fMRI signal + - Neurometrika

NMRData AnalysisPracticalitiesMRIPsychophysiologicalLaboratoryExperimentalTaskHemodynamicsFunctionalNeuroanatomyExperimental DataAnalysis DesignComparingBrainsComputationalNeuroanatomy

Overview of SPMImagetime-seriesSpatialSmoothingKernelDesignmatrixStatistical parametric map(SPM)RealignmentSmoothingGLM: GeneralLinear Model“SpatialNormalisation”StatisticalinferenceGaussianfield theoryStandardizedAnatomicalTemplateParameterestimatesp

Overview of SPMImagetime-seriesSpatialSmoothingKernelDesignmatrixStatistical parametric map(SPM)Realignment“SpatialNormalisation”SmoothingGLM: GeneralLinear ModelPreprocessing Modeling InferenceStatisticalinferenceGaussianfield theoryStandardizedAnatomicalTemplateParameterestimatesp

Lecture 1Optimizing Data AcquisitionThis is the attempt to get the best quality images that you canfrom your scanner, subject, and experiment.

Lecture 2Preprocessing: RealignmentPrimarily, this is to compensate, on a global level, for the factthat living subjects move. It is “realigning” images withsimilarly collected images---images like themselves.

Lecture 2Preprocessing: Spatial SmoothingThis is almost always done, because the size of the activatedregion is typically larger than a single voxel. It benefits thestatistical analysis in a number of ways...in most experiments.

Lecture 2Preprocessing: Spatial NormalizationPutting brain data in a “standard” 3-dimensional coordinatespace permits group analysis and reporting findings.Realignment can be associated with “spatial normalization whengoing between imaging modalities (PET & fMRI; T1 & T2*)

Lecture 3Modeling for First Level EstimationThis is the search for “task-related activity”in each individual subject.

Overview of SPMImagetime-seriesSpatialSmoothingKernelDesignmatrixStatistical parametric map(SPM)Realignment“SpatialNormalisation”SmoothingGLM: GeneralLinear ModelPreprocessing Modeling InferenceStatisticalinferenceGaussianfield theoryStandardizedAnatomicalTemplateParameterestimatesp

effects of interesterror varianceeffects of no interestThe Jewelwe seek!ExplainedVarianceThe UnexplainedVariance,called “noise”effects of intereststatistic = -----------------------------error variance

Lecture 4Modeling for Second Level EstimationExperimentalDesign“Second Level” modeling can refer to many things.Most commonly, it refers to group analyses of various kinds.

Lecture 5Inference and Critical ThresholdsNo StatisticsHere!Inference and “critical thresholds” are a core part of the historyof the SPM software package. “Gaussian Random Field” theorywas an attempt to deal with the problem of multiple comparisonswithout invoking the overly stringent Bonferonni correction.

View 1:A series of volumes(scans, 3D “images”)Useful for seeking spatio-temporalpatterns, e.g., via PCA, ICA, etc.View 2:Multiple voxel time seriesUseful for seeking temporalpatterns in each individualvoxel of the volumes.Two Views on fMRI DataStandard hypothesis-driven statisticalanalysis (e.g., GLM) goes withview 2 since it is appliedindependently for each voxel timecourse, i.e.,“voxel-wise” statistical analysis.

Inference: Use Uncertainty of Parameter Estimatesa)b)noise noisenoisenoiseMean 2Mean 1RestStimRestStim• Same mean values in Rest and Stim condition in cases a) and b)Simple subtraction (difference = Mean 2 – Mean 1) yields same result !Simple subtraction does NOT consider variance of means (noise)• Statistical analysis (e.g., t test) relates estimate (mean difference) touncertainty of estimate = pooled variance of values within conditions, i.e.

From Experimental Timing to Expected fMRI Time CourseNeural pathway Hemodynamics MR scannerConvolution kernel

The General Linear Model (GLM)β 0×X 0fMRI signal= β 1×X 1+++residualsTimeβ 2×X 2Timedesign matrixTimeThe design matrix models expected signal time courses for individual conditions (effects of interest), butmay also include additional predictor time courses (confounds, effects of no interest) helping to improveexplaining the data. In this example, two main predictors are defined (X 1and X 2). A “constant” predictor(X 0) is required to fit the base level of the signal time course. The expected time courses are obtained byconvolving the condition “box-car” functions (red / green) with the assumed impulse response (HRF, twogammafunction).

