### Synopsis

The general linear model (GLM) is one of the most
commonly used methods to analyse task-based fMRI data. This talk outlines the basic
concepts of the GLM, how it is used to study block and event-related paradigms and
associated statistical analysis, as well as some example applications. The talk
will then describe some limitations of a GLM, and briefly outline alternative methods
to study task-based fMRI paradigms, such as the phase-encoding or travelling-wave
method, and independent component analysis.

### Target Audience

Cognitive
neuroscientists, neuroradiologists, clinicians and imaging scientists who
currently utilize fMRI and MR physicists and engineers developing new fMRI
methodologies.### Outcome and Objectives

To understand use of the GLM for task-based fMRI analysis, be able to describe linear regression and a design matrix, describe limitations of the GLM in fMRI analysis, and outline alternative methods for task-based analysis.### Purpose

The General Linear Model (GLM) was introduced in the 1990s to analyze task-based fMRI data[1-4], and remains the analysis method most commonly used by the fMRI community [5]. The GLM applies multiple regression analysis to estimate the BOLD effect size in terms of a linear combination (weighted sum) of several reference functions. The model functions are assumed to have known shapes, but their amplitudes are unknown and need to be estimated. The method is commonly used to model the BOLD time course to different experimental conditions, from simple block designs to more complex multi-condition rapid-event related designs. The term “General” means that we model all the effects that are involved in our data together for all the voxels and time series present in the BOLD data. Specifically the effects of interest (hypothesized BOLD activation) and confounds (e.g., age of the subject involved in the data collection) are modelled together and each effect becomes a regressor in the model. The term “Linear” refers to the assumption that the BOLD data is a superposition (addition/subtraction) of weighted known data. An alternative method to the GLM for the study of sensory systems is the “phase-encoding” or “travelling wave” method [6, 7] which uses Fourier analysis to define topographic maps of the cortical responses. The most common application is retinotopic mapping of the visual cortex. ### Methods

GLM: Mathematically, the GLM can be expressed as a matrix equation Y = Xβ+ε, where Y represents the voxel time series, a column vector (of N time points); X represents the design matrix, where regressors modelling the effects and confounds (both of length N) are stacked into M different columns; β refers to the parameter estimate, a column vector (M elements), one element (effect size) for each modelled regressor; and ε refers to the residual time series, i.e. any unmodelled fluctuation, a column vector (of N time points). The parameter estimation process finds the least-square solution to this matrix equation, i.e. the β that minimizes the residual (ε). Each regressor of the design matrix is used as an explanatory variable so multiple effects within the experiment can be included and considered separately, including confounds such as the realignment parameters. The predicted time course modelling the effects is typically obtained by convolution of the box-car or event-related stimulus delivery with a standard haemodynamic response function (two-gamma HRF or single-gamma HRF). The resulting β parameter estimate is compared with the uncertainty in its estimation to result in a T value = β /standard error (β), which is then converted into a probability or Z statistic. In the travelling wave method, stimuli are periodic and cycle through sensory space producing a periodic response across the brain maps. Thus different locations on the map will differ by the phase of the response, as opposed to the amplitude or frequency of the response. ### Results

A GLM design matrix is used to generate BOLD activations that accurately represent fMRI signal changes in the specific regions of interest in the brain. As well as images of Z values that describe how strongly each voxel is related to each effect of interest, parameter estimates can be compared to test whether one effect of interest is more ‘‘relevant’’ than another. In the travelling-wave analysis, for each point in cortex, the harmonic function that best correlates with the fMRI time series is measured. The Fourier Transform gives the amplitude and phase at each frequency, with the amplitude informing on the reliability of the signal, and so signal coherence. The phase of the best-fitting harmonic informs about the location in the sensory space. ### Discussion

The GLM is widely used for analysing task-based BOLD fMRI data. However, it has certain limitations including multiple comparison problems and noise estimation assumptions. Further the haemodynamic response function (HRF) may vary between subjects, for different brain regions. An alternative fMRI analysis method is the use of data-driven fMRI analyses, such as Independent Component Analysis (ICA), which are not constrained by a fixed model-based hypothesis and prior constraints on the shape of the HRF [8]. ICA methods therefore might detect responses that would not have been revealed by a GLM analysis, but the challenging step is to identify the BOLD components that are relevant. Travelling wave methods have been used to demonstrate at least 18 human visual areas identified based on their retinotopic maps [6, 7]. In hearing, tonotopic maps of sound frequencies have been produced [9]. In the tactile modality, travelling wave methods have been used to map the somatotopic representation of the hand [10, 11] and topographic mapping of the motor cortex. The limitation with travelling wave designs is that they are not well-suited to investigate overlapping neural representations. ### Conclusion

Both GLM and travelling-wave fMRI analysis methods can be robustly used in basic neuroscience applications, but the limitations of each should be considered.### Acknowledgements

No acknowledgement found.### References

Below are some important
references on the utilization of the GLM in fMRI analysis and the use of
travelling wave analysis.

1. Friston,
K.J., et al., Characterizing dynamic
brain responses with fMRI: a multivariate approach. Neuroimage, 1995. 2(2): p. 166-72.

2. Friston, K.J., et al., Characterizing evoked hemodynamics with
fMRI. Neuroimage, 1995. 2(2): p.
157-65.

3. Friston, K.J., et al., Analysis of fMRI time-series revisited.
Neuroimage, 1995. 2(1): p. 45-53.

4. Worsley, K.J. and K.J. Friston, Analysis of fMRI time-series
revisited--again. Neuroimage, 1995. 2(3):
p. 173-81.

5. Poline, J.B. and M. Brett, The general linear model and fMRI: does love
last forever? Neuroimage, 2012. 62(2):
p. 871-80.

6. Engel, S.A., et al., fMRI of human visual cortex. Nature,
1994. 369(6481): p. 525.

7. Sereno, M.I., et al., Borders of multiple visual areas in humans
revealed by functional magnetic resonance imaging. Science, 1995. 268(5212): p. 889-93.

8. McKeown, M.J., et al., Spatially independent activity patterns in
functional MRI data during the stroop color-naming task. Proc Natl Acad Sci
U S A, 1998. 95(3): p. 803-10.

9. Da Costa, S., et al., Human primary auditory cortex follows the
shape of Heschl's gyrus. J Neurosci, 2011. 31(40): p. 14067-75.

10. Huang, R.S. and M.I. Sereno, Dodecapus: An MR-compatible system for
somatosensory stimulation. Neuroimage, 2007. 34(3): p. 1060-73.

11. Sanchez-Panchuelo, R.M.,
et al., Mapping Human Somatosensory
Cortex in Individual Subjects With 7T Functional MRI. Journal of
Neurophysiology, 2010. 103(5): p.
2544-2556.