Virtual Brain Inference (VBI), a flexible and integrative toolkit for efficient probabilistic inference on whole-brain models

Understanding the complex dynamics of the brain and their neurobiological underpinnings, with the potential to advance precision medicine (Falcon et al., 2016; Tan et al., 2016; Vogel et al., 2023; Williams and Whitfield Gabrieli, 2025), is a central goal in neuroscience. Modeling these dynamics provides crucial insights into causality and mechanisms underlying both normal brain function and various neurological disorders (Breakspear, 2017; Wang et al., 2023b; Ross and Bassett, 2024). By integrating the average activity of large populations of neurons (e.g. neural mass models; Wilson and Cowan, 1972; Jirsa and Haken, 1996; Deco et al., 2008; Jirsa et al., 2014; Montbrió et al., 2015; Cook et al., 2022) with information provided by structural imaging modalities (i.e. connectome; Honey et al., 2009; Sporns et al., 2005; Schirner et al., 2015; Bazinet et al., 2023), the whole-brain network modeling has proven to be a powerful tractable approach for simulating brain activities and emergent dynamics as recorded by functional imaging modalities (such as (s)EEG, MEG, and fMRI; Sanz-Leon et al., 2015; Schirner et al., 2022; Amunts et al., 2022; D’Angelo and Jirsa, 2022; Patow et al., 2024; Hashemi et al., 2025).

The whole-brain models have been well-established in network neuroscience (Sporns, 2016; Bassett and Sporns, 2017) for understanding the brain structure and function (Ghosh et al., 2008; Honey et al., 2010; Park and Friston, 2013; Melozzi et al., 2019; Suárez et al., 2020; Feng et al., 2024; Tanner et al., 2024) and investigating the mechanisms underlying brain dynamics at rest (Deco et al., 2011; Wang et al., 2019; Ziaeemehr et al., 2020; Kong et al., 2021), normal aging (Lavanga et al., 2023; Zhang et al., 2024), and also altered states such as anesthesia and loss of consciousness (Barttfeld et al., 2015; Hashemi et al., 2017; Luppi et al., 2023; Perl et al., 2023b). This class of computational models, also known as virtual brain models (Jirsa et al., 2010; Sanz Leon et al., 2013; Sanz-Leon et al., 2015; Schirner et al., 2022; Jirsa et al., 2023; Wang et al., 2024), has shown remarkable capability in delineating the pathophysiological causes of a wide range of brain diseases, such as epilepsy (Jirsa et al., 2017; Proix et al., 2017; Wang et al., 2023b), multiple sclerosis (Wang et al., 2024; Mazzara et al., 2025), Alzheimer’s disease (Yalçınkaya et al., 2023; Perl et al., 2023a), Parkinson’s disease (Jung et al., 2022; Angiolelli et al., 2025), neuropsychiatric disorders (Deco and Kringelbach, 2014; Iravani et al., 2021), stroke (Allegra Mascaro et al., 2020; Idesis et al., 2022), and focal lesions (Rabuffo et al., 2025). In particular, they enable the personalized simulation of both normal and abnormal brain activities, along with their associated imaging recordings, thereby stratifying between healthy and diseased states (Liu et al., 2016; Patow et al., 2023; Perl et al., 2023a) and potentially informing targeted interventions and treatment strategies (Jirsa et al., 2017; Proix et al., 2018; Wang et al., 2023b; Jirsa et al., 2023; Hashemi et al., 2025). Although there are only a few tools available for forward simulations at the whole-brain level, for example the brain network simulator The Virtual Brain (TVB; Sanz Leon et al., 2013), there is a lack of tools for addressing the inverse problem, that is finding the set of control (generative) parameters that best explains the observed data. This study aims to bridge this gap by addressing the inverse problem in large-scale brain networks, a crucial step toward making these models operable for clinical applications.

Accurately and reliably estimating the parameters of whole-brain models remains a formidable challenge, mainly due to the high dimensionality and nonlinearity inherent in brain activity data, as well as the non-trivial effects of noise and network inputs. A large number of previous studies in whole-brain modeling have relied on optimization techniques to identify a single optimal value from an objective function, scoring the model’s performance against observed data (Wang et al., 2019; Kong et al., 2021; Cabral et al., 2022; Liu et al., 2023). This approach often involves minimizing metrics such as the Kolmogorov-Smirnov distance or maximizing the Pearson correlation between observed and generated data features such as functional connectivity (FC), functional connectivity dynamics (FCD), and/or power spectral density (PSD). Although fast, such a parametric approach results in only point estimates and fails to capture the relationship between parameters and their associated uncertainty. This limits the generalizability of findings and hinders identifiability analysis, which explores the uniqueness of solutions. Furthermore, optimization algorithms can easily get stuck in local extrema, requiring multi-start strategies to address potential parameter degeneracies. These additional steps, while necessary, ultimately increase the computational cost. Critically, the estimation heavily depends on the form of the objective function defined for optimization (Svensson et al., 2012; Hashemi et al., 2018). These limitations can be overcome by employing Bayesian inference, which naturally quantifies the uncertainty in the estimation and statistical dependencies between parameters, leading to more robust and generalizable models. Bayesian inference is a principal method for updating prior beliefs with information provided by data through the likelihood function, resulting in a posterior probability distribution that encodes all the information necessary for inferences and predictions. This approach has proven essential for understanding the intricate relationships between brain structure and function (Hashemi et al., 2021; Lavanga et al., 2023; Rabuffo et al., 2025), as well as for revealing the pathophysiological causes underlying brain disorders (Hashemi et al., 2023; Yalçınkaya et al., 2023; Wang et al., 2024; Wang et al., 2024; Hashemi et al., 2025; Hashemi et al., 2024).

