Roving paradigm and event related potentials

The empirical data used in this study comprised trial-wise mismatch responses, acquired during a roving oddball experiment with electroencephalography (EEG) from eight healthy subjects in a previously published study [25], [26]. Twelve healthy volunteers (aged 24–34, 4 female) listened passively to a structured sequence of 1600 pure sine tones adapted from [27]. Subjects sat in front of a computer screen and were instructed to ignore the tones and press a button whenever there was a change in the luminance of the fixation cross. The structure of the stimulus sequences is illustrated in Figure 1 (lower right panel). For each subject, the stimulus sequence was structured into approx. 250 trains of a varying number of identical tones, each of which was followed by a train of tones with a different frequency. In other words, the same tone was repeated several times and then changed to a new tone. This lead to two types of events: tone repetition and tone change. The probabilities of trains with zero to ten tone repetitions were 2.5%, 2.5%, 3.75%, 3.75%, 12.5%, 12.5%, 12.5%, 12.5%, 12.5%, 12.5%, and 12.5%. The tone frequencies were , and they occurred with equal probability in a pseudorandom order. Tones lasted for and were presented at a constant stimulus onset asynchrony of for 15 minutes using headphones.

In this study, we quantified the MMN by subtracting the average of waveforms elicited by the sixth presentation of a tone (the standard) from the waveform elicited by its first presentation (the deviant). In other words, we compared responses to physically identical stimuli presented in different contexts (i.e. after different stimulus trains). This avoids confounding factors that would have arisen had we used a classical oddball or mismatch negativity paradigm [28] for our single-trial analysis (e.g., differences in physical stimulus properties between standards and deviants and differences in the degree to which the standard was expected [27]).

A framework for modelling single-trial responses

This section introduces our modelling framework for single-trial responses. In terms of notation, we denote vectors by lower case bold letters, matrices by upper case bold letters, and scalars and functions by lower case italics (except for variables like the free-energy for which there are notational conventions in the literature). Vector and matrix elements can be scalars, vectors, or matrices, and they are referred to via subscripts (e.g., denotes the t^th element of vector , and denotes the j^th element of the k^th row of matrix ).

Models of single-trial responses can be cast in a general dynamic state-space framework that models the measurements as manifestations of internal states which cannot be observed directly. The internal states evolve according to an evolution function mapping an internal state and some sensory input to the ensuing state. The internal states generate neurophysiological signals in response to sensory input according to a response function . These are scaled and combined according to a linear observation model with regression coefficients and corrupted by Gaussian measurement noise . Both the evolution function and the response function may depend on parameters and have the following general form:(1)Together with the prior density , the evolution function and the response function define a generative model of the measurements:(2)This framework is based on [34] and enables inferences about (hidden) computational processes and representations from neurophysiological measurements. It is particularly powerful in conjunction with model comparison methods such as random-effects Bayesian model selection [35] and model space partitioning (i.e., inference on model families [36]). Given competing models of learning and inference, Bayesian model inversion and comparison can be used to infer the nature of the underlying process and its relationship to the measured responses. The resulting posterior model probabilities assess each model's relative explanatory power in a way that balances fit and complexity such that the comparison between any two models is valid irrespective of their relative complexity.

Computational models of the mismatch negativity

We applied the framework introduced in the previous section to formalize five competing theories of the MMN by formulating thirteen models () of measured trial-wise MMN amplitudes elicited by tone sequences . Each of the five theories summarized in the introduction (predictive coding, novelty detection, model adjustment, change detection, and adaptation) explains the MMN as originating from a particular process operating on some neural state or cognitive representation . We modelled these processes and representations as well as the resulting neural responses which we interpret as local field potentials. Since the EEG signal is a linear mixture of local field potentials, we use a general linear model to map predicted neuronal activity to MMN amplitude; this is expressed by Eq. (3) where are the unknown regression coefficients, and the trial-wise values of define the design matrix:(3)Note that this is an equation for a single electrode (we generalize it to multiple electrodes in Eq. (13)).

