Beta series¶

This covers the conceptual background of beta series analysis; check out Usage for more on how to use NiBetaSeries.

Introduction¶

Beta series track the event-to-event hemodynamic fluctuations modeled in task functional magnetic resonance imaging (task fMRI). Beta series fills an important analytical gap between measuring hemodynamic fluctuations in resting-state fMRI and measuring regional activations via cognitive subtraction in task fMRI.

Conceptual Background¶

Jesse Rissman [docs-5] was the first to publish on beta series correlations, describing their usage in a working memory task. In this task, participants saw a cue, a delay, and a probe, all occurring within a short time period. The cue was presented for one second, a delay occurred for seven seconds, and a probe was presented for one second. Given that the blood-oxygen-level-dependent (BOLD) response takes approximately six seconds to reach its peak, and generally takes over 20 seconds to return to baseline, we can begin to see a problem. The events within the events occur too close to each other to discern what brain responses are related to encoding the cue, the delay, or the probe. To discern how the activated brain regions form networks, Rissman computed beta series correlations. Instead of having a single regressor to describe all the cue events, a single regressor for all the delay events, and a single regressor for all the probe events (as is done in traditional task analysis), there is an individual regressor for every event in the experiment. For example, if your experiment has 40 events, each with a cue, delay, and probe event, the model will have a total of 120 regressors, fitting a beta (i.e., parameter) estimate for each event. Once you calculate a beta estimate for each event of a given type (e.g., cue), you will have a four-dimensional dataset where each volume represents the beta estimates for a particular event.

Having one regressor per event in a single model is known as “least squares- all” (LSA). This method, however, has limitations in the context of fast event-related designs (e.g., designs where the events occur between 3-6 seconds apart on average). Since each event has its own regressor, events that occur very close in time are collinear (e.g., are very overlapping).

Jeanette Mumford [docs-4] derived a solution for the high collinearity observed in least squares- all by using another type of regression known as “least squares- separate” (LSS). Instead of having one general linear model (GLM) with a regressor per event, least squares- separate implements a GLM per event with only two regressors: 1) one for the event of interest, and 2) one for every other event in the experiment. This process reduces the collinearity of the regressors and creates a more valid estimate for each trial, but also combines all other conditions within the design matrix, which will reduce model fit.

Benjamin Turner [docs-6] improved upon the LSS method by retaining the original conditions in the design matrix. In this updated version, the individual trial’s design matrix is almost the same as the original design matrix, except that the trial is separated out into its own regressor.

Benjamin Turner [docs-6] also adapted the LSS method by combining it with finite impulse response (FIR) modeling, in which each of a set of temporal delays following each trial is modeled as an impulse function in order to characterize the shape of the BOLD response, as the finite BOLD response- separate (FS) model. The FS model showed promise as a method for decoding, as it does not make any assumptions about the shape of the BOLD response.

NiBetaSeries can use the updated “least squares- separate” method, the “finite BOLD response- separate” method, or the original “least squares- all” method.

Mathematical Background¶

(1)¶\[\begin{equation} Y = X\beta + \epsilon \end{equation}\]

The above equation is the general linear model (GLM) presented using matrix notation. \(Y\) represents the time-series we are attempting to explain (for a given voxel). \(X\) typically represents the trial(s) of interest for which you would like to have an estimate. For example, I would like to know how the brain responds to red squares, so \(X\) represents the brain response at all time points when a red square was presented. The \(\beta\) assumes any value that minimizes the squared error between the modeled data and the actual data, \(Y\). Finally, \(\epsilon\) (epsilon) refers to the error that is not captured by the model. In a typical GLM, event-to-event betas may be averaged for a given event type and the variance is treated as noise. However, those event-to-event fluctuations may also contain important information the typical GLM will ignore/penalize. With a couple modifications to the above equation, we arrive at calculating a beta series.

(2)¶\[\begin{equation} Y = X_1\beta_1 + X_2\beta_2 + . . . + X_n\beta_n + \epsilon \end{equation}\]

With the beta series equation, a beta is estimated for every event, instead of for each event type (or whatever logical grouping). This yields a series of event betas for a single event type. This operation is completed for all voxels, giving us as many lists of betas as there are voxels in the data. Essentially, this returns a 4-D dataset where the fourth dimension represents the number of events instead of time (as the fourth dimension is represented in resting state). Analogous to resting state data, we can perform correlations between the voxels to discern which voxels (or which aggregation of voxels) covary with other voxels.

