Before we introduce a real dataset, let us first consider a simulated simple EEG/ERP study to illustrate the possibilities of the deconvolution approach. For this, let’s imagine a typical stimulus-discrimination task with two conditions (Fig. 1): Participants are shown pictures of faces or houses and asked to classify the type of stimulus with a button press. Because this response is speeded, motor activity related to the preparation and execution of the manual response will overlap with the activity elicited by stimulus onset. Furthermore, we also assume that the mean reaction time (RTs) differs between the conditions, as it is the case in most experiments. In our example, if face pictures are on average classified faster than houses pictures (Fig. 1C), then a different overlap between stimulus- and response-related potentials will be observed in the two conditions. Importantly, as Fig. 1F shows, this will result in spurious conditions effects due to the varying temporal overlap alone, which can be easily mistaken for genuine differences in the brain’s processing of houses and faces.

Human faces are also complex, high-dimensional stimuli with numerous properties that are difficult to perfectly control and orthogonalize in any given study. For simplicity, we assume here that the average luminance of the stimuli was not perfectly matched between conditions and is slightly, but systematically, higher for faces than houses (Fig. 1D). From previous studies, we know that the amplitude of the P1 visually-evoked potential increases as a non-linear (log) function of the luminance of the presented stimulus (Halliday, 1982), and thus we also simulate a logarithmic effect of luminance on the P1 of the stimulus-aligned ERP (Fig. 1E), which creates another spurious condition difference (Fig. 1G) in addition to that of varying response times (Fig. 1F).

Figures 1G and 1H show the same data modeled with unfold. Fortunately, with deconvolution modeling, we can not only remove the overlap effect (Fig. 1G), but by including luminance as a non-linear predictor, we simultaneously also control the influence of this covariate (Fig. 1H). How this is done is explained in more detail in the following.

Deconvolution methods for EEG have existed for some time (Hansen, 1983; Eysholdt & Schreiner, 1982), but most older deconvolution approaches show severe limitations in their applicability. They are either restricted to just two different events (Hansen, 1983; Zhang, 1998), require special stimulus sequences (Eysholdt & Schreiner, 1982; Marsh, 1992; Delgado & Ozdamar, 2004; Jewett et al., 2004; Wang et al., 2006), rely on semi-automatic, iterative methods like ADJAR (Woldorff, 1993) that can be slow or difficult to converge (Talsma & Woldorff, 2004; Kristensen, Rivet & Guérin-Dugué, 2017b), or were tailored for special applications. In particular, the specialized RIDE algorithm (Ouyang et al., 2011; Ouyang, Sommer & Zhou, 2015) offers a unique feature in that it is able to deconvolve time-jittered ERP components even in the absence of a designated event marker. However, while RIDE has been successfully used to separate stimulus- and response-related ERP components (Ouyang et al., 2011; Ouyang, Sommer & Zhou, 2015), it does not support continuous predictors and is intended for a small number of overlapping events.

In recent years, an alternative deconvolution method based on the linear model has been proposed and successfully applied to the overlap problem (Lütkenhöner, 2010; Dandekar et al., 2012a; Litvak et al., 2013; Spitzer, Blankenburg & Summerfield, 2016; Kristensen, Rivet & Guérin-Dugué, 2017a, 2017b; Sassenhagen, 2018; Cornelissen, Sassenhagen & Võ, 2019; Coco, Nuthmann & Dimigen, 2018; Bigdely-Shamlo et al., 2018; Dimigen & Ehinger, 2019). This deconvolution approach was first applied extensively to fMRI data (Dale & Buckner, 1997) where the slowly varying BOLD signal overlaps between subsequent events. However, in fMRI, the shape of the BOLD response is well-known and this prior knowledge allows the researcher to use model-based deconvolution. If no assumptions about the response shape (i.e., the kernel) are made, the approach used in fMRI is closely related to the basic linear deconvolution approach discussed below.

With deconvolution techniques, overlapping EEG activity is understood as the linear convolution of experimental event latencies with isolated neural responses (Fig. 1B). The inverse operation is deconvolution, which recovers the unknown isolated neural responses given only the measured (convolved) EEG and the latencies of the experimental events (Fig. 1H). Deconvolution is possible if the subsequent events in an experiment occur with varying temporal overlap, in a varying temporal sequence, or both. In classical experiments, stimulus-onset asynchronies and stimulus sequences can be varied experimentally and the latencies of motor actions (such as saccades or button presses) also vary naturally. This varying overlap allows for modeling of the unknown isolated responses, assuming that the overlapping signals add up linearly. More specifically, we assume (1) that the electrical fields generated by the brain sum linearly (a justified assumption, see Nunez & Srinivasan, 2006) and (2) that the overlap, or interval between events, does not influence the computations occurring in the brain—and therefore the underlying waveforms (see also Discussion).