In this context, simulation-based inference (SBI; Cranmer et al., 2020; Gonçalves et al., 2020; Hashemi et al., 2023; Hashemi et al., 2024) has gained prominence as an efficient methodology for conducting Bayesian inference in complex models where traditional inference techniques become inapplicable. SBI leverages computational simulations to generate synthetic data and employs advanced probabilistic machine learning methods to infer the joint distribution over parameters that best explain the observed data, along with associated uncertainty. This approach is particularly well-suited for Bayesian inference on whole-brain models, which often exhibit complex dynamics that are difficult to retrieve from neuroimaging data with conventional estimation techniques. Crucially, SBI circumvents the need for explicit likelihood evaluation and the Markovian (sequential) property required in sampling. Markov chain Monte Carlo (MCMC; Gelman et al., 1995) is the gold-standard nonparametric technique and asymptotically exact for sampling from a probability distribution. However, for Bayesian inference on whole-brain models given high-dimensional data, the likelihood function becomes intractable, rendering MCMC sampling computationally prohibitive. SBI offers significant advantages, such as parallel simulation while leveraging amortized learning, making it effective for personalized inference from large datasets (Hashemi et al., 2024). Amortization in artificial neural networks refers to the idea of reusing learned computations across multiple tasks or inputs (Gershman and Goodman, 2014). Amortization in Bayesian inference refers to the process of training a shared inference network (e.g. a neural network) with an intensive upfront computational cost, to perform fast inference across many different observations. Instead of re-running inference for each new observation, the trained model can rapidly return posterior estimates, significantly reducing computational cost at test time. Following an initial computational cost during simulation and training to learn all posterior distributions, subsequent evaluation of new hypotheses can be conducted efficiently, without additional computational overhead for further simulations (Hashemi et al., 2023). Importantly, SBI sidesteps the convergence issues caused by complex geometries that are often encountered when using gradient-based MCMC methods (Betancourt and Girolami, 2013; Betancourt et al., 2014; Hashemi et al., 2020). It also substantially outperforms approximate Bayesian computation (ABC) methods, which rely on a threshold to accept or reject samples (Sisson et al., 2007; Beaumont et al., 2009; Gonçalves et al., 2020). Such a likelihood-free approach provides us with generic inference on complex systems as long as we can provide three modules:

1. A prior distribution, describing the possible range of parameters from which random samples can be easily drawn, that is $\overrightarrow{\theta }\sim p(\overrightarrow{\theta })$.

2. A simulator in computer code that takes parameters as input and generates data as output, that is $\overrightarrow{x}\sim p(\overrightarrow{x}\mid \overrightarrow{\theta })$.

3. A set of low-dimensional data features, which are informative of the parameters that we aim to infer.

These elements prepare us with a training data set $\{({\overrightarrow{\theta }}_i,{\overrightarrow{x}}_i){\}}_{i=1}^{N_{sim}}$ with a budget of $N_{sim}$ simulations. Then, using a class of deep neural density estimators, such as masked autoregressive flows (MAFs; Papamakarios and Pavlakou, 2017) or neural spline flows (NSFs; Durkan et al., 2019), we can approximate the posterior distribution of parameters given a set of observed data, that is $p(\overrightarrow{\theta }\mid {\overrightarrow{x}}_{obs})$. Therefore, a versatile toolkit should be flexible and integrative, adeptly incorporating these modules to enable efficient Bayesian inference over complex models.

