Regression Spline Mixed Models for Analyzing EEG Data and Event-Related PotentialsKaren E. Nielsen and Rich Gonzalez

Department of Statistics, University of Michigan, Ann Arbor, MI

EEG BackgroundElectroencephalography (EEG) is the measurement of electrical activity of the brain viaelectrodes placed on the scalp. Neuronal action potentials within the brain result in electricalpatterns which are detected by the scalp electrodes.

Figure 1 : The EEG cap sends data to a computer to be viewed in real time or analyzed later.

Since these electrodes are placed on the scalp and not the brain directly, and because theelectrical potential of a single neuron is too small to capture individually, EEGs are noisyrepresentations of brain activity. This presents a unique set of challenges to researchers.Statisticians are accustomed to working with noisy data, and thus possess skills that mayprovide valuable new insights into neuronal data.

Event-Related PotentialsThis work focuses on Event-Related Potentials (ERPs). AnERP is a brain response to a timelocked stimulus and is measuredusing EEG. ERPs are generallyrecorded in the first second afterstimulus presentation, and anERP waveform consists of one ora few components. Some of thesecomponents are well known, andvariance in amplitude within eachcomponent is important duringanalysis. For example, a P300 isa positive deflection of voltagethat occurs at approximately 300milliseconds after a novel stimu-lus is presented. Historically, thiswaveform was inspected duringpolygraph tests for lie detection.

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Figure 2 : Sample ERP data with known components

Notice here that we have broken from convention - in most EEG literature, negative voltages areplotted upwards on the y axis. We are plotting positive values upwards throughout this work.

Mixed-EffectsModelsMixed-effects models (also known as multilevel models; varying-intercept, varying-slopemodels; or hierarchical models) take into account the variation between groups. These modelsinclude fixed effects which do not change across groups, and random effects which do. Moreformally, we can write:

Yij = XTijB + ZT

ijβi + ei (1)

for the outcome Yij of individuals i and groups j where Xij is a design matrix for fixed effects, Zijis a design matrix for random effects, the coefficients βi ∼ N(0, G), and errors ei ∼ N(0, Ri).

In the context of ERP research, we can consider the channels to be groups within an individual’swaveform (here), or each individual to represent a group of trials contributing to apopulation-level waveform, or we can create a hierarchical model that includes both. Any ofthese models can include additional variables, such as age, gender, or disease status.

Regression Splines in ContextRegression splines are one option for fitting EEG and ERP waveforms. Landmark points, suchas local maxima and minima, make natural choices for knots. Smoothing splines are also anoption, but resulting waveforms vary substantially based on choice of tuning parameter.Ideally, a functional form could be defined so that coefficients have meaningful interpretationsto researchers.

A natural cubic spline g(t) for tmin 6 t 6 tmax with r knots τ = (τ1, ..., τr) ′ such thattmin < τ1 < ... < τr < tmax satisfies:I g(t) is a piecewise cubic polynomial of degree 6 3 on each [tmin, τ1], ..., [τr, tmax].I g(t), g ′(t), and g ′′(t) are continuous on [tmin, tmax].I g(d)(tmin) = g(d)(tmax) = 0 for d = r + 1, ..., 2r. That is, g(t) is linear beyond the knots.

A regression spline model can be written:

Qi =r∑

k=1

ciksk(ti) + ei (2)

where sk(x) are basis functions and ci = (ci1, ..., cir)′ are unknown coefficients.

Here, we find basis functions for each observation (one channel for one trial) and allow these tohave both fixed and random contributions in the model. Thus, Qi carries a (suppressed) labelfor its channel, j, and is part of both Xij and Zij in equation 1 (along with any other variables ofinterest).

ExampleTo demonstrate the use of Regression Spline Mixed Models (RSMMs) in the context of ERPs, wefit a model to a single individual’s 8-channel EEG over a 1400-millisecond window using thelme4 package in R, which finds Restricted Maximum Likelihood (REML) estimates of thecoefficients for the spline basis functions.

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Fixed Effects Spline Curve for Channels 3−10 For 1 Individual

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Figure 3 : Individual’s fixed effect curve fit to multi-channel data

Below, curves representing the random effects for each channel are shown along with the fixedeffect curve. We can see that, while the fixed effect curve is a representative summary of theoverall data, each channel has its own features.

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Figure 4 : Individual’s random (colors) and fixed (black) effect curves fit to multi-channel data

Example ContinuedAs shown in Figure 4, each channel has its own features. Since regression splines place aparticular set of assumption on the waveform, it is important to assess the fit of curves atseveral levels in the model hierarchy. Below, data from each channel is shown with the overallfixed effect and individual random effect curves.

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Figure 5 : Observations from each channel with curves for overall fixed (solid black) and random (dashed) effects.

In addition to a visibly representative fit to the subject’s EEG, this modeling approach allows usto calculate standard errors, and thus confidence bands, on the waveform at various levels.

FutureWork

I Establish a standardized processing stream for ERP data.

I Use random effects for baselining EEG/ERP data.

I Develop a formal test for identifying “uncertain” responses in arrow flanker and similarexercises.These responses are currently categorized at “correct” or “incorrect” and removed by hand.

I Develop a set of functional forms or biologically relevant bases which can be fit to data withinterpretable and scientifically meaningful coefficients.

I Explore the scientific accuracy of a “random walk” analogy to the paths of electrical potentialsin the brain.

References

I Bates, D., Machler, M., Bolker, B., & Walker, S. Fitting Linear Mixed-Effects Models using lme4.(2014), 151.

I Edwards, L. J., Stewart, P. W., MacDougall, J. E., & Helms, R. W. A method for fitting regressionsplines with varying polynomial order in the linear mixed model. Statistics in Medicine,25(2005), 513-527.

I Fitzmaurice, G. Longitudinal Data Analysis. Chapman & Hall, 2009.

I Gelman, A. & Hill, J. Data Analysis Using Regression and Multilevel/Hierarchical Models.Cambridge University Press, 2007.

I Luck, S. An Introduction to the Event-Related Potential Technique. The MIT Press, 2005.

I Nurnberger, G. Approximation by Spline Functions. Springer, 1989.

I Wand, M. P. A Comparison of Regression Spline Smoothing Procedures. ComputationalStatistics. 15(2000), 443-462.

contact: karenen@umich.edu