Markov chains as a proxy for the predictive memory representations underlying mismatch negativity

Abstract

Events not conforming to a regularity inherent to a sequence of events elicit prediction error signals of the brain such as the Mismatch Negativity (MMN) and impair behavioral task performance. Events conforming to a regularity lead to attenuation of brain activity such as stimulus-specific adaptation (SSA) and behavioral benefits. Such findings are usually explained by theories stating that the information processing system predicts the forthcoming event of the sequence via detected sequential regularities. A mathematical model that is widely used to describe, to analyze and to generate event sequences are Markov chains: They contain a set of possible events and a set of probabilities for transitions between these events (transition matrix) that allow to predict the next event on the basis of the current event and the transition probabilities. The accuracy of such a prediction depends on the distribution of the transition probabilities. We argue that Markov chains also have useful applications when studying cognitive brain functions. The transition matrix can be regarded as a proxy for generative memory representations that the brain uses to predict the next event. We assume that detected regularities in a sequence of events correspond to (a subset of) the entries in the transition matrix. We apply this idea to the Mismatch Negativity (MMN) research and examine three types of MMN paradigms: classical oddball paradigms emphasizing sound probabilities, between-sound regularity paradigms manipulating transition probabilities between adjacent sounds, and action-sound coupling paradigms in which sounds are associated with actions and their intended effects. We show that the Markovian view on MMN yields theoretically relevant insights into the brain processes underlying MMN and stimulates experimental designs to study the brain's processing of event sequences.

Keywords: Markov model; adaptation; memory; mismatch negativity; perception; predictive processing; regularity representation.

Figure 1

(A–C) Examples of 2-state Markov chains. Upper row: excerpts from event sequences; middle row: directed graphs of the Markov chains. S1 and S2 denote the states, the numbers at the arrows denote the probability of a transition from the state from which the arrow emerges to the state the arrow points to. Note that transitions with a probability of 0.0 are not shown for ease of display. Lower row: transition probability matrices. The probabilities in each row must add up to 1.0.

Figure 2

Analogy between the transition probability matrix of the Markov model (left) and the predictive regularity representations of the MMN theory (right). In a Markov model, the next state of the system is computed by multiplying the current vector state with the transition probability matrix, in MMN theory the next sound is predicted on the basis of the detected regularities.

Figure 3

Two-tone classical oddball paradigm with a frequent standard (green) and an infrequent deviant (red) tone. Deviants elicit the MMN, which may be regarded as a genuine mismatch (prediction error) signal and/or the expression of differential adaptation to feature-specific neurons underlying the N1 (resulting in a small standard-N1 and a large deviant-N1). (A) Only frequently presented standard sounds are regarded as constituting the rule in the tone sequence; so only one type of transition has to be modeled; the respective transition probability matrix specifying the one transition is shown below. (B) The stochastic process of the two-tone sequence is characterized by a 2-state Markov model with four transitions (three are actually shown): each could be of possible relevance for the experiment; the respective transition probability matrix specifies the four possible transitions.

Figure 4

(A) Exemplary sound sequence of Coy et al. (2022) and prototypical results [MMN, RT (reaction time)]; unpredictable deviants elicit full amplitude MMN, predictable deviants smaller MMN (e.g., Sussman and Winkler, 2001) and, when deviants are targets, shortened RT (cf. Coy et al., 2022); according to the Markovian view, standards that create a deviant-repetition rule violation should elicit MMN and/or P3a. (B) Second-order transition probability matrix (S = standard, L = Low pitch deviant, H = High pitch deviant; SL = standard followed by Low deviant, …); e.g., cell(1,2) denotes the probability (0.1036) that an SS pair is followed by an L sound (low deviant); reaction times for deviants are listed in brackets; second deviants with high transition probability [i.e., HH, cell(3,3)] have by far the shortest reaction time. (C) First-order Markov chain that generates a highly similar tone sequence to the second-order Markoc chain in (B) but which represents a rather different proxy for the predictive memory underlying MMN.

Figure 5

(A) Two-tone alternating paradigm, where a low and a high pitch tone are presented alternatingly, occasionally a high- or a low-pitch tone is repeated. Violations of the alternation rule usually elicit MMN (e.g., Nordby et al., 1988; Alain et al., 1994; Winkler and Czigler, 1998; Horváth et al., 2001). (B) Tone pairs are presented in ascending (high transition probability) or descending (low transition probability) pitch. Violations of the prevailing rule elicit MMN (e.g., Saarinen et al., 1992; Paavilainen et al., 1999).

Figure 6

Two tones (green, blue) are played frequently, and two other tones (ocher, red) ralely. (A) In a “stochastic” world red tones elicit MMN/∆N1, while ocher tones do not as they belong to the distribution spanned by the standard tones (cf. Garrido et al., 2013; Schröger and Roeber, 2021). (B) In a “deterministic” world where the two standard tones ocher and blue alternate and become predictable, ocher (and red) tones elicit MMN; if the pitch separation between the two standard tones (green, blue) increases, deviant (ocher) tones elicit ∆N1 but still no MMN (cf. Schröger and Roeber, 2021).

Figure 7

(A) In the Tone regularity condition, participants are instructed to press the left and right buttons equiprobably; each button press produces low (green) or a high (red) pitch tone. Most of the left and most of the right button presses produce a high pitch tone, but sometimes they produce a low pitch tone (deviant). The deviant tones (violating tone regularity) elicit MMN. (B) In the Intention condition, participants are instructed to generate the low tone with the left and the high tone with the right button equiprobably. In 20% of the trials a button press produces the “wrong” tone (i.e., high instead of low and vice versa). Although those deviants do not violate a tone regularity, they elicit MMN. This is because these wrong tones violate the intended action effect. The upper part of the figure shows excerpts of the sequences (adapted from Korka et al., 2019). The middle part shows the directed graphs of the Markov chains. The lower parts show the respective transition probability matrices.