This module provides definitions of various hemodynamic response functions (hrf).
In particular, it provides Gary Glover’s canonical HRF, AFNI’s default HRF, and a spectral HRF.
The Glover HRF is based on:
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This paramaterization is from fmristat:
http://www.math.mcgill.ca/keith/fmristat/
fmristat models the HRF as the difference of two gamma functions, g1 and g2, each defined by the timing of the gamma function peaks (pk1, pk2) and the fwhms (width1, width2):
raw_hrf = g1(pk1, width1) - a2 * g2(pk2, width2)
where a2 is the scale factor for the g2 gamma function. The actual hrf is the raw hrf set to have an integral of 1.
fmristat used pk1, width1, pk2, width2, a2 = (5.4 5.2 10.8 7.35 0.35). These are parameters to match Glover’s 1 second duration auditory stimulus curves. Glover wrote these as:
y(t) = c1 * t**n1 * exp(t/t1) - a2 * c2 * t**n2 * exp(t/t2)
with n1, t1, n2, t2, a2 = (6.0, 0.9, 12, 0.9, 0.35). The difference between Glover’s expression and ours is because we (and fmristat) use the peak location and width to characterize the function rather than n1, t1. The values we use are equivalent. Specifically, in our formulation:
>>> n1, t1, c1 = gamma_params(5.4, 5.2)
>>> np.allclose((n1-1, t1), (6.0, 0.9), rtol=0.02)
True
>>> n2, t2, c2 = gamma_params(10.8, 7.35)
>>> np.allclose((n2-1, t2), (12.0, 0.9), rtol=0.02)
True
Parameters for gamma density given peak and width
TODO: where does the coef come from again.... check fmristat code
From a peak location and peak fwhm, determine the parameters (shape, scale) of a Gamma density:
f(x) = coef * x**(shape-1) * exp(-x/scale)
The coefficient returned ensures that the f has integral 1 over [0,np.inf]
Parameters : | peak_location : float
peak_fwhm : float
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Returns : | shape : float
scale : float
coef : float
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