Selection of Efficient Type 1, Index 1 Sequences

A continuous carry-over experiment presents a continuous stream of stimuli to the subject, arranged such that every stimulus appears both before and after every other. Type 1, Index 1 sequences provide this first-order counterbalancing. In addition, T1I1 sequences are blocked, providing insensitivity to order effects across the entire period of stimulus presentation. This is unlike m-sequences, which have equal power at all frequencies, and thus contain experimental variance below the fundamental frequency of stimulus repetition of stimuli and therefore in the elevated noise range of fMRI. This wiki page describes the selection of T1I1 sequences that are optimized for use in BOLD fMRI.

Suppose we wish to present these 16 shapes (Drucker and Aguirre 2009) in a continuous carry-over experiment:

We might use this T1I1 sequence to determine the order of presentation:

 1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17
17 12 14 11 13  3 16  1 15  6  2  7  4  9  5  8 10
10  2 14  4 17  6  1  3  5  9 13  7 15  8 11 16 12
12  6 10 14  3 17 11 15  2 13  8 16  5  4  1  9  7
 7  9 16 13  1  4  6  8 12  5 15  3  2 11 10 17 14
14  5  2  9  6 15  7  3  8 13 11 17 16 10  1 12  4
 4  2  5 10 13  6 16  9 11  3  7 12 15  1 14 17  8
 8  5  1 11 14 16  7 10  4 12 17 15  9  3  6 13  2
 2  8 14 12  9  1  5 11  4  7 13 10 16  6 17  3 15
15 17 13 16  8  1 10  5 14  2  6  4  3  9 12  7 11
11  7  5 12  3 10 15 13  4 14  8  6  9  2 17  1 16
16  4 13  5  3 11  8 15 10  7  2 12  1 17  9 14  6
 6 12  2 15 11  9  4 10  8  3  1  7 16 14 13 17  5
 5  7  1  8  2 16 11  6 14  9 17  4 15 12 10  3 13
13  9 15  4  8 17  7 14 10 12 11  2  1  6  5 16  3
 3 14  7  6 11  1 13 12  8  4 16 15  5 17  2 10  9
 9  8  7 17 10  6  3 12 16  2  4 11  5 13 15 14  1

It is completely arbitrary, so far, which stimulus we treat as "number 1", which as "number 2", and so on. However, it turns out that from a practical standpoint, this matters.

In an fMRI experiment, we construct covariates which act as predictors of variations in neural activity over time. Statistical inference is supported by measuring the degree to which the observed BOLD fMRI signal is modeled by those predictors.

The BOLD signal does not, however, equally reflect all types of neural activity. Specifically, it is filtered by the hemodynamic response function (HRF), which attenuates variation in neural activity that occurs at higher temporal frequencies (i.e., the HRF is a low-pass temporal filter). The extent to which a temporal pattern of neural activity does pass through the HRF can be described as the Efficiency of the sequence.

The variance in our signal has a power spectrum; we wish to select a sequence such that as much power as possible is concentrated in the portion which will be passed by the BOLD filter, since any power that is at higher frequencies will be lost.

In this figure, in blue is the power spectrum of a sequence with high Efficiency; in red, a sequence with poor Efficiency. The green curve is the power spectrum of the hemodynamic filter. Note that the desirabe sequence concentrates much more of its power at frequencies which will be passed by the filter.

To measure the relative Efficiency of different sequences, we must assume a shape for the HRF, as well as a presentation time for each stimulus.

In addition, we must provide an expected degree of response to each stimulus to be studied. Different stimuli may be optimized to detect the "direct" effect of each stimulus (that is, the amplitude of response evoked by the stimulus in theoretical isolation as compared to a reference condition), or the "carry-over" effect, which is the modulatory effect of a stimulus upon the following stimulus (e.g., neural adaptation). All T1I1 sequences provide essentially identical Efficiency for measurement of direct effects (see here for a description of the effect of timing and null-trial duration upon Efficiency). Therefore, we focus here upon the identification of sequences that are Efficient for the measurement of carry-over (adaptation) effects.

To begin, we require a hypothesis regarding the modulatory effect of each stimulus upon each other stimulus. These relationships can be captured in a carry-over matrix (see Aguirre 2007). If we are studying neural habituation effects, we may be willing to hypothesize that the degree of neural habituation and therefore carry-over will be propotional to the perceptual similarity of the stimuli (Drucker and Aguirre 2009). Therefore, the Efficiency of a sequence in a carry-over experiment can be measured with respect to a similarity matrix. For the example stimuli above, the perceptual similarity is closely approximated by their Euclidean distance from one another within the perceptual grid. The following figure illustrates the perceptual similarity matrix for these 16 stimuli, with their relative similarity expressed as a grayscale intensity:

Given the perceptual distance between each stimulus, we can create a predictor which models the recovery from neural adaptation as being proportional to the perceptual distance between each subsequent stimulus.:

This sequence is a prediction of neural response. We wish to maximize the amount of the variance in this neural signal that passes through the HRF. We have a particular mathematical structure imposed on us by the T1I1 sequence, but the sequence is silent on the topic of which stimulus goes with which label. Some assignments of stimuli to sequence number will produce temporal patterns of neural activity that better pass through the HRF (see the figure above).

A brute force search is used to identify sequences with higher Efficiency for a given similarity matrix (and an assumed shape of the HRF and stimulus duration). We try all the possible permutations (if n < ~9) or some reasonable subset thereof. Each time, a new mapping of stimuli to labels is tried, and the resulting sequence is convolved with the HRF:

Efficiency is defined as the ratio of the variance in the filtered signal to the variance in the original sequence; if there is no change in the signal, Efficiency is 1; if there is no variance whatsoever remaining, Efficiency is 0. We have shown that this search can produce three-fold improvements in Efficiency over the expected Efficiency of a randomly selected sequence.

We have written code to assist in this process.

function [alleffs, BestSeqs, bestEs, BestVec, BestFiltVec, allseqs, matchseqs] = EvaluateSeqs(par,SimMat)
% Evaluate mapping of sequence labels for maximum efficiency
% given a set of similarity matrices.
% par.perms         = how many permutations to try
% par.TrialDuration = length of stimulus in secs
% par.HRFLength     = length of convolving function in secs
% par.ResolutionHz  = resolution to convolve at in Hz
% par.seqfile       = file generated by "seqgen -q"
% par.seq           = if present, sequence (ignore seqfile)
% par.HRFFile       = convolution function file
% par.HRFFileHz     = resolution of convolving function file in Hz
% par.BlankLength   = number of blanks to insert for each blank
% par.numSeqs       = number of top n sequences to return
% par.weneed        = how many matching seqs to return

We have provided a set of premade_sequences for several types of stimulus spaces and for a particular stimulus presentation time.