The benefits of this approach are numerous: The experimental design is not restricted to special stimulus sequences, multiple regression allows modeling of an arbitrary number of different events, and the mathematical properties of the linear model are well understood.

The classic massive univariate linear model, without overlap correction, is applied to epoched EEG data and can be written as: μi,τ=Xiβτ with yi,τ ∼normal(μi,τ,στ)

Here, X is the design matrix. It has i rows (each describing one instance of an event of type e) and c columns (each describing the status of one predictor).

Furthermore, let τ be the “local time” relative to onset of the event (e.g., −100 to +500 sampling points). μ_i,τ is the expected (average) EEG signal measured after event i that we wish to predict at a given time point τ relative to the event onset. β is a vector of unknown parameters that we wish to estimate for each time point in the epoched EEG data. Importantly, therefore, this approach fits a separate linear model at each time point τ. A single entry in the design matrix will be referred by lowercase x_i,c.

In contrast, with linear deconvolution we enter the continuous EEG data into the model. We then make use of the knowledge that each observed sample of the continuous EEG can be described as the linear sum of (possibly) several overlapping event-related EEG responses. Depending on the latencies of the neighboring events, these overlapping responses occur at different times τ relative to the current event (see Fig. 2). That is, in the example in Fig. 2, where the responses of two types of events, A and B, overlap with each other, the observed continuous EEG at time point t of the continuous EEG recording can be described as follows: EEGt=25=1βA,1+0βA,2+0βA,3+0βA,4+1βA,5+0βB,1+0βB,2+0βB,3+1βB,4+0βB,5

In the example in Fig. 2, the spontaneous EEG at time-point t is modeled as the linear sum of a to-be-estimated response to the first instance of event type A at local time τ = 5 (i.e., from the point of view of EEG(t) this instance occurred 5 time samples before), another response to the second instance of event type A at local time τ = 0 (i.e., this instance just occurred at t), and another response to the instance of event type B at local time τ = 4.

The necessary design matrix to implement this model, X_dc, will span the duration of the entire EEG recording. It can be generated from any design matrix X by an algorithm we will call time expansion in the following. In this process, each predictor in the original design matrix will be expanded to several columns, which then code a number of “local” time points relative to the event onset. An example for a time-expanded design matrix is shown in Fig. 2.

The process to create the time-expanded design matrix X_dc is illustrated in Fig. 2. In the following sections, we will describe the construction of X_dc more formally.

Let t be the time of the continuous EEG signal y, which keeps increasing throughout the experiment. τ is still the local time, that is, the temporal distance of an EEG sample relative to an instance of event e. Let i be the instance of one such event. X_i is therefore the accompanying row of the design matrix X which specifies the predictors for each event of type e. The design matrix X consists of multiple columns c, each representing one predictor (for which we want to estimate the accompanying β).

X_dc can be constructed from multiple concatenated, time-expanded square diagonal matrices G with size τ one for each instance of the event e. For the purpose of illustration, it is helpful to construct the design matrix first for just a single predictor and a single instance of a single type of event (e.g., a manual response). Afterwards, we will add multiple predictors, then multiple instances of a single event type and finally multiple different event types (e.g., stimuli and responses).

Spline regression is a method to estimate non-linear relationships between a continuous predictor and an outcome. In the simple case of a single predictor it can be understood as a type of local smoother function. For a detailed introduction to spline regression we recommend Harrell (2015) or Wood (2017). We will now outline how we can use spline predictors to model non-linear effects within the linear regression framework. We follow the definition of a generalized additive model (GAM) by Wood (2017, pp. 249–250): μi=Xiβ+∑jzjfj(xi)

The sum ∑jzjfj(xi) represents a basis set with j unknown parameters z (analog to the β vector). The time-indices were omitted here. The most common example for such a basis set would be polynomial expansion. Using the polynomial basis set with order three would results in the following function: μi=Xiβ+z1x1+z2x2+z3x3

However, due to several suboptimalities of the polynomial basis (Runge, 1901) we will make use of the cubic B-spline basis set instead. Spline regression is conceptually related to the better-known polynomial expansion, but instead of using polynomials, one uses locally bounded functions. In other words, whereas a polynomial ranges over the whole range of the continuous predictor, a B-spline is restricted to a local range.

