1. Computational and Systems Biology
  2. Neuroscience
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Multivariate stochastic volatility modeling of neural data

  1. Tung D Phan  Is a corresponding author
  2. Jessica A Wachter
  3. Ethan A Solomon
  4. Michael J Kahana  Is a corresponding author
  1. University of Pennsylvania, United States
Research Article
Cite this article as: eLife 2019;8:e42950 doi: 10.7554/eLife.42950
11 figures, 4 tables and 1 additional file

Figures

Empirical characteristics of iEEG.

(A) Sample raw iEEG time series during a resting state (count down non-task) period for subject R1240T. (B) Detrended iEEG timeseries after removing autoregressive components. (C) Empirical variance timeseries calculated using a rolling-window of size 20. (D) Distribution of empirical variance with the blue curve showing the estimated empirical density using kernel density estimation and the red curve showing the best-fitting log-normal density to the data.

https://doi.org/10.7554/eLife.42950.002
Task design and analysis.

(A) Subjects performed a verbal free-recall task which consists of three phases: (1) word encoding, (2) math distraction, and (3) retrieval. (B) 96 Participants were implanted with depth electrodes in the medial temporal lobe (MTL) with localized subregions: CA1, CA3, dentate gyrus (DG), subiculum (Sub), perirhinal cortex (PRC), entorhinal cortex (EC), or parahippocampal cortex (PHC). (C) To construct a directional connectivity network, we applied the MSV model to brain signals recorded from electrodes in the MTL during encoding. We analyzed the 1.6 s epochs during which words were presented on the screen. The network reflects directional lag-one correlations among the implied volatility timeseries recorded at various MTL subregions. (D) The upper row shows a sample of an individual patient’s raw voltage timeseries (blue) recorded from two electrodes during a word encoding period of 1.6 s, and the lower row shows their corresponding implied volatility timseries (gray) estimated using the MSV model.

https://doi.org/10.7554/eLife.42950.003
Correlation between volatility and spectral power over a frequency range from 3 to 180 Hz.

We fit a Gaussian process model to estimate the functional form of the correlation function between volatility and spectral power (solid blue line). The 95% confidence bands were constructed from 96 subjects (dashed gray lines). The red line shows the null model. We observe a significantly positive correlation between volatility and spectral power, and the correlation increases with frequency.

https://doi.org/10.7554/eLife.42950.004
Classification of subsequent memory recall.

(A) Average AUC of the classifier trained on spectral power across 42 subjects with at least three sessions of recording (blue). The red line indicates the average AUC of the classifier trained on volatility. (BΔAUC = AUCvolAUCpower as a function of frequency estimated by using a Gaussian regression model (dashed gray lines indicate 95% confidence bands). The red line shows the null model. We observe that the classifier trained on volatility performs at least as well as the one trained on spectral power across the frequency spectrum. We find that functional form of ΔAUC is significantly different from the function (χ112=42, P < 10-5) using a Gaussian process model, suggesting that the difference in performance between the volatility classifier and the spectral power classifier is significant.

https://doi.org/10.7554/eLife.42950.005
MTL directional connectivity network.

The MTL electrodes were divided into four subregions: hippocampus (Hipp.), parahippocampal cortex (PHC), entorhinal cortex (EC), and perirhinal cortex (PRC). The directional connectivity from region I to region J, CIJ=1|I||J|iI,jJ𝜷ij, was calculated by averaging the entries of the sub-matrix of the regression coefficient matrix 𝜷, whose rows and columns correspond to region I and J respectively. We computed the contrast between the directional connectivity of recalled and non-recalled events: ΔIJ=CIJR-CIJNR for each subject. Solid lines show significant (FDR-corrected) connections between two regions and dashed lines show trending but insignificant connections. Red indicates positive changes and blue indicates negative changes. The directional connectivity from Hipp. to PRC is significant (adj. P < 0.01) and the reverse directional connectivity is also significant (adj. P < 0.05).

https://doi.org/10.7554/eLife.42950.006
Appendix 3—figure 1
Time length analysis.

For k ranging from 30 to 180 with 30 increments and for each T=k(J2+2J), we simulated 30 datasets according to the data-generating process specified by the MSV model. Then, we estimated the connectivity matrix 𝜷MSV for each dataset. We assess the performance of the MSV model in estimating the true connectivity matrix 𝜷true using the log-determinant error metric, log(|det(𝜷MSV-1𝜷true-IJ)|). The figure shows the average performance at each time length with the standard error bars.

https://doi.org/10.7554/eLife.42950.012
Appendix 3—figure 2
Volatility Timeseries Recovery.

The red lines show the true simulated log-volatility timeseries for the first 1000 time points for two channels . The blue timeseries show the estimated log-volatility time series using the MSV model and the gray timeseries are the 95% posterior confidence intervals. The figure demonstrates that the MSV model can estimate the latent log-volatility timeseries well.

https://doi.org/10.7554/eLife.42950.013
Appendix 4—figure 1
Model fit plots for a hippocampal electrode.

The upper panels show the detrended iEEG signals using an AR(p) model for encoding periods of a list of words. The lower panels show the associated estimated latent volatility processes. The red lines indicate the average volatility during the encoding period. Red words are later recalled and blue words are not recalled.

https://doi.org/10.7554/eLife.42950.015
Appendix 6—figure 1
Cross-validation scheme.

