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Mathematical framework for large-scale brain network modeling in The Virtual Brain
Open accessAbstract
In this article, we describe the mathematical framework of the computational model at the core of the tool The Virtual Brain (TVB), designed to simulate collective whole brain dynamics by virtualizing brain structure and function, allowing simultaneous outputs of a number of experimental modalities such as electro- and magnetoencephalography (EEG, MEG) and functional Magnetic Resonance Imaging (fMRI). The implementation allows for a systematic exploration and manipulation of every underlying component of a large-scale brain network model (BNM), such as the neural mass model governing the local dynamics or the structural connectivity constraining the space time structure of the network couplings. Here, a consistent notation for the generalized BNM is given, so that in this form the equations represent a direct link between the mathematical description of BNMs and the components of the numerical implementation in TVB. Finally, we made a summary of the forward models implemented for mapping simulated neural activity (EEG, MEG, sterotactic electroencephalogram (sEEG), fMRI), identifying their advantages and limitations.
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“Models can involve highly parameterized simulations of regional activity (Hansen et al., 2015; Messe et al., 2014; Sanz-Leon et al., 2015), or sparser analytic predictions of FC using graph metrics of mutual communication (Goni et al., 2014; Grayson et al., 2016) (although, both approaches appear to be comparably effective; see Figure 1D).”
From Development of large-scale functional networks from birth to adulthood: A guide to the neuroimaging literature (2017) · cites this paper as Sanz-Leon et al., 2015
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“BNM is typically large-scale whole brain network model comprising of up to 1,000 network nodes (Honey et al., 2007, 2009; Deco et al., 2013a,b; Sanz-Leon et al., 2015).”
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“The Simulation module of the VBI pipeline is designed to be easily interchangeable, allowing researchers to replace it with other simulators, such as TVB ( Sanz-Leon et al., 2015 ), Neurolib ( Cakan et al., 2023 ), Brian ( Stimberg et al., 2019 ), and Brainpy ( Wang et al., 2023a ).”
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Self-citationModel-based estimation of intra-cortical connectivity using electrophysiological data
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“Therefore, the space-time structure of the brain connectivity (Sanz-Leon et al., 2015) must be accounted for to capture completely the physiologically accurate connectivity.”
BackgroundAn, S., Fousek, J., Kiss, Z. H. T., Cortese, F., van der Wijk, G., McAusland, L. B., Ramasubbu, R., Jirsa, V., & Protzner, A. B. (2021) · NeuroImage · doi:10.1016/j.neuroimage.2021.118848 · Cited by 17
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“Integrating these models into whole-brain dynamic simulators like The Virtual Brain (TVB) marks a major advancement in multiscale modelling, allowing for more detailed and reliable simulations that effectively bridge the micro and macro scales [32].”
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“The large-scale brain network model (Equation 1) is fully compatible with previous TVB descriptions (Sanz-Leon et al., 2015; Spiegler & Jirsa, 2013; Spiegler et al., 2016).”
MethodResultSelf-citationMultisensory integration in the mouse cortical connectome using a network diffusion model
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“The Virtual Brain enables the complete workflow from brain images to simulation (TVB; Sanz-Leon et al., 2013, 2015).”
BackgroundMethodSelf-citationErhardt, A. H., Mardal, K., & Schreiner, J. (2020) · Journal of Computational Neuroscience · doi:10.1007/s10827-020-00746-5 · Cited by 13
“and seizures have been considered in Jirsa et al. (2014), Sanz-Leon et al. (2015), Jirsa et al. (2017), Breakspear (2017), Olmi et al. (2019), and Lopes et al. (2020).”
BackgroundNeural field simulator: two-dimensional spatio-temporal dynamics involving finite transmission speed
Nichols, E. J., & Hutt, A. (2015) · Frontiers in Neuroinformatics · doi:10.3389/fninf.2015.00025 · Cited by 13
“The Virtual Brain (Sanz Leon et al., 2013) allows to simulate networks of neural mass models to reproduce global brain activity.”
