Mixed Eects Spectral Vector Autoregressive Model With Application to Brain Connectivity

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Mixed Eﬀects Spectral Vector

Autoregressive Model: With Application to

Brain Connectivity

Anastasiia Malinovskaia

Department of Statistics, King Abdullah University of Science and

Technology (KAUST)

October 7, 2022

Abstract

The primary goal of this paper is to develop a method that quantiﬁes how activity

in one brain region can explain future activity in another region. Here, we propose

the mixed eﬀects spectral vector-autoregressive (ME-SpecVar) model to investigate

diﬀerences in dynamics of dependence in a brain network between healthy children

and those who diagnosed with ADHD. Speciﬁcally, ME-SpecVar model will be used

to formally test for signiﬁcant connectivity structure obtained using ﬁltered EEG

signals in delta, theta, alpha, beta, and gamma frequency bands. Suggested model

allows one stage procedure for deriving Granger causality in common group structure

and variation of subject speciﬁc random eﬀects in diﬀerent frequency oscillations.

The model revealed novel results and showed more signiﬁcant connections in all fre-

quency bands and especially in slow waves in control group. In contrast, children with

ADHD shared a pattern of diminished connectivity and variability of random eﬀects.

The results are consistent with previous ﬁndings about decreased anterior-posterior

connectivity in children with ADHD. Moreover, the novel ﬁnding is that most di-

verse subject speciﬁc eﬀective connectivity parameters in healthy children belong to

parietal-occipital region that is associated with conscious visual attention.

Keywords: ADHD; Eﬀective connectivity; Linear mixed eﬀects model; Spectral decompo-

sition. Vector autoregressive model;

arXiv:2210.03017v1 [stat.AP] 16 Sep 2022

1 Introduction

1.1 Eﬀective connectivity

The goal in this paper is to develop a statistical model and tool that will be used to inves-

tigate diﬀerences in brain eﬀective connectivity between children diagnosed with attention

deﬁcit and hyperactivity disorder (ADHD) and healthy controls. There are several chal-

lenges to this problem. First, the measure of brain eﬀective connectivity needs to go beyond

the usual cross-correlation which captures some form of dependence but is inadequate for

characterizing brain networks. Here, we will explore other meaningful and interpretable

measures of eﬀective connectivity (EC). The second challenge is that brain responses vary

between people within groups and between groups. Thus, we have to develop a statistical

model that accounts for both between-subject variation and between-group diﬀerences of

EC. Lastly, it is important to study cross-relationships within diﬀerent oscillatory activity

of brain signals. Here, a novel statistical model was developed primarily for analyzing

connectivity using electroencephalograph (EEG) signals collected from the scalp. Elec-

troencephalography (EEG) and functional magnetic resonance imaging (fMRI) are the

main non-invasive methods for collecting information reﬂecting underlying brain functions.

Based on the type of the data and research question, diﬀerent models for eﬀective connec-

tivity can be chosen.

Measures of eﬀective connectivity can be roughly classiﬁed into three classes: Granger

causality (Granger 1980), information-theoretic measures (Schreiber 2000) and dynamic

causal modelling (Friston 2003). Dynamic causal modelling is not an exploratory model

because it assumes certain a-priori input parameters which are used for speciﬁc hypothesis

testing. It captures non-linear interactions between brain regions, however the network

size is limited (Friston 2003). Transfer entropy (TE) is model-free example of information

ﬂow-based theories. Kullback–Leibler divergence is a key moment in the model and it

treats conditional probabilities of coupled regions (Hinrichs et al. 2008). Using TE, the

analysis is applied without prior regions selection on whole-brain electrodes and the imple-

mentation is robust of model misspeciﬁcation (Good et al. 2009), however it can require

suﬃciently large number of time points. Other conventional way of characterizing eﬀective

connectivity is Granger Causality based on vector autoregressive models. Partial directed

coherence (Sameshima & Baccal´a 1999) and directed transfer function (Korzeniewska et al.

2003) are frequency domain extensions and hence they give distinct characterizations of EC

across various frequency bands. One major limitation of this approach is that it captures

only linear interactions, meanwhile EEG signals contain non-linear dependencies (Lopes da

Silva et al. 1989). In addition, whole brain electrodes implementation of VAR models is

problematic due to the curse of dimensionality and overparameterization. Implementing

the VAR model with rchannels will have O(r2) number of parameters which can be com-

putationally costly. Thus, penalized regression techniques and dimension reduction can be

used to alleviate the problem (Tibshirani 1996, Hu et al. 2019).

