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**SINPASE fMRI course Dr Cyril Pernet, University of Edinburgh**

Dr Gordon Waiter, University of Aberdeen

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**Overview Matlab® environment: images are matrices**

MRI and fMRI: image format and softwares Computational Neuro-anatomy: theory Computational Neuro-anatomy: pratice Statistics: theory fMRI Single level analysis in practice fMRI Random effects analysis Other software - visualization Data provided by the FIL:

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**Matlab® environment: images are matrices**

Read data and do basic stuffs

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Matlab (1) Command window: A = 3+5 Workspace: whos History Browser

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**Matlab (2) load MRI_3D look in the workspace**

size(MRI_3D) returns the dimensions (here nb of voxels) imagesc(MRI_3D(:,:,54)) All rows All columns ‘column’ 54 on dim3 imagesc(MRI_3D(:,:,54)') imagesc(flipud(MRI_3D(:,:,54)')) colormap('gray')

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Matlab (3) MRI images are matrices (tables) with 3 dimensions, but it can be 4 dimensions (fMRI) or more. Operations on matrices A = [ ]; B = A' (transpose) A*B → matrix multiplication sum of row*columns C = [ ]; A+C → addition / subtraction work by cell A.*C → multiplication by cell uses .* (and ./) Exercise: subtract slice 54 from slice 55 and image it imagesc(flipud(MRI_3D(:,:,55)'-MRI_3D(:,:,54)'))

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Matlab (4) Exercise: Make a script (Matlab Editor) to look at the all volume using an axial view Possible functions to use: for, imagesc, squeeze, pause … and the matlab help Loop For/End for z = 1:108 do this and that for each z end

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**Matlab (5) for z = 50:220 imagesc(squeeze(MRI_3D(:,z,:)))**

colormap('gray') title(['this is the slice ',num2str(z)]) pause(0.1) end

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**MRI and fMRI: image format and software**

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**Image format DICOM format (.dcm)**

‘Standard’ format coming from all scanners Stands for Digital Imaging and Communications in Medicine Part 10 of the DICOM standard describes a file format for the distribution of images A single DICOM file contains both a header (which stores information about the patient's name, the type of scan, image dimensions, etc), as well as all of the image data (which can contain information in three dimensions). Manufacturers tend to output in DICOM but also put lots of useful information in the ‘private’ part of the header

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**Image format Analyze format (.img .hdr)**

Analyze is an image processing program, written by The Biomedical Imaging Resource at the Mayo Foundation. There are two Analyze formats. One, by much the more common, is Analyze 7.5 (this is one format used by SPM), the other is Analyze AVW, the format used in the latest version of the Analyze program An Analyze (7.5) format image consists of two files, and image (.img) and a header file (.hdr). The .img file contains the numbers that make up the information in the image. The .hdr file contains information about the img file, such as the volume represented by each number in the image (voxel size) and the number of pixels in the X, Y and Z directions. This header contains fields of text, floating point, integer and other information.

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**Image format NIfti format (.img .hdr or .nii)**

Stands for Neuroimaging Informatics Technology Initiative (The National Institute of Mental Health and the National Institute of Neurological Disorders and Stroke) Facilitates inter-operation of functional MRI data analysis software packages Headers now include - affine coordinate definitions relating voxel index (i,j,k) to spatial location (x,y,z); codes to indicate spatio-temporal slice ordering for FMRI; - "Complete" set of bit data types; - Standardized way to store vector-valued datasets over 1-4 dimensional domains; - etc

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1. Importing Data Matlab DICOM tools in the image processing toolbox, but also plenty of free software including SPM Type SPM select fMRI Import from the directory 3D_dicom Save as one file

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1. Importing Data

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**1. Importing Data Select the newly imported image**

Surf to the front, near the eye Check orientation Check the voxel size!! edit spm_defaults defaults.analyze.flip = 1; % input data left = right

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1. Importing Data Now simply double click on the nii file – this should bring MRICron Again surf to the front near the eyes the white spot is at a different location ?? Check outside the brain MRICron is telling you are on the right side BE AWARE OF THE ORIENTATION

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**Computational Neuro-anatomy: theory**

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**fMRI time-series kernel Anatomical reference Slice timing and**

Realignment smoothing Statistics normalisation Anatomical reference

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Slice timing: Why? During the scanning, slices of the brain are acquired every TR (x sec) and one wants to correct for this delay between slices. Data can be acquired in ascending/descending order, in this case one realigns first, otherwise one would apply the same time correction to voxels possibly coming from different slices, i.e. acquired at different time. Often one acquires first slices 1, 3, 5, 7, 9 … then 2, 4,6, 8, … (interleaved mode), in this case slice timing is done first otherwise the realignment would move voxels possibly coming from different slices, i.e. acquired at different time (max TR/2), onto the same plane and the slice timing would then be wrong.

