init_unfold

# Introduction

Subjects are exploring a face stimulus. Using an eyetracker and the EEG-Eye toolbox we add markers of saccade onsets into our EEG-Data. The data were cleaned for eye-movement related artefacts using ICA.

In this tutorial we try to disentangle stimulus related ERPs and (micro)saccade related ERPs.

EEG = pop_loadset('C:\Users\behinger\Dropbox\deconv_olaf\workshop\data\facestudy_integrated.set')

# Defining the design

cfgDesign = [];

cfgDesign.eventtypes = {{'saccade'},{'S121','S122','S123'}}; % we model the saccadeonset and the stimulus

cfgDesign.formula = {'y ~ 1 + spl(sac_amplitude,10)','y~1'}; % 10 splines are generated; One needs to be careful to not overfit the data, regularization or cross validation can help here.

EEG = uf_designmat(EEG,cfgDesign);

The term spl(sac_amplitude,10) creates 10 splines over our predictor.

We can visualise the set of splines using:

spl = EEG.unfold.splines{1};

subplot(2,1,1),histogram(spl.paramValues,100),title('Histogram of saccadic amplitudes')

splineSet = spl.splinefunction(linspace(0,5,500),spl.knots);

subplot(2,1,2),plot(linspace(0,5,500),splineSet),xlabel('saccadic amplitude')

The default setting is to put the splines at points of the quantiles of the variable. This ensures that in regions with a lot of data, lots of splines are generated and in regions with sparse data, broad splines are used. One could also put the splines in a linear or logarithmical way or define the knotsequence manually.

To understand better what basis function do, it might be helpful to have a look at the sorted designmatrix:

uf_plotDesignmat(EEG,'sort',1)

The sorted designmatrix shows overlapping predictors, these are the splines for saccade amplitude. There is one predictor removed (between sac_amp 1.5 & 2.9). This is because we have an intercept in the model and else the model would be overcomplete. Not that the empty events (all grey rows) are events that are not modeled.

cfgTimeexpand = [];

cfgTimeexpand.timelimits = [-.3,.6];

EEG = uf_timeexpandDesignmat(EEG,cfgTimeexpand);

EEG = uf_continuousArtifactExclude(EEG,struct('winrej',winrej)); % we find data using a simple threshold tool. More complex algorithms or manual cleaning is recommended!

EEG= uf_glmfit(EEG);

# Plot the results

Currently the spline-predictor is still separated in intercept + 9 spline-betas. The relationship for e.g. the P100 can be visualized. We start with a simple investigation of the raw betas, without undoing the spline-basisset

ufresult= uf_condense(EEG);

sac_amp_ix = cellfun(@(x)~isempty(x),(strfind({ufresult.param.name},'sac_amplitude'))); % get all betas related to sac_amplitude

timeix = get_min(0.115,ufresult.times); % somewhere around the p100

splinevalue = [ufresult.param(sac_amp_ix).value];

figure;

plot(splinevalue,squeeze(ufresult.beta(1,timeix,sac_amp_ix)),'-o'),xlabel('saccade amplitude')

This plot is still lacking the intercept (a DC-offset) and of course, these betas represent only weights to the above plotted set of basisfunctions.

In order to get a better picture, let's multiply the betas with the basis function to get an estimate in the predictor-value-domain.

ufresult= uf_predictContinuous(ufresult,'predictAt',{{'sac_amplitude',linspace(0.3,5,100)}});

sac_amp_ix = cellfun(@(x)~isempty(x),(strfind({ufresult.param.name},'sac_amplitude'))); % get all betas related to sac_amplitude

y = ufresult.beta(1,timeix,sac_amp_ix);

% we could add the intercept; it is a constant offset

%y = y + ufresult.beta(1,timeix,1);

plot([ufresult.param(sac_amp_ix).value],squeeze(y)),xlabel('saccade amplitude'),title(sprintf('Non-linear relationship between ERP and saccadic amplitude @ %.2f s',ufresult.times(timeix)))

It is clear from this plot, that there is a strongly non-linear relationship between saccade amplitude and ERP at around 100ms.

It's important to remember here, that above 3° saccade amplitude only very few saccades remain. It is therefore important to not overinterpret this downward trend!

To really drive home the point we add the weighted splines to the plot:

ufresult= uf_condense(EEG);

sacX =linspace(0.3,5,500); % at which points to evaluate?

splineSet = spl.splinefunction(sacX,spl.knots);

beta = squeeze(ufresult.beta(1,timeix,1:end-1)); % end-1 because of the stimulus event predictor

splineEvaluated = bsxfun(@times,splineSet,beta'); % weight the basis functions by the betas, but don't add them

plot(sacX,splineEvaluated); hold all;

plot(sacX,sum(splineEvaluated,2),'k','LineWidth',1.5), % add the basis functions at each saccade-amplitude to get the modelfit

xlabel('saccade amplitude'),title(sprintf('Non-linear relationship between ERP and saccadic amplitude @ %.2f s',ufresult.times(timeix)))

As one can also see, the modelfit is a bit rough at the edges. this is to be expected because there is not much data for very small and very large splines.

% The default parameters plot the predictor at its quantiles

ufpredict = uf_predictContinuous(ufresult); % evaluate the splines at the quantiles

uf_plotParam(ufpredict,'channel',1,'add_average',1);

Important notes to this plot:

- The lines represent saccade-amplitudes at the quantiles of the saccade amplitude distribution.
- There are outliers/bad estimates at the extremes. The yellow-curve has a very high noise level. This is because the last spline is only informed by a small amount of trials, it is usually highly unstable.

In order to get a more representative picture, we can utilize the 'predictAt' option in the plot and put a linear spacing excluding the extremata

ufpredict= uf_predictContinuous(ufresult,'predictAt',{{'sac_amplitude',linspace(0.5,3,10)}});

ufmarginal = uf_addmarginal(ufpredict);

ufmarginal.effect_only = ufpredict.beta; % we are adding the predictor without the added intercept to show the difference

uf_plotParam(ufmarginal,'channel',1,'plotParam','sac_amplitude');

In the top row, the output of uf_addmarginal shows what happens after we add the mean value of all other predictors (in this case only the intercept). The lower plot can be seen as 'difference' waves