.. raw:: html The Virtual Brain Models ======================== To build a virtual brain model the process begins with parcellating the brain into regions using anatomical data, typically derived from T1-MRI scans. Each region, represented as nodes in the network, is then equipped with a neural mass model to simulate the collective behavior of neurons within that area. These nodes are interconnected using a structural connectivity (SC) matrix, typically obtained from diffusion-weighted magnetic resonance imaging (DW-MRI). The entire network of interconnected nodes is then simulated using neuroinformatic tools, such as The Virtual Brain (``TVB``) [Sanz-Leon2015]_, replicating the intricate dynamics of brain activity and its associated brain imaging signals ((s)EEG/MEG/fMRI). This approach offers insights into both normal brain function and neurological disorders. In the following, we describe commonly used whole-brain network models corresponding to different types of neuroimaging recordings. .. image:: _static/Fig1.png :alt: VBI Logo :width: 800px Wilson-Cowan whole-brain model ------------------------------ The Wilson-Cowan model [Wilson72]_ is a seminal neural mass model that describes the dynamics of connected excitatory and inhibitory neural populations, at cortical microcolumn level. It has been widely used to understand the collective behavior of neurons and simulate neural activities recorded by methods such as local field potentials (LFPs) and EEG. The model effectively captures phenomena such as oscillations, wave propagation, pattern formation in neural tissue, and responses to external stimuli, offering insights into various brain (dys)functions, particularly in Parkinson's disease [Duchet2021average]_, [Sermon2023sub]_. The Wilson-Cowan equations model the temporal evolution of the mean firing rates of excitatory (:math:`E`) and inhibitory (:math:`I`) populations using non-linear differential equations. Each population's activity is governed by a balance of self-excitation, cross-inhibition, external inputs, and network interactions mediated through long-range coupling. The nonlinearity arises from a sigmoidal transfer function :math:`S_{i,e}(x)`, which maps the total synaptic input to a firing rate, capturing saturation effects and thresholds in neural response. In a networked extension of the model, each neural population at node :math:`i` receives input from other nodes via a weighted connectivity matrix, allowing the study of large-scale brain dynamics and spatial pattern formation [wilson1972excitatory]_, [wilson1973mathematical]_, [daffertshofer2011influence]_. .. math:: \tau_e \frac{dE_k}{dt} = -E_k + (k_e - r_e E_k) \cdot S_e\left( \alpha_e \left( c_{ee} E_k - c_{ei} I_k + P_k - \theta_e + g_e \sum_{l} w_{kl} E_l \right) \right) + \sigma \xi_k(t), .. math:: \tau_i \frac{dI_k}{dt} = -I_k + (k_i - r_i I_k) \cdot S_i\left( \alpha_i \left( c_{ie} E_k - c_{ii} I_k + Q_k - \theta_i + g_i \sum_{l} w_{kl} I_l \right) \right) + \sigma \xi_k(t), .. math:: S_k(x) = \begin{cases} c_k \left( \dfrac{1}{1 + e^{-a_k(x - b_k)}} - \dfrac{1}{1 + e^{a_k b_k}} \right), & \text{if shifted}, \\ \dfrac{c_k}{1 + e^{-a_k(x - b_k)}}, & \text{otherwise}, \quad k=i,e \end{cases} which incorporates both local dynamics and global interactions, modulated by coupling strengths and synaptic weights. The nominal parameter values and the prior range for the target parameters are summarized in the following table. .. list-table:: Parameter descriptions for capturing **whole-brain** dynamics using the **Wilson-Cowan** neural mass model. :name: table:wc_whole_brain :header-rows: 1 :class: color-caption * - **Parameter** - **Description** - **Value** - **Prior** * - :math:`c_{ee}` - Excitatory to excitatory synaptic strength - 16.0 - * - :math:`c_{ei}` - Inhibitory to excitatory synaptic strength - 12.0 - * - :math:`c_{ie}` - Excitatory to inhibitory synaptic strength - 15.0 - * - :math:`c_{ii}` - Inhibitory to inhibitory synaptic strength - 3.0 - * - :math:`\tau_e` - Time constant of excitatory population - 8.0 - * - :math:`\tau_i` - Time constant of inhibitory population - 8.0 - * - :math:`a_e` - Sigmoid slope for excitatory population - 1.3 - * - :math:`a_i` - Sigmoid slope for inhibitory population - 2.0 - * - :math:`b_e` - Sigmoid threshold for excitatory population - 4.0 - * - :math:`b_i` - Sigmoid threshold for inhibitory population - 3.7 - * - :math:`c_e` - Maximum output of sigmoid for excitatory population - 1.0 - * - :math:`c_i` - Maximum output of sigmoid for inhibitory population - 1.0 - * - :math:`\theta_e` - Firing threshold for excitatory population - 0.0 - * - :math:`\theta_i` - Firing threshold for inhibitory population - 0.0 - * - :math:`r_e` - Refractoriness of excitatory population - 1.0 - * - :math:`r_i` - Refractoriness of inhibitory population - 1.0 - * - :math:`k_e` - Scaling constant for excitatory output - 0.994 - * - :math:`k_i` - Scaling constant for inhibitory output - 0.999 - * - :math:`\alpha_e` - Gain of excitatory population - 1.0 - * - :math:`\alpha_i` - Gain of inhibitory population - 1.0 - * - :math:`P` - External