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**Realignment: Why? Subjects will always move in the scanner.**

movement may be related to the tasks performed. When identifying areas in the brain that appear activated due to the subject performing a task, it may not be possible to discount artefacts that have arisen due to motion. The sensitivity of the analysis is determined by the amount of residual noise in the image series, so movement that is unrelated to the task will add to this noise and reduce the sensitivity.

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**Normalization: Why? Inter-subject averaging**

extrapolate findings to the population as a whole increase activation signal above that obtained from single subject increase number of possible degrees of freedom allowed in statistical model Enable reporting of activations as co-ordinates within a known standard space e.g. MNI

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**Smoothing: Why? Smoothing is used for 3 reasons:**

Potentially increase signal to noise (Depends on relative size of smoothing kernel and effects to be detected - Matched filter theorem: smoothing kernel = expected signal - Practically FWHM 3· voxel size ; May consider varying kernel size if interested in different brain regions (e.g. hippocampus -vs- parietal cortex)) Inter-subject averaging. Increase validity of SPM. In SPM, smoothing is a convolution with a Gaussian kernel, and the Kernel is defined in terms of FWHM (full width at half maximum).

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**Computational Neuro-anatomy: practice**

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**Single subject processing**

Protocol – Imaging parameters spm_data_set\Auditory_data_block_design Each acquisition consisted of 64 contiguous slices (3mm3). Acquisition took 6.05s, with the scan to scan repeat time (TR) set to 7s. 96 acquisitions were made (TR=7s) from a single subject, in blocks of 6, giving 16 42s blocks. The condition for successive blocks alternated between rest and auditory stimulation, starting with rest. Auditory stimulation was bi-syllabic words presented binaurally at a rate of 60 per minute. The functional data starts at acquisition 4, image fM00223_004.

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2. Slice timing

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2. Slice timing Session – Specify files --> select all functional images --> 96 files Number of slices? TR? TA? Slice order? Reference slice? OUTPUT: ‘a’ images .. afM00XX.img / .hdr 64 7 6.05 [64:-2:1, 64-1:-2:1] 2

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3. Realignment

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**New session Select the slice timed images afM00XX.img**

For each session during the scanning, create a session !!

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Creates the mean of all realigned EPI images – you do not have to write the data, parameters (which comes from the ‘estimate’) are kept in the header

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**created in your directory**

- the mean image a spm.ps .txt file = translation and rotation parameters

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4. Normalize We could normalize on the EPI template but since we have the subjects’ anatomical scan, we can use it to normalize on the T1 template. compute T1 MNI T1 template apply T1 like EPI EPI Realign and normalize Mean image Coregistration

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4a. Coregister

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**4a. Coregister Target image? mean EPI – this doesn’t change**

Source image? High resolution T1 (nsM00223_002.img) We want this to be like the EPI Other images? nothing here Reslice option interpolation trilinear is fine, could improve using b-spline (come back to this latter)

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4a. Coregistrer Output: rnsM00223_002.img

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4b. Normalization

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**4b. Normalize Data New Subject**

Source Image rns...img (compute from the coregistered T1) Images to write all the af…img (normalize all EPI images) we can add the mean.img we can add the rns…img (normalize the T1) Estimation options: Template image SPM5\Template\T1.nii Writing options: interpolation

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4b. Normlization Output: wa…img

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4c. Check the data Check Reg Several images at once

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4c. Check the data Select wmean…img T1 template

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5. Smoothing Select the normalized images Smooth at 6 mm Output: s…img

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**A word on interpolation**

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**Writing down the images**

What is interpolation? Interpolation is a process for estimating values that lie between known data points exit spm and open/run the script called interp_ex.m z is a 2D matrix and one interpolates between the ‘points’ defined by z – there are many options, here we use nearest neighbor, bilinear, and bicubic interpolation methods – observe the difference in the results

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**5. Writing down the images**

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**Writing down the images**

Why is it better to compute (estimate) everything 1st and then apply? While one can compute and write the realigned data and compute and write the normalized one, I suggest here to estimate the realignment parameters and then normalize. The realignment only uses affine transformation (translations and rotations in x, y ,z) and this info is stored in the header. Then we compute how to normalize an image (translation, rotation, zoom, shear and non-linear warping) and multiply the two transformation matrix – as a result a voxel (and its’ contend) is ‘cut’ only 1 time vs. twice.