This basis set is constructed using the De-Casteljau algorithm (De Casteljau, 1959) implemented by Luong (2016). It is a basis set that is strictly local: Each basis function is non-zero only over a maximum of 5 other basis functions (for cubic splines; see Wood, 2017, p. 204).

Multiple terms can be concatenated, resulting in a GAM: μi=Xiβ+∑jzjfj(x1i)+∑jzjfj(x2i)

If interactions between two non-linear spline predictors are expected, we can also make use of two-dimensional splines: μi=Xiβ+∑jzjfj(x1i,x2i)

In the unfold toolbox, 2D-splines are created based on the pairwise products between z_1,j and z_2,j. Thus, a 2D spline between two spline-predictors with 10 spline-functions each would result in 100 parameters to be estimated.

The number of basis functions to use for each non-linear spline term is usually determined by either regularization or cross-validation. Cross-validation is difficult in the case of large EEG datasets (see also Discussion) and we therefore recommend to the number of splines (and thus the flexibility of the non-linear predictors) prior to the analysis.

In the previous section, we made use of time-expansion to model the overlap. For this, we used the so-called stick function approach (also referred to as FIR or dummy-coding). That is, each time-point relative to an event (local time) was coded using a 1 or 0 (in the case of dummy-coded variables), resulting in staircase patterns in the design matrix (cf. Figs. 2 and 3A). However, this approach is computationally expensive. Due to the high sampling rate of EEG (typically 200–1,000 Hz), already a single second of estimated ERP response requires us to estimate 200–1,000 coefficients per predictor. Therefore, some groups started to use other time basis sets to effectively smooth the estimated rERPs (Litvak et al., 2013; but see Smith & Kutas, 2015b).

We will discuss two examples here: The time-Fourier set (Litvak et al., 2013; Spitzer, Blankenburg & Summerfield, 2016) and—newly introduced in this paper—the time-spline set. In the time-spline set, adjacent local time coefficients are effectively combined according to a spline set (Fig. 3B). Splines are a suitable basis function because EEG signals are smooth and values close in time have similar values. The number of splines chosen here defines the amount of smoothing of the resulting deconvolved ERP. The same principle holds for the time-Fourier set. Here, we replace the stick-function set with a truncated Fourier set (Fig. 3C). Truncating the Fourier set at high frequencies effectively removes high frequencies from the modeled ERP and can therefore be thought of as a low-pass filter. A benefit of using (truncated) temporal basis functions rather than the simple stick functions is that fewer unknown parameters need to be estimated, at the cost of temporal precision. It is therefore possible that this results in numerically more stable solutions to the linear problem, because we are constraining the solution space. Low-pass filtering and downsampling of the data followed by using the stick function deconvolution might result in better control of the spectral properties.

To our knowledge, no existing toolbox supports non-linear, spline-based general additive modeling of EEG activity. Also, we were missing a toolbox that solely focuses on deconvolving signals and allowed for a simple specification of the model to be fitted, for example, using the commonly used Wilkinson notation (as also used, e.g., in R).

A few other existing EEG toolboxes allow for deconvolution analyses, but we found that each has their limitations. Plain linear modeling (including second-level analyses) can be performed using the LIMO toolbox (Pernet et al., 2011), but this toolbox does not support deconvolution analyses or spline regression. To our knowledge, five toolboxes support some form of deconvolution: SPM (Litvak et al., 2013; Penny et al., 2006), the rERP extension for EEGLAB (Burns et al., 2013), pyrERP (Smith, 2013), mTRF (Crosse et al., 2016), and MNE (Gramfort et al., 2014). SPM allows for deconvolution of linear responses using Fourier temporal basis sets. However, in order to make use of these deconvolution functions, quite a bit of manual coding is needed. The rERP extension for EEGLAB and the pyrERP toolbox for Python both allow for estimation of linear models and deconvolution; however, both toolboxes appear not to be maintained anymore; rERP is currently non-functional (for current MATLAB versions) and no documentation is available for pyERP. The MNE toolbox is a general-purpose Python-based EEG processing toolbox that supports both deconvolution and massive univariate modeling. It is actively maintained and some basic tutorials are available. The mTRF toolbox is a special type of deconvolution toolbox designed to be used with continuous predictors (e.g., auditory streams) that last over the whole continuous EEG recording (see Discussion).

As a start, we need a data structure in EEGLAB format (Delorme & Makeig, 2004) that contains the continuous EEG data and event codes. In traditional EEG experiments, events will typically be stimulus and response triggers, but many other types of events are also possible (e.g., voice onsets, the on- or offsets of eye or body movements etc.). In most cases, the EEG data entered into the model should have already been corrected for biological and technical artifacts (e.g., ocular artifacts, scalp muscle EMG, or power line noise), for example, with independent component analysis (ICA).

