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## YNIMG-07913; No. of pages: 20; 4C: NeuroImage xxx (2010) xxx–xxx Contents lists available at ScienceDirect

### Citations

4303 |
Estimating the Dimension of a Model
- Schwarz
- 1978
(Show Context)
Citation Context ...f Bayesian networks. Ramsey et al. (2010) introduced an “independent multisample greedy equivalence search” algorithm (IMaGES) for fMRI data. This method uses the Bayesian information criterion (BIC; =-=Schwarz, 1978-=-) for automatic scoring of Markov equivalence classes of directed acyclic graphs (DAGs). The restriction to DAGs means, however, that IMaGES only returns acyclic (feed-forward) graphs of effective con... |

1858 |
Investigating Causal Relations by Econometric Models and Cross-Spectra
- Granger
- 1969
(Show Context)
Citation Context ...e of DCM, which deals with dynamic models, in relation to approaches that do not (see Valdés-Sosa et al., 2010 for a full discussion). Other schemes that use dynamic graphs include Granger causality (=-=Granger, 1969-=-) and Dynamic Bayesian Networks (DBN: e.g., Burge et al., 2009; Rajapakse and Zhou, 2007). However, there is a growing appreciation that Granger causality may not be appropriate for fMRI time-series (... |

1817 | Bayes factors
- Kass, Raftery
- 1995
(Show Context)
Citation Context ...ave seen no edges survive model selection. However, there was little evidence for the graph with four connections relative to graphs with fewer connections (with log-Bayes factors of less than three; =-=Kass and Raftery, 1995-=-). In short, even with real data, the post hoc model selection proposed for network discovery appears to identify anti-edges, provided one pays attention to the relative evidence for alternative model... |

1390 |
Neural mechanisms of selective visualattention
- Desimone, Duncan
- 1995
(Show Context)
Citation Context ...onal space. This reflects the strength of the coupling between these nodes and more generally the tight functional integration between visual and prefrontal areas during visual attention tasks (e.g., =-=Desimone and Duncan, 1995-=-; Gazzaley et al., 2007). Note that this characterisation of the network is insensitive to the sign of connections. Before concluding, we now provide an exemplar analysis that can only be pursued usin... |

673 |
Causation, prediction, and search
- Spirtes, Glymour, et al.
- 1993
(Show Context)
Citation Context ...is includes reciprocal connections between two nodes. It is 539 worthwhile noting structural causal modelling based on Bayesian 540 networks (belief networks or directed acyclic graphical models; 541 =-=Spirtes et al, 2000-=-; Pearl, 2009) generally deal with directed acyclic 542 graphs; although there are treatments of linear cyclic graphs as 543 models of feedback (Richardson and Spirtes, 1999). Furthermore, 544 analyse... |

609 |
Complex brain networks: Graph theoretical analysis of structural and functional systems
- Bullmore, Sporns
- 2009
(Show Context)
Citation Context ...e connections on a mesoscopic scale); it may also reflect the fact that we deliberately chose regions that play an integrative (associational) role in cortical processing (c.f., hubs in graph theory; =-=Bullmore and Sporns, 2009-=-). There is an interesting structure to the anti-edges that speaks to the well known segregation of dorsal and ventral pathways in the visual system (Ungerleider and Haxby, 1994): The missing connecti... |

416 |
Dynamic Causal Modeling.
- Friston, Harrison, et al.
- 2003
(Show Context)
Citation Context ...abstract 48 Historically, Dynamic Causal Modelling (DCM) has been portrayed 49 as a hypothesis-led approach to understanding distributed neuronal 50 architectures underlying observed brain responses (=-=Friston et al., 2003-=-). 51 Generally, competing hypotheses are framed in terms of different 52 networks or graphs, and Bayesian model selection is used to quantify the 53 evidence for one network (hypothesis) over another... |

361 |
Predictive coding in the visual cortex: A functional interpretation of some extra-classical receptive-field effects [see comments
- Rao, Ballard
- 1999
(Show Context)
Citation Context ...Friston et al. / NeuroImage xxx (2010) xxx–xxx 17 comfortably with predictive coding accounts of brain function, which emphasise the importance of predictions that are generated in a topdown fashion (=-=Rao and Ballard, 1999-=-; Friston, 2005). 1096 1097 1098 Discussion 1099 Fig. 12. The selected graph in anatomical space and functional space: This figure shows the graph selected (on the basis of the posterior probabilities... |

