Phases(:,:,1, idx_g),nNodes,Tmax); % simulated FCD
[~, ~, ksP(idx_g)]=kstest2(pcS(:),pcD(:));
fprintf(1, 'DONE.\n\n');
end
end
end %subjects