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 parametrization 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
SPM canonical HRF dispersion derivative, values for time values t
This is the canonical HRF dispersion derivative function as used in SPM.
It is the numerical difference between the HRF sampled at time t, and values at t for another HRF shape with a small change in the peak dispersion parameter (peak_disp in func:spm_hrf_compat).
SPM canonical HRF derivative, HRF derivative values for time values t
This is the canonical HRF derivative function as used in SPM.
It is the numerical difference of the HRF sampled at time t minus the values sampled at time t -1
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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SPM HRF function from sum of two gamma PDFs
This function is designed to be partially compatible with SPMs spm_hrf.m function.
The SPN HRF is a peak gamma PDF (with location peak_delay and dispersion peak_disp), minus an undershoot gamma PDF (with location under_delay and dispersion under_disp, and divided by the p_u_ratio).
Parameters : | t : array-like
peak_delay : float, optional
peak_disp : float, optional
under_delay : float, optional
under_disp : float, optional
p_u_ratio : float, optional
normalize : {True, False}, optional
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Returns : | hrf : array
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Notes
See spm_hrf.m in the SPM distribution.
SPM canonical HRF, HRF values for time values t
This is the canonical HRF function as used in SPM