The 13 models are derived in detail below. After formalizing two traditional phenomenological MMN theories (the change detection hypothesis and the adaptation hypothesis), we formalize three current theories of the MMN using Bayesian information processing models based on the free-energy principle. These models assume that the brain represents probabilistic beliefs about its environment whose evolution approximates Bayes optimal learning and perception according to the free-energy principle [37]. The predictive coding, the model adjustment, and the novelty detection theories were formalized by extending this core assumption by response models of different neural sub-processes of the belief updates prescribed by the free-energy principle. Overall, our model space is structured hierarchically, as shown in Figure 2. First, our 13 models can be grouped into five model families that correspond to the five MMN theories introduced above: change detection (), adaptation (), prediction error (), novelty (), and model adjustment (). The models within each family assume the same internal representation and the same evolution function, but differ in their response functions. Second, these model families can be grouped into two super-families: phenomenological models () and information processing models (). The latter are formulated within a Meta-Bayesian framework [34] and build upon the free-energy principle [37]. Table 1 summarizes all computational models, and the notation used to describe them is summarized in Table 2.

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Figure 2. Hierarchical structure of the model space: models, theories, and frameworks.

The MMN models developed in this article can be organized into a tree structure. The leaves at the bottom of the tree represent individual models of trial-wise MMN amplitudes, and the nodes above represent sets of models (model families). The nodes at the third level represent modelling frameworks. Three theories (the prediction error hypothesis, the novelty detection hypothesis, and model adjustment hypothesis) were formalized under the framework of the free-energy principle (). This framework explicitly models information processing, which makes it fundamentally different from phenomenological explanations (), such as change detection and adaptation models.

https://doi.org/10.1371/journal.pcbi.1002911.g002

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Table 1. This table lists the response models of our 13 computational models of trial-wise MMN amplitudes.

https://doi.org/10.1371/journal.pcbi.1002911.t001

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Table 2. Explanation of the variables in our computational models of trial-wise MMN amplitudes.

https://doi.org/10.1371/journal.pcbi.1002911.t002

Predictive coding, model adjustment, and novelty detection.

Predictive coding, model adjustment, and novelty detection are formalized by models based on the free-energy principle (). These models explain the MMN as an electrophysiological manifestation of the neural mechanisms that approximate Bayes-optimal perception and learning of sensory regularities. Figure 3 illustrates that these models are structured into two components: a Bayesian observer and a response function. This instantiates our general dynamic state-space framework: The internal states represent the Bayesian observer's probabilistic beliefs, and the response functions map belief updates to neural responses. The Bayesian observer is shared by all information processing models; it is their response functions (summarized in Table 1) that differentiate them into models of predictive coding, novelty detection, or model adjustment. As shown in Figure 3, the beliefs of the Bayesian observer evolve according to an evolution function that depends on the observer's mental model. The following two subsections introduce this mental model and the evolution function respectively, and the third subsection introduces the response functions. The notation used to describe the Bayesian observer model is summarized in Table 2.

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Figure 3. Structure of free-energy based models of the MMN.

Our free-energy models of trial-wise MMN amplitudes ( in Figure 2) are cast within the general dynamic state-space framework formulated in Equation (1). In contrast to the phenomenological models, the internal states () represent probabilistic beliefs about the environment and evolve according to approximate Bayesian inference by free-energy minimization (). All of these models share the Bayesian observer defined by the evolution function and the probabilistic mental model , but differ in their response functions . The graph in the innermost box shows the mental model as a probabilistic graphical model (with arrows indicating conditional dependencies). The random variables in circles are sensory inputs (), tone categories (), and transition probabilities (). This mental model determines how subjects perceive, learn about and predict tone sequences. Please see Table 2 for an explanation of the mathematical notation.