We begin the modeling process by specifying the model formula and by generating the corresponding design matrix X. In the unfold toolbox, models are specified using the intuitive Wilkinson notation (Wilkinson & Rogers, 1973) also commonly used in R, the Matlab statistics toolbox, and python StatsModels. For example, for the hypothetical face/house experiment depicted in Fig. 1, we might define the following model: EEG ∼ 1 + cat(is_face) + luminance

More generally, we can also specify more complex formulas, such as: EEG ∼ 1 + cat(predictor1)+ predictor2 + spl(predictor3,5)

Here, cat() specifies that the predictor1 should be dummy encoded as a categorical variable or factor rather than treated as a continuous variable. If a variable is already dummy-coded as 0/1 it is not strictly necessary to add the cat() command, but it would be necessary to specify multi-level categorical variables (3 levels or more). In contrast, predictor2 should be modeled as continuous linear covariate and predictor3 as a non-linear spline predictor. In the formula, a plus sign (+) means that only the main effects will be modeled. Interactions between predictors are added by replacing the + with a * or a :, depending on whether all main effects and interactions should be modeled (*), or only the interactions (:). In unfold, the type of coding (dummy/treatment/reference or effect/contrast/sum coding) can be selected. If the default treatment coding is used, the predictors will represent the difference to the intercept term (coded by the 1). The reference level of the categorical variable and the ordering of the levels is determined alphabetically or can be specified by the user manually. The toolbox also allows to specify different formulas for different events. For example, stimulus onset events can have a different (e.g., more complex) formula than manual response events.

Once the formula is defined, the design matrix X is time-expanded to X_dc and now spans the duration of the entire EEG recording. Subsequently for each channel, the equation (EEG = X_dc * b + e) is solved for “b”, the betas, which correspond to subject-level rERP waveforms. For example, in the model above, for which we used treatment coding, the intercept term would correspond to the group-average ERP. The other betas, such as those for cat(is_face), will capture the partial effect of that particular predictor, corresponding to a difference wave in traditional ERPs (here: face-ERP minus house-ERP).

In the same linear model, we can simultaneously model brain responses evoked by other experimental events, such as button presses. Each of these other event types can be modeled by its own formula. In our face/house example, we would want to model the response-related ERP that is elicited by the button press at the end of the trial, because this ERP will otherwise overlap to a varying degree with the stimulus-ERP. We do this by defining an additional simple intercept model for all button press events. In this way, the ERP evoked by button presses will be removed from the estimation of the stimulus ERPs. The complete model would then be: EEGfix∼1+cat(is_face)+spl(luminance,5) {for stimulus onset events}EEGbutton∼1 {for manual button press events}

As explained earlier, many influences on the EEG are not strictly linear. In addition to linear terms, one can therefore use cubic B-splines to perform spline regression, an approach commonly summarized under the name GAM. An illustration of this approach is provided in Fig. 5. In the unfold toolbox, spline regression can be performed by adding spl() around the predictor name, as for predictor3 in the formula above, which specifies a model using 5 B-splines instead of a continuous linear predictor. We can model covariates as non-linear predictors: EEG∼1+spl (A, 5)+2dspl (B, C, 5)+circspl (D, 5, 0, 360)

With this formula, the effect “A” would be modeled by a basis set consisting of five splines. We would also fit a 2D spline between continuous variable B and C with five splines each. In addition, we would fit a circular spline based on covariate D using five splines with the limits 0° and 360° being the wrapping point of the circular spline.

In unfold, three spline functions are already implemented. For B-splines we use the de Casteljau algorithm implemented by Bruno Luong. For interactions between spline-modeled covariates, first the default spline function is used on each predictor to generate n splines. Then the resulting vectors are elementwise multiplied with each other, generating n² final predictors. For cyclical predictors such as the angle of a saccadic eye movement (which ranges, e.g., from 0 to 2π), it is possibly to use cyclical B-splines, as explained above. These are implemented based on code from patsy, a python statistical package (https://patsy.readthedocs.io) which follows an algorithm described in Wood (2017, pp. 201–205). For maximal flexibility, we also allow the user to define custom spline functions. This would also allow to implement other basis sets, for example, polynomial expansion.