For each subject, we train our classifier on all (yellow) but one session and test the performance on the hold-out session (green) and repeat the procedure for each session.

https://doi.org/10.7554/eLife.42950.018
Appendix 6—figure 2
Sample ROC curves (gray).

The black dashed lines indicate ROC curves of at-chance classifiers. The area under the ROC curve (AUC) is a measure of the performance of the classifier across a spectrum of decision thresholds. An AUC of 1 indicates that the classifier can classify recalled items perfectly. An AUC of 0.5 indicates that the classifier is at chance, that is as good as a random coin flip.

https://doi.org/10.7554/eLife.42950.019
Author response image 1
Resting state vs.Encoding Connectivity Contrast.
https://doi.org/10.7554/eLife.42950.024

Tables

Key resources table
Reagent, type
(species) or
resource
DesignationSource
or reference
IdentifiersAdditional
Information
Software and algorithmAvants et al., 2008http://picsl.upenn.edu/software/antsadvanced normalization tool
Software and algorithmYushkevich et al., 2015https://www.nitrc.org/projects/ashsashs
Software and algorithmsklearncPedregosa et al., 2011https://scikit-learn.org/stable/
Software and algorithmThis paperhttp://memory.psych.upenn.edu/Electrophysiological_Datacustom processing scripts
Software and algorithmPTSAThis paperhttps://github.com/pennmem/ptsa_newprocessing pipeline for
reading in iEEG
Appendix 2—table 1
Parameter Recovery.
https://doi.org/10.7554/eLife.42950.010
TruthMSV
DatasetSNRChannelαb1b2b3b4b5σαb1b2b3b4b5σ
10.1613.390.850.200.000.000.000.173.390.860.190.00−0.040.000.17
23.600.000.88−0.100.000.000.193.600.000.87−0.10−0.00−0.010.19
33.550.000.000.870.300.000.193.53−0.00−0.000.850.290.010.20
43.510.000.000.000.710.000.123.510.01−0.000.010.69−0.030.14
53.380.000.000.000.000.800.153.380.00−0.01−0.010.060.790.15
20.2713.600.950.200.000.000.000.213.930.950.19−0.000.000.000.24
23.740.000.90−0.100.000.000.253.82−0.000.90−0.100.00−0.000.25
33.890.000.000.930.300.000.293.81−0.000.000.930.300.000.29
43.050.000.000.000.910.000.283.04−0.000.010.010.910.000.28
53.960.000.000.000.000.950.223.97−0.000.000.00−0.000.940.22
30.4213.250.520.200.000.000.000.423.250.510.200.01−0.020.010.43
23.640.000.57−0.100.000.000.463.630.010.55−0.140.020.020.46
33.180.000.000.580.300.000.313.180.030.010.560.320.030.31
43.440.000.000.000.650.000.413.450.00−0.00−0.000.640.000.41
53.260.000.000.000.000.600.363.26−0.02−0.000.00−0.010.600.36
  1. We generated three datasets with different signal-to-noise ratios. The observed multivariate time-series 𝐲t was simulated according to the data-generating process specified by the MSV model with pre-specified parameters (truth). We then applied the MSV model to the simulated series 𝐲t to recover the parameters of the MSV model. In this simulation study, the non-zero off-diagonal entries of the matrix 𝜷 were fixed across datasets. The diagonal elements of 𝜷 were generated from a uniform distribution on [0.7,0.9], [0.9,1.0], and [0.5,0.7] respectively. The volatilities of volatility of the electrodes were generated from a uniform distribution on [0.1,0.2], [0.2,0.3], and [0.3,0.5] respectively.

Appendix 7—table 1
intra-MTL directional connectivity in the left hemisphere.
https://doi.org/10.7554/eLife.42950.021
Region I region JMean ΔIJSeTNPadj. P
Hipp PRC−0.0440.013−3.494480.0010.009**
PRC Hipp−0.0600.022−2.667480.0110.045*
Hipp EC0.0100.0320.312140.7680.953
EC Hipp−0.0770.067−1.158140.2850.569
PHC PRC−0.0060.040−0.146160.8890.953
PRC PHC−0.0370.028−1.348160.2120.564
PRC EC0.0010.0220.061210.9530.953
EC PRC0.0050.0300.168210.8720.953
Appendix 7—table 2
intra-MTL directional connectivity in the right hemisphere.
https://doi.org/10.7554/eLife.42950.022
Region I region JMean ΔIJSeTNPadj. P
Hipp PRC−0.0100.020−0.471400.6450.838
PRC Hipp−0.0160.029−0.575400.5740.838
Hipp EC0.0110.0300.361140.7330.838
EC Hipp0.0540.0471.144140.2900.838
Hipp PHC−0.0270.032−0.837150.4320.838
PHC Hipp0.0440.0351.232150.2540.838
PRC EC0.0200.0520.378140.7220.838
EC PRC−0.0020.081−0.020140.9850.985
  1. *: P<0.05, **: P<0.01, adjusted p-values were calculated using the Benjamini-Hochberg procedure.

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