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Cited by 356Neural dynamics and brain functionFunctional Brain Connectivity StudiesAdvanced Neuroimaging Techniques and Applications

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Mathematical framework for large-scale brain network modeling in The Virtual Brain

Open access
Sanz-Leon, Paula, Knock, Stuart A., Spiegler, Andreas, Jirsa, Viktor K.·2015·NeuroImage
DOI 10.1016/j.neuroimage.2015.01.002
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Abstract

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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Figures

Image 1
CC-BY-NC-ND · From the PDF
Fig. 1 Anatomical parcellation lateral and medial views. The black area corresponds to the corpus callosum.
Fig. 1 Anatomical parcellation lateral and medial views. The black area corresponds to the corpus callosum.CC-BY-NC-ND · From the PDF
Fig. 2 The 74 ROIs demonstration dataset. a) Weights matrix. Its values are integer numbers according to the scale used in the CoCoMac database. b) Distances or tract lengths matrix. The values are expressed in millimeters and correspond to white matter fi ber lengths reconstructed by tractography algorithms.
Fig. 2 The 74 ROIs demonstration dataset. a) Weights matrix. Its values are integer numbers according to the scale used in the CoCoMac database. b) Distances or tract lengths matrix. The values are expressed in millimeters and correspond to white matter fi ber lengths reconstructed by tractography algorithms.CC-BY-NC-ND · From the PDF
Fig. 3 a) A generic neural mass model with 2 neural masses and two state variables per neural mass. b) To avoid the need for multiple subscripts, the following notation is sometimes used Φ i ={ φ n =1} i , where the subindex i only refers to the mass or population; and Ψ ≡ [ Φ 1 , Φ 2 ] represents the activity of the neural mass model. This denotes the fact that the neural masses at a node are in general interconnected through their fi rst state variable, or have two masses with one state variable as in the Wilson and Cowan model. c) A reader friendly diagram network diagram. This representation emphasizes the functional character of each population represented in the NMM. The notation [ Φ e , Φ i ] is often found in the models descriptions.
Fig. 3 a) A generic neural mass model with 2 neural masses and two state variables per neural mass. b) To avoid the need for multiple subscripts, the following notation is sometimes used Φ i ={ φ n =1} i , where the subindex i only refers to the mass or population; and Ψ ≡ [ Φ 1 , Φ 2 ] represents the activity of the neural mass model. This denotes the fact that the neural masses at a node are in general interconnected through their fi rst state variable, or have two masses with one state variable as in the Wilson and Cowan model. c) A reader friendly diagram network diagram. This representation emphasizes the functional character of each population represented in the NMM. The notation [ Φ e , Φ i ] is often found in the models descriptions.CC-BY-NC-ND · From the PDF
Fig. 4 This illustration depicts two relevant spatial scales at which structural connectivity is de fi ned. The human cortex is represented by a triangular mesh (lower panel). The density of local connections, often associated with intracortical fi bers, is represented with a kernel function, W ξ ( z ), of the exponential family. Furthermore, the cortex is divided into a fi nite number of patches according to an anatomical parcellation (upper right). The connections between parcels are represented in the connectome U v =2. Time-delays are introduced via long-range connections { ujk , τ jk }.
Fig. 4 This illustration depicts two relevant spatial scales at which structural connectivity is de fi ned. The human cortex is represented by a triangular mesh (lower panel). The density of local connections, often associated with intracortical fi bers, is represented with a kernel function, W ξ ( z ), of the exponential family. Furthermore, the cortex is divided into a fi nite number of patches according to an anatomical parcellation (upper right). The connections between parcels are represented in the connectome U v =2. Time-delays are introduced via long-range connections { ujk , τ jk }.CC-BY-NC-ND · From the PDF
Fig. 5 Schematic representation of a one dimensional chain of neural masses. Consider that the two sites, j and k , belong to two distinct cortical regions. The local connectivity kernel in this representation is assumed to be translationally and rotationally invariant, and therefore is often called homogeneous connectivity (Jirsa, 2009; Spiegler and Jirsa, 2013) or background connectivity (Bojak et al., 2011). It then follows that the probability of short-range interactions between j and k and their local neighborhoods is the same in both sites. Long-range interactions between j and k happen through the weights ( u ) and delays ( τ ) of the heterogeneous connectivity. Note that in this case the τ jk = τ kj while ujk is not necessarily equal to ukj .