297 | Hierarchical bayesian inference in the visual cortex
- Lee, Mumford
(Show Context)
Citation Context ...e have the 1010 unique opportunity to exploit asymmetries in reciprocal connections 1011 and revisit questions about hierarchical organisation (e.g., Capalbo et 1012 al., 2008; Hilgetag et al., 2000; =-=Lee and Mumford, 2003-=-; Reid et al., 1013 2009). There are many interesting analyses that one could consider, 1014 given a weighted (and signed) adjacency matrix. Here, we will 1015 illustrate a simple analysis of function... |

259 | A theory of cortical responses.
- Friston
- 2005
(Show Context)
Citation Context ...mage xxx (2010) xxx–xxx 17 comfortably with predictive coding accounts of brain function, which emphasise the importance of predictions that are generated in a topdown fashion (Rao and Ballard, 1999; =-=Friston, 2005-=-). 1096 1097 1098 Discussion 1099 Fig. 12. The selected graph in anatomical space and functional space: This figure shows the graph selected (on the basis of the posterior probabilities in the previou... |

251 |
Applications of centre manifold theory
- Carr
- 1981
(Show Context)
Citation Context ...how quickly flow changes with position. We now appeal (heuristically) to the centre manifold theorem and synergetic treatments of high-dimensional, self-organising systems (Ginzburg and Landau, 1950; =-=Carr, 1981-=-; Haken, 1983); see De Monte et al (2003), Melnik and Roberts, 2004 and Davis, 2006, for interesting examples and applications. Namely, we make the assumption that the eigenvalues λk = U− k IUk associ... |

225 |
Dynamics of blood flow and oxygenation changes during brain activation: the balloon model
- Buxton, Wong, et al.
- 1998
(Show Context)
Citation Context ... maps neuronal activity to observed hemodynamic 141 responses. This component has been described in detail many times 142 previously and rests on a hemodynamic model (subsuming the 143 Balloon model; =-=Buxton et al., 1998-=-; Friston et al, 2003; Stephan et al., 144 2007) and basically corresponds to a generalised (nonlinear) 145 convolution. In this paper, we will focus exclusively on the neuronal 146 model, because the... |

187 |
On the computational architecture of the neocortex. II. The role of cortico-cortical loops. Biol Cybernet
- Mumford
- 1992
(Show Context)
Citation Context ...03). From a functional point of view, modern theories of brain function that appeal to the Bayesian brain, call on reciprocal message passing between units encoding predictions and prediction errors (=-=Mumford, 1992-=-; Friston, 2008). Others theories that rest on reciprocal connections include belief propagation algorithms and Bayesian update schemes that have been proposed as metaphors for neuronal processing (De... |

178 | Probable networks and plausible predictions - a review of practical Bayesian methods for supervised neural networks
- MacKay
- 1995
(Show Context)
Citation Context ...rue value of zero, under the optimal model (see the central black dot in the upper right panel). This reflects the fact that this form of model selection implements automatic relevance determination (=-=MacKay, 1995-=-), by virtue of optimising the model evidence with respect to model hyperparameters; in this instance, the shrinkage priors prescribed by an adjacency matrix. Interestingly, there was a mild shrinkage... |

162 |
Synergetics: An Introduction: Nonequilibrium Phase Transitions and SelfOrganization
- Haken
- 1983
(Show Context)
Citation Context ...flow changes with position. We now appeal (heuristically) to the centre manifold theorem and synergetic treatments of high-dimensional, self-organising systems (Ginzburg and Landau, 1950; Carr, 1981; =-=Haken, 1983-=-); see De Monte et al (2003), Melnik and Roberts, 2004 and Davis, 2006, for interesting examples and applications. Namely, we make the assumption that the eigenvalues λk = U− k IUk associated − with e... |

158 | Modulation of connectivity in visual pathways by attention: Cortical interactions evaluated with structural equation modelling and fMRI.
- Buchel, Friston
- 1997
(Show Context)
Citation Context ...rate psychophysiological interactions, structural equation modelling, multivariate autoregressive models, Kalman filtering, variational filtering, DEM and Generalised Filtering (Friston et al., 1997; =-=Büchel and Friston, 1997-=-, 1998;Fristonetal., 2003, 2008, 2010; Harrison et al., 2003; Stephan et al., 2008; Li et al., 2010). Data were acquired from a normal subject at two Tesla using a Magnetom VISION (Siemens, Erlangen) ... |

128 | Intrinsic functional connectivity as a tool for human connectomics: theory, properties, and optimization. - Dijk, Hedden, et al. - 2010 |