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**Statistical modelling: theory**

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**Statistical Parametric Map Anatomical reference**

Parameter estimates Design matrix fMRI time-series kernel Slice timing and Realignment General Linear Model model fitting statistic image Multiple comparisons correction smoothing normalisation Statistical Parametric Map Anatomical reference corrected p-values

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General Linear Model Regression, t-tests, ANOVAs, AnCovas … are all instances of the same linear modelling. Regression: - simple: searching to explain the data (y) by a single predictor (x) such as y = βx + b – multiple: searching to explain the data (y) by several predictors (x1, x2, …) such as y = β1x1 + β2x2 + b Linear models can be solved by the least squares method, i.e. one looks for a coefficient (Beta) that minimizes the error, i.e. the difference between the model (βx+b) and the data (y)

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**General Linear Model Dummy coding:**

Instead of a continuous variable, we have categorical variables for each data point (y) we have a 2 groups, i.e. 1 predictor (x) regarding the group that we code like and we still use the equation y = βx + b ( = t-test) or we can have several groups/conditions (x1 = , x2 = , x1x2= ) such as y = β1x1 + β2x2 + β12x1x2 b (ANOVA)

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General Linear Model Using a matrix notation we can write any models like Y = Xβ + e Y a vector for each data point, X the design matrix where each column is a vector representing groups/conditions/continuous predictor β a vector (length = nb of column of X) of the coefficients to apply on X in order the minimize e the error (what is not modeled/explained) Matrix algebra offers a unique solution for all models: β = (XTX)-1 XT Y using pseudoinverse in matlab: betas = pinv(X’*X)*X’*Y

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**General Linear Model Exercise 1: multiple regression with 4 covariates**

Load the data called reg_eg.mat: load ('reg_eg') Y = reg_eg(:,1); X = reg_eg(:,2:6); imagesc(X); X Y=X*B+e B = pinv(X'*X)*X'*Y; Yhat = X*B; plot(Y); hold on; plot(Yhat,'r') ss_total = norm(Y - mean(Y)).^2; ss_effect = norm(Yhat - mean(Yhat)).^2; ss_error = ss_total – ss_effect; Rsquare = ss_effect/ss_total f = (ss_effect/(rank(X)-1)) / (ss_error/(length(Y)-rank(X)))

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**General Linear Model Exercise 2: ANOVA with 4 groups**

Clear all; close all; clc; load(‘anova_eg’); Y = anova_eg(:,1); X = anova_eg(:,2:6); imagesc(X); Y=X*B+e B = pinv(X'*X)*X'*Y; Yhat = X*B; plot(Y); hold on; plot(Yhat,'r') ss_total = norm(Y - mean(Y)).^2; ss_effect = norm(Yhat - mean(Yhat)).^2; ss_error = ss_total – ss_effect; Rsquare = ss_effect/ss_total f = (ss_effect/(rank(X)-1)) / (ss_error/(length(Y)-rank(X))) EVERY ANALYSIS DEPENDS ON X AND THE DOF

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**General Linear Model Application to fMRI: massive univariate approach**

For each voxel of the brain we have a time series (data points Y) and we know what happened during this time period (experimental conditions X). We also know that after a stimulus or a response, the blood flow increases peaking at 5sec then decreases (hemodynamic response) We thus model the data such as Y(t) = [u(t) h(τ)] β + e(t) where u represent the occurance of a stimulus or a response and h(τ) the hemodynamic response with τ the peristimulus time

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**General linear convolution model**

X(y) = u(t) h(τ) u = [ ]; h = spm_hrf(1) X = conv(u,h);

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**General linear convolution model**

y(t) = X(t)β + e(t) X as now two conditions u1 and u2 … And we search the beta parameters to fit Y X Y

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**General linear convolution model**

= β1 + β2 + u + e

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**General linear convolution model**

β1 β2 + e u =

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**fMRI characteristics which may increase error**