Fig. 5 Schematic representation of a one dimensional chain of neural masses. Consider that the two sites, j and k , belong to two distinct cortical regions. The local connectivity kernel in this representation is assumed to be translationally and rotationally invariant, and therefore is often called homogeneous connectivity (Jirsa, 2009; Spiegler and Jirsa, 2013) or background connectivity (Bojak et al., 2011). It then follows that the probability of short-range interactions between j and k and their local neighborhoods is the same in both sites. Long-range interactions between j and k happen through the weights ( u ) and delays ( τ ) of the heterogeneous connectivity. Note that in this case the τ jk = τ kj while ujk is not necessarily equal to ukj .CC-BY-NC-ND · From the PDF
Fig. 6 Local coupling schemes de fi ne how and which neighboring populations of different kind affect one another. a) As presented in Wilson and Cowan (1973) and b) Steyn-Ross et al. (1999) for neural fi eld models. van Rotterdam et al. (1982) used different connectivity kernels based on a Laplace distribution for the inhibitory and excitatory populations in spatially distributed model (Lopes da Silva et al., 1974).
Fig. 6 Local coupling schemes de fi ne how and which neighboring populations of different kind affect one another. a) As presented in Wilson and Cowan (1973) and b) Steyn-Ross et al. (1999) for neural fi eld models. van Rotterdam et al. (1982) used different connectivity kernels based on a Laplace distribution for the inhibitory and excitatory populations in spatially distributed model (Lopes da Silva et al., 1974).CC-BY-NC-ND · From the PDF
Fig. 7 Three different functions that are used to model the local connectivity kernel. (a) Gaussian, (b) Laplace and (c) Mexican hat functions.
Fig. 7 Three different functions that are used to model the local connectivity kernel. (a) Gaussian, (b) Laplace and (c) Mexican hat functions.CC-BY-NC-ND · From the PDF
Fig. 8 This fi gure illustrates the geodesic distance computed around one vertex, with panels a -c zooming in to show more detail. The colored contours are 1 mm apart from each other up to a maximum distance of 20 mm (red). The cortical surface has been removed from panel (d) to help reveal the highly convoluted geometry of the cortex along which the geodesic distance is computed.
Fig. 8 This fi gure illustrates the geodesic distance computed around one vertex, with panels a -c zooming in to show more detail. The colored contours are 1 mm apart from each other up to a maximum distance of 20 mm (red). The cortical surface has been removed from panel (d) to help reveal the highly convoluted geometry of the cortex along which the geodesic distance is computed.CC-BY-NC-ND · From the PDF
Fig. 9 Agaussian distribution was used to de fi ne the local connectivity kernel: W 1,1( z ), with σ 1,1 =10mmandthecutoffdistance zc =20mm.Forsimplicity in this fi gure, the isocontours of the kernel are plotted for only one vertex. Zooming in from panel (a) to (d), shows that the amplitude at the vertex itself is zero, because self-connections are already represented in the local model. Whom, ς ( z ), are computed for each vertex in the mesh up to a cutoff distance zc .
Fig. 9 Agaussian distribution was used to de fi ne the local connectivity kernel: W 1,1( z ), with σ 1,1 =10mmandthecutoffdistance zc =20mm.Forsimplicity in this fi gure, the isocontours of the kernel are plotted for only one vertex. Zooming in from panel (a) to (d), shows that the amplitude at the vertex itself is zero, because self-connections are already represented in the local model. Whom, ς ( z ), are computed for each vertex in the mesh up to a cutoff distance zc .CC-BY-NC-ND · From the PDF
Fig. 10 General coupling scheme for long-range connections. Afferent signals are related to the excitatory population activity, however this activity may target inhibitory populations in other connected nodes ( e → e ) ( e → i ).