To address the need for widely applicable, reliable, and efficient parameter estimation from different (source-localized) neuroimaging modalities, we introduce Virtual Brain Inference (VBI), a flexible and integrative toolkit for probabilistic inference at whole-brain level. This open-source toolkit offers fast simulation through just-in-time (JIT) compilation of various brain models in different programming languages (Python/C++) and devices (CPUs/GPUs). It supports space-efficient storage of simulated data (HDF5/NPZ/PT), provides a memory-efficient loader for batched data, and facilitates the extraction of low-dimensional data features (FC/FCD/PSD). Additionally, it enables the training of deep neural density estimators (MAFs/NSFs), making it a versatile tool for inference on neural sources corresponding to (s)EEG, MEG, and fMRI recordings. VBI leverages high-performance computing, significantly enhancing computational efficiency through parallel processing of large-scale datasets, which would be impractical with current alternative methods. Although SBI has been used for low-dimensional parameter spaces (Gonçalves et al., 2020; Wang et al., 2024; Baldy et al., 2024), we demonstrate that it can scale to whole-brain models with high-dimensional unknown parameters, as long as informative data features are provided. VBI is now accessible on the cloud platform EBRAINS (https://ebrains.eu), enabling users to explore more realistic brain dynamics underlying brain (dys)functioning using Bayesian inference.

In the following sections, we will describe the architecture and workflow of the VBI toolkit and demonstrate the validation through a series of case studies using in silico data. We explore various whole-brain models corresponding to different types of brain recordings: a whole-brain network model of Wilson-Cowan (Wilson and Cowan, 1972), Jansen-Rit (Jansen and Rit, 1995; David and Friston, 2003), and Stuart-Landau (Selivanov et al., 2012) for simulating neural activity associated with EEG/MEG signals, the Epileptor (Jirsa et al., 2014) related to stereoelectro-EEG (sEEG) recordings, and Montbrió (Montbrió et al., 2015), and Wong-Wang (Wong and Wang, 2006; Deco et al., 2013) mapped to fMRI BOLD signals. Although these models represent source signals and could be applied to other modalities (e.g. Stuart-Landau representing generic oscillatory dynamics), we focused on their capabilities to perform optimally in specific contexts. For instance, some are better suited for encephalographic signals (e.g. EEG/MEG) due to their ability to preserve spectral properties, while others have been used for fMRI data, emphasizing their ability to capture dynamic features such as bistability and time-varying functional connectivity.

VBI workflow

Figure 1 illustrates an overview of our approach in VBI, which combines virtual brain models and SBI to make probabilistic predictions on brain dynamics from (sourc-localized) neuroimaging recordings. The inputs to the pipeline include the structural imaging data (for building the connectome), functional imaging data such as (s)EEG/MEG, and fMRI as the target for fitting, and prior information as a plausible range over control parameters for generating random simulations. The main computational costs involve model simulations and data feature extraction. The output of the pipeline is the joint posterior distribution of control parameters (such as excitability, synaptic weights, or effective external input) that best explains the observed data. Since the approach is amortized (i.e. it learns across all combinations in the parameter space), it can be readily applied to any new data from a specific subject.

Figure 1

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In the first step, non-invasive brain imaging data, such as T1-weighted MRI and Diffusion-weighted MRI (DW-MRI), are collected for a specific subject (Figure 1A). T1-weighted MRI images are processed to obtain brain parcellation, while DW-MRI images are used for tractography. Using the estimated fiber tracts and the defined brain regions from the parcellation, the connectome (i.e. the complete set of links between brain regions) is constructed by counting the fibers connecting all regions. The SC matrix, with entries representing the connection strength between brain regions, forms the structural component of the virtual brain which constrains the generation of brain dynamics and functional data at arbitrary brain locations (e.g. cortical and subcortical structures).

Subsequently, each brain network node is equipped with a computational model of average neuronal activity, known as neural mass models (see Figure 1B and Materials and methods). They can be represented in the generic form of a dynamical model as $\overrightarrow{\dot{x}}=f(\overrightarrow{x},\overrightarrow{\theta },I_{input})$, with the system variables $\overrightarrow{x}$ (such as membrane potential and firing rate), the control parameters $\overrightarrow{\theta }$ (such as excitability), and the input current $I_{input}$ (such as stimulation). This integration of mathematical mean-field modeling (neural mass models) with anatomical information (connectome) allows us to efficiently analyze functional neuroimaging modalities at the whole-brain level.

To quantify the posterior distribution of control parameters given a set of observations, $p(\overrightarrow{\theta }\mid \overrightarrow{x})$, we first need to define a plausible range for the control parameters based on background knowledge $p(\overrightarrow{\theta })$, that is a simple base distribution known as a prior. We draw random samples from the prior and provide them as input to the VBI simulator (implemented by Simulation module) to generate simulated time series associated with neuroimaging recordings, as shown in Figure 1C. Subsequently, we extract low-dimensional data features (implemented by Features module), as shown in Figure 1D for FC/FCD/PSD, to prepare the training dataset $\{({\overrightarrow{\theta }}_i,{\overrightarrow{x}}_i){\}}_{i=1}^{N_{sim}}$ , with a budget of $N_{sim}$ simulations. Then, we use a class of deep neural density estimators, such as MAF or NSF models, as schematically shown in Figure 1E, to learn all the posterior $p(\overrightarrow{\theta }\mid \overrightarrow{x})$. Finally, we can readily sample from $p(\overrightarrow{\theta }\mid \overrightarrow{x_{obs}})$, which determines the probability distribution in parameter space that best explains the observed data.

