Hands on brain dynamics on the connectome sinergia summer school: 2021 Oct 11-12

Tutorial 6: Computational modelling

Overview

Teaching: 15 min
Exercises: 45 min
Questions

  • What is a Stuart-Landau oscillator (Hopf oscillator)?

  • How can a model based on coupled oscillators simulate the dynamics of a simple network?

  • How can such a model be used to characterize certain dynamical aspects of brain activity?

Objectives

  • Generating local dynamics: single-node oscillator

  • Visualize and describe the dynamics of a single-node oscillating system near a Hopf bifurcation.

  • Change the regime (bifurcation parameter) of the oscillator to generate different local dynamics.

  • From local to global: network of coupled oscillators

  • Understand how different oscillators can be linked to obtain a network of coupled oscillators and simulate data.

  • Characterize changes in certain aspects of network dynamics as a function of different model parameters (coupling parameter, connectome).

  • Using a model of coupled oscillators to characterize brain dynamics

  • Understand the role of the model parameters to constrain the model with real data.

  • Fit the model with empirical resting-state human fMRI data and characterize certain aspects of brain activity.

  • Understand the use, applicability and limitations of whole-brain models.

Introduction

Computational brain network models have emerged as a powerful tool to investigate the dynamics of the human brain. In a broad sense, modelling refers to idealizing (or simplifying while keeping the essential ingredients) the processes that generate the observed phenomena in a real system. Theoretical models are often applied to study complex non-linear systems, such as the brain, in order to investigate the interplay between known dynamical and structural features, e.g. combining SC with local dynamics to generate resting-state FC. For this, it is required to explain the relevant observable features and to ensure a robust interpretation of the models’ parameters to link them back to biological variables. Thus, theoretical models need to achieve a trade-off between simplicity and richness to explain the mechanisms underlying complex biological systems.

In this tutorial, a relatively simple whole-brain model will be introduced based on a set of coupled oscillators near a Hopf bifurcation. This model is a deterministic model with a bottom-up approach that has been used to describe the brain’s rsfMRI network activity in different experimental contexts. The model assumes that the brain’s resting-state activity emerges from the interaction between brain regions in an interconnected neuroanatomical network. Furthermore, the local dynamics are modelled by Stuart-Landau oscillators, which allow us to study the phase and amplitude interactions in large networks. This whole-brain model has been successfully applied to simulate the network non-linear dynamics occurring at the ultra-slow scale of resting-state BOLD signals. Furthermore, the model global and regional parameters obtained from the model can discriminate between brain states, thereby improving our understanding of brain network and local alterations in different brain states.

Overline

  1. We will focus on understanding the key properties of a Hopf oscillator. In particular, we will investigate the role of its bifurcation parameter, which describes if the system presents oscillatory or noisy activity.

  2. We will study the behaviour of a network of coupled Hopf oscillators in a simple simulated network. The complexity of the network structure and dynamics will be altered by changing the underlying connectivity matrix and the global coupling parameter.

  3. Finally, the previously introduced whole-brain model will be used to fit empirical data of resting state fMRI brain activity. The model parameters will be used to interpret certain aspects of the underlying dynamics.

Key Points