BackgroundMethodOn the Influence of Structural Connectivity on the Correlation Patterns and Network Synchronization
Nazemi, P. S., & Jamali, Y. (2019) · Frontiers in Computational Neuroscience · doi:10.3389/fncom.2018.00105 · Cited by 13
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Luppi, A. I., Mediano, P. A. M., Rosas, F. E., Allanson, J. E., Pickard, J. D., Williams, G. B., Craig, M. M., Finoia, P., Peattie, A. R. D., Coppola, P., et al. (2021) · bioRxiv (Cold Spring Harbor Laboratory) · doi:10.1101/2021.02.14.431140 · Cited by 12
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“There are a number of existing and well-developed CNMM simulator libraries currently avail-608 able (e.g. TVB (Ritter et al., 2013; Sanz-Leon et al., 2015), SPM-DCM, Brain Dynamics Toolbox (Heitmann et al., 2018), PyRates Gast et al. (2019), NeuroLib Cakan et al. (2021)), that allow simulations not…”
BackgroundModeling the role of the thalamus in resting-state functional connectivity: Nature or structure
Cabrera-Álvarez, J., Doorn, N., Maestú, F., & Susi, G. (2023) · PLoS Computational Biology · doi:10.1371/journal.pcbi.1011007 · Cited by 11
“SC matrices served as the skeleton for the BNMs implemented in TVB [97] where 374”
MethodYang, Y., Tang, S., Wang, X., Zhen, Y., Zheng, Y., Zheng, H., Liu, L., & Zheng, Z. (2023) · Communications Biology · doi:10.1038/s42003-023-05497-4 · Cited by 10
“…have been proposed to explore how the FC network is coupled with the SC network, ranging from the simplest one-to-one mapping 9 using statistical correlations to more sophisticated biophysical models 10,11 that derive functional connectivity from large-scale simulations of neural activity dynamics.”
MethodCabrera-Álvarez, J., Stefanovski, L., Martin, L., Susi, G., Maestú, F., & Ritter, P. (2024) · eNeuro · doi:10.1523/eneuro.0345-23.2023 · Cited by 11
“BNM. SC matrices served as the skeleton for the BNMs implemented in the virtual brain (Sanz-Leon et al., 2015) where regional signals were simulated using JR NMMs (Jansen and Rit, 1995).”
MethodKarimi, F., Steiner, M., Newton, T., Lloyd, B., Cassara, A. M., de Fontenay, P., Farcito, S., Triebkorn, P., Beanato, E., Wang, H., et al. (2025) · Journal of Neural Engineering · doi:10.1088/1741-2552/adb88f · Cited by 13
“TVB [23, 27] is an open-source neuroinformatics platform and simulation engine for whole-brain network modeling.”
InfluentialSelf-citationZhang, X., Lu, Z., Zhang, T., Li, H., Wang, Y., & Tao, Q. (2021) · Frontiers in Neuroscience · doi:10.3389/fnins.2021.727394 · Cited by 11
“Unlike clinical medicine (Sanz-Leon et al., 2015), EEG modeling launched by BCI usually adopted the neuronal population level-based modeling approach and has been more concerned with the theoretical qualitative simulation.”
MethodShaheen, H., Pal, S., & Melnik, R. (2022) · Brain Multiphysics · doi:10.1016/j.brain.2022.100058 · Cited by 12
Multiscale modeling of neuronal dynamics in hippocampus CA1
Tesler, F., Lorenzi, R. M., Ponzi, A. P. D., et al. (2024) · Frontiers in Computational Neuroscience · doi:10.3389/fncom.2024.1432593 · Cited by 11
TVB-EduPack—An Interactive Learning and Scripting Platform for The Virtual Brain
Matzke, H., Schirner, M., Vollbrecht, D., et al. (2015) · Frontiers in Neuroinformatics · doi:10.3389/fninf.2015.00027 · Cited by 11
“Brain connectivity as defined in the context of brain network models (Sanz-Leon et al., 2015) comprises the complete space-time structure of the large-scale connectome, i.e., time delays and connection strengths between brain regions.”
BackgroundSelf-citationSmall, S. L. (2024) · Ageing Research Reviews · doi:10.1016/j.arr.2024.102632 · Cited by 10
On the validity of electric brain signal predictions based on population firing rates
Ness, T. V., Tetzlaff, T., Einevoll, G. T., & Dahmen, D. (2025) · PLoS Computational Biology · doi:10.1371/journal.pcbi.1012303 · Cited by 10
Extracting Dynamical Understanding From Neural-Mass Models of Mouse Cortex
Siu, P. H., Müller, E. J., Zerbi, V., Aquino, K. M., & Fulcher, B. (2022) · Frontiers in Computational Neuroscience · doi:10.3389/fncom.2022.847336 · Cited by 9
“For a given system of coupled ODEs defined above, dynamics were simulated using The Virtual Brain (TVB) (23, 39), yielding simulated time series for each region.”
