A model-based approach to trial-by-trial p300 amplitude fluctuations

Abstract

It has long been recognized that the amplitude of the P300 component of event-related brain potentials is sensitive to the degree to which eliciting stimuli are surprising to the observers (Donchin, 1981). While Squires et al. (1976) showed and modeled dependencies of P300 amplitudes from observed stimuli on various time scales, Mars et al. (2008) proposed a computational model keeping track of stimulus probabilities on a long-term time scale. We suggest here a computational model which integrates prior information with short-term, long-term, and alternation-based experiential influences on P300 amplitude fluctuations. To evaluate the new model, we measured trial-by-trial P300 amplitude fluctuations in a simple two-choice response time task, and tested the computational models of trial-by-trial P300 amplitudes using Bayesian model evaluation. The results reveal that the new digital filtering (DIF) model provides a superior account of the trial-by-trial P300 amplitudes when compared to both Squires et al.'s (1976) model, and Mars et al.'s (2008) model. We show that the P300-generating system can be described as two parallel first-order infinite impulse response (IIR) low-pass filters and an additional fourth-order finite impulse response (FIR) high-pass filter. Implications of the acquired data are discussed with regard to the neurobiological distinction between short-term, long-term, and working memory as well as from the point of view of predictive coding models and Bayesian learning theories of cortical function.

Keywords: Bayesian surprise; P300; digital filtering; event-related brain potentials; predictive surprise; single trial EEG.

Figure 1

Block diagram of the new digital filter (DIF) model with input g_k(n) in (10) and output P_k(n) in (9), digital filter transfer functions H(f) with 0 ≤ f ≤ f_s/2, a stimulus presentation rate of f_s (= 2/3 Hz), and probability normalizing constant 1/C.

Figure 2

Block diagram of a first-order infinite impulse response length (IIR) filter with transfer function H_S(f ) equivalent to (11) or (12). Elements denote a delay of one trial. At the adder element ⊕, updating of the weighted output γ_S·c_S,k(n − 1) of the last trial n − 1 with the weighted input (1 − γ_S)·g_k(n − 1) of the last trial results in the current output c_S,k(n). Note that via γ_S·c_S,k(n − 1), all preceding inputs (and outputs) influence the current output, though for the short-term memory, the influence of trials not in the recent past is negligible.

Figure 3

Block diagram of a first-order infinite impulse response length (IIR) filter with transfer function H_L(f,n) equivalent to (13) or (14). Elements denote a delay of one trial. At the adder element ⊕, updating of the weighted output γ_L,n−1·c_L,k(n − 1) of the last trial n − 1 with the weighted input (1 − γ_L,n−1)·g_k(n − 1) of the last trial results in the current output c_L,k(n). Note that via γ_L,n−1·c_L,k(n − 1), all preceding inputs (and outputs) influence the current output.

Figure 4

Amplitude responses of the long-term (H_L(f,n), dash-dotted and solid low-pass curves), short-term (H_S(f ), dashed low-pass curve) and alternation (H_Δ(f ), dashed high-pass curve) filters of the DIF model as a function of the input signal frequency f (logarithmic scale!). The dynamic long-term filter (dash-dotted and solid curves) is shown for n = 1 and n = N = 192.

Figure 5

Grand-average waveforms (A,B) and topographic maps (C,D) of P300 amplitudes. (A,C) Probability effect on P300 amplitudes in the [0.3, 0.7] probability category. (C) The probability maps show the scalp topography of the rare-frequent difference wave in the [0.3, 0.7] probability category at various points in time (276–396 ms, divided into five windows of 24 ms each). (B,D) Sequence effect on P300 amplitudes in the [0.5, 0.5] probability category. Note that sequences of four successive stimuli are illustrated; a signifies a particular stimulus (b the other one). Note further that the two solid traces, originating from the abaa and the baba sequences, respectively, show reversed P300 amplitudes. Specifically, for the single -b- sequence abaa, the P300 waveform lies amongst those from dual -bb- sequences, whereas for the dual -bb- sequence baba, the P300 waveform appears indistinguishable from the waveforms from single -b- sequences. As further detailed in the Discussion, this amplitude reversal is attributed to the disconfirmation of alternation expectation in the abaa sequence, was well as to the confirmation of alternation expectation in the baba sequence. (D) Sequence maps show the scalp topography of the bbba-aaaa difference wave in the [0.5, 0.5] probability category at various points in time (292–412 ms, divided into five time windows of 24 ms each).

Figure 6

Log-Bayes factors, ln(BF_{DIF − XXX}), under variation of the model parameters. The upper contour always shows ln(BF_{DIF − MAR}), the lower one ln(BF_{DIF − SQU}). The “V” marks the parameter combination with the maximum log-Bayes factor, cf. Table 3. (A) Variation of the free short-term parameters β_S and α_S. (B) Variation of the free long-term parameters τ₁ and τ₂. In order to keep the lower contour visible, the log-Bayes factors for the 10 smallest values for τ₂ are not displayed. (C) Variation of the free alternation parameters γ_Δ,2 and α_Δ.

Figure 7

Timeline plots for one exemplary participant. (A,C,E) Probability category [0.5, 0.5]. (B,D,F) Probability category [0.3, 0.7]. Green symbols denote the stimuli s(n) ∈ {1,2} as they occurred. (A,B) Subjective probabilities P_k=1(n) and expectancies E_k=1(n) from (5), (7), and (9) for MAR, SQU, and DIF, respectively, for stimulus s(n) = 1. (C,D) Subjective probabilities P_k=s(n)(n) and expectancies E_k=s(n)(n) for the actually presented stimulus k = s(n). (E,F) The measured P300 estimates Y_ℓ(n) and the model-based P300 estimates ${\hat{Y}}_{\ell }\left(n\right)$ of the MAR, SQU, and DIF models, respectively.

Figure 8

Tree diagrams of measured P300 estimates Y_ℓ(n) and model-based P300 estimates ${\hat{Y}}_{\ell }\left(n\right)$ as a function of the sequence of preceding stimuli. Within each order (0–3), the stimulus sequence is labeled, and related sequences are connected by lines. (A) For both stimuli on probability category [0.5, 0.5]. (B) For the frequently occurring stimulus b on probability category [0.3, 0.7]. (C) For the rarely occurring stimulus a on probability category [0.3, 0.7].

Figure 9

Block diagram of the fourth-order finite impulse response (FIR) filter H_Δ(f ). Elements represent a delay of one trial, the multipliers γ_Δ,i compose the filter coefficients and C_Δ constitutes a normalizing constant.

Figure A1

Illustration of the equivalence of the digital filter (14) (Figure 3) to the count function as in (13). (A) Block diagram of the DIF model’s long-term memory filter of Figure 3 and (14) with input signal g_k(n) and output signal c_L,k(n). (B) Block diagram of a filter equivalent to (A), where the multiplier (1 − γ_{L,n − 1}) has been moved to the right. (C) Block diagram of a filter equivalent to (B), where the multiplier (1 − γ_{L,n − 1}) has been moved even further to the right. (D) Block diagram of a filter equivalent to (C), where the multiplier γ_{L,n − 1}/(1 − γ_{L,n − 1}) has been moved to the right of the delay unit in the lower branch.

Figure A2

The dynamics of the coefficients γ_L,n (solid) and β_L,n (dashed) over trials n = 1, …, N, with N = 192.