Gain & scanner drift Variations of signal amplitude for each new brain volume and between scanning sessions → Proportional & Grand-mean scaling of data → High-pass filtering in design matrix Serial temporal correlations breathing, heartbeat: activity at one time point correlates with other times → adjust error term Temporal and shape uncertainty & slice timing delays: model derivatives of HRF

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**Single subject stat modelling: practice**

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5. Single subject stats

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5. Single subject stats Directory select a directory to store your data Timing parameters Unit for the design: scan Interscan interval: 7 (TR) Microtime onset: 3 (TA = 6.05 and we slice timed on the middle slice) Data and Design ‘New Subject/Session’ Scans: images swaf…4.img to waf…99.img

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**5. Single subject stats Data and Design ‘New Subject/Session’**

Scans: start at 4 to avoid artefacts .. (you should always discard initial scans) = 96 files Conditions New condition Name: ‘activation’ Onsets: 6:12:84 (= every 12 scans from 6 to 84) Durations: 6 Multiple regressors: select the realignment parameters (.txt file) The rest is implicitly modelled (goes into the mean)

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5. Single subject stats Activations Movements Mean Model summary

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5. Single subject stats

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5. Single subject stats

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5. Single subject stats

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5. Single subject stats

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5. Single subject stats

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5. Single subject stats

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5. Single subject stats

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**5. Single subject stats Contrast estimate (effect size)**

Fitted response (model or Y-error)

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**5. Single subject stats SPM outputs**

beta000X.img /hdr: betas values in each voxel con000X.img /hdr: contrast (combination of betas) spmT000X.ing /hdr: statistics image Also mask.img / hdr: area analyzed by SPM (binary image) RessMS: residual mean square RPV: Resels-Per-Voxel image; image of roughness

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**Multiple comparisons correction: theory**

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**What Problem? . . . 4-Dimensional Data**

1,000 4-Dimensional Data 1,000 multivariate observations, each with 100,000 elements 100,000 time series, each with 1,000 observations Massively Univariate Approach 100,000 hypothesis tests Massive MCP! 3 2 1

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**Solutions for MCP Height Threshold Familywise Error Rate (FWER)**

Chance of any false positives; Controlled by Bonferroni & Random Field Methods False Discovery Rate (FDR) Proportion of false positives among rejected tests Set level statistic Bayes Statistics

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**From single univariate to massive univariate**

Univariate stat Functional neuroimaging 1 observed data Many voxels 1 statistical value Family of statistical values Type 1 error rate (chance to be wrong rejecting H0) Family-wise error rate Null hypothesis Family-wise null hypothesis

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Height Threshold Choose locations where a test statistic Z (T, 2, ...) is large to threshold the image of Z at a height z The problem is how to choose this threshold z to exclude false positives with a high probability (e.g. 0.95)? To control for family wise error on must take into account the nb of tests

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**Bonferroni 10000 Z-scores ; alpha = 5%**

alpha corrected = ; z-score = 4.42 100 voxels 100 voxels

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**Bonferroni 10000 Z-scores ; alpha = 5%**

2D homogeneous smoothing – 100 independent observations alpha corrected = ; z-score = 3.29 100 voxels 100 voxels

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**Random Field Theory 10000 Z-scores ; alpha = 5%**

Gaussian kernel smoothing – How many independent observations ? 100 voxels 100 voxels

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Random Field Theory RFT relies on theoretical results for smooth statistical maps (hence the need for smoothing), allowing to find a threshold in a set of data where it’s not easy to find the number of independent variables. Uses the expected Euler characteristic (EC density) 1 Estimation of the smoothness = number of resel (resolution element) = f(nb voxels, FWHM) 2 ! expected Euler characteristic = number of clusters above the threshold 3 Calculation of the threshold

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Random Field Theory The Euler characterisitc can be seen as the number of blobs in an image after thresholding At high threshold, EC = 0 or 1 per resel: E[EC] pFWE E[EC] = R · (4 loge 2) · (2)−2/3 · Zt · e−1/2 Z2t for a 2D image, more complicated in 3D

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Random Field Theory For 100 resels, the equation gives E[EC] = for a threshold Z of 3.8, i.e. the probability of getting one or more blobs where Z is greater than 3.8 is 0.049 100 voxels 100 voxels If the resel size is much larger than the voxel size then E[EC] only depends on the nb of resels othersize it also depends on the volume, surface and diameter of the search area (i.e. shape and volume matter)