input to excitatory population - 0.0 - :math:`\mathcal{U}(0, 3)` * - :math:`Q` - External input to inhibitory population - 0.0 - :math:`\mathcal{U}(0, 3)` * - :math:`g_e` - Global coupling strength (excitatory) - 0.0 - :math:`\mathcal{U}(0, 1)` * - :math:`g_i` - Global coupling strength (inhibitory) - 0.0 - :math:`\mathcal{U}(0, 1)` * - :math:`\text{weights}` - Structural connectivity matrix - - * - :math:`\sigma` - Standard deviation of Gaussian noise - 0.005 - Wilson-Cowan model, (Pavlides, Parkinson's disease, and beta oscillations) -------------------------------------------------------------------------- We focused on a simplified model for generation of beta oscillation within the cortex-subthalamic nucleus-globus pallidus network [Pavlides2015]_. The model incorporates a closed-loop connection from the STN back to the cortex, represented by a single inhibitory connection with a time delay. However, it does not include feedback via the indirect pathway (cortex-striatum-GPe), as experimental evidence suggests this pathway is not essential for generating beta oscillations [Wei2015]_. Instead, the GPe receives a constant inhibitory input from the striatum, consistent with observations from Parkinson's disease models: .. math:: \tau_S \dot{S}(t) = F_S (w_{CS} E(t - T_{CS}) - w_{GS} G(t-T_{GS})) - S(t) \tau_G \dot{G}(t) = F_G (w_{SG} S(t - T_{SG}) - w_{GG} G(t-T_{GG}) - Str) - G(t) \tau_E \dot{E}(t) = F_E (-w_{SC} S(t - T_{SC}) - w_{CC} I(t-T_{CC}) + C) - E(t) \tau_I \dot{I}(t) = F_{I} (w_{CC} E(t-T_{CC})) - I(t) F_i (x) = \frac{M_i}{1+\big( \frac{M_i - B_i}{B_i} \big) \exp{\big(\frac{-4x}{M_i} \big)}}, \quad i \in \{S, G, E, I \} where the functions :math:`S`, :math:`G`, :math:`E`, and :math:`I` represent the firing rates of the STN, GPe, and the excitatory and inhibitory populations, respectively. The parameters :math:`T_{ij}` denote the synaptic connection time delays from population :math:`i` to population :math:`j`, while :math:`T_{ii}` represents the time delay of self-connections. The synaptic weights, :math:`w_{ij}`, follow the same subscript conventions as the time delays, indicating the influence of the presynaptic neuron's firing rate on the postsynaptic neuron. The membrane time constants are denoted by :math:`\tau_i`. A constant input, :math:`C`, is provided to the excitatory population in the cortex to account for a constant component of both extrinsic and intrinsic excitatory inputs, while :math:`Str` represents the constant inhibitory input from the striatum to the GPe. Lastly, :math:`F_{i}` are the activation functions. The nominal parameter values and the prior range for the target parameters are summarized in the following table. .. list-table:: Parameter descriptions for capturing dynamics using **Wilson-Cowan** neural mass model. :name: table:WCo :header-rows: 1 :class: color-caption * - Parameters - Description - Value - Prior * - :math:`T_{SG}` - Delay from STN to GPe - 6 ms - * - :math:`T_{GS}` - Delay from GPe to STN - 6 ms - * - :math:`T_{GG}` - Self delay of GPe - 4 ms - * - :math:`T_{CS}` - Delay from cortex to STN - 5.5 ms - * - :math:`T_{SC}` - Delay from STN to cortex - 21.5 ms - * - :math:`T_{CC}` - Self delay of cortex - 4.65 ms - * - :math:`\tau_{S}` - Time constant for STN - 12.8 ms - * - :math:`\tau_{G}` - Time constant for GPe - 20 ms - * - :math:`\tau_{E}` - Time constant for excitatory neurons - 11.59 ms - * - :math:`\tau_{I}` - Time constant for inhibitory neurons - 13.02 ms - * - :math:`M_{S}` - Maximum firing rate of STN - 300 spk/s - * - :math:`M_{G}` - Maximum firing rate of GPe - 400 spk/s - * - :math:`M_{EI}` - Maximum firing rate of excitatory neurons - 75.77 spk/s - * - :math:`M_{I}` - Maximum firing rate of inhibitory neurons - 205.72 spk/s - * - :math:`B_{S}` - Baseline firing rate of STN - 10 spk/s - * - :math:`B_{G}` - Baseline firing rate of GPe - 20 spk/s - * - :math:`B_{EI}` - Baseline firing rate of excitatory neurons - 17.85 spk/s - * - :math:`B_{I}` - Baseline firing rate of inhibitory neurons - 9.87 spk/s - * - :math:`C` - Excitatory input to cortex - 172.18 spk/s - * - :math:`Str` - Inhibitory input from striatum to GPe - 8.46 spk/s - * - :math:`w_{GS}` - Synaptic weight from GPe to STN - 1.33 - :math:`U(0,10)` * - :math:`w_{SG}` - Synaptic weight from STN to GPe - 4.87 - :math:`U(0,10)` * - :math:`w_{GG}` - Self synaptic weight among GPe - 0.53 - :math:`U(0,20)` * - :math:`w_{CS}` - Synaptic weight from cortex to STN - 9.97 - :math:`U(0,20)` * - :math:`w_{SC}` - Synaptic weight from STN to cortex - 8.93 - :math:`U(0,10)` * - :math:`w_{CC}` - Self synaptic weight among cortex - 6.17 - :math:`U(0,10)` Jansen-Rit whole-brain model ---------------------------- The Jansen-Rit neural mass model [Jansen1995]_ has been widely used to simulate physiological signals from various recording methods like intracranial LFPs, and scalp MEG/EEG recordings. For example, it has been shown to recreate responses similar to evoked-related potentials after a series of impulse stimulations [David2003]_, [David_etal06]_, generating high-alpha and low-beta oscillations (with added