MethodStenroos, P., Guillemain, I., Tesler, F., Montigon, O., Collomb, N., Stupar, V., Alain Destexhe, A., Coizet, V., Olivier David, A., & Barbier, E. (2023) · eLife · doi:10.7554/elife.90318 · Cited by 13
Clarifying the structure of serious head and spine injury in youth Rugby Union players
Sasaki, K., Sato, H., Nakamura, A., et al. (2020) · PLoS ONE · doi:10.1371/journal.pone.0235035 · Cited by 10
Phase and amplitude dynamics of coupled oscillator systems on complex networks
Woo, J. H., Honey, C. J., & Moon, J.-Y. (2020) · Chaos An Interdisciplinary Journal of Nonlinear Science · doi:10.1063/5.0031031 · Cited by 10
Embedding Task-Based Neural Models into a Connectome-Based Model of the Cerebral Cortex
Ulloa, A., & Horwitz, B. (2016) · Frontiers in Neuroinformatics · doi:10.3389/fninf.2016.00032 · Cited by 10
“…the nodes in Hagmann’s connectome is represented as a Wilson-Cowan model of excitatory (E) and inhibitory (I) neuronal populations, as described in Sanz Leon et al. (2015): where S E and S I are sigmoid functions described by c EE , c EI , c II , c IE are the connections within the single…”
BackgroundMethodInfluentialVirtual Brain for neurological disease modeling
Bernard, C., & Jirsa, V. (2016) · Drug Discovery Today Disease Models · doi:10.1016/j.ddmod.2017.05.001 · Cited by 10
Self-citationBiophysically inspired mean-field model of neuronal populations driven by ion exchange mechanismsPreprint
Rabuffo, G., Bandyopadhyay, A., Calabrese, C., Gudibanda, K., Depannemaecker, D., Takarabe, L. M., Saggio, M. L., Desroches, M., Ivanov, A. I., Linne, M., et al. (2021) · bioRxiv (Cold Spring Harbor Laboratory) · doi:10.1101/2021.10.29.466427 · Cited by 10
Self-citationHashemi, M., Vattikonda, A. N., Jha, J., Ší́p, V., Woodman, M., Bartoloméi, F., & Jirsa, V. (2022) · medRxiv · doi:10.1101/2022.06.02.22275860 · Cited by 10
“To build a whole-brain network model, the brain regions (network nodes) are defined using a parcellation scheme and a set of mathematical equations is placed at each network node to model the regional brain activity (Sanz-Leon et al., 2015; Jirsa et al., 2017).”
MethodSelf-citationMitjans, A. G., Paz-Linares, D., Lopez-Naranjo, C., Gonzalez, A. A., Li, M., Wang, Y., Reyes, R. G., Bringas-Vega, M. L., Minati, L., Evans, A. C., et al. (2023) · NeuroImage · doi:10.1016/j.neuroimage.2023.120137 · Cited by 10
Recent Progress in Brain Network Models for Medical Applications: A Review
Ye, C., Zhang, Y., Chen, R., & Ma, T. (2024) · Health Data Science · doi:10.34133/hds.0157 · Cited by 9
“The foundational approaches of TVB software are extensively described among multiple publications [66,67,70,71] as well as in the online documentation (docs.thevirtualbrain.org).”
BackgroundHas, A. C., Chen, H., Sati, P., & Gao, W. (2024) · Communications Biology · doi:10.1038/s42003-024-07160-y · Cited by 11
A next generation neural mass model with neuromodulation
Depannemaecker, D., Duprat, C., Casagrande, G., Saggio, M. L., Athanasiadis, A. P., Angiolelli, M., Carbonell, C. S., Wang, H. E., Petkoski, S., Sorrentino, P., et al. (2026) · Journal of Computational Neuroscience · doi:10.1007/s10827-026-00920-1 · Cited by 12
“In order to capture macroscopic brain wide effects and allow for comprehensive parameter exploration, the said model ought to provide a level of description (Depannemaecker et al., 2023b; Depannemaecker, 2024) able to capture relevant features observed in human data (Angiolelli et al., 2024; Agouram et al., 2024) in a…”
Self-citationA Neural Mass Model for Abnormal Beta-Rebound in Schizophrenia
Byrne, Á., Coombes, S., & Liddle, P. F. (2019) · Springer series in cognitive and neural systems · doi:10.1007/978-3-030-18830-6_3 · Cited by 11
Bénar, C., Grova, C., Jirsa, V., & Lina, J. (2019) · Espace ÉTS (ETS) · Cited by 9
Self-citationA Robust Modular Automated Neuroimaging Pipeline for Model Inputs to TheVirtualBrain
Frazier-Logue, N., Wang, J. L., Wang, Z., Sodums, D. J., Khosla, A., Samson, A. D., McIntosh, A. R., & Shen, K. (2022) · Frontiers in Neuroinformatics · doi:10.3389/fninf.2022.883223 · Cited by 11
“TheVirtualBrain (TVB) is an open source software platform for large-scale network modelling (Sanz Leon et al., 2013; Sanz-Leon et al., 2015), where models can be personalized to an individual using a wide range of neuroimaging modalities.”
