Temporal
basis functions offer useful constraints on the form of the estimated response
that retain (i) the flexibility of FIR models and (ii) the efficiency of single
regressor models. The specification of
these constrained FIR models involves setting up stimulus functions *x*(*t*) that model expected neuronal changes
[*e.g.* boxcars of epoch-related
responses or spikes (delta functions) at the onset of specific events or
trials]. These stimulus functions are
then convolved with a set of basis functions _{}of peri-stimulus time *u*, that model the HRF, in some linear
combination. The ensuing regressors are
assembled into the design matrix. The
basis functions can be as simple as a single canonical HRF (middle), through to
a series of delayed delta functions (bottom).
The latter case corresponds to a FIR model and the coefficients
constitute estimates of the impulse response function at a finite number of
discrete sampling times. Selective
averaging in event-related fMRI (Dale and Buckner 1997) is mathematically
equivalent to this limiting case.