Fig. 10 General coupling scheme for long-range connections. Afferent signals are related to the excitatory population activity, however this activity may target inhibitory populations in other connected nodes ( e → e ) ( e → i ).CC-BY-NC-ND · From the PDF
Fig. 11 Reproducing Fig. 2 of FitzHugh (1961). a) Excitable regime; b) oscillatory regime; c) Van der Pol oscillator. Heun's method was used for the numerical integration; integration time step d t =2 -4 . The intrinsic frequency is approximately 6.5 Hz and 8 Hz in b) and c) respectively. Black lines are the traces of the variable V , while red lines are those of W .
Fig. 11 Reproducing Fig. 2 of FitzHugh (1961). a) Excitable regime; b) oscillatory regime; c) Van der Pol oscillator. Heun's method was used for the numerical integration; integration time step d t =2 -4 . The intrinsic frequency is approximately 6.5 Hz and 8 Hz in b) and c) respectively. Black lines are the traces of the variable V , while red lines are those of W .CC-BY-NC-ND · From the PDF
Fig. 12 The topology of the phase plane characterizes the initial dynamics of the individual nodes in a BNM. a) Excitable con fi guration. If a =2thedynamics correspond to an oscillator (limit cycle). b) Bistable con fi guration. c) Saddle Node on the Invariant Circle (SNIC) con fi guration.
Fig. 12 The topology of the phase plane characterizes the initial dynamics of the individual nodes in a BNM. a) Excitable con fi guration. If a =2thedynamics correspond to an oscillator (limit cycle). b) Bistable con fi guration. c) Saddle Node on the Invariant Circle (SNIC) con fi guration.CC-BY-NC-ND · From the PDF
Fig. 13 Effect of global coupling strength on local dynamics. First row: phase planes and exemplary trajectories of the initial local dynamics as per Table 8; from left to right columns correspond to excitable, bistable and SNIC con fi gurations, respectively. The second row shows the panels with the individual node traces for every regime in the case of an uncoupled network (the slope of the coupling function is zero). On the left subpanels, the black lines represent the traces of the V variable, while red lines are those of the W variable. On the right subpanels, the light blue traces are individual trajectories of V versus W , while the blue line is the average trajectory over nodes. Because all the nodes had the same model parameters, it is not possible to distinguish individual time-series. In the third row the network was weakly coupled by setting the global coupling strength to 0.0042 and the individual node traces are drawn in a lighter shade of black ( V ) or red ( W ) while the solid colored lines represent the average time-series over nodes. Lastly, in the fourth row the global coupling strength was increased by an order of magnitude to 0.042 and each node's time-series start to noticeably deviate from the initial local dynamical regime.
Fig. 13 Effect of global coupling strength on local dynamics. First row: phase planes and exemplary trajectories of the initial local dynamics as per Table 8; from left to right columns correspond to excitable, bistable and SNIC con fi gurations, respectively. The second row shows the panels with the individual node traces for every regime in the case of an uncoupled network (the slope of the coupling function is zero). On the left subpanels, the black lines represent the traces of the V variable, while red lines are those of the W variable. On the right subpanels, the light blue traces are individual trajectories of V versus W , while the blue line is the average trajectory over nodes. Because all the nodes had the same model parameters, it is not possible to distinguish individual time-series. In the third row the network was weakly coupled by setting the global coupling strength to 0.0042 and the individual node traces are drawn in a lighter shade of black ( V ) or red ( W ) while the solid colored lines represent the average time-series over nodes. Lastly, in the fourth row the global coupling strength was increased by an order of magnitude to 0.042 and each node's time-series start to noticeably deviate from the initial local dynamical regime.CC-BY-NC-ND · From the PDF
Fig. 14 Network stability maps for a) the 74 ROIs matrix (global coupling strength: a Γ ϵ [0.012, 0.042], conduction speed: c 2 ∈ [1, 14]; and b) averaged dataset from DTI ( a Γ ∈ [0.2, 0.9], c 2 ∈ [1, 14]). The metric displayed is the variance computed over all the datapoints after discarding the fi rst 500 ms (e.g., each element of the map represents one simulation). Only the traces of the fi rst state variable, V , were used. The y-axis of the temporal traces is between [ -4, 4].