There is one final concept to cover in order to understand how the betas are estimated in NiBetaSeries. You can model individual betas using a couple different strategies; “least squares- all” (LSA) estimation represented in the above equation (2), or “least squares- separate” (LSS) estimation, in which each event receives its own GLM. The advantage of LSS comes from reducing the collinearity between closely spaced events. In LSA, if events occurred close in time, it would be difficult to model whether the fluctuations should be attributed to one event or the other. LSS reduces this ambiguity by only having two regressors: one for the event of interest and another for every other event. This reduces the collinearity between regressors and makes each beta estimate more reliable.

import numpy as np

# the design of the brain response.
# each row represents a time point.
# each column represents a trial.
# the trials overlap each other.
X = np.array([[1, 0, 0, 0],
              [1, 1, 0, 0],
              [0, 1, 1, 0],
              [0, 0, 1, 1]])

# the trial in the order they were seen
trial_types = np.array(["red", "blue", "red", "blue"])

# the indices of the blue and red trials
blue_idxs = np.where(trial_types == 'blue')[0]
red_idxs = np.where(trial_types == 'red')[0]

# the observed brain data (transposed so data points are in one column)
Y = np.array([[2, 1, 5, 3]]).T

# least squares- all (LSA)
# there is one beta estimate per trial
lsa_betas, _, _, _ = np.linalg.lstsq(X, Y)

# least square separate (LSS)
lss_betas = []
# for each trial...
for index, trial_type in enumerate(trial_types):
    # select the trial (column) of interest
    X_interest = X[:, index]

    # find the indices of the "rest" of the trials of the same type
    # and the indices of the other trial type
    if trial_type == "blue":
        idxs = blue_idxs
        other_idxs = red_idxs
    elif trial_type == "red":
        idxs = red_idxs
        other_idxs = red_idxs

    # the "rest" of the trials of the same type
    non_target_idx = np.delete(idxs, np.where(idxs != index))[0]
    X_nontarget = X[:, non_target_idx]

    # select all the other trials (columns)
    # and sum over them to create a single column
    X_other = X[:, other_idxs].sum(axis=1)

    # combine the two columns such that:
    # the first column is the trial of interest
    # the second column represents all other trials
    X_trial = np.vstack([X_interest, X_nontarget, X_other]).T
    # solve for the beta estimates
    betas, _, _, _ = np.linalg.lstsq(X_trial, Y)
    # add the beta for the trial of interest to the list
    lss_betas.append(betas[0][0])

This python code demonstrates LSA (line 24) and LSS where each event is given its own GLM model. Note the GLM model written in python (line 55) has the form as the equation at the beginning of “Mathematical Background” (1), but X (specifically X_trial) has the particular representation of one column being the trial of interest (X_interest), another column where the trials are the same type as the trial of interest (X_nontarget), and finally a column with the other trial type (X_other). Generally, there will be as many columns as there are trial types plus a column for the trial of interest.

Relationship to Resting-State Functional Connectivity¶

Beta series connectivity analysis is similar to resting-state functional connectivity (time-series correlations) because the same analyses typically applied to resting-state data can ostensibly be applied to beta series. At the core of both resting-state functional connectivity and beta series we are working with a vector of numbers at each voxel. We can correlate, estimate regional homogeneity, perform independent components analysis, or perform a number of different analyses with the data in each voxel. However, beta series deviates from the time-series correlations used for resting-state analysis in two important ways. First, you can do cognitive subtraction using beta series. Since there is no explicit task in resting state, there are no cognitive states to compare. Second, the interpretations of resting-state connectivity and beta series differ. Resting state measures the unmodelled hemodynamic fluctuations that occur without explicit stimuli or task. Beta series, on the other hand, measures the modelled hemodynamic fluctuations that occur in response to an explicit stimulus. Both resting-state analyses and beta series may measure intrinsic connectivity (e.g., the functional structure of the brain independent of task), but beta series may also measure the task-evoked connectivity (e.g., connectivity between regions that is increased during some cognitive process).

