Modelling temporal structurehelper.ipam.ucla.edu/publications/mbi2008/mbi2008_7665.pdf · frequency...

• temporal noise: modelling temporal autocorrelation

• temporal signal: FLOBS HRF optimal basis functions• temporal signal: HRF deconvolution

• spatiotemporally structured signal / noise: ICA• “functional grand-plan”: integrating ICA+GLM

Modelling temporal structure(in noise and signal)

Mark Woolrich, Christian Beckmann*, Salima Makni & Steve SmithFMRIB, Oxford *Imperial/FMRIB

frequency

powerColoured / autocorrelated

White / independent

Even after high-pass filtering, FMRI noise has

extra power at low frequencies (positive

autocorrelation or temporal smoothness)

Uncorrected, this causes:

- biased stats ( increased false positives)

- decreased sensitivity

Non-independent/Autocorrelation/Coloured FMRI noise

FMRIB’s Improved Linear Modelling (FILM)

• FILM is used to fit the GLM voxel-wise in FEAT• Deals with the autocorrelation locally and uses prewhitening

FILM estimates autocorrelation by looking at the residuals of the GLM fit:

residuals = Y −Xβ̂

Y = Xβ + "


1) Fit the GLM and estimate the autocorrelation on the residuals

frequency

power

Power vs. freq in the residuals



frequency

2) Spatially and spectrally smooth the data

frequency

power


power



frequency


frequency

power

3) Construct prewhitening filter to “undo” autocorrelation

frequency

power


power



frequency


frequency

power

3) Construct prewhitening filter to “undo” autocorrelation

4) Apply filter to data and design matrix and refit

frequency

power

power

power

frequency

Dealing with Variations in Haemodynamics

• The haemodynamic responses vary between subjects and areas of the brain

• How do we allow haemodynamics to be flexible but remain plausible?

Reminder: the haemodynamic response function (HRF) describes the BOLD response to a short burst of neural activity

Temporal Derivatives

• A very simple approach to providing HRF variability is to include (alongside each EV) the EV temporal derivative

• Including the temporal derivative of an EV allows for a small shift in time of that EV

• This is based upon a first order Taylor series expansion

EVTemporal derivative

datamodel fit without

derivative

model fit with derivative

• We need to allow flexibility in the shape of the fitted HRF

Parameterise HRF shape and fit shape parameters to the data

Needs nonlinear fitting - HARD

Using Parameterised HRFs

Using Basis Sets

• We need to allow flexibility in the shape of the fitted HRF

Parameterise HRF shape and fit shape parameters to the data

We can use linear basis sets to span the space of expected HRF shapes

Needs nonlinear fitting - HARD Linear fitting (use GLM) - EASY

How do HRF Basis Sets Work?

Different linear combinations of the basis functions can be used to create different HRF shapes

+ -0.1*+ 0.3* 1.0* =

basis fn 1 basis fn 2 basis fn 3 HRF

How do HRF Basis Sets Work?

Different linear combinations of the basis functions can be used to create different HRF shapes

+ -0.1*+ 0.3* 1.0* =

+ 0.5*+ -0.2* 0.7* =

basis fn 1 basis fn 2 basis fn 3 HRF

FMRIB’s Linear Optimal Basis Set (FLOBS)

Using FLOBS we can:

• Specify a priori expectations of parameterised HRF shapes

• Generate an appropriate basis set

Generating FLOBs

(1) Take samples of the HRF

Generating FLOBs

(2) Perform SVD (3) Select the top eigenvectors as the optimal basis set

“Canonical HRF”

Generating FLOBs


“Canonical HRF”

temporal derivative

Generating FLOBs


The resulting basis set can then be used in FEAT

“Canonical HRF”dispersion derivative temporal

derivative

Woolrich et al. , TMI, 2004

Bayesian Inference

⊗Neural Activity

*A +BOLD FMRI data

Gaussian noise

HRF

Priors on HRF parameters

p(A, c, m1,m2 . . . |Y )Joint posterior distribution

Infer using MCMCWoolrich et al. , TMI, 2004

Bayesian Inference

⊗Neural Activity

*A +BOLD FMRI data

Gaussian noise

HRF

Priors on HRF parameters

p(A|Y ) =∫

p(A, c, m1,m2 . . . |Y )dcdm1dm2 . . .