123 |
Toward discovery science of human brain function
- Biswal, Mennes, et al.
- 2010
(Show Context)
Citation Context ...oblems due to combinatorics on 1246 connections and computational overhead. We envisage that this 1247 approach could be useful in analysing resting-state studies (Damoi1248 seaux and Greicius, 2009; =-=Biswal et al., 2010-=-; Van Dijk et al., 2010) or 1249 indeed any data reporting unknown or endogenous dynamics (e.g. 1250 sleep EEG). Although we have illustrated the approach using region 1251 specific summaries of fMRI ... |

115 |
Causal inference and causal explanation with background knowledge.
- Meek
- 1995
(Show Context)
Citation Context ...s there are finessed functional connectivity analyses that use partial correlations (e.g., Marrelec et al., 2006, 2009; Smith et al., 2010). Indeed, the principal aim of structural causal modelling ( =-=Meek, 1995-=-; Spirtes 2000; Pearl, 2009) is to identify these conditional independencies. An anti-edge requires that the effective connectivity between two nodes in a DCM is zero. This is enforced by a prior on t... |

114 | Comparing dynamic causal models. - Penny, Stephan, et al. - 2004 |

113 |
On the theory of superconductivity
- Ginzburg, Landau
- 1950
(Show Context)
Citation Context ...flow in state-space; i.e., how quickly flow changes with position. We now appeal (heuristically) to the centre manifold theorem and synergetic treatments of high-dimensional, self-organising systems (=-=Ginzburg and Landau, 1950-=-; Carr, 1981; Haken, 1983); see De Monte et al (2003), Melnik and Roberts, 2004 and Davis, 2006, for interesting examples and applications. Namely, we make the assumption that the eigenvalues λk = U− ... |

110 | Predicting human resting-state functional connectivity from structural connectivity. - Honey, Sporns, et al. - 2009 |

98 |
Constraints on cortical and thalamic projections: The no-strong-loops hypothesis.
- Crick, Koch
- 1998
(Show Context)
Citation Context ...son to 1019 bottom-up influences, the net effects of top-down connections on 1020 their targets are inhibitory (e.g., by recruitment of local lateral Q10 connections; cf, Angelucci and Bullier, 2003; =-=Crick and Koch, 1998-=-). 1022 Theoretically, this is consistent with predictive coding, where top1023 down predictions suppress prediction errors in lower levels of a 1024 hierarchy (e.g., Summerfield et al., 2006; Friston... |

95 | On the phase reduction and response dynamics of neural oscillator populations
- Brown, Moehlis, et al.
- 2004
(Show Context)
Citation Context ... parameters ζp but have in 277 mind a single circular (phase) variable (see Fig. 1), such that the rate 278 of change ζ˙ 1 : = ζ˙ reflects the instantaneous frequency of an 279 oscillating mode (cf., =-=Brown et al., 2004-=-; Kopell and Ermentrout, 280 1986; Penny et al., 2009). If we define xi : = ζ˙ i as the frequency of 281 Q4 the i-th node and ωi : = ˙ω i as fluctuations in that frequency, Eq. (3) 282 tells us that (... |

85 |
The functional logic of cortical connections.
- Zeki, Shipp
- 1988
(Show Context)
Citation Context ...graph is large. To finesse this problem we can assume all connections in the brain are directed and reciprocal. This (bidirectional coupling) assumption rests on longstanding anatomical observations (=-=Zeki and Shipp, 1988-=-) that it is rare for two cortical areas to be connected in the absence of a reciprocal connection (there are rare but important exceptions in subcortical circuits). More recently, this notion was con... |

80 | From attractor to chaotic saddle: a tale of transverse instability, Nonlinearity 9 - Ashwin, Buescu, et al. - 1996 |

64 | How good is good enough in path analysis of fMRI data? - Bullmore - 2002 |

59 |
Reaching beyond the classical receptive field of V1 neurons: horizontal or feedback axons?
- Angelucci, Bullier
- 2003
(Show Context)
Citation Context ...e to suggest that, in comparison to 1019 bottom-up influences, the net effects of top-down connections on 1020 their targets are inhibitory (e.g., by recruitment of local lateral Q10 connections; cf, =-=Angelucci and Bullier, 2003-=-; Crick and Koch, 1998). 1022 Theoretically, this is consistent with predictive coding, where top1023 down predictions suppress prediction errors in lower levels of a 1024 hierarchy (e.g., Summerfield... |

