##
**Evolutionary state-space model and its application to time-frequency analysis of local field potentials.**
*(English)*
Zbl 1453.62724

Summary: We propose an evolutionary state-space model (E-SSM) for analyzing high-dimensional brain signals, the statistical properties of which evolve over the course of a nonspatial memory experiment. Under the E-SSM, brain signals are modeled as mixtures of components (e.g., an AR(2) process) with oscillatory activity at predefined frequency bands. To account for the potential nonstationarity of these components (because brain responses can vary throughout an experiment), the parameters are allowed to vary over epochs. Compared with classical approaches, such as independent component analyses and filtering, the proposed method accounts for the entire temporal correlation of the components and accommodates nonstationarity. For inference purposes, we propose a novel computational algorithm based on a Kalman smoother, maximum likelihood, and blocked resampling. The E-SSM model is applied in simulation studies and applied to multi-epoch local field potential (LFP) signal data, collected from a nonspatial (olfactory) sequence memory task study. The results confirm that our method captures the evolution of the power of the components across different phases in the experiment, and identifies clusters of electrodes that behave similarly with respect to the decomposition of different sources. These findings suggest that the activity of electrodes does change over the course of an experiment in practice. Thus, treating these epoch recordings as realizations of an identical process could lead to misleading results. In summary, the proposed method underscores the importance of capturing the evolution in brain responses over the study period.

### MSC:

62P10 | Applications of statistics to biology and medical sciences; meta analysis |

62M15 | Inference from stochastic processes and spectral analysis |

62M10 | Time series, auto-correlation, regression, etc. in statistics (GARCH) |

### Keywords:

autoregressive model; brain signals; state-space models; spectral analysis; time-frequency analysis### References:

[1] | Allen, T. A., Salz, D. M., McKenzie, S. and Fortin, N. J. (2016). Nonspatial sequence coding in ca1 neurons.The Journal of Neuroscience36, 1547-1563. |

[2] | Brillinger, D. (1964). A frequency approach to the techniques of principal components, factor analysis and canonical variates in the case of stationary time series. InInvited Paper, Royal Statistical Society Conference, Cardiff Wales. |

[3] | Busch, N. A., Dubois, J. and VanRullen, R. (2009). The phase of ongoing eeg oscillations predicts visual perception.Journal of Neuroscience29, 7869-7876. |

[4] | Buzsaki, G. (2006).Rhythms of the Brain. Oxford University Press. · Zbl 1204.92017 |

[5] | Cheng, Q., Gao, X., Martin, R. et al. (2014). Exact prior-free probabilistic inference on the heritability coefficient in a linear mixed model.Electronic Journal of Statistics8, 3062- 3076. · Zbl 1308.62050 |

[6] | Delorme, A. and Makeig, S. (2004). Eeglab: an open source toolbox for analysis of singletrial eeg dynamics including independent component analysis.Journal of Neuroscience Methods134, 9-21. |

[7] | Deuschl, G. (1999).Recommendations for the Practice of Clinical Neurophysiology: Guidelines of the International Federation of Clinical Neurophysiology. Elsevier. |

[8] | Douc, R., Moulines, E., Olsson, J. and van Handel, R. (2011). Consistency of the maximum likelihood estimator for general hidden markov models.The Annals of Statistics39, 474- 513. · Zbl 1209.62194 |

[9] | Einevoll, G. T., Pettersen, K. H., Devor, A., Ulbert, I., Halgren, E. and Dale, A. M. (2007). Laminar population analysis: estimating firing rates and evoked synaptic activity from multielectrode recordings in rat barrel cortex.Journal of Neurophysiology97, 2174-2190. |

[10] | Epstein, C. L. (2005). How well does the finite fourier transform approximate the fourier transform?Communications on Pure and Applied Mathematics58, 1421-1435. · Zbl 1079.65139 |

[11] | Fiecas, M. and Ombao, H. (2016). Modeling the evolution of dynamic brain processes during an associative learning experiment.Journal of the American Statistical Association111, 1440- 1453. |

[12] | Gao, X., Shahbaba, B. and Ombao, H. (2018). Modeling binary time series using gaussian processes with application to predicting sleep states.Journal of Classification35, 549- 579. · Zbl 1422.62219 |

[13] | Gao, X., Shen, W., Hu, J., Fortin, N. and Ombao, H. (2019). Modeling local field potentials with regularized matrix data clustering. In2019 9th International IEEE/EMBS Conference on Neural Engineering (NER), 597-602. IEEE. |

