Causality in Time Series - ClopiNet

Causal analysis of fMRIthe SNR of BOLD fMRI in general is very good in comparison to electrophysiologicaltechniques like EEG and MEG.Although the balloon model and its variations have played an important role indescribing the transient features of the fMRI response and inferring neuronal activity,simplified ways of representing the BOLD signal responses are very often used. Mostprominent among these is a linear finite impulse response (FIR) convolution with asuitable kernel. The most used single convolution kernel characterizing the ‘canonical’hemodynamic reponse is formed by a superposition of two gamma functions (Glover,1999), the first characterizing the initial signal increase, the second the later negativeundershoot (Figure 2B, solid line):h(t) = m 1 t τ 1e (−l 1t) − cm 2 t τ 2e (−l 2t)m i = max (︁ t τ ie (−l it) )︁ (6)With times-to-peak in seconds τ 1 = 6, τ 2 = 16, scale parameters l i (typically equal to 1)and a relative amplitude of undershoot to peak of c = 1/6.Often, the canonical two-gamma HRF kernel is augmented with one or two additionalorthogonalized convolution kernels: a temporal derivative and a dispersionderivative. Together the convolution kernels form a flexible basis function expansionof possible HRF shapes, with the temporal derivative of the canonical accounting forvariation in the response delay and the dispersion derivative accounting for variations intemporal response width (Henson et al., 2002; Liao et al., 2002). Thus, the linear basisfunction representation is a more abstract characterization of the HRF (i.e. further awayfrom the physiology) that still captures the possible variations in responses.It is an interesting property of hemodynamic processes that, although they are characterizedby a large overcompensating reaction to neuronal activity, their effects arehighly local. The locality of the hemodynamic reponse to neuronal activity limits theactual spatial resolution of fMRI. The path blood inflow in the brain is from large arteriesthrough arterioles into capillaries where exchange with neuronal tissue takes placeat a microscopic level. Blood outflow takes place via venules into the larger veins. Themain regulators of blood flow are the arterioles that are surrounded by smooth muscle,although arteries and capillaries are also thought to be involved in blood flow regulation(Attwell et al., 2010). Different hemodynamic parameters have different spatialresolutions. While CBV and CBF changes in all compartments but mostly venules,oxygenation changes mostly in the venules and veins. Thus, the achievable spatial resolutionwith fMRI is limited by its specificity to the smaller arterioles and venules andmicroscopic capillaries supplying the tissue, rather than the larger supplying arteriesdraining veins. The larger vessels have a larger domain of supply or extraction and, as aconsequence, their signal is blurred and mislocalized with respect to active tissue. Here,physiology and physics interact in an important way. It can be shown theoretically – bythe effects of thermal motion of spin diffusion over time and the distance of the spinsto deoxy-Hb – that the origin of the BOLD signal in SE sequences at high main fieldstrengths (larger than 3T) is much more specific to the microscopic vasculature than tothe larger arteries and veins. This does not hold for GRE sequences or SE sequences81