https://doi.org/10.1371/journal.pcbi.1002911.g003

Bayesian model selection

Above, we have derived 13 different models predicting the trial-wise MMN amplitudes during our roving oddball experiment. These models differ in numerous ways, conceptually and mathematically. For example, the evolution function of the change detection models has no free parameters whereas the evolution function of free-energy models has 3 free parameters (see Table 2). Critically, because model fit increases monotonically with model complexity, the relative plausibility of these models cannot simply be established based on how well they fit the data. Generally, the true desideratum of model comparison, the generalizability of a model, cannot be determined from fit measures alone; instead, model comparison needs to assess the trade-off between model fit and model complexity [65], [66]. From a Bayesian perspective, this is provided by the (log) model evidence (i.e., the log probability of the data given a model) which corresponds to the negative surprise about the data and represents a principled measure of the balance between model fit and model complexity. Here, we used a Bayesian model selection (BMS) procedure at the group level that treats models as random effects in the population and can successfully deal with population heterogeneity and outliers [35]. As input, this procedure requires the log-evidence of each model considered, for each subject separately. In the following, we describe how these log-evidences were obtained, detailing the likelihood function and priors that underlie the computation of the log-evidence for individual models and subjects.

As EEG signals result from a linear superposition of local electrophysiological responses, one can use a general linear model to map the predictions of local field potentials ( in Table 2) to measured trial-wise MMN amplitudes. In each subject and for each model considered, we modelled the data matrix of trial-wise MMN amplitudes across all trials and across all selected electrodes as follows:

Let denote the vector of MMN amplitudes at a selected electrode . We regard each as noisy observations of an electrode-specific linear mixture of evoked neuronal responses that reflect the trial-by-trial evolution of internal states. For each response model described above, we therefore apply the following multivariate Bayesian linear regression model with conjugate priors to each subject's data:(13)Here, denotes the design matrix that was created by replacing the non-constant columns of (cf. Eq. 3) by their z-transforms, are the regression coefficients for the k^th electrode, and is the standard deviation of measurement errors at the k^th electrode. When inverting this model, we used uninformative Gaussian priors on the regression coefficients and uninformative Gamma priors on the error precisions; for details see Section 5 in Text S1.

Note that we are not interested in the regression coefficients but in each model's log-evidence . Given the likelihood function and priors described above, the log-model evidences were computed by Monte-Carlo integration (see Section 5 in Text S1 for details). Based on the log model-evidences, we estimated the posterior probability of each model by a Bayesian random effects analysis at the group level [35] with a uniform prior on models. For comparing the model families described in Figure 2 Bayesian inference on partitions of model-space [36] was performed to compute the posterior probability of each model family, where denotes the data across all pre-defined electrodes and subjects. This approach can easily deal with families of different size (i.e., different numbers of models per family). In brief, unbiased family-level inference requires uniform (flat) priors over families, and this was achieved by setting each model's “prior count“ (i.e. the parameters of the Dirichlet prior on model probabilities) to 1 over the size of the respective model family; see [36] for details. Inference on model families used Gibbs sampling with two million samples per family. Finally, we computed the exceedance probability [35] for each model and model family, i.e., the probability that this model (family) was more likely to have generated the data than any other model (family).

Models and theories of the MMN

In the Models and Methods section, we derived five classes of models describing how the MMN may reflect the computational processes that govern learning and perception during the roving oddball experiment. Three of the five model classes were derived from the free-energy principle and correspond to formal representations of three contemporary theories of the MMN; i.e., predictive coding, novelty detection, and model adjustment. These models explain the MMN as arising from prediction error signals, surprise or adjustments to model parameters, respectively. Furthermore, we formalized two traditional theories of the MMN: the change detection and adaptation theory. The resulting model space comprised 13 models in five families (see Figure 2). In all models, we have connected the (hidden) processes of perception and learning to measured EEG responses via different response models and a linear electromagnetic forward model. In this section, we assess the relative plausibility of these models and model families using posterior model probabilities and exceedance probabilities computed by Bayesian model selection (BMS) as detailed above. The resulting posterior distributions will be presented as figures, and the main text will report inferences based on those distributions in terms of exceedance probabilities.