[14] | Gao, X., Shen, W. and Ombao, H. (2018). Regularized matrix data clustering and its application to image analysis.arXiv preprint arXiv:1808.01749. |

[15] | Gerrard, P. and Malcolm, R. (2007). Mechanisms of modafinil: a review of current research. Neuropsychiatric Disease and Treatment3, 349. |

[16] | Guo, Y. (2011). A general probabilistic model for group independent component analysis and its estimation methods.Biometrics67, 1532-1542. · Zbl 1274.62783 |

[17] | Hamilton, J. D. (1994).Time Series Analysis. Princeton university press, Princeton. · Zbl 0831.62061 |

[18] | Jiru, A. R. (2008).Relationships between Spectral Peak Frequencies of a Causal AR (P) Process and Arguments of Roots of the Associated ar Polynomial. PhD thesis, San Jose State University. |

[19] | Lukemire, J., Wang, Y., Verma, A. and Guo, Y. (2018). Hint: A toolbox for hierarchical modeling of neuroimaging data.arXiv preprint arXiv:1803.07587. |

[20] | Makarova, J., Ibarz, J. M., Makarov, V. A., Benito, N. and Herreras, O. (2011). Parallel readout of pathway-specific inputs to laminated brain structures.Frontiers in Systems Neuroscience,5, 77. |

[21] | Makarova, J., Ortu˜no, T., Korovaichuk, A., Cudeiro, J., Makarov, V. A., Rivadulla, C. and Herreras, O. (2014). Can pathway-specific lfps be obtained in cytoarchitectonically complex |

[22] | 1582GAO ET AL. structures?Frontiers in Systems Neuroscience8, 66. |

[23] | Michel, C., Lehmann, D., Henggeler, B. and Brandeis, D. (1992). Localization of the sources of eeg delta, theta, alpha and beta frequency bands using the fft dipole approximation. Electroencephalography and Clinical Neurophysiology82, 38-44. |

[24] | Mitzdorf, U. (1985).Current Source-Density Method and Application in Cat Cerebral Cortex: Investigation of Evoked Potentials and EEG Phenomena. American Physiological Society. |

[25] | Prado, R. and Lopes, H. F. (2013). Sequential parameter learning and filtering in structured autoregressive state-space models.Statistics and Computing23, 1-15. · Zbl 1322.62219 |

[26] | Shumway, R. H. and Stoffer, D. S. (2013).Time Series Analysis and its Applications. Springer Science & Business Media. · Zbl 0942.62098 |

[27] | Wang, Y. and Guo, Y. (2018). A hierarchical independent component analysis model for longitudinal neuroimaging studies.arXiv preprint arXiv:1808.01557. |

[28] | Wang, Y., Ting, C.-M., Gao, X. and Ombao, H. (2019). Exploratory analysis of brain signals through low dimensional embedding. In2019 9th International IEEE/EMBS Conference on Neural Engineering (NER), 997-1002. IEEE. |

[29] | Wang, Y., Ting, C.-M. and Ombao, H. (2016). Exploratory analysis of high dimensional time series with applications to multichannel electroencephalograms.arXiv preprint arXiv:1610.07684. |

[30] | Whitmore, N. W. and Lin, S.-C. (2016). Unmasking local activity within local field potentials (lfps) by removing distal electrical signals using independent component analysis.NeuroImage132, 79-92. |

[31] | Zhang, K. and Hyv¨arinen, A. (2011). A general linear non-gaussian state-space model: Identifiability, identification, and applications. InJMLR Workshop and Conference Proc., Asian Conf. on Machine Learning, 113-128. |

[32] | Department of Statistics, University of California, Irvine, 2226 Donald Bren Hall, Irvine, CA |

[33] | 92697-1250, USA. |

[34] | E-mail: xgao2@uci.edu |

[35] | Department of Statistics, University of California, Irvine, 2206 Donald Bren Hall, Irvine, CA |

[36] | 92697-1250, USA. |

[37] | E-mail: weinings@uci.edu |

[38] | Department of Statistics, University of California, Irvine, 2224 Donald Bren Hall, Irvine, CA |

[39] | 92697-1250, USA. |

[40] | E-mail: babaks@uci.edu |

[41] | University of California, Irvine, 106 Bonney Research Laboratory Building Department of Neu |

[42] | robiology and Behavior, Irvine, CA 92697, USA. |

[43] | E-mail: norbert.fortin@uci.edu |

[44] | Statistics Program, 4700 King Abdullah University of Science and Technology (KAUST) Thuwal, |

[45] | 23955-6900, Kingdom of Saudi Arabia, USA. |

[46] | E-mail: hernando. |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.