# modalities.fmri.hrf¶

## Module: modalities.fmri.hrf¶

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:

@article{glover1999deconvolution,
title={{Deconvolution of impulse response in event-related BOLD fMRI}}, author={Glover, G.H.}, journal={NeuroImage}, volume={9}, number={4}, pages={416–429}, year={1999}, publisher={Orlando, FL: Academic Press, c1992-}

}

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), and c1, c2 being 1/max(t**n1 * exp(t/t1)), 1/max(t**n2 * exp(t/t2). 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


## Functions¶

nipy.modalities.fmri.hrf.ddspmt(t)

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).

nipy.modalities.fmri.hrf.dspmt(t)

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

nipy.modalities.fmri.hrf.gamma_expr(peak_location, peak_fwhm)
nipy.modalities.fmri.hrf.gamma_params(peak_location, peak_fwhm)

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 Location of the peak of the Gamma density peak_fwhm : float FWHM at the peak shape : float Shape parameter in the Gamma density scale : float Scale parameter in the Gamma density coef : float Coefficient needed to ensure the density has integral 1.
nipy.modalities.fmri.hrf.spm_hrf_compat(t, peak_delay=6, under_delay=16, peak_disp=1, under_disp=1, p_u_ratio=6, normalize=True)

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 vector of times at which to sample HRF. peak_delay : float, optional delay of peak. under_delay : float, optional delay of undershoot. peak_disp : float, optional width (dispersion) of peak. under_disp : float, optional width (dispersion) of undershoot. p_u_ratio : float, optional peak to undershoot ratio. Undershoot divided by this value before subtracting from peak. normalize : {True, False}, optional If True, divide HRF values by their sum before returning. SPM does this by default. hrf : array vector length len(t) of samples from HRF at times t.

Notes

See spm_hrf.m in the SPM distribution.

nipy.modalities.fmri.hrf.spmt(t)

SPM canonical HRF, HRF values for time values t

This is the canonical HRF function as used in SPM