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False discovery Rate Whereas family wise approach corrects for any false positive, the FDR approach aim at correcting among positive results only. 1. Run an analysis with alpha = x% 2. Sort the resulting data 3. Threshold the resulting data to remove the false positives (theoretical problem: threshold any voxels whatever their spatial positions)

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False discovery Rate Signal+Noise FEW correction FDR correction

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**Levels of inference: theory**

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**Levels of inference 3 levels of inference can be considered:**

Voxel level (prob associated at each voxel) Cluster level (prob associated to a set of voxels) Set level (prob associated to a set of clusters) The 3 levels are nested and based on a single probability of obtaining c or more clusters (set level) with k or more voxels (cluster level) above a threshold u (voxel level): Pw(u,k,c) Both voxel and cluster levels need to address the multiple comparison problem. If the activated region is predicted in advance, the use of corrected P values is unnecessary and inappropriately conservative – a correction for the number of predicted regions (Bonferroni) is enough

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Levels of inference Set level: we can reject H0 for an omnibus test, i.e. there are some significant clusters of activation in the brain. Cluster level: we can reject H0 for an area of a size k, i.e. a cluster of ‘activated’ voxels is likely to be true for a given spatial extend. Voxel level: we can reject H0 at each voxel, i.e. a voxel is ‘activated’ if exceeding a given threshold

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Inference in practice

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**Levels of inference and MCC**

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**From single subjects to random effects: theory**

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**From single subjects to random effects**

Why random effect (also called pseudo-mixed)? Basic stats: compute the mean for each condition for each subjects and do the stats on these means (inter-subject variance) Random effect: compute the beta parameters for each condition (intra-subject variance) and do the stats on these beta parameter (inter-subject variance)

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**From single subjects to random effects: practice**

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Face data set Repetition priming experiment performed using event-related fMRI 2x2 factorial study with factors `fame‘ and `repetition' where famous and non-famous faces were presented twice against a checkerboard baseline Four event-types of interest; first and second presentations of famous and non-famous faces, which are denoted N1, N2, F1 and F2 TR=2s TA = 1.92s 24 descending slices (64x64 3x3mm2) 3mm thick with a 1.5mm gap

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**Single subject modelling**

Here we consider a more sophisticated model in which we use the hrf and its derivatives. In the folder, spm_data_set\Face_data_event_related_design\ single_subject\Preprocessed, you can find the smoothed, normalized, realigned, and slice timed images Try to model by yourself

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**Single subject modelling**

Stimulus Onsets Times sots.mat Cond 1 to 4 are F1, F2, N1, N2 with the respective onset time sot{1} sot{2} sor{3} and sot{4}

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**Single subject modelling**

Directory stats Timing parameters: Units (sec), Interscan interval (2), Microtime resolution (24), Microtime onset (12) Data and design ‘New Subject/Session’: Scan (all EPI swa…img), Condition create 4 conditions, each time enter name and sot (F1 sot{1} F2 sot {2} N1 sot{3} N2 sot{4}), Multiple regressors: enter the .txt file from realignment Factorial Design create 2 factors (famous, level =2, and repetition, level =2) Basis function: Canonical HRF, model derivatives (time and dispersion)

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**Single subject modelling**

Result select the SPM.mat Because we specified the factorial design, the variance has been partitioned in a specific way and contrasts are already there Select contrast nb 5

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**Single subject modelling**

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**Single subject modelling**

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**Single subject modelling**

Define a new F contrast called ‘effect of interest’ (any effect of any regressor and combinations) ok done

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**Single subject modelling**

Plot contrast estimates and 90% CI select effect of interest N2 F1 F2 N1

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**Single subject modelling**

Plot event-related response / fitted response

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**Single subject modelling**

In the matlab workspace Y = fitted response, y = adjusted response Exercise: plot for 4 fitted responses on 1 figure Each time save using e.g. N1 = Y; then N2 = Y; .. then plot(N1); hold on; plot(N2,’--’) …

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Multi-subjects All data were analyzed and are stored in the folder Face_data_event_related_design\multisubjects\cons_informed We have all the con images from 12 subjects, data were modelled with Famous and Non famous faces (2 conditions only) with the hrf and the two derivatives (12*6 files)

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Random effects

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**Random effects Analyze Faces vs baseline**