The General Linear Model (GLM)β 0×X 0fMRI signal= β 1×X 1+++residualsTimeβ 2×X 2Timedesign matrixTime...The expected time courses are obtained byconvolving the condition “box-car” functions(red / green) with the assumed impulseresponse (HRF, two-gamma function).

The General Linear Model (GLM)Time= β 0× + β 1× + β 2× +TimefMRI signaldesign matrixresidualsThe time courses of the signal, predictors and residuals can be rearranged in column form with timerunning from top to bottom. This view is helpful to understand the matrix form of the General LinearModel.

The General Linear Model (GLM)estimate!β 0× β 1×β 2×Time= + ++fMRI signal= datadesign matrix= model(defined by YOU)residuals= error

The General Linear Model (GLM)estimate!β 0× β 1×β 2×Time= + ++fMRI signal= datadesign matrix= model(defined by YOU)residuals= errorThe GLM time courses, especially the design matrix is often shown in a graphical view using a black-towhitecolour range for coding (expected) signal intensities.

Time= + + +fMRI signal= datadesign matrix= model(defined by YOU)residuals= errorTime=+++fMRI signal= datadesign matrix= model(defined by YOU)residuals= error

With many thanks for slides& images to:FIL Methods group

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Convolve stimulusfunction with a canonicalhemodynamic responsefunction (HRF):⊗Stimulus timing shown in REDStimulus timing convolved with the “canonical” HRF Shown in GREENMR data shown in BLUE

Stimulus timing convolved with the“canonical” HRF...What is the use, in most analyses, of “canonicalHRF” function?It refers to the fact that the SAME “HRF” is usedFOR ALL voxels in the brain,FOR ALL stimulus types,FOR ALL subjects!

Stimulus timing convolved with the“canonical” HRF...One can use the “Finite Impulse Response (FIR)”model to estimate the HRF individually: for eachvoxel in the brain, for each stimulus type, for eachsubject.In practice, full FIR analysis is rarely done. Butestimating the dispersion and delay of the HRFindividually on a voxel basis, and then treating thoseas “covariates of no interest”, is more commonlyuseful.

The General Linear Model (GLM)

The GLM in Matrix Notationobserved variable,fMRI time coursedesignmatrixbeta weights,coefficientserrors,residualspredictor,explanatory variable,regressor, covariate,basis functionanother predictor...one of “k” functions tohelp fit the data in theGLM

The GLM in Matrix Notation

The GLM in Matrix NotationObserved fMRI signal:Predicted fMRI signal:Prediction error:

Fitting a GLM

Observed fMRI signalPredicted fMRI signalresidualsb 0 = 101.9design matrixGLM – Example, block design

Observed fMRI signalPredicted fMRI signalresidualsb 0= 101.6+b 1= 1.8design matrixGLM - Example

Observed fMRI signalPredicted fMRI signalresidualsb 0= 101.0+b 1= 4.5design matrixGLM - Example

Observed fMRI signalPredicted fMRI signalresidualsb 0= 100.2+b 1= 3.4+b 2= 5.5GLM - Exampledesign matrix

Lessons from the GLM Example• To obtain optimal GLM fits, all known effects should be modelled inthe design matrix.• Non-modelled effects are moved to the residuals, substantiallyincreasing the error variance. This leads to poor fits and reducedstatistical power (the error variance is one component of the standarderror of contrasts).• Inspection of the residuals is a good diagnostic to assess thegoodness-of-fit of a GLM: If one observes prominent, low-frequencyfluctuations, it is likely that not all effects have been modelled or thatthe modelled time courses do not fit to the data. The latter might befor example, due to delayed fMRI responses with respect to predictors.• Confound predictors may be added to improve the fit, for example,predictors capturing low-frequency drifts (Fourier basis functions orDiscrete Cosine Transform, DCT, basis functions), if drifts are notremoved already during data preprocessing.

Functional MRI WorkshopsIntroduction to the GLM in fMRI Data AnalysisAn Overview ofData Processing Tasksin the Analysis offMRI-based ExperimentsRobert L. Savoy, Ph.D.The Athinoula A. Martinos Center for Biomedical ImagingHyperVision, Inc.e-mail: Robert.L.Savoy@alum.mit.edu