Figure 2 depicts the structure of the VBI toolkit, which consists of three main modules. The first module, referred to as the Simulation module, is designed for fast simulation of whole-brain models, such as Wilson-Cowan (Wilson-Cowan model), Jansen-Rit (Jansen-Rit model), Stuart-Landau (Stuart-Landau oscillator), Epileptor (Epileptor model), Montbrió (Montbrió model), and Wong-Wang (Wong-Wang model). These whole-brain models are implemented across various numerical computing libraries such as Cupy (GPU-accelerated computing with Python), C++ (a high-performance systems programming language), Numba (a JIT compiler for accelerating Python code), and PyTorch (an open-source machine learning library for creating deep neural networks).

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The second module, Features, provides a versatile tool for extracting low-dimensional features from simulated time series (see Comprehensive feature extraction). The features include, but are not limited to, spectral, temporal, connectivity, statistical, and information theory related features, and the associated summary statistics. The third module focuses on Inference, that is training the deep neural density estimators, such as MAF and NSF (see Simulation-based inference), to learn the joint posterior distribution of control parameters. See Figure 2—figure supplement 1 and Figure 2—figure supplement 2 for benchmarks comparing CPU/GPU and MAF/NSF performances, and Figure 2—figure supplement 3 for the estimation of the global coupling parameter across different whole-brain network models, evaluated under multiple configurations.

In the following, we demonstrate the capability of VBI for inference on the state-of-the-art whole-brain network models using in silico testing, where the ground truth is known. We apply this approach to simulate neural activity and associated measurements, including (s)EEG/MEG and fMRI, while also providing diagnostics for the accuracy and reliability of the estimation. Note that for (s)EEG/MEG neuroimaging, we perform inference at the regional level rather than at the sensor level, whereas for fMRI, it is mapped using the Balloon-Windkessel model (see The Balloon-Windkessel model). The results presented are based on synthetic data generated using a set of predefined parameters, referred to as the ground truth, randomly selected within biologically plausible ranges and incorporating a certain level of heterogeneity.

Whole-brain network of Wilson-Cowan model

We first demonstrate inference on the whole-brain network model of the Wilson-Cowan (see Equation 1), which is capable of generating a wide range of oscillatory dynamics depending on the control parameters. Specifically, we estimate the bifurcation parameters $P_i\in {\mathbb{R}}^{88}$, representing the external input to each excitatory population, and the global excitatory coupling parameter $g_e$. Figure 3A and B present the observed and predicted EEG-like signals, represented by the activity of excitatory populations $E$ across regions, and Figure 3C and D show the corresponding power spectral density (PSD), as data features. Figure 3E and F illustrate the inferred posterior distributions for parameters $P_i$ and $g_e$, respectively, given $\theta =\{g_e,P_i\}\in {\mathbb{R}}^{89}$. For training, we conducted 250 k random simulations from uniform priors $g_e\sim \mathcal{U}(0,3)$ and $P_i\sim \mathcal{U}(0,3)$. After approximately 2 hr of training using MAF density estimators, posterior sampling was completed within a few seconds. Due to the large number of simulations and informativeness of data features, we achieved accurate estimations of the high-dimensional and heterogeneous control parameters. Ground-truth values (shown in green) are well recovered, leading to close agreement between observed and predicted PSDs of the signals. Finally, Figure 3G reports the posterior shrinkage and z-score metrics used to evaluate the quality of the parameter estimation. The results indicate that the inferred posteriors are both precise and well-centered around the ground-truth values, as reflected by high shrinkage and low z-scores. See Figure 3—figure supplement 1 for estimation over other configurations. Moreover, Figure 3—figure supplement 2, and Figure 3—figure supplement 3, show the estimations by ignoring the spatial information in the data features, indicating the higher accuracy of NSF, though with substantially more computational cost for training compared to MAF.

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Whole-brain network of Jansen-Rit model

Then, we demonstrate the inference on heterogeneous control parameters in the whole-brain network of Jansen-Rit (see Equation 2), commonly used for modeling EEG/MEG data, for example in dementia and Alzheimer’s disease (Triebkorn et al., 2022; Stefanovski et al., 2019). Figure 4A and B show the observed and predicted EEG signals, given by $y_{1i}-y_{2i}$ at each region, while Figure 4C and D illustrate the observed and predicted features such as PSD, respectively. Figure 4E and F show the estimated posterior distributions of synaptic connections $C_i$, and the global coupling parameter $G$, respectively, given the set of unknown parameters $\overrightarrow{\theta }=\{G,C_i\}\in {\mathbb{R}}^{89}$. Here we conducted 50 k random simulations with samples drawn from uniform priors $G\in \mathcal{U}(0,5)$ and $C_i\in \mathcal{U}(100,650)$. After approximately 45 min of training (MAF density estimator), the posterior sampling took only a few seconds. With such a sufficient number of simulations and informative data features, VBI shows accurate estimation of high-dimensional heterogeneous parameters (given the ground truth, shown in green), leading to a strong correspondence between the observed and predicted PSD of EEG/MEG data. Figure 4G displays the shrinkage and z-score as the evaluation metrics, indicating an ideal Bayesian estimation for $C_i$ parameters, but not for the coupling parameter $G$. This occurred because the network input did not induce a significant change in the intrinsic frequency of activities at the regional level, resulting in diffuse uncertainty in its estimation for this model.