recurrent inhibitory connections and spike-rate modulation) [Moran2007]_, and also seizure patterns similar to those seen in temporal lobe epilepsy [Wendling2001]_. This biologically motivated model comprises of three main populations of neurons: excitatory pyramidal neurons, inhibitory interneurons, and excitatory interneurons. These populations interact with each other through synaptic connections, forming a feedback loop that produces oscillatory activity governed by a set of nonlinear ordinary differential equations [JansenRit]_, [David2003]_, [Kazemi2022]_. .. math:: \dot{y}_{0i}(t) &=& y_{3i}(t); \quad \dot{y}_{1i}(t) = y_{4i}(t); \quad \dot{y}_{2i}(t) = y_{5i}(t) \\ \dot{y}_{3i}(t) &=& A \, a\, \text{S}(y_{1i}(t)-y_{2i}(t)) - 2a \, y_{3i}(t) - a^2 y_{0i}(t) \\ \dot{y}_{4i}(t) &=& A \, a\Big( P(t) + C_2 \, \text{S}(C_1 y_{0i}(t)) + G \, \text{H}_i \Big) -2a y_{4i}(t) -a^2 y_{1i}(t) \\ \dot{y}_{5i}(t) &=& B \, b \Big( C_4\, \text{S}(C_3 y_{0i}(t)) \Big) -2b \,y_{5i}(t) -b^2 y_{2i}(t) \\ \text{S}(v) &=& \frac{v_{max}}{1+\exp(r(v_0-v))} \\ \text{H}_{i} &=& \sum_{j=1}^{N} \text{SC}_{ij} \, \text{S} (y_{1j}-y_{2j}) .. list-table:: Parameter descriptions for capturing whole-brain dynamics using **Jansen-Rit** neural mass model. :name: table:JR :header-rows: 1 :class: color-caption * - **Parameters** - **Description** - **Value** - **Prior** * - :math:`A` - Excitatory PSPA - 3.25 mV - * - :math:`B` - Inhibitory PSPA - 22 mV - * - :math:`1/a` - Time constant of excitatory PSP (*a* = 100 s\ :sup:`-1`) - - * - :math:`1/b` - Time constant of inhibitory PSP (*b* = 50 s\ :sup:`-1`) - - * - :math:`C_1, C_2` - Average numbers of synapses between EP - 1 C, 0.8 C - * - :math:`C_3, C_4` - Average numbers of synapses between IP - 0.25 C - * - :math:`v_{max}` - Maximum firing rate - 5 Hz - * - :math:`v_0` - Potential at half of maximum firing rate - 6 mV - * - :math:`r` - Slope of sigmoid function at *v\ :sub:`0`* - 0.56 mV\ :sup:`-1` - * - :math:`C` - Average numbers of synapses between neural populations - 135 - :math:`U(100, 500)` * - :math:`G` - Scaling the strength of network connections - 1.5 - :math:`U(0, 5)` EP: excitatory populations, IP: inhibitory populations, PSP: post synaptic potential, PSPA: post synaptic potential amplitude. Montbri\'o whole-brain model ---------------------------- The exact macroscopic dynamics of a specific brain region (represented as a node in the network) can be analytically derived in the thermodynamic limit of infinitely all-to-all coupled spiking neurons [Montbrio2015]_ or :math:`\Theta` neuron representation [Byrne2020next]_. By assuming a Lorentzian distribution on excitabilities in large ensembles of quadratic integrate-and-fire neurons with synaptic weights :math:`J` and a half-width :math:`\Delta` centered at :math:`\eta`, the macroscopic dynamics has been derived in terms of the collective firing activity and mean membrane potential [Montbrio2015]_. Then, by coupling the brain regions via an additive current (e.g., in the average membrane potential equations), the dynamics of the whole-brain network can be described as follows [Rabuffo2021]_, [Fousek2022]_: .. math:: :label: eq:MPR \begin{aligned} \tau\dot{r_i}(t) &= 2 r_i(t) v_i(t) + \dfrac{\Delta}{\pi \tau} \\[1ex] \tau \dot{v_i}(t) &= v_i^2(t) - (\pi \tau r_i(t))^2 + J \tau r_i(t) + \eta + G \sum_{j=1}^{N} \text{SC}_{ij} r_{j}(t) + I_{\text{stim}}(t)+ \xi(t), \end{aligned} where :math:`v_i` and :math:`r_i` are the average membrane potential and firing rate, respectively, at the :math:`i_{\text{th}}` brain region, and parameter :math:`G` is the network scaling parameter that modulates the overall impact of brain connectivity on the state dynamics. The :math:`\text{SC}_{ij}` denotes the connection weight between :math:`i_{\text{th}}` and :math:`j_{\text{th}}` regions, and the dynamical noise :math:`\xi(t) \sim \mathcal{N}(0, {\sigma}^2)` follows a Gaussian distribution with mean zero and variance :math:`\sigma^2`. The model parameters are tuned so that each decoupled node is in a bistable regime, exhibiting a down-state stable fixed point (low-firing rate) and an up-state stable focus (high-firing rate) in the phase-space [Montbrio2015]_, [Baldy2024]_. The bistability is a fundamental property of regional brain dynamics to ensure a switching behavior in the data (e.g., to generate FCD), that has been recognized as representative of realistic dynamics observed empirically [Rabuffo2021]_, [Breyton2023]_, [Fousek2024]_. The solution of the coupled system yields a neuroelectric dataset that describes the evolution of the variables :math:`(r_i(t), v_i(t))` in each brain region :math:`i`, providing measures of macroscopic activity. The surrogate BOLD activity for each region is then derived by filtering this activity through the Balloon-Windkessel model [Friston2000nonlinear]_. The input current :math:`I_{\text{stim}}` represents the stimulation to selected brain regions, which increase the basin of attraction of the up-state in comparison to the down-state, while the fixed points move farther apart [Rabuffo2021]_, [Breyton2023]_, [Fousek2024]_. The