In this tutorial we will discuss NiPy’s specification of a typical event-related fMRI model.
This involves:
This design is the canonical “faces” vs. “objects” type design. For an event-related design, we can model the experiment as
Formally, this can be thought of as realization of a marked point
process that says we observe 10 points in the space where E is the set of all event types. Practically speaking, we can read
this as saying that our experiment has 10 events, occurring at times
with event types
.
Typically, the events occur in groups, say odd events are labelled a, even ones b. We might rewrite this as
This type of experiment can be represented by two counting processes
defined as
These delta-function responses are effectively events of duration 0 and infinite height.
For block designs, we might also allow event durations, which fit
nicely into the counting processes .
Suppose that the presentations above each had a duration of 20 seconds, the counting processes could look like
Though the experiment can be represented in terms of the pair , it is a little easier, and more common in neuroimaging applications to work with the derivatives
Some experiments do not fit well into this “event-type” paradigm but are,
rather, more continuous in nature. For instance, a rotating checkerboard,
for which orientation, contrast, are functions of experiment time t.
This experiment can be represented in terms of a state vector .
The neuronal model is a model of the activity as a function of t at a neuron
x given the experimental model . For instance, one model could be
This states that the neuronal response is a delta function, with identical heights for each trial type of a, and b, respectively. An alternative model is for the height of each event to decay exponentially immediately after each stimulus presentation. Mathematically, this can be represented by convolution with an exponential kernel as
Another model, perhaps less plausible scientifically, might keep delta functions, but have the height of the spikes be a function of experiment time, perhaps decreasing exponentially.
This model states that the neuronal activity decreases exponentially in time within each event type, with a time scale specific to each group.
Note that each of these neuronal models are linear operators of the pair
though some have nonlinear parameters, i.e.
the timeconstants
. The inputs are timecourses, and the output
is a timecourse representing the neuronal activity at neuron x as a function
of experiment time t.
In our continuous example above, a reasonable neuronal model might be
Allowing for possible time shifts for both orientation and contrast, another model might be
Note that this model is linear in the pair , but has
two nonlinear parameters
.
A third model, could incorporate derivative information of
where .