(From: Takuya Ito's https://github.com/ito-takuya/HemodynamicResponseModeling modeling hemodynamic response functions of the BOLD signal using Windkessel-Balloon model)
Based on Friston KJ, Harrison L, Penny W. Dynamic causal modelling. Neuroimage 2003;19:1273–1302.
The remaining state variables of each region are biophysical states engendering the BOLD signal and mediate the translation of neuronal activity into hemodynamic responses. Hemodynamic states are a function of, and only of, the neuronal state of each region. These equations have been described elsewhere (Friston et al., 2000) and constitute a hemodynamic model that embeds the Balloon–Windkessel model (Buxton et al., 1998; Mandeville et al., 1999).
Takuya Ito implementation of Friston KJ et al. Dynamic causal modelling. Neuroimage 2003;19:1273-1302.
def balloonWindkessel(z, sampling_rate, alpha=0.32, kappa=0.65, gamma=0.41, tau=0.98, rho=0.34, V0=0.02):
"""
Computes the Balloon-Windkessel transformed BOLD signal
Numerical method (for integration): Runge-Kutta 2nd order method (RK2)
z: Measure of neuronal activity (space x time 2d array, or 1d time array)
sampling_rate: sampling rate, or time step (in seconds)
alpha: Grubb's exponent
kappa: Rate of signal decay (in seconds)
gamma: Rate of flow-dependent estimation (in seconds)
tau: Hemodynamic transit time (in seconds)
rho: Resting oxygen extraction fraction
V0: resting blood vlume fraction
RETURNS:
BOLD: The transformed BOLD signal (from neural/synaptic activity)
s: Vasodilatory signal
f: blood inflow
v: blood volume
q: deoxyhemoglobin content
"""
The HRF is the filter that couples neural responses to the metabolic-related changes in the MRI signal. HRF models are simply phenomenological.
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Guo-Rong Wu, Nigel Colenbier, Sofie Van Den Bossche, Kenzo Clauw, Amogh Johri, Madhur Tandon, Daniele Marinazzo. “rsHRF: A Toolbox for Resting-State HRF Estimation and Deconvolution.” Neuroimage, 2021, 244: 118591. [link]
Resting state HRF estimation from BOLD-fMRI signal. This toolbox is aimed to retrieve the onsets of pseudo-events triggering an hemodynamic response from resting state fMRI BOLD voxel-wise signal. It is based on point process theory, and fits a model to retrieve the optimal lag between the events and the HRF onset, as well as the HRF shape, using either the canonical shape with two derivatives, or a (smoothed) Finite Impulse Response.
Once that the HRF has been retrieved for each voxel, it can be deconvolved from the time series (for example to improve lag-based connectivity estimates), or one can map the shape parameters everywhere in the brain (including white matter), and use the shape as a pathophysiological indicator.