In our default B-spline implementation, the so-called knots define the peaks of the cubic splines, except for the boundaries (i.e., extremes) of the covariate (see Fig. 5B). The resulting number of splines are given by the number of knots minus 4 (4 due to the cubic nature, see Wood, 2017, p. 204), but the user only has to specify the number of basis functions, not the exact number of knots. The placement of knots (and therefore the number of splines) are critical parameters to appropriately model the predictor and to avoid over- or underfitting the data. The toolbox’s default knot placement is on the quantiles of the predictor, which will increase resolution of the splines in areas where there are a lot of data points and offers stronger smoothing in other areas, where predictor values are sparser (similar to Harrell, 2015, p. 26). This can be changed by users who want to use a custom sequence of knots. Generalized cross-validation or penalized regression could be used to narrow down the number of knots to be used but is computational expensive and currently not supported in unfold.

Temporal basis functions were introduced earlier. The stick-function approach, as also illustrated in Figs. 2 and 3A, is the default option in unfold. As alternatives, it is also possible to employ either a Fourier basis set or a set of temporal spline function. For example, for the time-expansion step, Litvak et al. (2013; Spitzer, Blankenburg & Summerfield, 2016) used a Fourier basis sets instead of stick-functions. Figure 6 compares simulation results for stick functions with those obtained with a Fourier basis set and a spline basis set in terms of the spectral components and the resulting filter artifacts. At this point, more simulation studies are needed to understand the effects of temporal basis sets on EEG data. We therefore follow the recommendation of Smith & Kutas (2015b) to use stick-functions for now.

If a predictor has a missing value in massive univariate regression models, it is typically necessary to remove the whole trial. One workaround for this practical problem is to impute (i.e., interpolate) the value of missing predictors. In the deconvolution case, imputation is even more important for a reliable model fit, because if a whole event is removed, then overlapping activity from this event with that of the neighboring events would not be accounted for. In unfold we therefore offer several algorithms to treat missing values: the dropping of events with missing information, or imputation by the marginal, mean, or median values of the other events.

Linear deconvolution needs to be performed on continuous, rather than epoched data. This creates challenges with regard to the treatment of intervals that contain EEG artifacts. The way to handle artifacts in a linear deconvolution model is therefore to detect—but not to remove—the intervals containing artifacts in the continuous dataset. For these contaminated intervals, the time-expanded design matrix (X_dc) is then blanked out, that is, filled with zeros, so that the artifacts do not affect the model estimation (Smith & Kutas, 2015b). If the data would have been cleaned prior to the time-expansion step, then events might have been removed that would overlap with clean data segments.

Of course, this requires the researcher to use methods for artifact correction that can be applied to continuous rather than segmented data (such as ICA). Similarly, we need methods that can detect residual artifacts in the continuous rather than epoched EEG. One example would be a peak-to-peak voltage threshold that is applied within a moving time window (shifted step-by-step across the entire recording). Whenever the peak-to-peak voltage within the window exceeds a given threshold, the corresponding interval would then be blanked out in the design matrix. Detecting artifacts in the continuous rather than segmented EEG also has some small additional benefit, because if the data of a trial is only partially contaminated, the clean parts can still enter the model estimation (Smith & Kutas, 2015b).

The unfold toolbox includes a function to remove artifactual intervals from the design matrix before fitting the model. In addition, we offer basic functionality, adapted from ERPLAB (Lopez-Calderon & Luck, 2014), to detect artifacts in the continuous EEG.

Solving for the betas is not always easy in such a large model. We offer several algorithms to solve for the parameters. The currently recommended one is LSMR (Fong & Saunders, 2011), an iterative algorithm for sparse least-squares problems. This algorithm allows to use very large design matrices as long as they are sparse (i.e., contain mostly zeroes) which is usually the case if one uses time-expansion based on stick-functions (cf. Figs. 2 and 3A).

However, especially with data containing a high level of noise, the tradeoff between bias and variance (i.e., between under- and overfitting) might be suboptimal, meaning that the parameters estimated from the data might be only weakly predictive for held-out data (i.e., they show a high variance and tend to overfit the data). Regularization is one way to prevent overfitting of parameter estimates. In short, regularization introduces a penalty term for high beta values, effectively finding a trade-off between overfit and out-of-sample prediction. The unfold toolbox allows the user to specify alternative solvers that use regularization. In particular, we include the glmnet-solver (Qian et al., 2013), which allows for ridge (L2-norm), lasso (L1, leads to sparse solutions) and elastic net regularization. The regularization parameter is automatically estimated using cross-validation but the elastic-net parameter (deciding between L1 and L2 norm) has to be specified manually. Procedures to regularize with linear deconvolution have recently been examined and validated by Kristensen, Rivet & Guérin-Dugué (2017a). Effects of regularization on noisy data are also depicted in Fig. 7, which compares deconvolution results for noisy simulated data with and without regularization. In this simulation we used strongly correlated predictors (r = 0.85), thereby increasing collinearity. Note that collinearity in itself is only a problem for very extreme cases; specifically, if the matrix becomes ill-conditioned, the parameter solution might be instable and the estimands will “explode.” Regularization can help in this situation. As can be seen in Fig. 7, the non-regularized estimates show strong variance (Figs. 7B and 7C), whereas the regularized estimates show strong bias (Figs. 7D and 7E), that is, the estimated effects are shrunk towards zero but, simultaneously, the variance of the estimate over time is greatly reduced. At this point, it is not yet clear whether and what type of regularization should be used for the standard analysis of EEG data, but we provide different solvers in unfold to facilitate future work on this topic. Please also see Kristensen, Rivet & Guérin-Dugué (2017a) for more simulation work.