NFTsim: Theory and Simulation of Multiscale Neural Field DynamicsPreprint
Sanz‐Leon, P., Robinson, P. A., Knock, S. A., Drysdale, P. M., Abeysuriya, R., Fung, P. K., RENNIE, C. J., & Zhao, X. (2017) · bioRxiv (Cold Spring Harbor Laboratory) · doi:10.1101/237032 · Cited by 9
Self-citationMultistability in the corticothalamic system
Sanz‐Leon, P., & Robinson, P. A. (2017) · Journal of Theoretical Biology · doi:10.1016/j.jtbi.2017.07.015 · Cited by 8
Self-citationEnhanced simulations of whole-brain dynamics using hybrid resting-state structural connectomes
Manos, T., Diaz-Pier, S., Fortel, I., Driscoll, I. F., Zhan, L., & Leow, A. (2023) · Frontiers in Computational Neuroscience · doi:10.3389/fncom.2023.1295395 · Cited by 8
PyRates—A code-generation tool for modeling dynamical systems in biology and beyond
Gast, R., Knösche, T. R., & Kennedy, A. (2023) · PLoS Computational Biology · doi:10.1371/journal.pcbi.1011761 · Cited by 10
“…sets it apart from dynamical system modeling frameworks such as COMSOL MultiPhysics [22], PyDS [23], PySD [24], Simupy [25], The Virtual Brain [26, 27], the Brain Dynamics Toolbox [28], or the Brain Modeling Toolkit [29], which provide a range of dynamical system analysis methods within a single…”
A Simple Human Brain Model Reproducing Evoked MEG Based on Neural Field Theory
Burlakov, E. O., Verkhlyutov, V. M., & Ushakov, V. (2021) · Studies in computational intelligence · doi:10.1007/978-3-030-91581-0_15 · Cited by 8
Linking connectomics and dynamics in the human brain
Stefanovski, L., Ghani, A., McIntosh, A. R., & Ritter, P. (2016) · e-Neuroforum · doi:10.1007/s13295-016-0027-1 · Cited by 8
Multiscale co-simulation design pattern for neuroscience applications
Kusch, L., Diaz-Pier, S., Klijn, W., et al. (2024) · Frontiers in Neuroinformatics · doi:10.3389/fninf.2024.1156683 · Cited by 7
Self-citationCapturing the emergent dynamical structure in biophysical neural models
Milinkovic, B., Barnett, L., Carter, O. L., Seth, A. K., & Andrillon, T. (2025) · PLoS Computational Biology · doi:10.1371/journal.pcbi.1012572 · Cited by 8
“In the biophysical model, functional integration is modulated by the global coupling parameter, influencing the strength of connections between regions and affecting local dynamics [9,60].”
InfluentialCortical origin of theta error signals
Herrera, B., Sajad, A., Errington, S. P., Schall, J. D., & Riera, J. J. (2023) · Cerebral Cortex · doi:10.1093/cercor/bhad367 · Cited by 6
Spatiotemporal brain complexity quantifies consciousness outside of perturbation paradigms
Breyton, M., Fousek, J., Rabuffo, G., Sorrentino, P., Kusch, L., Massimini, M., Petkoski, S., & Jirsa, V. (2025) · eLife · doi:10.7554/elife.98920.3 · Cited by 7
Self-citation
Metadata
- Authors
- Sanz-Leon, P., Knock, S. A., Spiegler, A., & Jirsa, V. K.
- Affiliations
- QIMR Berghofer Medical Research Institute (AU)
- Inserm (FR)
- Corresponding author
- Jirsa, Viktor K.
- Venue
- NeuroImage
- Publisher
- Elsevier BV
- Published
- 2015-05
- Volume
- 111
- Pages
- 385-430
- ISSN
- 1053-8119 (ISSN-L)
- OpenAlex
- W2074725578
- Semantic Scholar
- 23be65f2b2fee15432e3086c0d98a6610709848f
- Cited by
- 356
- References
- 240
- Citations by year
- 2026: 27 · 2025: 53 · 2024: 41 · 2023: 34 · 2022: 37 · 2021: 36 · 2020: 29 · 2019: 23 · 2018: 25 · 2017: 21 · 2016: 15 · 2015: 13
- FWCI
- 14.82
- Type
- journal article
- Language
- en
- Open access
- hybrid
- doi.org
- Licence
- Locations
- NeuroImage (publishedVersion, cc-by-nc-nd)
- PubMed (publishedVersion)
- Funders
- James S. McDonnell Foundation (FP7-ICT BrainScales - 2009-6)
- Topics
- Life Sciences › Neuroscience › Cognitive Neuroscience › Neural dynamics and brain function
- Life Sciences › Neuroscience › Cognitive Neuroscience › Functional Brain Connectivity Studies
- Health Sciences › Medicine › Radiology, Nuclear Medicine and Imaging › Advanced Neuroimaging Techniques and Applications
- Keywords
- The Virtual Brain, whole-brain dynamics, neural mass model, structural connectivity, forward model, EEG, MEG, functional MRI, stereoelectroencephalography, mathematical notation, computational neuroscience simulation
- MeSH terms
- Brain, Humans, Models, Neurological, Nerve Net, Functional Neuroimaging
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