Table 1 Schematic description of the PF algorithm

Input the BOLD fMRI data series

\(\mathbf{x }_t\)

Set the number of particles

N (\(=2000\) in our case)

Set starting point for coefficients values

\(\mathbf{a }_{t=0}=0\)

Start PF

for \(t=1:T\)

updating step

generate N particles from previous coefficients’ values through \(a_{ij}(t)=a_{ij}(t-1)+\nu _{ij}(t)\)

estimation step

predict the values of the observations at time t from values at time \(t-1\) with \(\mathbf{x }(t)=\mathbf{a }(t)\mathbf{x }(t-1)+{\varvec{\eta }}(t)\) compute the likelihood between predicted values and observed values with (6) normalize the weights and resample

End PF

end for on t