Figure 4 shows the results of BMS in terms of the posterior probabilities of all models considered. First, note that our “null” model (M₁, the first change detection model), the only model predicting the absence of trial-by-trial changes in MMN amplitudes, is not the best model. Contrary to the predictions of this model, the MMN amplitude appears to vary systematically over deviant trials. This suggests that the MMN is not simply a categorical response to regularity violation but context dependent, as predicted by trial-by-trial statistical learning. Notably, the best five models were all derived within the free-energy framework. Model M₆, which explains trial-wise changes in MMN amplitude as a manifestation of precision weighted prediction errors (on the hidden tone category), was best supported by our data (exceedance probability ). It was followed by three “model adjustment” models (M₁₀, M₁₁, M₁₃), each with exceedance probability . These models explain fluctuations in MMN amplitude as arising from a trial-wise adjustment of the parameters encoding posterior beliefs about the expected number of tone repetitions and the conditional transition probabilities. When examining the fit of the best model, we found that it accounted for 2.3% of the total variance of single-trial MMN amplitudes (across all subjects). The amount of variance explained was significant in each and every subject (p<0.01 in 6 subjects; p<0.02 in two subjects). To put this into perspective, this model-based explanation accounted for about 6.5 times as much variance as could be explained by a more conventional analysis, i.e., a linear regression model considering recent stimulus history (number of standards preceding the deviant).

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Figure 4. Posterior probabilities of the 13 MMN models.

The 13 MMN models were compared by their posterior probability given the trial-wise MMN amplitudes of all eight subjects. These posterior probabilities were computed by random effects Bayesian model selection at the group level. The bars are coloured according to the theory instantiated by each model. The model explaining trial-wise MMN amplitudes by precision weighted prediction errors on the unobservable tone category () had the highest posterior probability (). It is closely followed by three almost equally probable “model adjustment” models (), and the model explaining trial-wise MMN amplitudes by prediction errors on the observed log-frequency ().

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While the exceedance probability of the best model M₆ was about five times as large as the exceedance probability of our “null” model M₁, this was too small to yield an acceptably low probability of model selection error [67]. As the bar plot shows, the probability mass is concentrated on two model families (prediction error and model adjustment) but distributed over several models. Thus, BMS at the level of model families was more appropriate than comparing individual models. From a statistical perspective, this trades a reduced resolution of the hypothesis (model) space for increased statistical power. In other words, we move from asking which specific model is best to asking which of the five general MMN theories best explains the data, irrespective of their precise implementations (cf. Figure 2). This comparison of the five model families is summarized in Figure 5a. The most plausible MMN theory was the model adjustment theory (), followed by the prediction error theory ().

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Figure 5. Bayesian model comparison of the five MMN theories (a) and the two frameworks (b).

The bar plot in the upper panel (a) summarizes the comparison of the five model families in terms of their posterior probabilities. Each bar indicates the posterior probability of a particular MMN theory (i.e. ). The most plausible explanations of our trial-wise MMN data were provided by the model adjustment hypothesis () and the prediction error hypothesis (). The lower panel (b) shows the results of comparing phenomenological () vs. free-energy based models (); see Figure 2. It shows that our free-energy based models provide considerably more convincing explanations of our MMN data than traditional change detection or adaptation models ().

https://doi.org/10.1371/journal.pcbi.1002911.g005

Finally, we used BMS to examine whether the free-energy principle based models provide, in general, better explanations of the variability of single-trial MMN amplitudes than phenomenological models. This means we are now comparing only two families (Figure 2): the family of free-energy based models (predictive coding, novelty detection and model adjustment; ) and the family of more traditional phenomenological models (change detection and adaptation, ). Family-level BMS indicated that models based on the free-energy principle were considerably more convincing than phenomenological models; (see Figure 5b).