Many researchers use source reconstruction (e.g., MNE, LORETA) or blind source separation methods (e.g., ICA) to try to isolate the signal contributions of individual neural sources. In our framework, this can be understood as performing a spatial deconvolution of the signal that addresses the problem of volume conduction. Nevertheless, the activity time courses of each neural source may still overlap in time, for example, due to repeated stimulus presentations. To apply the deconvolution, it does not matter whether the input time series consist of raw EEG signals, or whether they are the result of spatial filtering (e.g., beamformer: Van Veen et al., 1997), blind source separation (Makeig et al., 1996; Delorme et al., 2012) or some other transformation (Cohen, 2017). To use our toolbox with other types of time series, the data has to be simply copied into the EEG.data matrix in EEGLAB. For convenience, we also offer a flag (“ica,” “true”), which allows the researcher to directly model ICA activations (stored in EEG.icaact), instead of the raw EEG. While we can only speculate about this issue at this point, it seem likely that a prior spatial decomposition of the data improves the performance and interpretability of the final, spatially and temporally deconvolved signals (Burwell et al., 2019).

The unfold toolbox offers the option to compute a mass univariate regression model on the same data using the exact same model but without correction for overlap. In our experience, running this model in addition to the linear deconvolution model can be helpful to understand the impact of overlap on the results. However, with this function, unfold can also be used as a standalone toolbox for Mass-Univariate modeling, for the (rare) cases in which an experiment does not involve any overlapping activity (e.g., from small saccades; Dimigen et al., 2009).

unfold offers multiple inbuilt functions to visualize rERP results (Fig. 8). We provide functions for marginal plots over splines and continuous variables, and functions to evaluate splines/continuous covariates at specific values. For the topographical output we make use of functions from the EEGVIS toolbox (Ehinger, 2018).

unfold focuses on two main things: linear deconvolution and (non)-linear modeling at the single-subject level. In contrast, the toolbox itself does not offer functions for group-level statistics. However, the betas for each participant can be easily exported as plain text (.csv) or as different MATLAB structures to apply statistics with other toolboxes. A tutorial to process unfold results using group-level permutation tests with the TFCE-toolbox (Mensen & Khatami, 2013) is provided in the online documentation.

Figure 9 shows a complete analysis script for the hypothetical face/house experiment introduced above (see Fig. 1). The complete analysis can be run with a few lines of code.

To create simulated data, we produced overlapped data using four different response shapes, shown in the first column of Fig. 10: (1) a boxcar function, (2) a Dirac delta function, (3) a simulated auditory ERP (the same as used by Lütkenhöner, 2010), and (4) random pink noise. We then simulated 5 s of continuous data, during which 18 experimental events happened. Intervals between subsequent events were randomly drawn from a normal distribution (M = 0.25 s, SD = 0.05 s). Convolving the simulated responses with the randomly generated event latencies produced the continuous overlapped signal depicted in the third column of Fig. 10. The last column of Fig. 10 shows the non-overlapped responses recovered by unfold (orange lines). For comparison, overlapped responses without deconvolution are plotted in dark red. As can be seen, unfold recovered the original response in all cases. The data of Fig. 1 were also simulated and then analyzed using our toolbox. Together, these simulations show that unfold successfully deconvolves heavily overlapping simulated signals.

Finally, we will also analyze a real dataset from a single participant who performed a standard ERP face discrimination experiment¹

¹The same example data was also analyzed in our accompanying paper (Dimigen & Ehinger, 2019), but with a different focus. Further details on this dataset are given in Dimigen et al. (2009) or Dimigen & Ehinger (2019).

