Graph Modeling of Macroscale Brain Activity

Department of Radiology and Biomedical Imaging, University of California at San Francisco, USA


Ashish Raj (

Eva M Palacios (

Pratik Mukherjee (


Tuesday, July 16th - Wednesday, July 17th 2019 (9:30am – 1:10pm)

Brief Description

One of the most important questions in computational neuroscience is how the brain's structural wiring gives rise to its function and its patterns of activity. Although numerical models of single neurons and local microscopic neuronal assemblies, ranging from simple integrate-and-fire neurons to detailed multi-compartment and multi-channel models have been proposed, it is unclear if these models can explain structure-function coupling at meso- or macroscopic scales. Therefore, there is a need for more direct models of structural network-induced neural activity patterns and a need for collecting various modeling efforts together in one place, in order to catalyze discussion of how these approaches can be combined to produce new models that are more effective than their individual constituent elements. In this workshop we have put together recent modeling efforts aimed towards filling the chasm between neuron-level and whole brain activity modeling. We focus largely but not exclusively on graph models for this purpose, since at the macroscopic scale, brain regions interact with each other via long range projection fibers – a network organization that is best addressed using graph theory. Graph theory will play an increasingly important role in attempts to understand the massive amounts of data generated by large collaborative projects such as the Human Connectome Project.

Tuesday July 16th (9:30am-1:10pm)

Theme 1: Biologically informed models of neuronal ensembles


Bill Lytton, SUNY Downstate Medical Center, New York, USA

Title: Biomimesis for computer simulation: multiscale modeling to connect micro, meso, and macro

Abstract: Modeling and simulation are widely acknowledged to be essential tools for understanding the brain and will be increasingly needed as innovative neurotechnologies are developed that add new molecular, cellular and circuit data. One advantage of neural simulation is the prospect of connecting hitherto incommensurable datasets from cell electro- and chemophysiology, up through microconnectomics, and on to the macroconnectomics of wide area networks. Bridging these scales will be required in order to understand brain function, and to reveal the many complementary neural codes that are involved.


Willem de Haan , VUmc, Amsterdam, The Netherlands

Title: Connectome-coupled neural mass model of macroscopic brain activity: the essential tool for activity-targeting treatment development in Alzheimer’s disease?

Abstract: In this presentation I will give a short overview of major achievements and dilemmas in AD network analysis (with an emphasis on neurophysiological techniques such as EEG and MEG), and suggest a way in which brain network research may get closer to obtaining a clinical role: the prediction of treatment strategies focused on the modification of brain activity and preservation of functional brain network status. An intriguing, relatively recent finding is pathological neuronal hyperactivity in early (preclinical) stage AD, in contrast to the later widespread slowing and dysconnectivity. Assuming that cognition benefits by optimization of functional network characteristics, preventing or diminishing this early, abnormal activity may have a palpable clinical impact, and there is strong evidence to support this. There are many ways to modify brain activity (e.g. medication, direct stimulation), but the question is: where and how do we stimulate or inhibit the brain to preserve normal network function? I will demonstrate a dynamic brain model based on neural masses coupled according to human (DTI-based) structural network topology. This way, effects of changes in neuronal activity parameters on the large-scale brain network structure and function can be examined. In the model, AD-resembling damage and virtual therapeutic interventions that target neuronal excitability are simulated simultaneously, leading to counterintuitive outcomes. The combination of neural mass modeling and graph theoretical analysis can offer a test environment in which positive and negative influences on brain activity and network integrity can be systematically explored, with the aim to find effective countermeasures against neurodegenerative network damage.

10:50-11:20am: Coffee Break


Prejaas Tewarie, 1) University of Nottingham, United Kingdom, 2) VU University Amsterdam, Netherlands

Title: Explaining frequency specific spatiotemporal patterns of electrophysiological networks.

Abstract: Electrophysiological networks as estimated with magnetoencephalography or electroencephalography are characterized by frequency specific spatial patterns. At the same time these networks can be characterized by their evolution in the time domain. I will explore how these spatiotemporal networks can emerge, from different angles 1) structural-functional network relationships; 2) concepts from multilayer network modeling; and 3) the use of neural mass modeling and linear stability analysis. With respect to structural-functional network relationships I will explore the role of eigenmodes of the structural network to explain emerging functional connectivity patterns in different frequency bands. A multilayer network model is subsequently used as another angle to explain the same phenomenon. Lastly, I will explore how neural mass modeling can help to explain the formation and dissolution of functional brain networks in the time domain.

Theme 2: Biologically informed graph models of brain activity (Part1)


Pratik Mukherjee , University of California, San Francisco, USA

Title: How white matter microstructure & connectivity can inform graph models of brain activity.