Fig. 14 Network stability maps for a) the 74 ROIs matrix (global coupling strength: a Γ ϵ [0.012, 0.042], conduction speed: c 2 ∈ [1, 14]; and b) averaged dataset from DTI ( a Γ ∈ [0.2, 0.9], c 2 ∈ [1, 14]). The metric displayed is the variance computed over all the datapoints after discarding the fi rst 500 ms (e.g., each element of the map represents one simulation). Only the traces of the fi rst state variable, V , were used. The y-axis of the temporal traces is between [ -4, 4].CC-BY-NC-ND · From the PDF
Fig. 15 Exemplary deterministic trajectories for the local dynamics presented in a) Ghosh et al. (2008a) and Knock et al. (2009), b) Sanz-Leon et al. (2013) and c) Spiegler and Jirsa (2013).
Fig. 15 Exemplary deterministic trajectories for the local dynamics presented in a) Ghosh et al. (2008a) and Knock et al. (2009), b) Sanz-Leon et al. (2013) and c) Spiegler and Jirsa (2013).CC-BY-NC-ND · From the PDF
Fig. 16 Phase-space and exemplary trajectory as per parameters in columns one to three from Table 11. a) Three steady states; b) fi ve steady states; and, c) a limit cycle.
Fig. 16 Phase-space and exemplary trajectory as per parameters in columns one to three from Table 11. a) Three steady states; b) fi ve steady states; and, c) a limit cycle.CC-BY-NC-ND · From the PDF
Fig. 17 Effect of the global coupling strength on the individual node dynamics, as in Fig. 13 but here for a network of Wilson and Cowan NMMs. In the fi rst row the initial single node dynamics are shown: a) two stable states separated by one unstable (default con fi guration in The Virtual Brain ); b) fi ve steady states and c) limit cycle. Second to fourth rows show the effect of increasing the global coupling strength (the coupling function is linear function with varying a Γ and b =0). Vertically a Γ takes the values of 0 (uncoupled network), 0.0042 (third row) and 0.042 (fourth row). Here, the black lines are the traces of E and red lines those of I . Solid black and red lines are the average time-series computed over nodes. On the right subpanels (PP), we present the trajectories in the E -I plane.
Fig. 17 Effect of the global coupling strength on the individual node dynamics, as in Fig. 13 but here for a network of Wilson and Cowan NMMs. In the fi rst row the initial single node dynamics are shown: a) two stable states separated by one unstable (default con fi guration in The Virtual Brain ); b) fi ve steady states and c) limit cycle. Second to fourth rows show the effect of increasing the global coupling strength (the coupling function is linear function with varying a Γ and b =0). Vertically a Γ takes the values of 0 (uncoupled network), 0.0042 (third row) and 0.042 (fourth row). Here, the black lines are the traces of E and red lines those of I . Solid black and red lines are the average time-series computed over nodes. On the right subpanels (PP), we present the trajectories in the E -I plane.CC-BY-NC-ND · From the PDF
Image 19
CC-BY-NC-ND · From the PDF
Fig. 19 Parametric curves with respect to the recurrent connection strength w . The left panel shows the graph of d S which allows us to identify the equilibria. An enlarged view of the region between solid horizontal black lines is displayed on the right panel. Here we can see that small changes in w have a signi fi cant impact on multistability of the system.
Fig. 19 Parametric curves with respect to the recurrent connection strength w . The left panel shows the graph of d S which allows us to identify the equilibria. An enlarged view of the region between solid horizontal black lines is displayed on the right panel. Here we can see that small changes in w have a signi fi cant impact on multistability of the system.CC-BY-NC-ND · From the PDF
Fig. 20 Variants of the Jansen and Rit model including connections and external interventions. The three neural masses of the model are linked by excitatory and inhibitory connections (arrow and circle respectively). First block: original model (Jansen and Rit, 1995); second block: Moran et al. (2007) included a recurrent connection in the inhibitory mass and the extrinsic input enters the spiny stellate population; third block: Aburn et al. (2012), went back to the classical model but the extrinsic inputs target both excitatory masses; fourth block: a simpli fi ed representation of the scheme presented by David et al. (2006) where extrinsic connections representing afferent signals from other sources affect the three masses and external perturbations such as noise only enter through the spiny stellate mass; the last block is the general scheme adopted by the authors to implement this model in The Virtual Brain .