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Note that relying on only alpha-peak while excluding other summary statistics, such as total power (i.e., area under the curve), leads to poor estimation of synaptic connections across brain regions (see Figure 4—figure supplement 1). This results in less accurate predictions of the PSD, with more dispersion in their amplitudes. This example demonstrates that VBI provides a valuable tool for hypothesis evaluation and improved insight into data features by uncertainty quantification and their impact on predictions.

Whole-brain network of Stuart-Landau oscillators

To demonstrate efficient inference on the whole-brain time delay from EEG/MEG data, we used a whole-brain network model of coupled generic oscillators (see Equation 3). This model could establish a causal link between empirical spectral changes and the slower conduction velocities observed in multiple sclerosis patients, resulting from immune system attacks on the myelin sheath (Wang et al., 2024; Mazzara et al., 2025). The parameter set to estimate is $\overrightarrow{\theta }=\{G,V\}\in {\mathbb{R}}^2$, consisting of the global scaling parameter $G$ and the averaged velocity of signal transmission $V$ . The training was performed using a budget of only 2 k simulations, which was sufficient due to the low dimensionality of the parameter space. Figure 5A illustrates the comparison between observed (in green) and predicted neural activities (in red). Figure 5B shows a close agreement between observed and predicted PSD signals, as the data feature used for training. Figure 5C and D provide visualizations of the posterior distributions for the averaged velocity $V$ and the global coupling $G$. In these panels, we can see a large shrinkage in the posterior (in red) from the uniform prior (in blue) centered around the true values (vertical green lines). Importantly, Figure 5E presenting the joint posterior distribution indicates a high correlation of $\rho =0.7$ between parameters $G$ and $V$. This illustrates the advantage of Bayesian estimation in identifying statistical relationships between parameters, which helps to detect degeneracy among them. This is crucial for causal hypothesis evaluation and guiding conclusions in clinical settings. Finally, Figure 5F illustrates the sensitivity analysis (based on the eigenvalues of the posterior distribution), revealing that the posterior is more sensitive to changes in $V$ compared to $G$. This highlights the relative impact of these parameters on the model’s posterior estimates.

Figure 5

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Whole-brain network of Epileptor model

Next, we demonstrate the inference on a whole-brain model of epilepsy spread, known as the Virtual Epileptic Patient (VEP; Jirsa et al., 2017; Hashemi et al., 2020), used to delineate the epileptogenic and propagation zone networks from the invasive sEEG recordings (see Equation 4). Here, we used a large value for system time constant $\tau =90ms$ to generate slow-fast dynamics in pathological areas, corresponding to seizure envelope at each brain region. Figure 6 demonstrates the inference of the set of inferred parameters $\overrightarrow{\theta }=\{G,{\eta }_i\}\in {\mathbb{R}}^{89}$, with the global scaling parameter $G$ and spatial map of epileptogenicity ${\eta }_i$, given $i\in \{1,2,...,N_n=88\}$ parcelled regions. Figure 6A and B show the observed and predicted envelope, respectively, at each brain region. Here, the whole brain regions are classified into two epileptogenic zones (in red, corresponding to high excitability), three propagation zones (in yellow, corresponding to excitability close to bifurcation), and the rest as healthy regions (in green, corresponding to low excitability). Figure 6C and D illustrate the observed and predicted data features as the total power energy per region, calculated as the area under the curve. Additionally, the seizure onset at each region was used as a data feature for training the MAF density estimator. From these panels, we observe accurate recovery of seizure envelopes in pathological regions. Figure 6E and F show that the posterior distribution of heterogeneous ${\eta }_i$, and global coupling parameter $G$, respectively, indicating 100% accurate recovery of the true values (in green). Figure 6G confirms the reliability and accuracy of the estimates through shrinkage and z-score diagnostics. With our efficient implementation, generating 10 k whole-brain simulations took less than a minute (using 10 CPU cores). The training took approximately 13 min to converge, while posterior sampling required only a few seconds. See Figure 6—figure supplement 1 for a similar analysis with a slower time scale separation ($\tau =10ms$). These results demonstrate an ideal and fast Bayesian estimation, despite the stiffness of equations in each region and the high dimensionality of the parameters. See Figure 6—figure supplement 2 and Figure 6—figure supplement 3 showing the accuracy and reliability of estimation under different levels of additive and dynamical noise. Note that for the VEP model, the total integration time is less than 100 ms, and due to the model’s stable behavior and a large time step integration, the simulation cost is significantly lower compared to other whole-brain models.

