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:


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:

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)
>>> n2, t2, c2 = gamma_params(10.8, 7.35)
>>> np.allclose((n2-1, t2), (12.0, 0.9), rtol=0.02)



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

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]


Location of the peak of the Gamma density


FWHM at the peak


Shape parameter in the Gamma density


Scale parameter in the Gamma density


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).


vector of times at which to sample HRF.

peak_delayfloat, optional

delay of peak.

under_delayfloat, optional

delay of undershoot.

peak_dispfloat, optional

width (dispersion) of peak.

under_dispfloat, optional

width (dispersion) of undershoot.

p_u_ratiofloat, 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.


vector length len(t) of samples from HRF at times t.


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