Image Processing Reference
In-Depth Information
Temporal basis functions
Temporal basis functions
Stimulus function
f i (
u
)
x
(
t
)
h ( t ) = b 1 f 1 ( u ) + b 2 f 2 ( u ) + ---
y ( t ) = Â b i f i ( u ) x ( t ) + e
i
Conventional model
f i
(
u
)
x
(
t
)
f
(
u
)
h
(
t
) = b f
(
u
)
Design matrix
y ( t ) = b f ( u ) x ( t ) + e
FIR model
h
(
t
) = b 1 d (
u 1 ) + b 2 d (
u 2 ) + ---
d (
u i
)
y ( t ) = Â b i x ( t u i ) + e
i
FIGURE 17.3
Temporal basis functions offer useful constraints on the form of the esti-
mated response that retain the flexibility of FIR models and the efficiency of single regressor
models. The specification of these models involves setting up stimulus functions
) 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
x
(
t
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 is mathematically equiv-
alent to this limiting case.
f
(
u
) of peristimulus time
i
The advantage of this approach is that it can partition differences among evoked
responses into differences in magnitude, latency, or dispersion, which can be
tested for using specific contrasts (20).
Temporal basis functions are important because they enable a graceful tran-
sition between conventional multilinear regression models with one stimulus
function per condition and FIR models with a parameter for each time point
following the onset of a condition or trial type. Figure 17.3 illustrates this
graphically (see figure legend). In summary, temporal basis functions offer useful
constraints on the form of the estimated response that retain the flexibility of
FIR models and the efficiency of single regressor models. The advantage of
using several temporal basis functions (as opposed to an assumed form for the
HRF) is that one can model voxel-specific forms for hemodynamic responses
and formal differences (e.g., onset latencies) among responses to different sorts
of events. The advantages of using basis functions over FIR models are that the
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