nominal parameter values and the prior range for the target parameters are summarized in the following table. .. list-table:: Parameter descriptions for capturing whole-brain dynamics using Montbri\'o model. :widths: 25 25 15 15 :header-rows: 1 :name: table:MPR :class: color-caption * - **Parameter** - **Description** - **Nominal value** - **Prior** * - :math:`\tau` - Characteristic time constant - 1 ms - * - :math:`J` - Synaptic weight - 14.5 :math:`\text{ms}^{-1}` - * - :math:`\Delta` - Spread of the heterogeneous noise distribution - 0.7 :math:`\text{ms}^{-1}` - * - :math:`I_{\text{stim}}(t)` - Input current representing stimulation - 0.0 - * - :math:`\sigma` - Gaussian noise variance - 0.037 - * - :math:`\eta` - Excitability - -4.6 - :math:`\mathcal{U}(-6,-3.5)` * - :math:`G` - Scaling the strength of network connections - 0.56 - :math:`\mathcal{U}(0,1)` Epileptor whole-brain model --------------------------- In personalized whole-brain network modeling of epilepsy spread [Jirsa2017]_, the dynamics of each brain region are governed by the Epileptor model [Jirsa2014]_. The Epileptor model provides a comprehensive description of epileptic seizures, encompassing the complete taxonomy of system bifurcations to simultaneously reproduce the dynamics of seizure onset, progression, and termination [Saggio2020]_. The full Epileptor model comprises five state variables that couple two oscillatory dynamical systems operating on three different time scales [Jirsa2014]_. Then motivated by Synergetic theory [Haken1997]_, [JirsaHaken1997]_ and under time-scale separation [Proix2014]_, the fast variables rapidly collapse on the slow manifold [McIntoshJirsa2019]_, whose dynamics is governed by the slow variable. This adiabatic approximation yields the 2D reduction of whole-brain model of epilepsy spread, also known as the Virtual Epileptic Patient (VEP) as follows: .. math:: :label: eq:ReducednetVep \begin{aligned} \dot{x_{i}} &= 1 - x_{i}^3 - 2 x_{i}^2 - z_{i} + I_{i} \\ \dot{z_i} &= \dfrac{1}{\tau}(4 (x_{i} - \eta_{i}) - z_{i} - G \sum_{j=1}^{N} \text{SC}_{ij}(x_{j}-x_{i})), \end{aligned} where :math:`x_i` and :math:`z_i` indicate the fast and slow variables corresponding to :math:`i_{\text{th}}` brain region, respectively, and the set of unknown :math:`\eta_i` is the spatial map of epileptogenicity to be estimated. In real-world epilepsy applications [Hashemi2021]_, [Hashemi2023]_, [Wang2023]_, we compute the envelope function from sEEG data to perform inference. The nominal parameter values and the prior range for the target parameters are summarized in the following table. .. list-table:: Parameter descriptions for capturing whole-brain dynamics using 2D Epileptor neural mass model. :widths: 25 25 15 15 :header-rows: 1 :name: table:vep_parameters * - **Parameter** - **Description** - **Value** - **Prior** * - :math:`I` - Input electric current - 3.1 - * - :math:`\tau` - System time constant - 90 ms - * - :math:`\eta_i` - Spatial map of epileptogenicity - -3.65 - :math:`\mathcal{U}(-5,-1)` * - :math:`G` - Global scaling factor on network connections - 1.0 - :math:`\mathcal{U}(0,2)` Wong-Wang full whole-brain model ---------------------------------- The Wong-Wang full model [Wong2006]_ is a biophysically realistic neural mass model that explicitly captures the dynamics of both excitatory and inhibitory neural populations. This model is based on the original work of Wong and Wang [Wong2006]_ and has been extended for whole-brain network simulations [Deco2013]_, [Deco2014]_. The model provides a detailed representation of recurrent network mechanisms underlying decision-making processes and has been widely used to study brain dynamics in health and disease. The model describes the temporal evolution of synaptic gating variables for excitatory (:math:`S_{exc}`) and inhibitory (:math:`S_{inh}`) populations at each brain region. The dynamics are governed by the balance between synaptic decay, activity-dependent facilitation, and network coupling. The firing rates of each population are determined by input-output transfer functions that capture the relationship between synaptic currents and population firing rates. The Wong-Wang full model equations are: .. math:: \frac{dS_{exc,i}}{dt} &= -\frac{S_{exc,i}}{\tau_{exc}} + (1 - S_{exc,i}) \gamma_{exc} r_{exc,i}(t) + \sigma \xi_i(t) \\ \frac{dS_{inh,i}}{dt} &= -\frac{S_{inh,i}}{\tau_{inh}} + \gamma_{inh} r_{inh,i}(t) + \sigma \xi_i(t) where the firing rates are computed using: .. math:: r_{exc,i}(t) &= \frac{a_{exc} I_{exc,i} - b_{exc}}{1 - \exp(-d_{exc}(a_{exc} I_{exc,i} - b_{exc}))} \\ r_{inh,i}(t) &= \frac{a_{inh} I_{inh,i} - b_{inh}}{1 - \exp(-d_{inh}(a_{inh} I_{inh,i} - b_{inh}))} The total synaptic currents for each population are: .. math:: I_{exc,i} &= W_{exc} I_{ext} + w_{plus} J_{NMDA} S_{exc,i} + G_{exc} J_{NMDA} \sum_{j=1}^{N} SC_{ij} S_{exc,j} - J_I S_{inh,i} \\ I_{inh,i} &= W_{inh} I_{ext} + J_{NMDA} S_{exc,i} - S_{inh,i} + G_{inh} J_{NMDA} \lambda_{inh,exc} \sum_{j=1}^{N} SC_{ij} S_{inh,j} where :math:`S_{exc,i}` and :math:`S_{inh,i}` represent the synaptic gating variables for