the neuron activity term \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x\left(t\right)$$\end{document}xt as a simple on–off model with series of delta functions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{x}\left( t \right)$$\end{document}x^t as2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \hat{x}\left( t \right) = \mathop \sum \limits_{\tau = 0}^{\infty } \delta \left( {t - \tau } \right). $$\end{document}x^t=∑τ=0∞δt-τ. Note that the delta functions modeling the events exist at random times, which essentially amounts to modeling the resting state data as an event-related paradigm with randomly occurring events. Then the HRF \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h\left(t\right)$$\end{document}ht was fitted using a canonical HRF (two gamma functions) and two derivatives (temporal derivative and dispersion derivative). The parameters of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h\left(t\right)$$\end{document}ht were allowed to vary for each time series. The approximation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{x} \left( t \right)$$\end{document}x⌣t of the latent neural signal can be obtained from the observed data using a Wiener filter as described below:3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{x} \left( t \right) = d\left( t \right) \otimes y\left( t \right), $$\end{document}x⌣t=dt⊗yt,
