Bayesian Multivariate Gaussian Random Walk Regression for ERP Estimation

The present analysis implements a multivariate Gaussian random walk (MGRW).

Model

The present model attempts to estimate event-related potentials (ERPs) across the whole epoch. To that aim, we use an MGRW prior to model the voltage changes across time plus a Gaussian noise parameter. Covariance is assumed as a diagonal matrix of ones. A normal distribution (likelihood) is used for the observed voltages with a half-normal distribution as error parameter.

Σ₁,...Σ_C = E × E diagonal matrix

g₁,... g_C ~ GRW(0,1), S × E

x₁,... x_C = Σ_cg_c^T

B = S × E × C matrix of all x

α_s ~ Normal(0, 0.05), S × 1

μ = α_s + B^T

σ ~ HalfNormal(0.05) + 1

y ~ Normal(μ, σ)

Where C = 4 mandarin tones (tone 1... tone 4), E = EEG electrodes (32), and S = number of samples (282, 100ms baseline, 1s epoch). Data comes from a tone detection oddball task (Tone 4 was the deviant target, 25% of total stimuli), completed by learners and non-learners of Chinese Mandarin. We fit two models, as described above, to data from each learners and non-learners.

We sampled the model using Markov chain Monte Carlo (MCMC) No U-turn sampling (NUTS) with 2000 tuning steps, 2000 samples, 4 chains. The model sampled well, with 1.01 > R̂ > 0.99; BFMIs > 0.9, and bulk ESS > 2000 for all parameters. Ranked trace plots ("trank plots") evidence excellent mixing of chains (see grw_learners/tranks/ and grw_learners/tranks/ folders).

Results

The estimates from learners indicate that the target tone (tone 4) induced a strong positive voltage deflection after ~200ms respect to the non-target tones at Pz (i.e., tone 4 induced a P3b). Image below shows the contrasts between tone 4 and each other tone from posterior distributions.

The estimates from non-learners indicate that the target tone (tone 4) induced a milder positive voltage deflection after ~200ms respect to the non-target tones at Pz. Image below shows the contrasts between tone 4 and each other tone from posterior distributions.

Predictions from the posterior for learners indicate more uncertainty but the P3b is still present. Image below shows contrasts between tone 4 and each other tone from Pz predictions.

Predictions from the posterior for non-learners also indicate more uncertainty but there is still a mild P3b. Image below shows contrasts between tone 4 and each other tone from Pz predictions.

Images below show posterior distributions from the learners’ model as scalp topographies (posterior of tone 4 minus all other tones combined).

5% highest density intervals (HDI)

Posterior means

95% highest density intervals (HDI)

Images below show posterior distributions from the non-learners model as scalp topographies (posterior of tone 4 minus all other tones combined).

5% highest density intervals (HDI)

Posterior means

95% highest density intervals (HDI)

Conclusion

The estimates show that there is a difference of P3b amplitude between learners and non-learners, but some uncertainty. The predictions indicate that the models are efficient. However, the current models lack a proper covariance matrix for electrodes, which is relevant to understand voltage variation across the scalp. Further development of these models is required.