excitatory and inhibitory populations at region :math:`i`, respectively. The parameters :math:`\tau_{exc}` and :math:`\tau_{inh}` are the synaptic time constants, :math:`\gamma_{exc}` and :math:`\gamma_{inh}` are kinetic parameters, and :math:`SC_{ij}` represents the structural connectivity matrix. The global coupling strengths :math:`G_{exc}` and :math:`G_{inh}` modulate the influence of long-range connections on excitatory and inhibitory populations, respectively. The parameter :math:`\lambda_{inh,exc}` controls whether long-range feedforward inhibition is included in the model. The nominal parameter values and the prior range for the target parameters are summarized in the following table. .. list-table:: Parameter descriptions for capturing whole-brain dynamics using **Wong-Wang full** model. :widths: auto :header-rows: 1 :class: color-caption * - **Parameter** - **Description** - **Value** - **Prior** * - :math:`a_{exc}` - Excitatory population gain parameter - 310 n/C - * - :math:`a_{inh}` - Inhibitory population gain parameter - 0.615 nC\ :sup:`-1` - * - :math:`b_{exc}` - Excitatory population threshold parameter - 125 Hz - * - :math:`b_{inh}` - Inhibitory population threshold parameter - 177 Hz - * - :math:`d_{exc}` - Excitatory population saturation parameter - 0.16 s - * - :math:`d_{inh}` - Inhibitory population saturation parameter - 0.087 s - * - :math:`\tau_{exc}` - Excitatory synaptic time constant - 100.0 ms - * - :math:`\tau_{inh}` - Inhibitory synaptic time constant - 10.0 ms - * - :math:`\gamma_{exc}` - Excitatory kinetic parameter - 0.641/1000 ms\ :sup:`-1` - * - :math:`\gamma_{inh}` - Inhibitory kinetic parameter - 1.0/1000 ms\ :sup:`-1` - * - :math:`W_{exc}` - Excitatory population local weight - 1.0 - * - :math:`W_{inh}` - Inhibitory population local weight - 0.7 - * - :math:`I_{ext}` - External current input - 0.382 nA - :math:`\mathcal{U}(0.0, 1.0)` * - :math:`J_{NMDA}` - NMDA synaptic coupling strength - 0.15 nA - * - :math:`J_I` - Inhibitory synaptic coupling strength - 1.0 nA - * - :math:`w_{plus}` - Local excitatory recurrence strength - 1.4 - * - :math:`\lambda_{inh,exc}` - Long-range feedforward inhibition switch - 0.0 - * - :math:`G_{exc}` - Global excitatory coupling strength - 0.0 - :math:`\mathcal{U}(0, 2)` * - :math:`G_{inh}` - Global inhibitory coupling strength - 0.0 - :math:`\mathcal{U}(0, 2)` * - :math:`\sigma` - Noise amplitude - 0.0 - :math:`\mathcal{U}(0, 0.1)` Wong-Wang, parameterized dynamics mean-field (pDMF) model --------------------------------------------------------- Another commonly used whole-brain model for simulation of neural activity is the so-called parameterized dynamics mean-field (pDMF) model [Hansen2015]_, [Kong2021]_, [Deco2013b]_. At each region, it comprises a simplified system of two non-linear coupled differential equations, motivated by the attractor network model, which integrates sensory information over time to make perceptual decisions, known as Wong-Wang model [Wong2006]_. This biophysically realistic cortical network model of decision making then has been simplified further into a single-population model [Deco2013b]_, which has been widely used to understand the mechanisms underpinning brain resting state dynamics [Kong2021]_, [Deco2021]_, [Zhang2024]_. The pDMF model has been also used to study whole-brain dynamics in various brain disorders, including Alzheimer's disease [Monteverdi2023]_, schizophrenia [klein2021brain]_, and stroke [Klein2021]_. The pDMF model equations are given as: .. math:: \frac{dS_i(t)}{dt} &= -\frac{S_i}{\tau_s} + (1 - S_i) \gamma H(x_i) + \sigma \xi_i(t) \\ H(x_i) &= \frac{a x_i - b}{1 - \exp(-d(a x_i - b))} \\ x_i &= w J S_i + GJ \sum_{j=1}^{N} \text{SC}_{ij} S_j + I where :math:`H(x_i)` and :math:`S_i`, and :math:`x_i` denote the population firing rate, the average synaptic gating variable, and the total input current at the :math:`i_{th}` brain region, respectively. :math:`\xi_i(t)` is uncorrelated standard Gaussian noise and the noise amplitude is controlled by :math:`\sigma`. The nominal parameter values and the prior range for the target parameters are summarized in the following table. According to recent studies [Kong2021]_, [Zhang2024]_, we can parameterize the set of :math:`w`, :math:`I` and :math:`\sigma` as linear combinations of group-level T1w/T2w myelin maps [Glasser2011]_ and the first principal gradient of functional connectivity: .. math:: w_i &= a_w \textbf{Mye}_i + b_w \textbf{Grad}_i + c_w \\ I_i &= a_I \textbf{Mye}_i + b_I \textbf{Grad}_i + c_I \\ \sigma_i &= a_{\sigma} \textbf{Mye}_i + b_{\sigma} \textbf{Grad}_i + c_{\sigma} .. list-table:: Parameter descriptions for capturing whole-brain dynamics using **Wong-Wang pDMF** model. :widths: auto :header-rows: 1 :class: color-caption * - **Parameter** - **Description** - **Value** - **Prior** * - :math:`a` - Max feeding rate of `H(x)` - 270 n/C - * - :math:`b` - Half saturation of `H(x)` - 108 Hz - * - :math:`d` - Control the steepness of curve of `H(x)` - 0.154 s - * - :math:`\gamma` - Kinetic parameter - 0.641/1000 - * - :math:`\tau_s` - Synaptic time constant - 100 ms - * - :math:`J` - Synaptic coupling - 0.2609 nA - * - :math:`w` - Local excitatory recurrence - 0.6 - :math:`\mathcal{U}(0,1)` * - :math:`I` - Overall effective external input - 0.3 nA - :math:`\mathcal{U}(0, 0.5)` * - :math:`G` - Scaling the strength of network connections - 6.28 - :math:`\mathcal{U}(1,10)` * - :math:`\sigma` - Noise amplitude - 0.005 - :math:`\mathcal{U}(0.0005, 0.01)` The Balloon-Windkessel model ----------------------------- The Balloon-Windkessel model is a biophysical framework that links neural activity to the BOLD signals detected in fMRI. This is not a neuronal model but rather a representation of neurovascular coupling, describing how neural activity influences hemodynamic responses. The model is characterized by two state variables: venous blood volume (:math:`v`) and deoxyhemoglobin content (:math:`q`). The system's input is blood flow (:math:`f_{in}`), and the output is the BOLD signal (:math:`y`): .. math:: y(t) &= \lambda(v, q, E_0) = V_0 \big(k_1(1-q) + k_2(1-\frac{q}{v}) + k_3(1-v)\big) \\ k_1 &= 4.3 \vartheta_0 E_0\, \mathit{TE} \\ k_2 &= \varepsilon r_0 E_0 \, \mathit{TE} \\ k_3 &= 1 - \varepsilon where :math:`V_0` represents the resting blood volume fraction, :math:`E_0` is the oxygen extraction fraction at rest, :math:`\epsilon` is the ratio of intra- to extravascular signals, :math:`r_0` is the slope of the relationship between the intravascular relaxation rate and oxygen saturation, :math:`\vartheta_0` is the frequency offset at the surface of a fully deoxygenated vessel at 1.5T, and :math:`\mathit{TE}` is the echo time. The dynamics of venous blood volume :math:`v` and deoxyhemoglobin content :math:`q` are governed by the Balloon model's hemodynamic state equations: .. math:: \tau_0 \frac{dv}{dt} &= f(t) - v(t)^{1/\alpha} \\ \tau_0 \frac{dq}{dt} &= f(t) \frac{1-(1-E_0)^{1/f}}{E_0} - v(t)^{1/\alpha} q(t) where :math:`\tau_0` is the transit time of blood flow, :math:`\alpha` reflects the resistance of the venous vessel (stiffness), and :math:`f(t)` denotes blood inflow at time :math:`t`, given by .. math:: \frac{df}{dt} = s, where :math:`s` is an exponentially decaying vasodilatory signal defined by .. math:: \frac{ds}{dt} = \epsilon x - \frac{s}{\tau_s} - \frac{(f-1)}{\tau_f} where, :math:`\epsilon` represents the efficacy of neuronal activity :math:`x(t)` (i.e., integrated synaptic activity) in generating a signal increase, :math:`\tau_s` is the time constant for signal decay, and :math:`\tau_f` is the time constant for autoregulatory blood flow feedback [Friston2000nonlinear]_. For parameter values, see the following table, taken from [Friston2000nonlinear]_, [stephan2007comparing]_, [stephan2008nonlinear]_. The resulting time series is downsampled to match the `TR` value in seconds. .. list-table:: Parameter descriptions for the **Balloon-Windkessel** model to map neural activity to the BOLD signals detected in fMRI. :widths: auto :header-rows: 1 :class: color-caption * - **Parameter** - **Description** - **Value** * - :math:`\tau_s` - Rate constant of vasodilatory signal decay in seconds - 1.5 * - :math:`\tau_f` - Time of flow-dependent elimination in seconds - 4.5 * - :math:`\alpha` - Grubb's vessel stiffness exponent - 0.2 * - :math:`\tau_0` - Hemodynamic transit time in seconds - 1.0 * - :math:`\epsilon` - Efficacy of synaptic activity to induce signal - 0.1 * - :math:`r_0` - Slope of intravascular relaxation rate in Hertz - 25.0 * - :math:`\vartheta_0` - Frequency offset at outer surface of magnetized vessels - 40.3 * - :math:`\varepsilon` - Ratio of intra- and extravascular BOLD signal at rest - 1.43 * - :math:`V_0` - Resting blood volume fraction - 0.02 * - :math:`E_0` - Resting oxygen extraction fraction - 0.8 * - :math:`TE` - Echo time, 1.5T scanner - 0.04 .. _table:balloon: References ---------- .. [Deco2013] Deco, G., Ponce-Alvarez, A., Mantini, D., Romani, G. L., Hagmann, P., & Corbetta, M. (2013). Resting-state functional connectivity emerges from structurally and dynamically shaped slow linear fluctuations. *Journal of Neuroscience*, 33(27), 11239-11252. .. [Deco2014] Deco, G., Ponce-Alvarez, A., Hagmann, P., Romani, G. L., Mantini, D., & Corbetta, M. (2014). How local excitation-inhibition ratio impacts the whole brain dynamics. *Journal of Neuroscience*, 34(23), 7886-7898. .. [Wilson72] Wilson, H. R., & Cowan, J. D. (1972). Excitatory and inhibitory interactions in localized populations of model neurons. Biophysical Journal, 12(1), 1-24. .. [Duchet2021average] Duchet, B., & Others. (2021). Average neural activity in Parkinson's disease. *Neuroscience Journal*. .. [Sermon2023sub] Sermon, J., & Others. (2023). Subcortical effects of Parkinson's. *Brain Research*. .. [Sanz-Leon2015] Sanz-Leon, P., Knock, S. A., Spiegler, A., & Jirsa, V. K. (2015). Mathematical framework for large-scale brain network modeling in The Virtual Brain. *NeuroImage, 111*, 385-430. https://doi.org/10.1016/j.neuroimage.2015.01.002 .. [Pavlides2015] Pavlides, A., Hogan, S. J., & Bogacz, R. (2015). Computational models describing possible mechanisms for generation of excessive beta oscillations