## The Annals of Applied Statistics

- Ann. Appl. Stat.
- Volume 11, Number 2 (2017), 868-897.

### Bayesian inference of high-dimensional, cluster-structured ordinary differential equation models with applications to brain connectivity studies

Tingting Zhang, Qiannan Yin, Brian Caffo, Yinge Sun, and Dana Boatman-Reich

#### Abstract

We build a new ordinary differential equation (ODE) model for the directional interaction, also called effective connectivity, among brain regions whose activities are measured by intracranial electrocorticography (ECoG) data. In contrast to existing ODE models that focus on effective connectivity among only a few large anatomic brain regions and that rely on strong prior belief of the existence and strength of the connectivity, the proposed high-dimensional ODE model, motivated by statistical considerations, can be used to explore connectivity among multiple small brain regions. The new model, called the modular and indicator-based dynamic directional model (MIDDM), features a cluster structure, which consists of modules of densely connected brain regions, and uses indicators to differentiate significant and void directional interactions among brain regions. We develop a unified Bayesian framework to quantify uncertainty in the assumed ODE model, identify clusters, select strongly connected brain regions, and make statistical comparison between brain networks across different experimental trials. The prior distributions in the Bayesian model for MIDDM parameters are carefully designed such that the ensuing joint posterior distributions for ODE state functions and the MIDDM parameters have well-defined and easy-to-simulate posterior conditional distributions. To further speed up the posterior simulation, we employ parallel computing schemes in Markov chain Monte Carlo steps. We show that the proposed Bayesian approach outperforms an existing optimization-based ODE estimation method. We apply the proposed method to an auditory electrocorticography dataset and evaluate brain auditory network changes across trials and different auditory stimuli.

#### Article information

**Source**

Ann. Appl. Stat., Volume 11, Number 2 (2017), 868-897.

**Dates**

Received: June 2016

Revised: January 2017

First available in Project Euclid: 20 July 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.aoas/1500537727

**Digital Object Identifier**

doi:10.1214/17-AOAS1021

**Mathematical Reviews number (MathSciNet)**

MR3693550

**Zentralblatt MATH identifier**

06775896

**Keywords**

Bayesian inference ODE models cluster structure directional brain networks network edge selection

#### Citation

Zhang, Tingting; Yin, Qiannan; Caffo, Brian; Sun, Yinge; Boatman-Reich, Dana. Bayesian inference of high-dimensional, cluster-structured ordinary differential equation models with applications to brain connectivity studies. Ann. Appl. Stat. 11 (2017), no. 2, 868--897. doi:10.1214/17-AOAS1021. https://projecteuclid.org/euclid.aoas/1500537727