The models of eﬀective connectivity are relevant and give insights about cognition. Most

of the cognitive functions are anatomically distributed across functionally specialized brain

regions. Mechanisms that coordinate neuronal assemblies are important for understanding

normal cognitive functioning and neuropathological conditions and diseases. The main

novel contribution in this paper is a model that accounts for eﬀective connectivity at

diﬀerent frequency bands to gain new insights about lead-lag dynamics and interconnections

of brain regions at corresponding frequencies. Results obtained from the model might

be linked to existing knowledge about particular frequency oscillations correlations with

cognitive functions and abnormalities.

1.2 Eﬀective connectivity in ADHD

Non-invasive collection of neuronal activity data is the main source of brain information

nowadays. Electroencephalogram (EEG), magnetoencephalogram (MEG) and fMRI tech-

niques are commonly used in neuroscience and clinical research. They reveal the aspects of

abnormal brain activity and develop diagnostics. EEG based eﬀective connectivity stud-

ies broadened understanding of epilepsy, Alzheimer disease, major depression disorder and

attention deﬁciency hyperactivity disorder (ADHD) (Blinowska et al. 2017, Saeedi et al.

2021, Parker et al. 2018). We focus on ADHD ﬁndings in the paper and we conducted

analysis for EEG collected from healthy and ADHD children provided by Motie Nasrabadi

et al. (2020).

Results of eﬀective connectivity studies on ADHD participants are based on diﬀerent

experimental conditions and methods, therefore they lack of consistent outcomes and con-

current patterns. In paper by Ueda and collegues (2020), children with ADHD had signiﬁ-

cant functional hyperconnectivity in the theta range during executive functional tasks but

not during the resting-state (in the delta to beta range) compared with children without

ADHD (Ueda et al. 2020). Another ﬁnding showed a resting state deﬁciency in connectiv-

ity in ADHD and overconnectivity within and between frontal hemispheres while stimulus

presentation (Murias et al. 2006). Chabot and Serfontein (1996) stated reduced parietal

and increased intrahemispheric coherence in frontal and central regions in children with

ADHD (Chabot & Serfontein 1996). The results of Ahmadlou & Adeli (2010) on func-

tional connectivity identiﬁed a signiﬁcant diﬀerence between the two groups in the T5

and O2 electrodes in the delta and theta frequencies. In the paper Ekhlasi et al. (2021)

posterior to anterior patterns of information ﬂow in theta frequency bands were disrupted

compared to healthy children. Information ﬂow between anterior regions and connections

from central and lateral parietal areas to Pz electrode were signiﬁcantly higher in healthy

individuals than in the ADHD group in the beta band.

In general, analysis showed abnormal hyperconnectivity for certain regions in speciﬁc

frequency bands and overall lack of connectivity compared to healthy controls. Current

approaches share several limitations. The main approaches do not account for subject spe-

ciﬁc variations within the group and an average is used to represent the group connectivity

in the vast majority of papers. Another observation indicates the lack of work that can

attribute brain eﬀective connectivity on oscillatory activity. Our proposed ME-SpecVAR

approach can overcome these limitations by using mixed eﬀects model for ﬁltered signals

in diﬀerent frequency bands.

1.3 Overview of the proposed model: Mixed Eﬀects Spectral

VAR (ME-SpecVAR)

Most statistical models of brain eﬀective connectivity are able to make statements about

diﬀerences between groups of patients (healthy vs disease) and between experimental con-

ditions. A standard approach for such kind of questions is to ﬁnd estimates for each subject

individually, and then average parameters inside each condition or group. Although, these

approaches can be ineﬀective because they do not account for between-subject variation.

Moreover, in some paradigms it could be essential to account for variation across sub-

jects. Linear mixed eﬀects vector autoregressive model (ME VAR) (Gorrostieta et al.

2012) tackles the accounting uncertainty of subject speciﬁc estimates and the authors pro-

posed method of single-stage estimation of random variation between subjects. Moreover,

Granger causality and connectivity diﬀerences between experimental conditions are cap-

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MixedEectsSpectralVectorAutoregressiveModel:WithApplicationtoBrainConnectivityAnastasiiaMalinovskaiaDepartmentofStatistics,KingAbdullahUniversityofScienceandTechnology(KAUST)October7,2022AbstractTheprimarygoalofthispaperistodevelopamethodthatquantieshowactivityinonebrainregioncanexplainfutureactiv...

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