Relationship to Traditional Task Analysis¶

Beta series is also similar to traditional task analysis because cognitive subtraction can be used in both. As with resting-state analysis, beta series deviates from traditional task analysis in several important ways. Say we are interested in observing how the brain responds to faces versus houses. The experimenter has a timestamp of exactly when and how long a face or house is presented. That timestamp information is typically convolved with a hemodynamic response function (HRF) to represent how the brain stereotypically responds to any stimulus resulting in a model of how we expect the brain to respond to places and/or faces. This is where traditional task analysis and beta series diverge. In traditional task analysis all the face events are estimated at once, giving one summary measure for how strongly each voxel was activated (same for house events). The experimenter can subtract the summary measure of faces from houses to see which voxels are more responsive to houses relative to faces (i.e., cognitive subtraction). In beta series analysis, each event is estimated separately and each voxel has as many estimates at there are events (which can be labelled as either face or house events). The experimenter can now reduce the series of estimates (a beta series) for each voxel into a summary measure such as correlations among regions of interest. The correlation map for faces can be subtracted from houses, giving voxels that are more correlated with the region of interest for houses relative to faces. Whereas traditional task analysis treats the variance of brain responses between events of the same type (e.g., face or house) as noise, beta series leverages this variance to make conclusions about which brain regions may communicate with each other during a particular event type (e.g., faces or houses).

Summary¶

Beta series is not in opposition to resting state or traditional task analysis; the methods are complementary. For example, network parcelations derived from resting state data can be used on beta series data to ascertain if the networks observed in resting state follow a similar pattern with beta series. Additionally, regions determined from traditional task analysis can be used as regions of interest for beta series analysis. Beta series straddles the line between traditional task analysis and resting-state functional connectivity, observing task data through a network lens.

Relevant Software¶

BASCO (BetA Series COrrelations) is a MATLAB program that also performs beta series correlations
pybetaseries is a python script that runs on files that have been processed by FSL’s FEAT

References¶

docs-1: Hunar Abdulrahman and Richard N. Henson. Effect of trial-to-trial variability on optimal event-related fMRI design: Implications for Beta-series correlation and multi-voxel pattern analysis. NeuroImage, 125:756–766, 2016. URL: http://dx.doi.org/10.1016/j.neuroimage.2015.11.009, doi:10.1016/j.neuroimage.2015.11.009.
docs-2: Josh Cisler, Keith Bush, and Steele Scott. A Comparison of Statistical Methods for Detecting Context- Modulated Functional Connectivity in fMRI. Neuroimage, 24:1042–1052, 2014. arXiv:NIHMS150003, doi:10.1016/j.neuroimage.2013.09.018.
docs-3: Martin Göttlich, Frederike Beyer, and Ulrike M. Krämer. BASCO: a toolbox for task-related functional connectivity. Frontiers in Systems Neuroscience, 9(September):1–10, 2015. URL: http://journal.frontiersin.org/Article/10.3389/fnsys.2015.00126/abstract, doi:10.3389/fnsys.2015.00126.
docs-4: Jeanette A. Mumford, Benjamin O. Turner, F. Gregory Ashby, and Russell A. Poldrack. Deconvolving BOLD activation in event-related designs for multivoxel pattern classification analyses. NeuroImage, 59(3):2636–2643, 2012. URL: http://dx.doi.org/10.1016/j.neuroimage.2011.08.076, doi:10.1016/j.neuroimage.2011.08.076.
docs-5: Jesse Rissman, Adam Gazzaley, and Mark D’Esposito. Measuring functional connectivity during distinct stages of a cognitive task. NeuroImage, 23(2):752–763, 2004. doi:10.1016/j.neuroimage.2004.06.035.
docs-6(1,2): Benjamin O. Turner, Jeanette A. Mumford, Russell A. Poldrack, and F. Gregory Ashby. Spatiotemporal activity estimation for multivoxel pattern analysis with rapid event-related designs. NeuroImage, 62(3):1429–1438, 2012. URL: http://dx.doi.org/10.1016/j.neuroimage.2012.05.057, doi:10.1016/j.neuroimage.2012.05.057.