59 | A statespace model of the hemodynamic approach: nonlinear filtering of BOLD signals.
- Riera, Watanabe, et al.
- 2004
(Show Context)
Citation Context ...ssentially, this converts the problem of 389 inferring hidden states into a problem of inferring the parameters 390 (coefficients) of temporal basis functions modelling unknown hidden 391 states (cf. =-=Riera et al., 2004-=-). This rests on reformulating Eq. (4) to give 392 ˙x = Ax + ω = Ax + Cu ωðÞ t ij = ∑j CijuðÞ t j : Here, u(t) j:j=1, …, J is the j-th temporal basis function. In what follows, we use a discrete cosin... |

58 |
The dynamic brain: from spiking neurons to neural masses and cortical fields.
- Deco, VK, et al.
- 2008
(Show Context)
Citation Context ...t fluctuations that are specific to 307 each node. 308 Generative models of network activity 309 To simplify the model of responses distributed over n nodes, we 310 adopt a mean-field assumption (see =-=Deco et al., 2008-=-). This simply 311 means that the dynamics of one node are determined by the mean or 312 average activity in another. Intuitively, this is like assuming that each 313 neuron in one node ‘sees’ a suffi... |

49 |
Bayesian spiking neurons I: inference
- Deneve
- 2008
(Show Context)
Citation Context ...92; Friston, 2008). Others theories that rest on reciprocal connections include belief propagation algorithms and Bayesian update schemes that have been proposed as metaphors for neuronal processing (=-=Deneve, 2008-=-). Despite this strong motivation for introducing symmetry constraints on the adjacency matrix, it should be noted that the assumption of 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 59... |

46 | Greater than the sum of its parts: a review of studies combining structural connectivity and resting-state functional connectivity,” Brain Structure and Function, - Damoiseaux, Greicius - 2009 |

46 | Hierarchical models in the brain
- Friston
(Show Context)
Citation Context ...ctional point of view, modern theories of brain function that appeal to the Bayesian brain, call on reciprocal message passing between units encoding predictions and prediction errors (Mumford, 1992; =-=Friston, 2008-=-). Others theories that rest on reciprocal connections include belief propagation algorithms and Bayesian update schemes that have been proposed as metaphors for neuronal processing (Deneve, 2008). De... |

44 | Estimating brain functional connectivity with sparse multivariate autoregression. - Valdés-Sosa, Sánchez-Bornot, et al. - 2005 |

40 |
Key role of coupling, delay, and noise in resting brain fluctuations
- Deco, VK, et al.
- 2009
(Show Context)
Citation Context ...the ultra slow fluctuations seen in fMRI may reflect a modulation of fast synchronised activity at the neuronal level that may be a principal determinant of observed BOLD signal (Kilner et al., 2005; =-=Deco et al., 2009-=-; de Pasquale et al., 2010). From the point of 1185 1186 1187 1188 1189 1190 1191 1192 1193 1194 1195 1196 1197 1198 1199 1200 1201 1202 1203 1204 1205 1206 1207 1208 1209 1210 1211 Please cite this a... |

37 |
Identifying the brain's most globally connected regions.
- Cole, Pathak, et al.
- 2010
(Show Context)
Citation Context ...portant work in this area has looked at the efficiency of various correlation schemes and Granger causality, when identifying the sparsity and connectivity structure of real and simulated data (e.g., =-=Cole et al., 2010-=-; Gates et al., 2010; Smith et al., 2010). Finally, discovery of causal network structure from neuroimaging data has also been pursued in the context of Bayesian networks. Ramsey et al. (2010) introdu... |

36 | Partial correlation for functional brain interactivity investigation in functional MRI. - Marrelec, Krainik, et al. - 2006 |

34 | Ten simple rules for dynamic causal modeling. - Stephan, Penny, et al. - 2010 |

33 | Temporal dynamics of spontaneous MEG activity in brain networks. - Pasquale, Penna, et al. - 2010 |

33 |
The identification of interacting networks in the brain using fMRI: model selection, causality and deconvolution,”
- Roebroeck, Formisano, et al.
- 2009
(Show Context)
Citation Context ... way to identify conditional dependencies and, by scoring all possible models, discover the underlying dependency graph. Note that this can, in theory, finesse so called missing region problem (c.f., =-=Roebroeck et al., 2009-=-; Daunizeau et al., in press) that can arise when a connection is inferred that is actually mediated by common input. This is because an exhaustive model search will preclude a false inference of cond... |