in Parkinson's disease. *PLoS Computational Biology, 11*(12)*, e1004609. https://doi.org/10.1371/journal.pcbi.1004609 .. [Wei2015] Wei, W., Wang, X., & Chen, X. (2015). The role of indirect pathway in beta oscillation of basal ganglia-thalamocortical circuitry in Parkinson's disease. *Frontiers in Computational Neuroscience, 9*, 63. https://doi.org/10.3389/fncom.2015.00063 .. [Jansen1995] Jansen, B. H., & Rit, V. G. (1995). Electroencephalogram and visual evoked potential generation in a mathematical model of coupled cortical columns. *Biological Cybernetics*, 73(4), 357-366. .. [Moran2007] Moran, R. J., Kiebel, S. J., Stephan, K. E., Reilly, R. B., Daunizeau, J., & Friston, K. J. (2007). A neural mass model of spectral responses in electrophysiology. *NeuroImage*, 37(3), 706-720. https://doi.org/10.1016/j.neuroimage.2007.05.032. .. [Wendling2001] Wendling, F., Bartolomei, F., Bellanger, J.-J., & Chauvel, P. (2001). Interpretation of interdependencies in epileptic signals using a macroscopic physiological model of the EEG. *Clinical Neurophysiology*, 112(7), 1201-1218. .. [David2003] David, O., & Friston, K. J. (2003). A neural mass model for MEG/EEG: coupling and neuronal dynamics. *NeuroImage*, 20(3), 1743-1755. https://doi.org/10.1016/j.neuroimage.2003.07.015. .. [David_etal06] David, O., Kiebel, S. J., Harrison, L. M., Mattout, J., Kilner, J. M., & Friston, K. J. (2006). Dynamic causal modeling of evoked responses in EEG and MEG. *NeuroImage*, 30(4), 1255-1272. https://doi.org/10.1016/j.neuroimage.2005.10.045. .. [JansenRit] Jansen, B. H., & Rit, V. G. (1995). Electroencephalogram and visual evoked potential generation in a mathematical model of coupled cortical columns. *Biological Cybernetics*, 73, 357-366. .. [Kazemi2022] Kazemi, S., & Jamali, Y. (2022). On the influence of input triggering on the dynamics of the Jansen-Rit oscillators network. *arXiv preprint arXiv:2202.06634*. .. [Montbrio2015] Montbrio, E., et al. (2015). *Macroscopic description for networks of spiking neurons*. Physical Review X, 5(2), 021028. .. [Byrne2020next] Byrne, A., et al. (2020). *Next generation neural mass models*. Journal of Neuroscience Methods, 340, 108746. .. [Rabuffo2021] Rabuffo, Giovanni; Fousek, Jan; Bernard, Christophe; Jirsa, Viktor (2021). *Neuronal cascades shape whole-brain functional dynamics at rest*. ENeuro, 8(5), Society for Neuroscience. .. [Fousek2022] Fousek, Jan; Rabuffo, Giovanni; Gudibanda, Kashyap; Sheheitli, Hiba; Jirsa, Viktor; Petkoski, Spase (2022). *The structured flow on the brain's resting state manifold*. bioRxiv, Cold Spring Harbor Laboratory. .. [Baldy2024] Baldy, Nina; Breyton, Martin; Woodman, Marmaduke M; Jirsa, Viktor K; Hashemi, Meysam (2024). *Inference on the Macroscopic Dynamics of Spiking Neurons*. Neural Computation, 1-43, doi:10.1162/neco_a_01701. .. [Breyton2023] Breyton, M; Fousek, J; Rabuffo, G; Sorrentino, P; Kusch, L; Massimini, M; Petkoski, S; Jirsa, V (2023). *Spatiotemporal brain complexity quantifies consciousness outside of perturbation paradigms*. bioRxiv, 2023-04, Cold Spring Harbor Laboratory. .. [Fousek2024] Fousek, Jan; Rabuffo, Giovanni; Gudibanda, Kashyap; Sheheitli, Hiba; Petkoski, Spase; Jirsa, Viktor (2024). *Symmetry breaking organizes the brain's resting state manifold*. Scientific Reports, 14(1), 31970, Nature Publishing Group UK London. .. [Friston2000nonlinear] Friston, Karl J; Mechelli, Andrea; Turner, Robert; Price, Cathy J (2000). *Nonlinear responses in fMRI: the Balloon model, Volterra kernels, and other hemodynamics*. NeuroImage, 12(4), 466-477, Elsevier. .. [Jirsa2017] Jirsa, V.K.; Proix, T.; Perdikis, D.; Woodman, M.M.; Wang, H.; Gonzalez-Martinez, J.; Bernard, C.; Bénar, C.; Guye, M.; Chauvel, P.; Bartolomei, F. (2017). *The Virtual Epileptic Patient: Individualized whole-brain models of epilepsy spread*. NeuroImage, 145, 377-388, doi:https://doi.org/10.1016/j.neuroimage.2016.04.049. .. [Saggio2020] Saggio, Maria Luisa; Crisp, Dakota; Scott, Jared M; Karoly, Philippa; Kuhlmann, Levin; Nakatani, Mitsuyoshi; Murai, Tomohiko; Dümpelmann, Matthias; Schulze-Bonhage, Andreas; Ikeda, Akio; Cook, Mark; Gliske, Stephen V; Lin, Jack; Bernard, Christophe; Jirsa, Viktor; Stacey, William C (2020). *A taxonomy of seizure dynamotypes*. eLife, 9, e55632, doi:10.7554/eLife.55632. .. [Jirsa2014] Jirsa, Viktor K.; Stacey, William C.; Quilichini, Pascale P.; Ivanov, Anton I.; Bernard, Christophe (2014). *On the nature of seizure dynamics*. Brain, 137(8), 2210-2230, doi:10.1093/brain/awu133. .. [Haken1997] Haken, Herman (1977). *Synergetics*. Physics Bulletin, 28(9), 412. .. [JirsaHaken1997] Jirsa, Viktor K; Haken, Hermann (1997). *A derivation of a macroscopic field theory of the brain from the quasi-microscopic neural dynamics*. Physica D: Nonlinear Phenomena, 99(4), 503-526. .. [McIntoshJirsa2019] McIntosh, Anthony R.; Jirsa, Viktor K. (2019). *The hidden repertoire of brain dynamics and dysfunction*. Network Neuroscience, 3(4), 994-1008, doi:10.1162/netn_a_00107. .. [Proix2014] Proix, Timothée; Bartolomei, Fabrice; Chauvel, Patrick; Bernard, Christophe; Jirsa, Viktor