Marginal posterior distribution

• Inputs are raw paradigm (stimulation and “modulation”) timecourses

• Model based on Bilinear Dynamical Systems (Penny 2005), where modulatory input changes neural response to stimulation

• What’s new:• estimate HRF from data• full Bayesian inference on model, using VB

Makni, NeuroImage 2008

Temporal deconvolution of FMRI timecourses

UNCO

RREC

TED P

ROOF

307 VB does not guarantee the optimal global solution, that is308 why a good initialization of the different BDS model309 parameters is important. We initialize the HRF using the310 canonical HRF that is widely used in fMRI to model the311 response. The neuronal response is initialized using a simple

312linear least squares optimization. The neurodynamic para-313meters, the state and the space noise terms are all randomly314initialized. This initialisation is sufficient for our VB315algorithm to converge within 10–20 iterations, taking only3161.5–3 min per single time series of length 250 on a standard

Fig. 1. Results on simulated data using EM and VB (low SNR=11.5, Scan number=250). (a) Driving input v used to generate the data. (b, c) Modulatory inputs un(1) and un(2) used togenerate the data. (d, f) HRF using EM and VB, respectively. (e, g) neuronal response using EM and VB, respectively. In all cases dashed line is for the true value of the parameter andcontinuous line is for the estimated parameter.

4 S. Makni et al. / NeuroImage xxx (2008) xxx–xxx

ARTICLE IN PRESS

Please cite this article as: Makni, S., et al., Bayesian deconvolution fMRI data using bilinear dynamical systems, NeuroImage (2008),doi:10.1016/j.neuroimage.2008.05.052

UNCO

RREC

TED P

ROOF

307 VB does not guarantee the optimal global solution, that is308 why a good initialization of the different BDS model309 parameters is important. We initialize the HRF using the310 canonical HRF that is widely used in fMRI to model the311 response. The neuronal response is initialized using a simple

312linear least squares optimization. The neurodynamic para-313meters, the state and the space noise terms are all randomly314initialized. This initialisation is sufficient for our VB315algorithm to converge within 10–20 iterations, taking only3161.5–3 min per single time series of length 250 on a standard

Fig. 1. Results on simulated data using EM and VB (low SNR=11.5, Scan number=250). (a) Driving input v used to generate the data. (b, c) Modulatory inputs un(1) and un(2) used togenerate the data. (d, f) HRF using EM and VB, respectively. (e, g) neuronal response using EM and VB, respectively. In all cases dashed line is for the true value of the parameter andcontinuous line is for the estimated parameter.

4 S. Makni et al. / NeuroImage xxx (2008) xxx–xxx

ARTICLE IN PRESS

Please cite this article as: Makni, S., et al., Bayesian deconvolution fMRI data using bilinear dynamical systems, NeuroImage (2008),doi:10.1016/j.neuroimage.2008.05.052

To do the Bayes, either:• MCMC (computer takes ages)• Variational Bayes (maths takes ages)

Model-free Functional Data Analysis

• decomposes data into a set of statistically independent spatial component maps and associated time courses

• can perform multi-subject/ multi-session analysis

• fully automated (incl. estimation of the number of components)

• inference on IC maps using alternative hypothesis testing

MELODICMultivariate Exploratory Linear Optimised Decomposition

into Independent Components

EDA techniques for FMRI

• are mostly multivariate• often provide a multivariate linear decomposition:

space# m

aps=

time Scan #k

FMRI dataspatial maps

time

# mapsspace

Data is represented as a 2D matrix and decomposed into factor matrices (or modes)

Model Order Selection

• can estimate the model order from the Eigenspectrum of the data covariance matrix (corrected using Wishart random matrix theory)

• approximate the Bayesian evidence for the model order for a probabilistic PCA model (PPCA)

Minka, TR 514 MIT Media Lab 2000

Laplace approximation BIC AIC

C.F. Beckmann , J.A. Noble , S.M. SmithOxford Centre for Functional Magnetic Resonance Imaging of the Brain (FMRIB)

Medical Vision Laboratory, Department of Engineering, University of Oxfordemail: beckmann,steve @fmrib.ox.ac.uk

IntroductionAnalysing FMRI data using linear techniques like the

GLM, PCA or ICA can be understood as a form of di-

mensionality reduction where a small set of spatial maps

(Z-scores; IC maps; eigenimages) is sought that, together

with pre-specified (GLM) or estimated (PCA/ICA) time-

courses, represent the signal. The signal subspace is gen-

erally expected to be of lower dimensionality than the data,

e.g. in the case of simple block designs, the signal is implic-

itly assumed to be contained in a subspace roughly of size

less than or equal to the length of one stimulation cycle.

