Fig 1.
The naive oddball count (NOC) and the IB lossy compression models of the oddball sequence.
An illustration of the two compression models for the case of a past length of size N = 4 is shown. To be able to code any of the 16 possible sequences of 4 tones in memory that the subject heard, 4 bits of memory would be needed. (Left) The NOC model only keeps the number of oddball occurrences in the previous window in memory; i.e., the minimal sufficient statistic (see Methods). The upper plot shows the number of oddball tone occurrences n (filled red circles) and the number of standard tone occurrences N − n (empty blue circles) in the previous window for each trial, starting from trial N + 1. All plots are aligned to the stimulus sequence. The occurrence predictor (bottom plot) on each trial is n if a standard tone was played and N − n if an oddball tone was played. To be able to code the 5 alternatives of the past in memory (i.e., 0, 1, 2, 3 or 4 oddball tones out of the previous 4 tones), 2.32 bits of memory are required. (Right) The IB model keeps a fuzzy representation of the oddball counter in memory, which requires less memory usage than the NOC model due to its lower representation accuracy (with the accuracy controlled by the β parameter. see Methods for details). The upper plot illustrates a fuzzy representation m of the oddball occurrences n in the previous window, as given by IB for a high compression case (low β). The darker red represents higher p(m|n) probability. The two lower plots show two IB predictors of different compression levels (0.1 bits for the bottom plot, 2.31 bits for the middle plot). The surprise level on each trial is defined as -∑m p(m|n) log p(next tone|m) where the probabilities are defined by the IB solution for a specific compression level.
Fig 2.
The model for the oddball experiment and the EEG surprise measure.
(a) A two-tone oddball sequence is presented to the subject, illustrated by the red (oddball, high-tone) and blue (standard, low-tone) rectangles. The previous stimuli are processed and the subject holds an internal representation m of the past in memory. Based on m the subject holds a prediction of the next tone (high vs. low). The subject’s response to the tone at time t is dependent on the tone type and the incomplete memory m of the past. A fixed window size N of the past is considered on each trial (in this illustration N = 4). (b) The definition of the P300 area-under-the-curve (AUC): event-related potentials averaged over all oddball (red) and standard (blue) trials in electrode Cz are shown for a representative subject. The difference between the oddball and standard curves (solid black) was used to determine for each subject the zero-crossing points (t1 and t2, solid vertical lines) around the P300 peak (tpeak). The P300 AUC per trial was defined as the area between t1 and t2 on each trial. (c), The average trace of all oddball trials in each block is shown color-coded (for the same subject as in (b)). The oddball probability (OP) of each block is shown in the legend. (d), The average trace of all standard trials in each block is shown color-coded (as in (c)). (e), The mean normalized AUC of all oddball trials in each block averaged over all subjects are plotted as a function of the corresponding oddball probability of that block. The error bars indicate the standard error of the mean (SEM). (f) The same as (e) for standard trials.
Fig 3.
Subject-specific compression parameters extracted by single-trial analysis.
(a) Single-trial P300 AUC responses of a representative subject to standard tones (blue empty circles) and to oddball tones (red, filled circles) as a function of the number of occurrences (n) of the opposite tone in the preceding sub-sequence of N = 11 tones (the fitted N for that subject). The black-edged filled circles show the average response for each n. Single trials: weighted-R2 = 0.230, one-sided permutation test for the R2, 1000 permutations, p-value<0.001, 1145 data points. Mean values: R2 = 0.934 p-value = 3.14x10−7, 12 data points. (b) Single-trial P300 AUC responses to standard tones (blue, empty circles) and to oddball tones (solid red circles) as a function of the optimal IB surprise predictor (N = 11, β = 48.33) for the same subject as in (a). The black-edged solid circles show the average response for each surprise value. Single trials: weighted-R2 = 0.231, one-sided permutation test for the R2, 1000 permutations, P-value<0.001, 1145 data points. Mean values: R2 = 0.939 p-value = 2.22x10−7, 12 data points. (c) color-coded weighted-R2 values of the linear regression analysis for all tested IB compression parameters, for the same subject as in (a,b). The vertical axis denotes the memory window length N. The horizontal axis denotes the representation accuracy parameter β. The pair of model parameters (N, β) that achieved the highest R2 value (marked in black asterisk) was used for the remainder of the analysis. (d) Surprise-related analysis (SRA). The single-trial waveforms were averaged by the IB surprise associated with each trial and are shown color-coded according to the surprise level (shown in the legend). The average SRA waveforms exhibited a gradual increase in the P300 magnitude with the IB surprise level.
Fig 4.
Multi-subject comparison of the NOC and IB models.
(a) The mean normalized AUC of each subject is plotted as a function of the running probability (RP) of the opposite tone. The RP on each trial is defined as p = n/N, where n is the number of occurrences of the opposite tone (with respect to the current tone) in the preceding N tones in the sequence, and N is the fitted model parameter. R2 = 0.498, 409 data points, error DOF = 407, F-statistic vs. constant model: 404, p-value = 7.28 × 10−63. Inset: The mean responses across all subjects as a function of the RP, calculated using the data presented in (a). The error bars indicate the SEM. (b) The mean normalized AUC of each subject is plotted as a function of the IB surprise. R2 = 0.471, 445 data points, error DOF = 443, F-statistic vs. constant model: 394, p-value = 0. Inset: The mean responses across all subjects as a function of the IB surprise, calculated using the data presented in (b). The error bars indicate the SEM. (c) Each pair of bars shows the estimated effective capacity (in bits) for the optimal NOC model (blue bars) and for the optimal IB model (red bars) for each subject. The effective capacity for the NOC model is defined as the number of bits required to represent all possible values of occurrences from 0 to N; i.e., log2(N + 1). For the IB model the estimated capacity is defined as I(X; M); i.e., the mutual information between the past variable X = [0..N] and the representation variable M defined by IB. The colored map under each pair of columns shows the IB R2 map of that subject (as in Fig 3c).