Fig. 20 Variants of the Jansen and Rit model including connections and external interventions. The three neural masses of the model are linked by excitatory and inhibitory connections (arrow and circle respectively). First block: original model (Jansen and Rit, 1995); second block: Moran et al. (2007) included a recurrent connection in the inhibitory mass and the extrinsic input enters the spiny stellate population; third block: Aburn et al. (2012), went back to the classical model but the extrinsic inputs target both excitatory masses; fourth block: a simpli fi ed representation of the scheme presented by David et al. (2006) where extrinsic connections representing afferent signals from other sources affect the three masses and external perturbations such as noise only enter through the spiny stellate mass; the last block is the general scheme adopted by the authors to implement this model in The Virtual Brain .CC-BY-NC-ND · From the PDF
Fig. 21 A scheme of the modi fi ed Zetterberg -Jansen model currently implemented in The Virtual Brain .
Fig. 21 A scheme of the modi fi ed Zetterberg -Jansen model currently implemented in The Virtual Brain .CC-BY-NC-ND · From the PDF
Fig. 22 Trajectories of the excitatory mass on the ξ -η plane for the different modes in the SJ3D model. Each panel depicts the behavior of one mode, o , of the model. Multimodal model better represent the variability of a population of neurons. Notice that the phase planes are very similar, yet not identical, giving rise to more heterogenous local dynamics.
Fig. 22 Trajectories of the excitatory mass on the ξ -η plane for the different modes in the SJ3D model. Each panel depicts the behavior of one mode, o , of the model. Multimodal model better represent the variability of a population of neurons. Notice that the phase planes are very similar, yet not identical, giving rise to more heterogenous local dynamics.CC-BY-NC-ND · From the PDF
Fig. 23 Timeseries of the uncoupled network whose local dynamics are governed by the Stefanescu -Jirsa 3D model. The panels on the fi rst row show the trajectories of the excitatory mass on the ξ -η plane. Each column represents one mode, o , of the model. The lower rows display the traces of ξ (black) and α (red).
Fig. 23 Timeseries of the uncoupled network whose local dynamics are governed by the Stefanescu -Jirsa 3D model. The panels on the fi rst row show the trajectories of the excitatory mass on the ξ -η plane. Each column represents one mode, o , of the model. The lower rows display the traces of ξ (black) and α (red).CC-BY-NC-ND · From the PDF
Fig. 24 The same description as in Fig. 23 applies here. However, the global coupling strength was increased to 0.5, so synchronization among individual nodes is lost. The light black and red lines are the individual node time-series while, the solid lines are the average time-series computed over nodes.
Fig. 24 The same description as in Fig. 23 applies here. However, the global coupling strength was increased to 0.5, so synchronization among individual nodes is lost. The light black and red lines are the individual node time-series while, the solid lines are the average time-series computed over nodes.CC-BY-NC-ND · From the PDF
Fig. 25 a) Metastability map on the global coupling strength and speed plane. The colormap represents a variance metric referred to as metastability. b) Global variance map. Five exemplary datasets are displayed on the right panels. Despite the fact that the underlying time-series are the same for both maps, the resulting landscape captures different features of the global dynamics. Metastability, M , seems to better capture global synchronization, that is, the higher the value of M the more synchronized the activity across all the network nodes is, as displayed in panels A, B, C, D and E of Fig. 25a.
Fig. 25 a) Metastability map on the global coupling strength and speed plane. The colormap represents a variance metric referred to as metastability. b) Global variance map. Five exemplary datasets are displayed on the right panels. Despite the fact that the underlying time-series are the same for both maps, the resulting landscape captures different features of the global dynamics. Metastability, M , seems to better capture global synchronization, that is, the higher the value of M the more synchronized the activity across all the network nodes is, as displayed in panels A, B, C, D and E of Fig. 25a.CC-BY-NC-ND · From the PDF
Fig. 26 Larter -Breakspear. (a) V -W ; (b) V -Z and (c) W -Z planes.