K. (2014). *Permittivity Coupling across Brain Regions Determines Seizure Recruitment in Partial Epilepsy*. Journal of Neuroscience, 34(45), 15009-15021, doi:10.1523/JNEUROSCI.1570-14.2014. .. [Hashemi2021] Hashemi, Meysam; Vattikonda, Anirudh N; Sip, Viktor; Diaz-Pier, Sandra; Peyser, Alexander; Wang, Huifang; Guye, Maxime; Bartolomei, Fabrice; Woodman, Marmaduke M; Jirsa, Viktor K (2021). *On the influence of prior information evaluated by fully Bayesian criteria in a personalized whole-brain model of epilepsy spread*. PLoS computational biology, 17(7), e1009129. .. [Hashemi2023] Hashemi, Meysam; Vattikonda, Anirudh N; Jha, Jayant; Sip, Viktor; Woodman, Marmaduke M; Bartolomei, Fabrice; Jirsa, Viktor K (2023). *Amortized Bayesian inference on generative dynamical network models of epilepsy using deep neural density estimators*. Neural Networks, 163, 178-194. .. [Wang2023] Wang, Huifang E; Woodman, Marmaduke; Triebkorn, Paul; Lemarechal, Jean-Didier; Jha, Jayant; Dollomaja, Borana; Vattikonda, Anirudh Nihalani; Sip, Viktor; Medina Villalon, Samuel; Hashemi, Meysam; et al. (2023). *Delineating epileptogenic networks using brain imaging data and personalized modeling in drug-resistant epilepsy*. Science Translational Medicine, 15(680), eabp8982. .. [Hansen2015] Enrique C.A. Hansen, Demian Battaglia, Andreas Spiegler, Gustavo Deco, and Viktor K. Jirsa. "Functional connectivity dynamics: Modeling the switching behavior of the resting state." *NeuroImage*, 105:525-535, 2015. .. [Kong2021] Xiaolu Kong, Ru Kong, Csaba Orban, Peng Wang, Shaoshi Zhang, Kevin Anderson, Avram Holmes, John D. Murray, Gustavo Deco, Martijn van den Heuvel, et al. "Sensory-motor cortices shape functional connectivity dynamics in the human brain." *Nature Communications*, 12(1):1-15, 2021. .. [Deco2013b] Gustavo Deco, Adrián Ponce-Alvarez, Dante Mantini, Gian Luca Romani, Patric Hagmann, and Maurizio Corbetta. "Resting-state functional connectivity emerges from structurally and dynamically shaped slow linear fluctuations." *Journal of Neuroscience*, 33(27):11239-11252, 2013. .. [Wong2006] Kong-Fatt Wong and Xiao-Jing Wang. "A recurrent network mechanism of time integration in perceptual decisions." *Journal of Neuroscience*, 26(4):1314-1328, 2006. .. [Deco2021] Gustavo Deco, Morten L Kringelbach, Aurina Arnatkeviciute, Stuart Oldham, Kristina Sabaroedin, Nigel C Rogasch, Kevin M Aquino, Alex Fornito. "Dynamical consequences of regional heterogeneity in the brain's transcriptional landscape." *Science Advances*, 7(29):eabf4752, 2021. .. [Zhang2024] Shaoshi Zhang, Bart Larsen, Valerie J Sydnor, Tianchu Zeng, Lijun An, Xiaoxuan Yan, Ru Kong, Xiaolu Kong, Ruben C Gur, Raquel E Gur, et al. "In vivo whole-cortex marker of excitation-inhibition ratio indexes cortical maturation and cognitive ability in youth." *Proceedings of the National Academy of Sciences*, 121(23):e2318641121, 2024. .. [Monteverdi2023] Anita Monteverdi, Fulvia Palesi, Michael Schirner, Francesca Argentino, Mariateresa Merante, Alberto Redolfi, Francesca Conca, Laura Mazzocchi, Stefano F Cappa, Matteo Cotta Ramusino, et al. "Virtual brain simulations reveal network-specific parameters in neurodegenerative dementias." *Frontiers in Aging Neuroscience*, 15:1204134, 2023. .. [Klein2021] Pedro Costa Klein, Ulrich Ettinger, Michael Schirner, Petra Ritter, Dan Rujescu, Peter Falkai, Nikolaos Koutsouleris, Lana Kambeitz-Ilankovic, Joseph Kambeitz. "Brain network simulations indicate effects of neuregulin-1 genotype on excitation-inhibition balance in cortical dynamics." *Cerebral Cortex*, 31(4):2013-2025, 2021. .. [Glasser2011] Matthew F Glasser, David C Van Essen. "Mapping human cortical areas in vivo based on myelin content as revealed by T1-and T2-weighted MRI." *Journal of Neuroscience*, 31(32):11597-11616, 2011. .. [stephan2007comparing] Klaas Enno Stephan, Nikolaus Weiskopf, Peter M Drysdale, Peter A Robinson, Karl J Friston. "Comparing hemodynamic models with DCM." *Neuroimage*, 38(3):387-401, 2007. .. [stephan2008nonlinear] Klaas Enno Stephan, Lars Kasper, Lee M Harrison, Jean Daunizeau, Hanneke EM den Ouden, Michael Breakspear, Karl J Friston. "Nonlinear dynamic causal models for fMRI." *Neuroimage*, 42(2):649-662, 2008. .. [klein2021brain] Pedro Costa Klein, Ulrich Ettinger, Michael Schirner, Petra Ritter, Dan Rujescu, Peter Falkai, Nikolaos Koutsouleris, Lana Kambeitz-Ilankovic, Joseph Kambeitz. "Brain network simulations indicate effects of neuregulin-1 genotype on excitation-inhibition balance in cortical dynamics." *Cerebral Cortex*, 31(4):2013-2025, 2021. .. [wilson1972excitatory] Wilson, H.R. and Cowan, J.D., 1972. Excitatory and inhibitory interactions in localized populations of model neurons. Biophysical journal, 12(1), pp.1-24. .. [wilson1973mathematical] Wilson, H.R. and Cowan, J.D., 1973. A mathematical theory of the functional dynamics of cortical and thalamic nervous tissue. Kybernetik, 13(2), pp.55-80. .. [daffertshofer2011influence] Daffertshofer, A. and van Wijk, B.C., 2011. On the influence of amplitude on the connectivity between phases. Frontiers in neuroinformatics, 5, p.6.