33 | Comparing hemodynamic models with DCM.
- Stephan, Weiskopf, et al.
- 2007
(Show Context)
Citation Context ...esponds to a generalised (nonlinear) 145 convolution. In this paper, we will focus exclusively on the neuronal 146 model, because the hemodynamic part is exactly the same as 147 described previously (=-=Stephan et al., 2007-=-). Although we will focus 148 on neuronal systems, the following arguments apply to any complex 149 distributed system with coupled nonlinear dynamics. This means that 150 the procedures described lat... |

32 | Dynamics of a neural system with a multiscale architecture - Breakspear, Stam - 2005 |

29 |
Network participation indices: characterizing component roles for information processing in neural networks.
- Kotter, KE
- 2003
(Show Context)
Citation Context ...al circuits). More recently, this notion was confirmed in comprehensive analyses of large connectivity databases demonstrating a very strong tendency of cortico-cortical connections to be reciprocal (=-=Kötter and Stephan, 2003-=-). From a functional point of view, modern theories of brain function that appeal to the Bayesian brain, call on reciprocal message passing between units encoding predictions and prediction errors (Mu... |

29 |
Causality: Models, Reasoning and Inference, 2 nd edition,
- Pearl
- 2009
(Show Context)
Citation Context ...l connections between two nodes. It is 539 worthwhile noting structural causal modelling based on Bayesian 540 networks (belief networks or directed acyclic graphical models; 541 Spirtes et al, 2000; =-=Pearl, 2009-=-) generally deal with directed acyclic 542 graphs; although there are treatments of linear cyclic graphs as 543 models of feedback (Richardson and Spirtes, 1999). Furthermore, 544 analyses of function... |

28 | Effective connectivity: influence, causality and biophysical modeling. Neuroimage 58, 339–361. doi: 10.1016/j.neuroimage.2011.03.058 - Valdes-Sosa, Roebroeck, et al. - 2011 |

27 | Functional interactions between prefrontal and visual association cortex contribute to topdown modulation of visual processing. - Gazzaley, Rissman, et al. - 2004 |

27 | Comparing families of dynamic causal models.
- Penny, Stephan, et al.
- 2010
(Show Context)
Citation Context ...models over which 55 people search (the model-space) has grown enormously; to the extent 56 that DCM is now used to discover the best model over very large model57 spaces (e.g., Stephan et al., 2010; =-=Penny et al., 2010-=-). Here, we take this 58 discovery theme one step further and throw away prior knowledge 59 about the experimental causes of observed responses to make DCM 60 entirely data-led. This enables network d... |

26 |
Variational Bayesian identification and prediction of stochastic nonlinear dynamic causal models
- Daunizeau, Friston, et al.
- 2009
(Show Context)
Citation Context ... variants of 111 Dynamic Causal Modelling. 112 We have introduced several schemes recently that accommodate 113 fluctuations on hidden neuronal and other physiological states (Penny 114 et al., 2005; =-=Daunizeau et al, 2009-=-; Friston et al., 2010; Li et al., 2010). 115 This means that one can estimate hidden states generating observed 116 data, while properly accommodating endogenous or random fluctua117 tions. These bec... |

26 |
Network modelling methods for FMRI. NeuroImage
- Smith, Miller, et al.
- 2011
(Show Context)
Citation Context ...d be conditionally independent when conditioned on a third node. Having said this there are finessed functional connectivity analyses that use partial correlations (e.g., Marrelec et al., 2006, 2009; =-=Smith et al., 2010-=-). Indeed, the principal aim of structural causal modelling ( Meek, 1995; Spirtes 2000; Pearl, 2009) is to identify these conditional independencies. An anti-edge requires that the effective connectiv... |

24 |
Learning effective brain connectivity with dynamic
- Rajapakse, Zhou
- 2007
(Show Context)
Citation Context ...not (see Valdés-Sosa et al., 2010 for a full discussion). Other schemes that use dynamic graphs include Granger causality (Granger, 1969) and Dynamic Bayesian Networks (DBN: e.g., Burge et al., 2009; =-=Rajapakse and Zhou, 2007-=-). However, there is a growing appreciation that Granger causality may not be appropriate for fMRI time-series (e.g., Nalatore et al, 2007) and performs poorly in comparison to structural (non-dynamic... |

21 | Scale-free dynamics of global functional connectivity in the human brain. - Stam, Bruin - 2004 |