Fig. 26 Larter -Breakspear. (a) V -W ; (b) V -Z and (c) W -Z planes.CC-BY-NC-ND · From the PDF
Fig. 27 Larter -Breakspear. Metastability plane. The colormap range is [0, 8.8]. High values of metastability re fl ect fi xed point attractor dynamics. Looking at the bottom row of the parameter plane, for C =0.19, we can see that when δ V N 0.54 the system exhibits chaotic attractor dynamics.
Fig. 27 Larter -Breakspear. Metastability plane. The colormap range is [0, 8.8]. High values of metastability re fl ect fi xed point attractor dynamics. Looking at the bottom row of the parameter plane, for C =0.19, we can see that when δ V N 0.54 the system exhibits chaotic attractor dynamics.CC-BY-NC-ND · From the PDF
Fig. 28 Implemented HRF kernels with their default parameters. All the kernels have a length of 20 s. The blue one corresponds to the kernel derived from the Balloon model and whose derivation can be found in Appendix D. The parameters for these curves have been extracted from peer-reviewed articles. Note however that the HRFs shown in this fi gure exhibit substantial differences, and would yield very different results when convoluted with the same neural activity. Such discrepancies make us aware of the need to have a common and more realistic model of the BOLD signal.
Fig. 28 Implemented HRF kernels with their default parameters. All the kernels have a length of 20 s. The blue one corresponds to the kernel derived from the Balloon model and whose derivation can be found in Appendix D. The parameters for these curves have been extracted from peer-reviewed articles. Note however that the HRFs shown in this fi gure exhibit substantial differences, and would yield very different results when convoluted with the same neural activity. Such discrepancies make us aware of the need to have a common and more realistic model of the BOLD signal.CC-BY-NC-ND · From the PDF
Fig. 29 Lead fi eld coef fi cients of a single channel. In the left panel, the color gradient represents the values of the cortical sensitivity distribution of channel Oz. The resulting fi eld recorded by this channel is in fl uenced by the position and orientations of the distributed source dipoles. In the right panel, the same is displayed for the MEG lead fi eld matrix (channel A70 of a 248 sensor 4D-Neuroimaging MEG system). The seemingly wide spread of the MEG sensor sensitivity can be due to the longer distance to the scalp of MEG sensors with respect to that of EEG sensors. Also the colormap used in the EEG is not symmetric with respect to 0, and therefore negative values are more visible than the equivalent positive sensitivity values.
Fig. 29 Lead fi eld coef fi cients of a single channel. In the left panel, the color gradient represents the values of the cortical sensitivity distribution of channel Oz. The resulting fi eld recorded by this channel is in fl uenced by the position and orientations of the distributed source dipoles. In the right panel, the same is displayed for the MEG lead fi eld matrix (channel A70 of a 248 sensor 4D-Neuroimaging MEG system). The seemingly wide spread of the MEG sensor sensitivity can be due to the longer distance to the scalp of MEG sensors with respect to that of EEG sensors. Also the colormap used in the EEG is not symmetric with respect to 0, and therefore negative values are more visible than the equivalent positive sensitivity values.CC-BY-NC-ND · From the PDF
Fig. 31 Vertex redistribution schematic.
Fig. 31 Vertex redistribution schematic.CC-BY-NC-ND · From the PDF
Fig. 30 A schematic representing the operations of edge collapse (left and middle diagrams) and edge splitting (left and right-hand diagrams). An edge collapse or contraction operation can be viewed as moving 2 vertices, k and j , toward one another until they are in the same place. In the middle diagram, white circles are the original vertices and the black circle is the new vertex. The mesh has one less vertex and two less triangles.
Fig. 30 A schematic representing the operations of edge collapse (left and middle diagrams) and edge splitting (left and right-hand diagrams). An edge collapse or contraction operation can be viewed as moving 2 vertices, k and j , toward one another until they are in the same place. In the middle diagram, white circles are the original vertices and the black circle is the new vertex. The mesh has one less vertex and two less triangles.CC-BY-NC-ND · From the PDF
Fig. 33 Cortical meshes simpli fi ed using (a) iso2mesh -meshresample and (b) reducepatch methods, the coloring represents local curvature.