In ICA, it is common to whiten the data using singular value

decomposition. The data is often projected onto a set of

dominant eigenvectors in order to reduce the data dimen-

sionality. It is common practice to arbitrarily choose the

number of principal directions to retain such that the vari-

ance in the discarded directions is negligible [4]. In con-

trast, we present a probabilistic ICA model that will allow

us to address issues of model order or equivalently the num-

ber of latent source signals contained in the data by esti-

mating independent components in lower-dimensional sub-

spaces spanned by the dominant eigenvectors of the covari-

ance matrix of the data.

The choice of the number of components to extract is a

problem of model order selection. Overestimating the di-

mensionality results in a large number of spurious compo-

nents due to underconstrained estimation and a factoriza-

tion that will overfit the data, harming later inference and

dramatically increasing computational costs. Underestima-

tion, however, will discard valuable information and result

in suboptimal signal extraction.

We show that the accuracy of probabilistic ICA estimates

is consistent over a wide range of subspace dimensions be-

yond a value that appears to represent the ‘intrinsic’ dimen-

sionality of the data and discuss different approaches to de-

termining this dimensionality from the data prior to ICA

decomposition.

Probilistic ICAIn the probabilistic ICA model, the -dimensional data vec-

tors are modelled as a mixture of statistically in-

dependent latent sources which are linearly mixed by

and corrupted by additive noise such that

(1)

where denotes -dimensional random noise with zero

mean and unit variance.

In the presence of noise, the covariance matrix of the obser-

vations will be the sum of and the noise covariance [2]

i.e. will be of full rank and its rank will no longer equal

the rank of .

Determining a cutoff value for the eigenvalues of an initial

PCA using simplistic criteria like the reconstruction error

or predictive likelihood [4] that do not incorporate a noise

model will naturally predict that the accuracy steadily in-

creases with increased dimensionality. Many other infor-

mal methods have been proposed, the most popular choice

being the ”scree plot” where one looks for a ”knee” in the

plot of ordered eigenvalues that signifies a split between

significant and unimportant directions of the data.

This naturally raises the issues of sensitivity of ICA to the

subspace dimensionality and possible techniques to deter-

mine the optimal subspace dimensionality from the data.

Estimation Accuracy vs DimensionalityWe generated artificial FMRI data based on autoregres-

sive noise (where the AR coefficients were extracted from

real ‘null’ data ) and in vivo ‘null’ data plus additive sig-

nal as outlined in [1]. The 180-dimensional data was de-

composed using ICA after initially projecting the data into

lower dimensional subspaces (3-180 dimensions) by pro-

jection onto the dominant eigenvectors.

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false pos. rate

correlation

Figure 1: Spatio-temporal accuracy of ICA esti-

mates as a function of subspace dimensionality

After convergence, we calculated (i) the correlation be-

tween the best extracted and the ‘true’ activation time

courses (temporal accuracy) and (ii) the false positive / false

negative rates between thresholded IC maps and clusters of

voxels that have been used to generate the artificial data

(spatial accuracy) [1].

Figure 1 shows an example for data with two types of artifi-

cial activation introduced in a 10on/10off and 15on/15off

block design. These results suggest that the quality of

the estimates does not significantly improve (and actually

reduces in the case of temporal correlation) if more than

dimensions are retained.

Eigenspectrum AnalysisUnder the assumption of Gaussian noise, the sample co-

variance matrix has a Wishart distribution and we can

utilise results from random matrix theory [3] on the em-

pirical distribution function for the eigenvalues of

the covariance matrix of a random -dimensional ma-

trix . Suppose that as , then

almost surely, where the limiting distribu-

tion has a density

and where .

This can be used to obatin a modification to the scree-plot

where one compares the eigenspectrum of the observations

against the quantiles of the predicted cumulative distribu-

tion , i.e. against the expected eigenspectrum of a

randomGaussian matrix. Given the probabilistic model, we

project the data into a ‘signal’ subspace spanned by those

eigenvectors that violate the ‘null’ hypothesis of random

Gaussian signal in the data. Figure 2 shows an example

of the eigenspectrum for different artificial data sets and

the predicted eigenspectrum of a random Gaussian matrix.

Note, that the increase in AR dependency will render the

data to have more Eigenvectors that cannot be explained as

resulting from a random Gaussian distribution. Note also,

that artificial data based on true ‘null’ data differs signifi-

cantly from AR 16 noise.