Although a central fixation cross was presented prior to each trial and participants were instructed to avoid eye movements during the task, concurrent video-based eye-tracking revealed that participants executed at least one involuntary (micro)saccades during the vast majority of trials (see also Dimigen et al., 2009; Yuval-Greenberg et al., 2008). For the participant analyzed here, the median amplitude of these small saccades was 0.6° and most were aimed at the mouth region of the presented faces, which was most informative for the emotion discrimination task.

This means that our stimulus-locked ERPs are contaminated with two other processes: visually-evoked potentials (lambda waves) generated by the retinal image motion produced by the (micro)saccades (Gaarder et al., 1964; Dimigen et al., 2009) and motor processes related to preparing and executing the finger movement.

To disentangle these potentials with unfold, we specified three events: Stimulus onset, saccade onset, and button press. For this simple demonstration, we modeled both stimulus onsets and button press events using only an intercept term (y ∼ 1), that is, regardless of emotion. For the saccade onsets, we included both an intercept as well as saccade amplitude as a continuous predictor, because larger saccades are followed by larger lambda waves (Gaarder et al., 1964; Dimigen et al., 2009). Because this relationship is non-linear (Dandekar et al., 2012b) we used a set of 5 splines in the formula, y ∼ 1 + spl(saccade_amplitude,5). Brain responses were modeled in the time window from −1.5 to 1 s around each event. Before fitting the model, we removed all intervals from the design matrix in which the recorded activity at any channels differed by >250 μV within a window of 2 s.

Figure 11 presents the results for occipital electrode Oz and the signal both with (in red) and without (blue) the modeling and removal of overlapping activity. The large effect of overlapping activity can be clearly seen in the averaged ERP waveforms (top row in Figs. 11C–11E). In the corresponding panels below that, we see the color-coded single trial activity (erpimages), in which segments time-locked to one type of event (e.g., stimulus onset) were sorted by the latency of the temporally adjacent event (e.g., saccade onset). These panels clearly show the overlapping activity and how it was successfully removed by the deconvolution. In particular, we wish to highlight the substantial effect of overlap correction on the shape of both the stimulus-onset ERP (elicited by the faces) and the response-related ERP (elicited by the button press), despite the fact that average RT was relatively long (>800 ms) in this task. Microsaccades have an additional distorting effect (Dimigen et al., 2009). We can therefore easily imagine how without any overlap correction, differences in mean RT and microsaccade occurrence between conditions will create spurious condition effects in the stimulus-ERP. A more complex application where we correct for similar spurious effects in a natural reading EEG experiment with 48 participants is found in Dimigen & Ehinger (2019). The data and code to reproduce Fig. 11 can be found at https://osf.io/wbz7x/.

Linear deconvolution can be applied to many types of paradigms and data. As shown above, one application is to separate stimulus- and response-related components in traditional ERP studies (see also Ouyang et al., 2011; Ouyang, Sommer & Zhou, 2015). Deconvolution is also particularly useful with complex ERP designs that involve, for example, multimodal streams of visual, tactile, and auditory stimuli (Spitzer, Blankenburg & Summerfield, 2016). Deconvolution is also helpful in paradigms where it is problematic to find a neutral interval to place a baseline, for example, in experiments with fast tone sequences (Lütkenhöner, 2010). In ERP research, the interval for baseline correction is usually placed immediately before stimulus onset, but activity in this interval can vary systematically between conditions due to overlapping activity, for example, in self-paced paradigms (Ditman, Holcomb & Kuperberg, 2007). This problem can be solved by first deconvolving the signal and then applying the baseline subtraction to the resulting isolated responses.

It is also possible to add time-continuous signals as predictors to the design matrix (Lalor et al., 2006; Crosse et al., 2016). Examples for continuous signals that could be added as predictors include the luminance profile of a continuously flickering stimulus (Lalor et al., 2006; VanRullen & MacDonald, 2012), the sound envelope of an audio or speech signal (with temporal lags to model the auditory temporal response function Crosse et al., 2016), the participants’ gaze position or pupil size (from concurrent eye-tracking Dimigen & Ehinger, 2019), but also more abstract time series, such as predictions from a cognitive computational model. Including time-continuous covariates such as gait-signals, movement features, or environmental sounds could also improve the model fit in mobile EEG situations (Ehinger et al., 2014; Gramann et al., 2014). The resulting betas of time-continuous covariates are often named (multivariate) temporal response functions (mTRFs, but not consistently so, e.g., Broderick et al., 2018 call FIR-deconvolved betas TRFs). We propose to follow this nomenclature to distinguish it from (stick-function-fitted) rERPs as we have been using in this paper. Of course, one could combine both approaches and fit mTRFs and rERPs simultaneously.