For each subject we have the con images for the hrf, the 1st derivative and 2nd derivative A full analysis will thus be a repeated measure ANOVA (we have 3 measures per subject) SPM allows to correct for non-sphericity, i.e. it will take into account the correlation between regressors – here regressors are by inception correlated

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**Random effects Design Full factorial create a ‘new factor’**

Name: Basis functions Levels: Independence: Specify cells create 3 cells, 1 per level of the factor (cell 1 = con 3 to 14 / cell 2 = 15 to 26 / cell 3 = 27 to 38) Run Estimate 3 No

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Random effects In the contrast manager, enter a new F contrast: effect of interest (eye(3)) Evaluate and look at the result with a correction .05

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Random effects

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Random effects In the contrast manager, enter a new T contrast for the hrf only: 1 0 0 Evaluate and look at the result with a correction .05

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**Random effects Select an image to display (e.g. SPM /canonical/T1)**

Add blobs select the SPM.mat and the 2 contrasts

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**} Random effects The display is ‘surfable’**

MNI and voxel coordinates available Voxel value }

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**Random effects F Contrast [0 0 1 ] F Contrast [0 1 0]**

Duration dispersion F Contrast [0 1 0] Time dispersion (+/- 2 sec) Narrower response ( - = wider) Earlier response (- = delayed)

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**Random effect Paired t-test 12 pairs 3 vs 39 4 vs 40 …**

Independence: No Variance: equal

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Visualizing the data SPM and other software

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Rendering with SPM

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Segmentation

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Segmentation Input coregistered image in \spm_data_set\ Face_data_event_related_design\single_subject\Structural rs …img Output grey / white matter tissues bias corrected image affine normalization parameters (and inverse)

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Rendering with SPM Select the grey and white matter (c1 and c2) images and save the rendering

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Rendering with SPM Select the SPM.mat from the individual subject and then 2 sets / contrasts

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Rendering in SPM For random effects analysis you can use the render in SPM/render which is in the MNI space For individual subjects, best is to not normalize and make a render .. Results are better with smaller voxel size (i.e. you could interpolate the 1mm3) Overall, better to use slices / section – poor rendering capabilities although new plug in are appearing ..

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**Surface visualization with Caret**

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**What does it do? Surface visualization**

Display experimental data (activation maps, connectivity patterns, etc) View data in flexible combinations (cortical surface, flat maps, contours, outlines, etc) Surface manipulation and data analysis generate inflated maps, spherical maps, and flat maps probabilistic maps, surface based analysis On-line search for fMRI maps, comparisons, etc ..

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**The Software http://brainmap.wustl.edu/caret**

my experience is that it works better with linux / mac but windows does the job From David Van Essen lab – Washingtom University in St Louis – School of Medicine – Dpt of Anatomy & Neurobiology Most famous paper from this guy? Felleman, D.J. and Van Essen, D.C. (1991) Distributed hierarchical processing in primate visual cortex, Cerebral Cortex, 1: 1-47.

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**Caret File Open Spec File …**

In Caret\ HUMAN.COLIN.ATLAS\ LEFT_HEM\ …spec

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Caret

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Caret

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Caret Attributes Map Volume(s) to Surface(s) … leave the Metric option ticked and press next

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Caret Add Volumes From Disk select the spmT_0005.hdr file from the single subject analysis (Face data set)

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Caret Map to Caret

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Caret Chose your algorithm Average Voxel Neighbor Box Size 1 Next ..

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Caret

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Caret

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Caret

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**Individual visualization with Anatomist**

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Brain Visa / Anatomist Caret as well as BrainVisa / Anatomist works better with clean data … 1 st – data are usual better handled with isotropic voxels (same dimensions in x, y, z) better acquire isotropic voxels, at least for the T1 2 nd – there is usual a bias in one direction (often Z), i.e. for a given concentration of tissue, the gray scale is slightly different Solution: correct and resample My favourite tools for this: FSL

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**Quick Tour into FSL Depending on what you want to do**

ApplyXFM reinterpolation using the identity matrix FSL brings the interface BET / SUSAN / FSL View

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**Back to BrainVisa / Anatomist**

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Back to Anatomist

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Anatomist

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Resources

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**IdoImaging: http://idoimaging.com/index.shtml**

SPM: Caret: FSL: BrainVisa / Anatomist: Cambridge imaging wiki: Russ Poldracks’wiki on Matlab: Digital signal processing in general: Maths in general:

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