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Figure 2: Eigenvalues (blue) and predicted distri-

bution (red) for different artificial FMRI data sets

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Figure 3: Bayesian estimates of the intrin-

sic dimensionality: Laplace approximation

due to Minka [5](blue) and Raftery [6](green)

and Bayesian score function of Rajan &

Rayner [7](red).

Bayesian AnalysisAs an alternative to methods based on the expected eigen-

spectrum, the problem of model order can also be ap-

proched within the framework of Bayesian model selection.

Minka [5] develops a simple criterion based on the Laplace

approximation of the Bayesian evidence for the model or-

der and gives a review of other existing techniques, includ-

ing the related technique of Raftery [6] to calculate approx-

imate Bayes factors and the selection criterion of Rajan and

Rayner [7] derived for the probabilistic model (1) with as-

sumed Gaussian noise and unit variance of the sources. Fig-

ure 3 shows the normalized score function for all three tech-

niques. For simple AR noise, all estimators essentially pre-

dict similar dimensionality close to the results in figure 2.

For ‘real’ data, techniques based on Laplace approximation

of Bayesian evidence are similar to the modified scree plot

and predict a dimensionality of 92. The selection crite-

rion due to Rajan& Rayner, however, predicts a much lower

dimensionality (33) that matches the results from the analy-

sis of spatio-temporal accuracy of IC estimates. This differ-

ence appears to be due to the different model assumptions

for the source variances.

DiscussionProjecting the data into lower-dimensional signal subspaces

appears to not interfere with the accuracy to estimate pat-

terns of activation, provided a sufficient number of com-

ponents are extracted. The optimal dimensionality is hard

to estimate exactly and probabilistic models vary depend-

ing on the specific noise model. In all cases, however, the

amount of dimensionality reduction proposed well exceeds

the standard level commonly used for ICA. A high level

of reduction does not only significantly reduce computa-

tional demand, but also allows for an estimate of the noise-

subspace which can be used to infer the validity of IC esti-

mates in the signal subspace.

AcknowledgementsThe authors gratefully acknowledge funding from the UK

Medical Research Council.

References

[1] C.F. Beckmann, J.A. Noble, and S.M. Smith. Investigating the intrinsic dimen-

sionality of FMRI data for ICA. In Seventh Int. Conf. on Functional Mapping of

the Human Brain, 2001.

[2] R. Everson and S.J. Roberts. Inferring the eigenvalues of covariance matrices from

limited, noisy data. IEEE Trans. Signal Processing, 1998. to appear.

[3] I.M. Johnstone. On the distribution of the largest principal component. Technical

report, Department of Statistics, Stanford University, 2000.

[4] M. J. McKeown, S. Makeig, G. G. Brown, T. P. Jung, S. S. Kindermann, A. J. Bell,

and T. J. Sejnowski. Analysis of fMRI data by blind separation into independent

spatial components. Human Brain Mapping, 6(3):160–88, 1998.

[5] T. Minka. Automatic choice of dimensionality for PCA. Technical Report 514,

M.I.T. Media Laboratory – Perceptual Computing Section, 2000.

[6] A.E. Raftery. Approximate bayes factors and accounting for model uncertainty

in generalized linear models. Technical Report 255, Department of Statistics,

University of Washington, 1994.

[7] J. Rajan and P.J.W Rayner. Model order selection for the singular value decompo-

sition and the discrete Karhunen-Lo’eve transform using Bayesian approach. IEE

Vision, Image and Signal Processing, 144:123–166, 1997.

Address for correspondence: Oxford Centre for Functional Magnetic Resonance Imaging of the Brain (FMRIB), Department of Clinical Neurology, University of Oxford, John Radcliffe Hospital, Headington, Oxford OX3 9DU, UK

Probabilistic ICA

GLM analysis standard ICA (unconstrained)

Probabilistic ICA

GLM analysis probabilistic ICA

Probabilistic ICA

• designed to address the ‘overfitting problem’:

• tries to avoid generation of ‘spurious’ results

• high spatial sensitivity and specificity

GLM analysis probabilistic ICA

Applications

EDA techniques can be useful to

‣ investigate the BOLD response• estimate artefacts in the data • find areas of ‘activation’ which respond in a non-

standard way

• analyse data for which no model of the BOLD response is available

Investigate BOLD response

estimated signal time

course

standard hrf model

Applications


• investigate the BOLD response‣ estimate artefacts in the data • find areas of ‘activation’ which respond in a non-

standard way

• analyse data for which no model of the BOLD response is available

slice drop-outs

gradient instability

EPI ghost

high-frequency noise

head motion

field inhomogeneity

eye-related artefacts

eye-related artefacts

Wrap around

Structured Noise and the GLM

• ‘structured noise’ appears:• in the GLM residuals and inflate variance estimates

(more false negatives)

• in the parameter estimates (more false positives and/or false negatives)

• In either case lead to suboptimal estimates and wrong inference!