A fundamental assumption of traditional ERP averaging is that the shape of the underlying neural response is identical in all trials belonging to the same condition. Trials with short and long manual RTs are therefore usually averaged together. Similarly, with linear deconvolution modeling, we assume that the brain response is the same for all events of a given type. However, like in traditional ERP analyses, we also assume that the neural response is independent of the interval between two subsequent events (e.g., the interval between a stimulus and a manual response). This is probably a simplification, since neural activity will likely differ between trials with a slow or fast reaction.

A related assumption concerns sequences of events: processing one stimulus can change the processing of a following stimulus, for instance due to adaptation, priming, or attentional effects. We want to note that if such sequential effects occur often enough in an experiment, they can be explicitly modeled; for example, on could add an additional predictors coding whether a stimulus is repeated or not or whether it occurred early or late in a sequence of stimuli. We hope that the unfold toolbox will facilitate the analysis of simulations on these issues and also propose to analyze experiments where temporal overlap is experimentally varied.

Non-linear predictors can have considerable advantages over linear predictors. However, one issue that is currently unresolved is how to select an appropriate number of spline functions to model a non-linear effect without under- or overfitting the data. While automatic selection methods exist (e.g., based on generalized cross-validation, Wood, 2017), their high computational cost of repeatedly deconvolving the data precluded us from using these techniques. In the current implementation of unfold, we assume the same number of splines are needed for all parts of the response. But it is possible, for example, that with a constant number of splines the baseline interval is overfitted, whereas the true response is underfitted. Therefore, algorithms to find smoothing parameters need to take into account that the amount of smoothing changes throughout the response. Choosing the correct number of splines that neither overfit nor underfit the data is an important question to resolve, and again, we hope that the unfold toolbox will facilitate future simulation studies, new algorithms, and new experiments on this issue.

While all example analyses presented here were conducted in the time domain, it is also possible to model and deconvolve overlapping time-frequency representations with unfold (see also Litvak et al., 2013). One simple option is to enter the band-bass filtered EEG signal into the model (or alternatively bandpass filter the estimated betas, but be aware of boundary filter artefacts). This would model the evoked potentials. One could also estimate the instantaneous power for induced potentials. For instance Ossandón, König & Heed (2019) deconvolved the instantaneous power after band pass filtering the alpha band. But this assumes that the overlap is linear in power. Litvak et al. (2013) used an empirical approach based on model fit to decide between three possible power transformations (power, sqrt(power), and log(power)) and found slightly better model fit for the sqrt(power) transform. Alternatively, complex linear regression could be a solution (Hussin, Abdullah & Mohamed, 2010), but future work is needed here and we recommend more simulation studies prior to such time-frequency work.

In a traditional ERP analysis, the researcher has to set numerous analysis parameters, which will influence the final ERP results (e.g., filter settings, epoch length, baseline interval). Similarly, with deconvolution we have to make these and additional choices. Because the deconvolution approach is still in its infancy, there are not yet clearly established best-practices for all of the necessary settings. However, in the following, we discuss some basic and advanced parameters:

Basic settings/options 1)

Sampling rate. Deconvolution can be applied to data recorded at any sampling rate, but some researchers have temporally downsampled their data (e.g., to 100 Hz; Sassenhagen, 2018) before deconvolution. Aside from the obvious loss of temporal resolution that results from downsampling, we (anecdotally) have not observed a benefit for the stability of the estimation; in our experience, downsampling made the fitting process faster but not necessarily better.

In recent years, linear mixed-effects models (LMM, e.g., Gelman & Hill, 2007) have been slowly superseding traditional models like repeated-measures ANOVA or the two-stage hierarchical approach used here. LMMs allow to model the hierarchical structure of single-subject and group-level data directly and have several other advantages, for example, when analyzing unbalanced designs (Baayen, Davidson & Bates, 2007; Kliegl et al., 2010). Combining LMMs with linear deconvolution is theoretically possible (Ehinger, 2019). The main challenge is that one needs to fit all continuous EEG datasets of all participants at the same time. Thus, currently, the high computational cost of fitting such large models precludes us from taking advantage of mixed models. Nevertheless, recent progress with similarly large models (Wood et al., 2017) shows that the combination of LMMs with deconvolution modeling might be computationally feasible in future implementations.

Finally, it is also possible to model other types of overlapping psychophysiological signals with unfold, such as overlapping magnetic fields (MEG, Litvak et al., 2013), pupil dilations (Wierda et al., 2012; Gagl, Hawelka & Hutzler, 2011) or skin conductance responses (Bach et al., 2009).