Abstract: Diffusion MRI has made enormous strides over the past three decades and is now the primary method to probe the living human brain across orders of magnitude of scale: from tissue microarchitecture to the whole-brain structural connectome. Greatly improved gradient strength and speed, innovative pulse sequences, combined with novel biophysical modeling of diffusion MRI data, now enable the noninvasive measurement of biologically meaningful microstructural parameters such as axonal and dendritic density and fiber orientation dispersion. These granular diffusion MRI data are also integrated into advanced fiber tractography algorithms to generate more comprehensive and accurate structural connectomes. In this lecture, I review recent advances in white matter microstructural and connectomic mapping. First, I focus on mapping the anatomic embedding of the whole-brain structural network, which identifies special zones of vulnerability that are of particular importance in many white matter diseases across the lifespan, from premature infants to the elderly. Second, I review progress in applying spectral graph theory to decompose the human macroscale connectome into its fundamental “eigenmodes”. These structural eigenmodes provide a robust and parsimonious basis set with which to describe functional connectivity networks from fMRI and MEG, as well as to characterize the unique perturbations of brain network organization and activity caused by diverse neurological and psychiatric disorders. Finally, I explore how white matter microstructural parameters might be integrated with structural connectivity to better inform graph models of fast long-distance information transmission in the human brain.


Sebastien Naze, IBM Research, USA

Talk title: Sensitivity analysis of the connectome harmonics framework and its application to neurodegenerative disorders.

Abstract: Decomposition of the connectome into Laplacian eigenmodes has been suggested to relate to several neurodegenerative disorders such as dementia, Alzheimer and possibly Huntington’s disease (Raj et al., 2012). Recently, a new framework called connectome harmonics proposed that the default-mode network is also present in the low frequency modes of the Laplacian decomposition (Atasoy et al. 2016, 2017). In short, the discrete Laplacian operator is constructed by combining long-distance white-matter connectivity estimated from diffusion MRI data and local gray matter structural connectivity of the cortical surface. The resulting matrix is decomposed into Laplacian eigenmodes, which are called "connectome harmonics". Through a sensitivity analysis of the connectome harmonics computation that we incorporated into an existing open-source pre-processing pipeline (Proix et al., 2016), we describe how robust harmonic patterns can be obtained from diffusion MRI data and tractography. Specifically, we investigate how the balance of local versus long-range connectivity and the selective trimming of tracks based on their distance to cortical mesh affect the spatial patterns forming on the cortical surface. The impact of several framework parameters (cortical surface types and templates, mesh resolution, etc) are evaluated with respect to harmonics robustness. We find that the number of tracks retained to compute the graph Laplacian is an influential factor of the connectome harmonics reliability. Small changes in the shape of the cortical surface that occur during the downsampling of the cortical mesh can also alter the higher frequency harmonics, but low-frequency connectome harmonics are generally robust to processing method variations. We discuss that prospective disease structural changes, such as the atrophy of inter-hemispheric callosal fibers observed in certain pathologies such as Huntington’s disease (McColgan et al., 2017), can be applied to the benchmarked healthy connectome to predict functional deficits associated with the disease conditions.

Atasoy S, Donnelly I, Pearson J (2016) Human brain networks function in connectome-specific harmonic waves. Nature Communications 7:10340.

Atasoy S, Deco G, Kringelbach ML, Pearson J (2017) Harmonic Brain Modes: A Unifying Framework for Linking Space and Time in Brain Dynamics. The Neuroscientist:107385841772803.

McColgan P, Seunarine KK, Gregory S, Razi A, Papoutsi M, Long JD, Mills JA, Johnson E, Durr A, Roos RA (2017) Topological length of white matter connections predicts their rate of atrophy in premanifest Huntington’s disease. JCI insight 2.

Proix T, Spiegler A, Schirner M, Rothmeier S, Ritter P, Jirsa VK (2016) How do parcellation size and short-range connectivity affect dynamics in large-scale brain network models? NeuroImage 142:135–149.

Raj A, Kuceyeski A, Weiner M (2012) A network diffusion model of disease progression in dementia. Neuron 73:1204–1215.

Wednesday July 17th (9:30am-1:10pm)

Theme 2: Biologically informed graph models of brain activity (Part 2)


Ashish Raj, University of California, San Francisco, USA

Title: A linear spectral graph model of brain activity

Abstract: Although neural mass models that involve large coupled non-linear differential equations have previously been proposed for capturing these structure-function relationships, they can be unwieldy, difficult to compute, and are only accessible via simulations. Consequently, the essential minimal rules of organization and dynamics of the brain remain unknown. Furthermore, due to their nonlinear and stochastic nature, model parameter inference is typically ill-posed, computationally demanding and manifest with inherent identifiability issues. Here we present for the first time a linear and analytically accessible model that extends our prior work on linear network diffusion modeling that was capable of predicting a subject's resting state BOLD functional connectivity matrix from their SC Laplacian matrix. The extended model incorporates all frequencies of oscillations, is fast and in a closed form, given by the eigen-modes of the graph Laplacian. This simple graph model can predict both the spectral and spatial patterns of wide-band brain activity measurable on MEG. Intriguingly, the model is capable of correctly reproducing spatial gradients in the spectral response, despite having only global parameters. This suggests that the whole brain network topology might be the key mediator of macroscopic brain activity, rather than region-specific rhythm generators.