Level of representation

Finally, we asked which level of the processing hierarchy contributes most to the fluctuations in trial-wise MMN amplitudes. In other words, we examined whether response variations arise from lower auditory areas representing physical sound properties like frequency, or from higher areas that represent abstract temporal structure. For this purpose we re-partitioned the 13 models into two families according to whether they explain MMN generation in relation to a low-level auditory feature (sound frequency) or a high-level auditory feature (temporal structure). For the models based on the free-energy principle models the two levels of representation map onto the two levels of the mental model: sensory inputs and hidden sequence of tone categories (Figure 3). We assigned the free-energy based models that relate the MMN elicited by changes in sound frequency to the representation of sound frequencies to the first model family and those that relate it to the represented sequence of tone categories to the second. Furthermore, both the adaptation model and the change detection theory are formulated explicitly with regard to stimulus frequencies and are therefore assigned to the first model family. Overall, this resulted in the following two model families: and . Comparing these two model families yielded an exceedance probability of for , suggesting that the auditory MMN is more closely related to a representation of high-level auditory features, such as temporal structure, than to a representation of low-level features, such as sound frequency.

Single-trial MMN amplitudes are history-dependent

The models reported above were designed to predict the evolution of single-trial MMN amplitudes throughout the experiment. This was done to capture putative history-dependent effects. The models which did take into account such effects (i.e., free energy based models) were found to have higher evidence than models which did not (e.g., the various change detection models). One may ask, however, as did one of our reviewers, whether our single-trial approach was really necessary or whether it would have been sufficient to analyse the average MMN amplitude as a function of the number of preceding standards and the change in frequency. Here we provide a conventional analysis of variance to demonstrate that our data did contain history-dependent effects that would have been removed by conventional averaging. By history-dependent effects we mean that the MMN amplitude evoked by a deviant following a given number of standards and a given frequency change will differ depending on the tones that preceded the current sequence of standards. The mere number of such tones is a minimal definition that ignores the effects of their statistical structure, some of which are captured by our models. However, it allows for a conservative test of history-dependence, i.e., whether a 3-way analysis of variance (ANOVA) of trial-wise MMN amplitudes reveals interactions among three factors: (i) number of preceding standards, (ii) frequency difference, and (ii) time, i.e., the number of preceding trial sequences. We found significant main effects for the number of preceding standards and for frequency difference (Figure 6). More importantly, however, we found highly significant interaction effects, indicating that the effect of the number of preceding standards on MMN amplitude did not only depend on the frequency difference between standard and deviant () but also on the number of previous tone sequences (). This demonstrates that the trial-wise MMN amplitudes we recorded do indeed show history-dependent effects that would be removed by conventional averaging procedures.

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Figure 6. Trends in MMN amplitudes.

Figure 6a shows the average MMN amplitude as a function of the number of standards preceding the deviant. Figure 6b shows the average MMN amplitude as a function of the frequency of the deviant minus the frequency of the preceding standard (frequency change).

https://doi.org/10.1371/journal.pcbi.1002911.g006

Single-trial MMN amplitudes are informative about statistical learning

Our analyses suggested that stimulus history (i.e., previous tone sequences) affects the MMN in intricate ways. This was not only demonstrated by a simple ANOVA of single-trial MMN amplitudes, but, more importantly, by our systematic model comparisons which favoured free-energy based Bayesian information processing models that capture history-dependent effects. In particular, these models explain the dependence of the MMN on interactions between previous tone sequences and the current tone sequence in terms of trial-by-trial learning of statistical structure. Trial-by-trial statistical learning implies that the probabilistic expectation evoked by a given tone sequence is different for every presentation, and that each difference reflects what has been learned since the previous presentation. While traditional MMN studies have ignored trial-specific effects by averaging responses across deviant events, several studies have addressed sequential changes in the MMN across trials [19], [23], [25]–[27], [51]–[54], [68]–[70]. However, only [23] and [68] have completely avoided averaging procedures altogether. The results of this study and [23] question the frequent assumption that the MMN amplitude is constant throughout an experimental condition (i.e., for given tones and following a given number of standards). Instead, our results suggest that trial-by-trial changes in MMN amplitude are highly history-dependent and represent an informative index of statistical learning as the recording session proceeds. It is pleasing that [23] reached a similar conclusion, even though they studied mismatch potentials in a different modality (i.e., somatosensory) and with simpler models, but with source-reconstruction and a high temporal resolution. Thus, while averaging is a useful tool to increase the signal-to-noise ratio, single-trial data carry unique information about the processes of learning and perception that underlie the MMN.