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Whole-brain network of Montbrió model

Targeting the fMRI data, we demonstrate the inference on the whole-brain dynamics using Montbrió model (see Equation 5). Figure 7 demonstrates the inference on heterogeneous control parameters of the Montbrió model, operating in a bistable regime. Figure 7A and B show the observed and predicted BOLD time series, respectively, while Figure 7C and D illustrate the observed and predicted data features, such as the static and dynamical functional connectivity matrices (FC and FCD, respectively). Figure 7E and F show the estimated posterior distributions of excitability ${\eta }_i$ per brain region, and the global coupling parameter $G$. Figure 7G displays the reliability and accuracy of estimation through the evaluation of posterior shrinkage and z-score (see Equation 13 and Equation 14). See Figure 7—figure supplement 1 for estimation over different configurations of the ground-truth values in this model.

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Due to the large number of simulations for training and the informativeness of the data features (both spatio-temporal and functional data features), the results indicate that we achieved accurate parameter estimation and, consequently, a close agreement between the observed and predicted features of BOLD data. This required 500 k simulations for training, given the uniform priors $G\in \mathcal{U}(0,1)$ and ${\eta }_i\in \mathcal{U}(-6,-3.5)$. After approximately 10 h of training (of MAF density estimator), posterior sampling took only 1 min. Our results indicate that training the MAF model was two to four times faster than the NSF model. This showcase demonstrates the capability of VBI in inferring heterogeneous excitability, given the bistable brain dynamics, for fMRI studies. Note that removing the spatio-temporal features and considering only FC/FCD as the data features (see Figure 7—figure supplement 2) leads to poor estimation of the excitability parameter across brain regions (see Figure 7—figure supplement 3). Interestingly, accurate estimation of the only global coupling parameter, $\overrightarrow{\theta }=\{G\}\in {\mathbb{R}}^1$, from only FC/FCD requires around 100 simulations (see Figure 7—figure supplement 4 and Figure 7—figure supplement 5). See Figure 7—figure supplement 6 and Figure 7—figure supplement 7 showing the accuracy and reliability of estimation under different levels of additive and dynamical noise.

Whole-brain network of Wong-Wang model

Finally, in Figure 8, we show the inference on the so-called parameterized dynamics mean-field (pDMF) model, that is a whole-brain network model of the reduced Wong-Wang equation (see Equation 6 and Equation 7), comprising 10 control parameters: the global scaling of connections $G$ and the linear coefficients $(a_w,b_w,c_w,a_I,b_I,c_I,a_{\sigma },b_{\sigma },c_{\sigma })\in {\mathbb{R}}^9$. These parameters are introduced to reduce the dimension of whole-brain parameters as recurrent connection strength $w_i$, external input current $I_i$, and noise amplitude ${\sigma }_i$ for each region (in total, 264 parameters were reduced to 9 dimensions; see Equation 7). Here, we used summary statistics of both spatio-temporal and functional data features extracted from simulated BOLD data to train the MAF density estimator, with a budget of 200 k simulations. The training took around 160 min to converge, whereas posterior sampling took only a few seconds.

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The diagonal panels in Figure 8A show estimated posterior distributions (in red), along with the prior (in blue), and true values (in green). The upper diagonal panels illustrate the joint posterior distributions between parameters (i.e. statistical dependency between parameters). Figure 8B illustrates the observed and predicted BOLD time series, generated by true and estimated parameters (in blue and red, respectively). From Figure 8C, we can see a close agreement between the observed and predicted data features (FC/FCD matrices). Note that due to the stochastic nature of the generative process, we do not expect an exact element-wise correspondence between these features, but rather a match in their summary statistics, such as the mean, variance, and higher order moments (see Figure 8—figure supplement 1). Figure 8D shows the posterior z-score versus shrinkage, indicating less accurate estimation for the coefficients $c_w$, $c_I$, and $c_{\sigma }$ compared to others, as they are not informed by anatomical data such as the T1w/T2w myelin map and the first FC principal gradient (see Equation 7). This showcase demonstrates the advantage of Bayesian inference over optimization in assessing the accuracy and reliability of parameter estimation, whether informed by anatomical data.

Note that in the whole-brain network of the Wong-Wang model, the global scaling parameter $G$ and synaptic coupling $J$ exhibit structural non-identifiability, meaning their combined effects on the system cannot be uniquely disentangled (see Figure 8—figure supplement 2, and Figure 8—figure supplement 3). This is evident in the parameter estimations corresponding to selected observations, where the posterior distributions appear diffuse. The joint posterior plots reveal a nonlinear dependency (banana shape) between $G$ and $J$, arising from their product in the neural mass equation (see Equation 6). Such a nonlinear relationship between parameters poses challenges for deriving causal conclusions, as often occurs in other neural mass models. This is a demonstration of how Bayesian inference facilitates causal hypothesis testing without requiring additional non-identifiability analysis.