Alex Leow, MD PhD, University of Illinois, Chicago

Title: Can we vectorize the resting state and compute the speed of mind? Embedding brain dynamics using manifold learning and spectral graph theory with applications to fMRI and EEG.

Abstract: Modern functional magnetic resonance imaging (rs-fMRI) and quantitative electroencephalography (q-EEG) provide a wealth of information about the inherent functional connectivity of the human brain, both at rest and during tasks. However, understanding the nonlinear topology of brain dynamics remains a challenge. Here, I will discuss a class of novel graph embedding techniques, to study the nonlinear topology (i.e., the intrinsic geometry) of the rs-fMRI connectome, as well as EEG-derived state-space trajectories of the mind. These techniques are closely related to graph-cut techniques developed in spectral graph theory, as well as recent developments of maximum mean discrepancy (MMD)-based kernel methods defined in a reproducing kernel Hilbert space (RKHS).

10:50-11:30am: Coffee Break

Theme 3: Phenomenological graph theoretic models of activity


Roser Sala-Llonch, University of Barcelona, Barcelona, Spain

Title: Methodological choices related to the estimation of brain Functional Connectivity signals.

Abstract:Functional Connectivity (FC) signals extracted from fMRI can inform brain states and they have been useful to provide understanding of changes associated with age and disease. A common approach for whole-brain network modelling is to study functional dependencies between a set of regions or nodes, which can be defined using a variety of methodologies and at different granularity levels. Using these approaches, it has been possible to characterize the functional architecture of the brain and the interactions between and within the main large-scale networks. However, the estimation of FC signals depends on several methodological choices, including the preprocessing steps, the definition of nodes and the estimation of dependency measures. I will discuss some of the recent findings related with the discriminability and interpretability offered by FC features obtained using different methodological pipelines. For that, I will first present results from multivariate pattern analyses for classifying basic steady states. In addition, I will also discuss some of the applications of network modeling for understanding functional changes in aging and neurodegenerative diseases.

Sala-Llonch R, Smith SM, Woolrich M, Duff EP. (2019) Spatial parcellations, spectral filtering, and connectivity measures in fMRI: Optimizing for discrimination. Hum Brain Mapp. 2019 40(2):407-419.


Gorka Zamora-López, Pompeu Fabra University, Barcelona, Spain

Title: Model-based graph theory to the rescue: a new framework to interpret the relation between structural connectivity and function.

Abstract:The use of graph theory has been essential in the past two decades to investigate brain connectivity. However, classical graph theory comes with two main limitations which have restricted (and often obscured) the interpretation of results. (i) Graph theory was designed to study binary graphs. Adaptation to weighted networks has been clumsy and ad-hoc. And (ii) a very much ignored aspect, the definition of most graph metrics implicitly assume an underlying dynamic model, which is kept away from the eyes of the user and does not necessarily match the dynamical category of the real system under study. Here, we propose a generalization of graph theory in which the underlying dynamic model is explicit [1]. It allows to redefine graph metrics such that the link weights are a built-in aspect of the metric and temporal evolution becomes a natural characteristic of the analysis. We then apply this formalism to study brain connectivity [2]. The few model parameters are infered by estimating effective connectivity from resting-state dynamics, thus tuning the analysis to the particular system at hand. The framework allows to identify novel features of the information flow in the network, e.g., differentiated input and output roles of structural hubs, and identifying a time-scale separation between ‘early’ and ‘late’ integration motifs.

[1] M. Gilson, N.E. Kouvaris, G. Deco and G. Zamora-López "Framework based on communicability and flow to analyze complex network dynamics." Phys. Rev. E 97, 052301 (2018).

[2] M. Gilson, N.E. Kouvaris, et al. "Network analysis of whole-brain fMRI dynamics: A new framework based on dynamic communicability" Submitted.



Final Questions & Remarks

Target audience

This workshop will provide attendees with detailed knowledge of the area that spans fundamental concepts through to advanced applications. It is thus suitable for people with varying levels of experience. Graph theoretic analysis of neuroscientific data is inherently interdisciplinary, and provides a unitary framework that transcends measurement techniques, resolution scales and also different levels of complexity of the mathematical approaches. The presented integrated program is likely to be helpful in providing a comprehensive understanding of the field.

Our target audience includes neuroscientists with a background in the biological or psychological sciences with some prior exposure to graph theory, and individuals with a more quantitative computational background who have knowledge of neuronal models and are interested in how graph theory can be applied to characterize neural networks.