MMN amplitude as a function of the number of preceding standards

A number of previous studies reported that the MMN amplitude elicited by a change in sound frequency increases monotonically with the number of preceding standards [27], [51]–[54], [69], [70]. By contrast, we found a non-monotonic effect of the number of preceding standards on deviant response amplitude (see Figure 6a). The reason for this discrepancy may be that previous studies did not disentangle the contributions of the standard ERP and the deviant ERP (cf. [71]). In contrast, in this study, we operationalized the MMN with respect to a fixed standard ERP (see Models and Methods), so that changes in MMN amplitude reflected changes in the neural response to the deviant only. In summary, our results do not contradict previous findings on the relationship between the number of preceding standards and the MMN amplitude [27], [51]–[54], [69], [70] but complement them. Furthermore, our models based on the free-energy principle can explain why Haenschel et al. [27] observed a monotonic decay of the standard response with the number of standard repetitions, and they predict how stimulus history determines the effect of preceding standards on deviant response amplitude.

Implications for theories of the MMN

Our modelling results do not lend support to the adaptation hypothesis of the MMN [18] or the change detection interpretation of the memory trace hypothesis [72]. Instead, our results support explanations postulating that the brain maintains and constantly updates an internal model of its environment. For example, the model adjustment hypothesis [19] posits that auditory cortex maintains a model of the acoustic environment, and that stimulus-induced updates of this model are indexed by the MMN [20]. While the original proposal was of a conceptual nature, our present work formalizes this hypothesis by specifying how trial-wise changes in MMN reflect an approximation to Bayesian updating of a probabilistic mental model. The resulting models are consistent with the conclusion drawn by [23] that (somatosensory) mismatch potentials reflect perceptual learning. However, our analysis was more fine-grained in that it distinguished between three computational mechanisms that might underlie the perceptual learning that [23] indexed in terms of Bayesian surprise. Concretely, we distinguished between prediction error signalling, novelty detection, and model adjustment. Our results supported model adjustment and, to a lesser extent, prediction error signalling, but not novelty detection, even though it computes an approximation to (Shannon) surprise. We also distinguished between perceptual learning at the level of physical stimulus properties (sound frequency) and learning of abstract temporal structure and found strong evidence for the latter. In neurobiological terms, model adjustment might correspond to synaptic plasticity at top-down projections targeting pyramidal neurons in layers 2 and 3 (“prediction error units”) via NMDA receptors [3] (see Section 6 in Text S1). This would be consistent with the observation that pharmacological blockage of NMDA receptors diminishes the MMN [73]–[75].

Predictive coding formulations of free-energy minimization assign prediction errors a critical role in the update of posterior beliefs. When comparing all models individually, the best model was indeed one that explained trial-wise fluctuations in MMN amplitude as a function of precision weighted prediction errors (model M₆; Figure 4). However, its superiority over other models was marginal, and model comparison at the family-level (Figure 5a) did not support the hypothesis (proposed in [3]) that the MMN solely reflects precision weighted prediction errors. This suggests that while prediction error signalling may be essential for the free-energy minimization process underlying the MMN, it is probably not the sole determinant of trial-wise MMN amplitudes. Alternatively, our failure to find stronger evidence for the hypothesis that (precision weighted) prediction errors alone determine trial-wise MMN amplitudes may be due to some of our simplifying assumptions, as discussed in the next section.

Limitations

Overall, one should bear in mind that our inferences are primarily about rather abstract models or classes of models. Our free-energy based models, in particular, consider the outcomes of neuronal computations rather than their process. This is a necessary constraint on models of discrete trial-by-trial variations in responses; as opposed to continuous time models that would consider the precise time-course of neural responses over peristimulus time. This means that we have to assume that there is some aspect of neuronal activity or excitability that encodes the posterior beliefs associated with each oddball trial. However, the relationship between biophysical quantities like synaptic activity or gain, on the one hand, and posterior beliefs, predictions, and surprisal, on the other hand, are not specified explicitly in this sort of model. This means that it is difficult to make any strong statements about the neurobiology that implements any Bayesian inference.