Fig. 33 Cortical meshes simpli fi ed using (a) iso2mesh -meshresample and (b) reducepatch methods, the coloring represents local curvature.CC-BY-NC-ND · From the PDF
Fig. 34 Adepiction of mesh simpli fi cation, starting from a mesh with a) 2 18 vertices which was subsequently simpli fi ed to b) 2 14 vertices, while minimizing change in the surfaces other properties.
Fig. 34 Adepiction of mesh simpli fi cation, starting from a mesh with a) 2 18 vertices which was subsequently simpli fi ed to b) 2 14 vertices, while minimizing change in the surfaces other properties.CC-BY-NC-ND · From the PDF
Fig. 35 Distributions of surface properties, at the edge, triangle and vertex level, for a range of resolutions: (a) edge length; (b) triangle area; (c) vertex degree.
Fig. 35 Distributions of surface properties, at the edge, triangle and vertex level, for a range of resolutions: (a) edge length; (b) triangle area; (c) vertex degree.CC-BY-NC-ND · From the PDF
Fig. 37 Dynamics of the full Balloon -Windkessel model (BWF). Upper left panel: time-dependent in the induced perfusion signal s ; the gray line is the neural activity k 2 and k 3 scaled to fi t the axis. This highlights the fact that BOLD signal lags the rCBF signal by about a second. Lower left panel: changes in normalized blood fl ow ( f in ). Upper right panel: changes in normalized venous volume, v , (blue line) and normalized deoxyhemoglobin content ( q ) (dashed green line). Lower right panel: change in BOLD signal.
Fig. 37 Dynamics of the full Balloon -Windkessel model (BWF). Upper left panel: time-dependent in the induced perfusion signal s ; the gray line is the neural activity k 2 and k 3 scaled to fi t the axis. This highlights the fact that BOLD signal lags the rCBF signal by about a second. Lower left panel: changes in normalized blood fl ow ( f in ). Upper right panel: changes in normalized venous volume, v , (blue line) and normalized deoxyhemoglobin content ( q ) (dashed green line). Lower right panel: change in BOLD signal.CC-BY-NC-ND · From the PDF
Fig. 38 Simulated percentage change in BOLD signals to the neural drive u showninFig.37. The blue lines is the signal obtained with the full model, while the green line is the one obtained with the simpli fi ed version as de fi ned in Eq. (D.5). The main difference is the lack of undershoot in the simpli fi ed version.
Fig. 38 Simulated percentage change in BOLD signals to the neural drive u showninFig.37. The blue lines is the signal obtained with the full model, while the green line is the one obtained with the simpli fi ed version as de fi ned in Eq. (D.5). The main difference is the lack of undershoot in the simpli fi ed version.CC-BY-NC-ND · From the PDF
Fig. 39 Top panel: Simulated responses of percentage change if the BOLD signal to the neural drive presented in Fig. 37. Middle panel: The responses to the second impulse around 40 s. Lower panel: The responses to a wider stimulus around 70 s. Both the simpli fi ed and convolution-based model lack the undershoot preceding the peak. Furthermore, they exhibit a time advance with respect to the peak response time of the full Balloon model.
Fig. 39 Top panel: Simulated responses of percentage change if the BOLD signal to the neural drive presented in Fig. 37. Middle panel: The responses to the second impulse around 40 s. Lower panel: The responses to a wider stimulus around 70 s. Both the simpli fi ed and convolution-based model lack the undershoot preceding the peak. Furthermore, they exhibit a time advance with respect to the peak response time of the full Balloon model.CC-BY-NC-ND · From the PDF

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  1. Spiegler, 2015, The critical brain: dissipation and restin-state networks, PNAS

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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.
ORCID
Venue
NeuroImage
Publisher
Elsevier BV
Published
2015-05
Volume
111
Pages
385-430
ISSN
1053-8119 (ISSN-L)
DOI
10.1016/j.neuroimage.2015.01.002
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
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

About these checks

Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.

Statistics and arithmetic are recomputed, not judged.

Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.

What counts as a claim ›How we measure confidence ›How we check this ›

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