Synopsis
Task-based
fMRI data is often analysed using the General Linear Model (GLM). This talk introduces
this analytical approach starting from its basic concepts, benefits and
limitations. Examples will be given showing how it can be used in block and
event-related paradigms. Furthermore, the discussion will cover an introduction
to the flexible use of the GLM in forward (or encoding) modelling approaches of
task-based fMRI as the population receptive field (pRF) analysis.
Target audience
Cognitive
neuroscientists, imaging scientists, neuroradiologists, and clinicians who
currently utilize fMRI.
Outcome and Objectives
Understand
the use of GLM for task-based fMRI analysis, be able to describe its basis and
limitations and be aware of its application in forward (encoding) modelling
approaches.
Purpose
The
basis of the General Linear Model can be found in linear regression. Thanks to
its intuitive feel and simplicity it became one of the most common analysis methods
used in the fMRI community.
GLM
is normally used to model the blood-oxygen level dependant (BOLD) time courses
among several experimental conditions. In its simplest form the shape of the
impulse response function (the hemodynamic response function, HRF) is
assumed constant, but their amplitudes need to be estimated1,2,3. However, the same family of
models can be used also to estimate HRF shape, using deconvolution.
These
types of model are appealing as they allow to include multiple predictors that
might influence the data in a single and concise model. As such, predictors can
be included as variables of interest as well as potential confounds.
All
the included predictors are combined in a linear fashion (weighted sum) to
account for BOLD data, our single dependent variable.
GLM can be the final
step of a single-subject analysis pipeline but can also be used as the main
engine to test multiple predictions in a forward (encoding) modelling approach4,5.
Methods
In
general, GLM is formulated as a matrix equation Y = Xβ+ε. In this case Y
represents the measured BOLD signal over the length of the experimental run
(the dependent variable), X is a matrix with a collection of measurements (a
single or multiple independent variables). X can include variables of interest
(e.g. the experimental manipulation as well as nuisance variables as motion, slow
drifts, respiration etc.) β is a vector of parameters (weights) that need to be
estimated via least-square estimation and ε is a vector containing the residual
error between the observed and the estimated dependent variable (noise)1,2,3. ε is
assumed to be distributed following a normal distribution centred on zero. The
objective of the estimation is to minimize the sum of squares of the residuals.
The estimated values are obtained via a linear combination (weighted sum) of
each independent variable in X and the parameters in β.
The
same method can be used to iteratively test prediction based on forward
(encoding) modelling approach and infer the underlying tuning properties of the
single voxel.
Results
GLM
results generates the β parameters that best fit the observed data in terms of
ordinary least squares estimation. BOLD activations due to the experimental
stimulation is given by the β parameter associated with the experimental
stimulation. T scores are associated with each independent predictor in
X, indicating how reliably the measured response (dependent variable) is
associated with the individual predictor. Parameter estimates and T scores can
be compared to assess which predictor is more relevant to the dependent
variable.
Similarly, T scores
or other measures of goodness of fit (for example: R2) can be used
in a forward (encoding) modelling approach to assess which expected time series
better predicts the observed data4,5.
Discussion
GLM
can be used as an analysis tool to assess the influence of a given experimental
manipulation on a voxel time-series, while accounting for several nuisance
variables.
One
limiting factor of this tool is noise estimation. Noise (the residuals, ε) needs
to be distributed normally and centred on 0. Several factors can influence the
goodness of fit and the normality of the residuals (for example the local shape
of the HRF).
On the flipside, obtaining
a heavily skewed or non-normal noise distribution is informative, as it
suggests that other variables can further contribute to the goodness of fit. This
represents the basis of forward (encoding) models, where features are
explicitly modelled into an expected time series, and tested against the
observed data. Allowing to make an inference about the underlying
tuning properties at a single voxel level.
Conclusion
GLM
per se is a robust analysis method for fMRI research. When used in a forward
modelling approach with tailored experimental designs it becomes a powerful
tool to study tuning properties at the individual voxel level.
Acknowledgements
No acknowledgement found.References
1.
Friston, K.J., et al., Analysis of fMRI time-series revisited. Neuroimage,
1995. 2(1): p. 45-53.
2.
Worsley, K.J. and K.J. Friston, Analysis of fMRI time-series revisited--again.
Neuroimage, 1995. 2(3): p. 173-81.
3.
Poline, J.B. and M. Brett, The general linear model and fMRI: does love last
forever? Neuroimage, 2012. 62(2): p. 871-80.
4. Dumoulin,
S.O. and Wandell, B.A., 2008. Population receptive field estimates in human
visual cortex. Neuroimage, 39(2), pp.647-660.
5. Wandell
BA, Winawer J. Computational neuroimaging and population receptive fields.
Trends in cognitive sciences. 2015 Jun 1;19(6):349-57.