Furthermore, our models make several simplifying assumptions that may turn out to be false. First, there is still no conclusive evidence about how prediction errors are represented at the level of single neurons. Second, the assumption of a linear relationship between the encoded quantity and the MMN amplitude is simplistic and ignores potential nonlinearities. Third, all of our models represent the MMN by a single number (i.e., its peak amplitude), rather than by its waveform, thereby ignoring its temporal dynamics and spatial topography. Fourth, each of our models relates trial-wise MMN amplitudes to a single computational variable, whereas it is known that the MMN scalp potential is a mixture of signals from several brain areas with (presumably) different functional characteristics [29]–[31], [76]. Finally, while our results indicated that our neuronal adaptation model M₄ is insufficient to explain single-trial variations in MMN, we have not tested the fresh-afferent theory [13] that is based on stimulus specific adaptation. In future work, it would be useful to formulate this theory as models of stimulus specific adaptation [12], [13], [42] under the present framework and compare it to the computational models presented in this paper.

Relation to the Bayesian-brain hypothesis

Our models based on the free-energy principle link the MMN to the neuronal encoding of posterior beliefs that is postulated by the Bayesian brain hypothesis. According to this hypothesis, the brain represents probabilistic beliefs, and updates them in an (approximately) Bayesian fashion. Previous work along these lines has assumed that the support of probability distributions is partitioned into small bins and that each bin's probability mass is represented by the firing rate of dedicated neurons [77], [78], or that probability densities are approximated by a linear combination of basis functions [79]. In contrast to these high-dimensional representations, we have implicitly assumed a much simpler, low dimensional fixed-form approximation to the posterior density. Our predictors of electrophysiological responses are simple functions of posterior expectations on log-frequency, tone category and transition probabilities. These posterior expectations might be encoded by the average activities of neuronal populations, and the precision parameters that determine the relative weight assigned to prior beliefs and sensory evidence could be encoded by the strength of the recurrent connections of prediction error units [80] (see also Section 6 in Text S1). This representation is not motivated by sparseness, but by computational efficiency: It replaces the problem of computing the (potentially very high-dimensional) posterior probability density by optimizing the free-energy with respect to a small set of sufficient statistics. This variational Bayesian optimization rests on free-energy minimization [37] and proposes the minimization of prediction error as an explanation for stimulus-evoked transient neuronal responses such as the MMN [3], [63], [81]. The work presented in this paper is a step towards linking models of probabilistic neural coding and inference to neuronal signals that can be measured non-invasively in humans.

Potential future directions

Our present results were based on a single “roving oddball” EEG experiment that was originally designed for comparing dynamic causal models of interactions among cortical areas during the MMN [25]. In the future, it would be interesting to apply the approach presented here to other types of MMN paradigms. Additionally, one could use our models in conjunction with recent advances in design optimization that maximize the sensitivity of Bayesian model selection [67] to create an experiment that is optimal for discerning between the models selected by our analysis. In addition, our modelling and model comparison framework could be applied to source-reconstructed mismatch potentials to characterize functional differences between the brain areas jointly generating MMN scalp potentials.

Furthermore, the link between single-trial mismatch potentials, on the one hand, and statistical learning and perceptual inference, on the other hand, could be exploited to measure the temporal dynamics of how the brain learns the probabilistic structure of complex environments. This is an attractive prospect, given that the MMN is elicited not only in simple oddball experiments, but also in more complex experiments involving speech, language, music, and abstract features, as well as various other sensory modalities [14], [71], [82], [83]. Our modelling framework could also be used to probe disturbances of perceptual inference and learning in psychiatric conditions, such as schizophrenia [84]–[86]. In addition, future studies might use the meta-Bayesian approach [34] for inferring, from single-trial MMN amplitudes, subjects' prior beliefs about hidden temporal structure, which constitute the inductive biases [87] that endow the brain with its remarkable ability to discover complex sequential regularities.