finite impulse response filter
- finite to indicate that the response due to an impulse input goes to 0 with infinite time. As opposed to a filter that maybe stores some state.
- When you fit an FIR you're looking for the best convolution over the input to produce the output
0.1. use in fMRI
- Context: In fMRI task studies, we present stimuli and measure the brain's BOLD response. We want to model the brain's response to stimuli. The hemodynamic response lags behind the stimulus. So we want to have a model for the hemodynamic response function for each stimulus that reflects this. Then, for a voxel that sees different types of stimuli, we will estimate some weighting \(\beta\) between stimuli, and combine the HRFs to get the response of the voxel (see below). Here, the quantity of interest are the \(\beta\) s.
- Precisely, we want to estimate the response \(y\) which is a \(T \times 1\) dimensional vector where \(T\) is the time-steps. If there are \(C\) types of stimuli, our input (or design matrix) is \(X\) which is \(T \times C\) and our model is \(y=X \beta\) (see here) where \(\beta\) is our estimated weighting (\(C \times 1\)). Because the blood response lags, we model each stimulus response as \(D * h\), where this is convolution between the \(T \times 1\) stimulus and the \(K \times 1\) FIR of order \(K\).
- We could use a stereotyped HRF, which neuroscientists have crafted based on previous studies. Or we can use the non-parametric (not that it has literally no parameters, but that it can take any shape that a histogram can take, not just shapes in the family of, e.g., gamma functions) FIR model, that we fit jointly as we model the response.
- really simple gloss: our encoding model takes the stimulus as input and outputs the voxel activation as a time series. The brain response lags behind stimuli, so our model outputs should lag behind stimuli.
0.2. other fmri contexts
- I've also seen studies that use "FIR delays." These are usually machine learning works that need to align the stimulus to the response. I assume that they look at the estimated FIR coefficients and then make a determination based on their arg-max of the lag between stimuli and blood response.