This study introduces the VBI, a flexible and integrative toolkit designed to facilitate probabilistic inference on complex whole-brain dynamics using connectome-based models (forward problem) and SBI (inverse problem). The toolkit leverages high-performance programming languages (C++) and dynamic compilers (such as Python’s JIT compiler), alongside the computational power of parallel processors (GPUs), to significantly enhance the speed and efficiency of simulations. Additionally, VBI integrates popular feature extraction libraries with parallel multiprocessing to efficiently convert simulated time series into low-dimensional summary statistics. Moreover, VBI incorporates state-of-the-art deep neural density estimators (such as MAF and NSF generative models) to estimate the posterior density of control parameters within whole-brain models given low-dimensional data features. Our results demonstrated the versatility and efficacy of the VBI toolkit across commonly used whole-brain network models, such as Wilson-Cowan, Jansen-Rit, Stuart-Landau, Epileptor, Montbrió, and Wong-Wang equations placed at each region. The ability to perform parallel and rapid simulations, coupled with a taxonomy of feature extraction, allows for detailed and accurate parameter estimation from associated neuroimaging modalities such as (s)EEG/MEG/fMRI. This is crucial for advancing our understanding of brain dynamics and the underlying mechanisms of various brain disorders. Overall, VBI represents a substantial improvement over alternative methods, offering a robust framework for both simulation and parameter estimation and contributing to the advancement of network neuroscience, potentially, to precision medicine.

The alternatives for parameter estimation include optimization techniques (Hashemi et al., 2018; Wang et al., 2019; Kong et al., 2021; Cabral et al., 2022; Liu et al., 2023), approximate Bayesian computation (ABC) method, and MCMC sampling (Jha et al., 2022). Optimization techniques are sensitive to the choice of the objective function (e.g. minimizing distance error or maximizing correlation) and do not provide estimates of uncertainty. Although multiple runs and thresholding can be used to address these issues, such methods often fall short in revealing relationships between parameters, such as identifying degeneracy, which is crucial for reliable causal inference. Alternatively, a technique known as ABC compares observed and simulated data using a distance measure based on summary statistics (Beaumont et al., 2002; Beaumont, 2010; Sisson et al., 2018). It is known that ABC methods suffer from the curse of dimensionality, and their performance also depends critically on the tolerance level in the accepted/rejected setting (Gonçalves et al., 2020; Cranmer et al., 2020). The self-tuning variants of MCMC sampling have also been used for model inversion at the whole-brain level (Hashemi et al., 2020; Hashemi et al., 2021). Although MCMC is unbiased and exact with infinite runs, it can be computationally prohibitive, and sophisticated reparameterization methods are often required to facilitate convergence at whole-brain level (Jha et al., 2022; Baldy et al., 2023). This becomes more challenging for gradient-based MCMC algorithms, due to the bistability and stiffness of neural mass models. Tailored to Bayes’ rule, SBI sidesteps these issues by relying on expressive deep neural density estimators (such as MAF and NSF) on low-dimensional data features to efficiently approximate the posterior distribution of model parameters. Taking spiking neurons as generative models, this approach has demonstrated superior performance compared to alternative methods, as it does not require model or data features to be differentiable (Gonçalves et al., 2020; Baldy et al., 2024).

In previous studies, we demonstrated the effectiveness of SBI on virtual brain models of neurological (Hashemi et al., 2023; Wang et al., 2024; Mazzara et al., 2025; Mazzara et al., 2025), and neurodegenerative diseases (Hashemi et al., 2024; Yalçınkaya et al., 2023; Hashemi et al., 2025) as well as focal intervention (Rabuffo et al., 2025) and healthy aging (Lavanga et al., 2023). In this work, we extended this probabilistic methodology to encompass a broader range of whole-brain network models, highlighting its flexibility and scalability in leveraging diverse computational resources, from CPUs/GPUs to high-performance computing facilities.

Our results indicated that the VBI toolkit effectively estimates posterior distributions of control parameters in whole-brain network modeling, offering a deeper understanding of the mechanisms underlying brain activity. For example, using the Montbrió and Wong-Wang models, we achieved a close match between observed and predicted FC/FCD matrices derived from BOLD time series (Figures 7 and 8). Additionally, the Jansen-Rit and Stuart-Landau models provided accurate inferences of PSD from neural activity (Figures 4 and 5), while the Epileptor model precisely captured the spread of seizure envelopes (Figure 6). These results underscore the toolkit’s capability to manage complex, high-dimensional data with precision. Uncertainty quantification using VBI can illuminate and combine the informativeness of data features (e.g., FC/FCD) and reveal the causal drivers behind interventions (Lavanga et al., 2023; Hashemi et al., 2024; Rabuffo et al., 2025). This adaptability ensures that VBI can be applied across various (source-localized) neuroimaging modalities, accommodating different computational capabilities and research needs.

