Dear Dr Bellec,
Thank you for your answer..So I've some other questions:

• when you talk about "slow time drift", it means the same of my "linear trend"?
• What do you mean with "polynomial do not scale well with the length of the signal"? I would like to discuss about this problem in my thesis, so I'll be grateful if you could tell me something else.
• if I well understood, the code Niak_filter_tseries generates covariates (discrete cosines) that stand for the linear trend regressors. Can you show me some lines as example of code usage? I used the code with opt=struct('flag_mean',false,'tr',0.295,'hp',-Inf,'lp',Inf); and I obteined empty "extra" structure. Then I tried with lp=1 and I obteined EXTRA.TSERIES_DC_HIGH as a 2D array of time x 370 elements.Have I to regress all in the GLM? 

Thank you for your time.
Best,
Chiara D.S.

Dear Chiara, Slow time drifts is a loose term for any slow fluctuations in the time series. It includes linear, quadratic or cubic trend, and low frequencies (typically <0.01 Hz).  If you use a polynomial model for slow time drifts, say linear and quadratic, you will have two regressors regardless of the duration of your time series. If you have 5 mns of signal, the linear/quadratic trend  will correct for drifts at this time scale, while if you have an hour of signal thay will correct for comparatively much, much slower variations. And drifts at the 5 mns scale will essentially remain uncorrected. This is why in fmristat for example, a new set of linear/quadratic trend are introduced for every 5 mns or so time block. But this is very much of a hack. With standard filtering approach, the number of discrete cosines covering the [0 , 0.01] Hz frequency band will vary with the length of your signal: longer time series translate into higher spectral resolution, and so more covariates to regress. This approach naturally adjusts the complexity of the model for slow time drifts as a function of the duration of the signal.  The cosines corresponding to slow frequencies are in "extra.tseries_dc_low". I am surprised that you have 370 elements, but you also entered a TR of 0.295 seconds, so I assume this is some fancy multiband data with tons of time points, so it may make sense. I've added an example in the help of niak_filter_tseries, if you clone the head of niak on github.com/SIMEXP/niak. I've copy/pasted it below for your convenience:

tseries = randn([128,20]) + ([0:127]'/127)*rand([1 20]); % simulate time series with linear drift
opt.tr = 3;
opt.hp = 0.01;
opt.lp = Inf;
[tseries_f,extras] = niak_filter_tseries(tseries,opt);
ftseries = fft(tseries);
ftseries_f = fft(tseries_f);
figure
subplot(1,2,1)
slow_drifts = extras.tseries_dc_low * extras.beta_dc_low;
plot(0:3:(3*127),[tseries(:,1),tseries_f(:,1),slow_drifts(:,1)]);
legend({'example raw signal','example filtered signal','slow drift'})
subplot(1,2,2)
plot([0:1/(3*128):1/6],[mean(abs(ftseries(1:65,:)).^2,2) mean(abs(ftseries_f(1:65,:)).^2,2)]);
legend({'Average power spectrum','average filtered power spectrum'});

figure
plot(extras.tseries_dc_low)
title('the low-frequency discrete cosines');

I hope this helps,

Pierre

Thank you very much!
Best,
Chiara