Structured noise and GLM Z-stats bias

• Correlations of the noise time courses with ‘typical’ FMRI regressors can cause a shift in the histogram of the Z-statistics

• Thresholded maps will have wrong false-positive rate

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Denoising FMRI

• Example: left vs right hand finger tapping

before denoising

after denoisingJohansen-Berg et al.PNAS 2002

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Denoising FMRI


before denoising


LEFT - RIGHT contrast

Denoising FMRI


before denoising


Apparent variability

McGonigle et al.: 33 Sessions under motor paradigm

‘de-noising’ data by regressing out noise:reduced ‘apparent’ session variability

Applications


• investigate the BOLD response• estimate artefacts in the data • find areas of ‘activation’ which respond in a non-

standard way

‣ analyse data for which no model of the BOLD response is available

PICA on resting data

• perform ICA on null data and compare spatial maps between subjects/scans

• ICA maps depict spatially localised and temporally coherent signal changes

Example: ICA maps - 1 subject at 3

different sessions

Spatial characteristics

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(e) x=-4 y=-29 z=33

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(f) x=5 y=6 z=27

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(g) x=45 y=-42 z=47

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Medial visual cortex Lateral Visual Cortex

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Auditory system Sensori-motor system

Spatial characteristics

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Visuospatial system Executive control

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Visual Stream

Temporal deconvolution of ICA timecourses

• ‘What are the “task-related” components?• Use explicit time series model on the IC-

generated temporal modes (using BDS)(a)

(b) (c)

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Fig. 6. Results on the estimated time course for IC1 (using PICA approach), (a):thresholded (at p > 0.5) spatial IC map. (b): Estimated mean of the posteriordistribution of the neuronal response, the shaded regions represent the error bars.(c): Sample of the neuronal response using the estimated mean and covariancematrix of its posterior distribution. (d): Estimated mean of the Gaussian posteriordistribution of the HRF, the shaded regions represent the error bars. (e): VB-BDS fitto fMRI data (using the mean of the variable distributions), the time-series relativeto IC1 (dashed line) and VB-BDS model fit (continuous line).

B Model priors

First we choose for the parameters; a, b and d; zero mean Gaussian priorswith precisions equal to φa, φb and φd, respectively.

The parameter h is a non-parametric unknown function, and is given a zero

13

Example: BDS/ICA integration

• Estimate the mean of the posterior distribution over the neuronal response

(a)

(b) (c)

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• Estimate the mean of the Gaussian posterior distribution of the HRF

(a)

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• Assess the total model fit

(a)

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• posterior probability of neuronal response > 0 for the IC

Data

= + Noise

Predicted response

Estimated effect

Integrating GLM and ICAfor FMRI analysis

GLM

βi

Data

= + Noise

Predicted response

Estimated effect


GLM

βi

+ clear conclusions on a particular question

- results depend on the model

PICA

time

=

time

# maps

+ NoiseScan #k

FMRI data

space

time

space

spatial maps

# maps

Data Estimated spatially independent mapsEstimated artefact time courses


PICA

time

=

time

# maps

+ Noise

+ data driven and multivariate approach

- no knowledge about the fMRI paradigm is used

- can be hard to interpret activation results

Scan #k

FMRI data

space

time

space

spatial maps

# maps



GLM + ICA

space

# maps=

Scan #k

FMRI data

spatial maps

time

# mapsspace

+ Noise


time

Predicted response


GLM + ICA

space

# maps=

Scan #k

FMRI data

spatial maps

time

# mapsspace

+ Noise


time

• Stimulus model-based hypothesis testing • Adaptive model-free artefact modelling

- e.g. stimulus correlated motion, physiological noise, networks of spontaneous neuronal activity

Predicted response


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0

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10

auditory stimulusvisual stimulus

Time (s)

Modelled ICs

Example “model free” IC

Estimated HRFs

Time courseSpatial map

Time courseSpatial map

space

# maps

spatial maps

time

# maps

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