Note that there is no specific rule for determining the optimal number of simulations required for training. In general, a larger number of simulations, depending on the available computational resources, tends to improve the quality of posterior estimation. However, when using synthetic data, we can monitor the z-score and posterior shrinkage to assess the accuracy and reliability of the inferred parameters. This also critically depends on the parameter dimensionality. For instance, in estimating only global coupling parameter, a maximum of 300 simulations was used, demonstrating accurate estimation across models and different configurations (see Figure 2—figure supplement 3), except for the Jansen-Rit model, where coupling did not induce a significant change in the intrinsic frequency of regional activity. Importantly, the choice of data features is critical, and some factors (e.g. that lead to inaccurate feature calculation) can lead to the collapse of this method. For instance, high noise levels in observations or dynamical noise can compromise the accurate calculation of data features, undermining the inference process (see Figure 6—figure supplement 2, Figure 6—figure supplement 3, Figure 7—figure supplement 6, Figure 7—figure supplement 7). Identifying the set of low-dimensional data features that are relevant to the control parameters for each case study is another challenge in effectively applying SBI. Nevertheless, the uncertainty of the posterior informs us about the predictive power of these features. Statistical moments of time series could be effective candidates for most models. However, this poses a formidable challenge for inference from empirical data, as certain moments, such as the mean and variance, may be lost during preprocessing steps. The hyperparameter and noise estimation can also be challenging for SBI. Moreover, there is no established rule for determining the number of simulations for training, aside from relying on z-score values during in silico testing, as it depends on available computational resources.

Various sequential methods, such as SNPE (Greenberg et al., 2019), SNLE (Lueckmann et al., 2019), and SNRE (Durkan et al., 2020), have been proposed to reduce computational costs of SBI by iteratively refining the fit to specific targets. These approaches aim for more precise parameter estimation by progressively adjusting the model based on each new data set or subset, potentially enhancing the accuracy of the fit at the reduced computational effort. The choice of method depends on the specific characteristics and requirements of the problem being addressed (Lueckmann et al., 2021). Our previous study indicates that for inferring whole-brain dynamics of epilepsy spread, the SNPE method outperforms alternative approaches (Hashemi et al., 2023). Nevertheless, sequential methods can become unstable, with simulators potentially diverging and causing probability mass to leak into regions that lack prior support (Hashemi et al., 2023). In this study, we used single-round training to benefit from an amortization strategy. This approach brings the costs of simulation and network training upfront, enabling inference on new data to be performed rapidly (within seconds). This strategy facilitates personalized inference at the subject level, as the generative model is tailored by the SC matrix, thereby allowing for rapid hypothesis evaluation specific to each subject (e.g. in delineating the epileptogenic and propagation zones). Note that model comparison across different configurations or model structures, as well established in dynamic causal modeling (Penny et al., 2004; Penny, 2012; Baldy et al., 2025), has yet to be explored in this context.

Deep learning algorithms are increasingly gaining traction in the context of whole-brain modeling. The VBI toolkit leverages a class of deep generative models, called Normalizing Flows (NFs; Rezende and Mohamed, 2015; Kobyzev et al., 2019), to model probability distributions given samples drawn from those distributions. Using NFs, a base probability distribution (e.g. a standard normal) is transformed into any complex distribution (potentially multi-modal) through a sequence of invertible transformations. Variational autoencoders (VAEs; Kingma et al., 2014; Doersch, 2016) are a class of deep generative models to encode data into a latent space and then decode it back to reconstruct the original data. Recently, Sip et al., 2023 introduced a method using VAEs for nonlinear dynamical system identification, enabling the inference of neural mass models and region- and subject-specific parameters from functional data. VAEs have also been employed for dimensionality reduction of whole-brain functional connectivity (Perl et al., 2023b), and to investigate various pathologies and their severity by analyzing the evolution of trajectories within a low-dimensional latent space (Perl et al., 2023a). Additionally, Generative Adversarial Networks (GANs; Goodfellow et al., 2014; Creswell et al., 2018) have demonstrated remarkable success in mapping latent space to data space by learning a manifold induced from a base density (Liu et al., 2021). This method merits further exploration within the context of whole-brain dynamics. To fully harness the potential of deep generative models in large-scale brain network modeling, integrating VAEs and GANs into the VBI framework would be beneficial. This will elucidate their strengths and limitations within this context and guide future advancements in the field.

In summary, VBI offers fast simulations, taxonomy of feature extraction, and deep generative models, making it a versatile tool for model-based inference from different neuroimaging modalities, helping researchers to explore brain (dys)functioning in greater depth. This advancement not only enhances our theoretical understanding but also holds promise for practical applications in diagnosing and treating neurological conditions.

No new data were created or analyzed in this study. All code is available on GitHub (https://github.com/ins-amu/vbi copy archived at Ziaeemehr, 2025).

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