20 |
Dynamic connectivity in neural systems: theoretical and empirical considerations
- Breakspear
- 2004
(Show Context)
Citation Context ...levance for cortical 248 dynamics. Indeed, manifolds that arise from near symmetry in 249 coupled dynamical systems have been studied extensively as models 250 of synchronised neuronal activity (e.g. =-=Breakspear, 2004-=-; Breakspear 251 and Stam, 2005). 252 Usually, the centre manifold theorem is used to characterise the 253 dynamics on the centre manifold in terms of its bifurcations and 254 structural stability, th... |

20 | Dynamic changes of effective connectivity characterized by variable parameter regression and - Buchel, Friston - 1998 |

20 | Dynamic causal modeling and Granger causality. Comments on: The identification of interacting networks in the brain using fMRI: Model selection, causality and deconvolution. Neuroimage 2009; doi:10.1016/j.neuroimage.2009.09.031 - Friston |

20 | Automated discovery of linear feedback models.
- Richardson, Spirtes
- 1999
(Show Context)
Citation Context ...ected acyclic graphical models; 541 Spirtes et al, 2000; Pearl, 2009) generally deal with directed acyclic 542 graphs; although there are treatments of linear cyclic graphs as 543 models of feedback (=-=Richardson and Spirtes, 1999-=-). Furthermore, 544 analyses of functional connectivity (and of diffusion tensor imaging 545 data) only consider undirected graphs because the direction of the 546 influence between two nodes is not a... |

18 | Dynamic causal modelling of induced responses
- Chen, Kiebel, et al.
- 2008
(Show Context)
Citation Context ...eters 1184 increases quadratically with the number of nodes). Having said this, DCM is used routinely to invert models with thousands of free parameters (e.g. DCM for induced electromagnetic sources; =-=Chen et al., 2008-=-). One approach to large numbers of nodes (e.g., voxels) is to summarise distributed activity in terms of modes or patterns and then estimate the coupling among those patterns (cf, Chen et al., 2008; ... |

18 | 2006b) Mistaking a house for a face: Neural correlates of misperception in healthy humans. Cereb Cortex 16:500–508
- Summerfield, Egner, et al.
(Show Context)
Citation Context ...llier, 2003; Crick and Koch, 1998). 1022 Theoretically, this is consistent with predictive coding, where top1023 down predictions suppress prediction errors in lower levels of a 1024 hierarchy (e.g., =-=Summerfield et al., 2006-=-; Friston, 2008; Chen et al., 1025 2009). One might therefore ask which hierarchical ordering of the 1026 nodes maximises the average strength of forward connections relative to their backward homolog... |

16 | Self-organized criticality and scale-free properties in emergent functional neural networks. - Shin, Kim - 2006 |

15 | Criteria for optimizing cortical hierarchies with continuous ranges - Krumnack, Reid, et al. - 2010 |

13 |
Corticothalamic interactions in the transfer of visual information.
- Sillito, Jones
- 2002
(Show Context)
Citation Context ...g. 12). This is entirely 1092 sensible, given the greater abundance of backward connections 1093 anatomically, both within the cortical hierarchy and from cortex to 1094 subcortical structures (e.g., =-=Sillito and Jones, 2002-=-). Furthermore, the 1095 importance of backward connections or top-down influences fits The quest for discovering causal network structure has a long history, and automatic procedures for determining ... |

11 | Coherent regimes of globally coupled dynamical systems - Monte, d’Ovidio, et al. - 2003 |

10 | Bilinear dynamical systems - Penny, Ghahramani, et al. - 2005 |

10 | Nonlinear dynamic causal models for fMRI - Stephan, Kasper, et al. - 2008 |

9 |
Electroencephalography/functional MRI in human epilepsy: what it currently can and cannot do. Current opinion in neurology 20
- Laufs, Duncan
- 2007
(Show Context)
Citation Context ...f neuronal activity is the most prescient for fMRI responses. This is because it is generally assumed that fMRI signals scale with the predominant frequency of neuronal activity (Kilner et al., 2005; =-=Laufs and Duncan, 2007-=-; Rosa et al., 2010). We now turn to how different nodes are coupled and see how a separation of fast and slow dynamics in a distributed network of nodes provides a model for network dynamics. We will... |

9 |
Mitigating the effects of measurement noise on granger causality
- Nalatore, Ding, et al.
- 2007
(Show Context)
Citation Context ...Dynamic Bayesian Networks (DBN: e.g., Burge et al., 2009; Rajapakse and Zhou, 2007). However, there is a growing appreciation that Granger causality may not be appropriate for fMRI time-series (e.g., =-=Nalatore et al, 2007-=-) and performs poorly in comparison to structural (non-dynamic) approaches based upon partial correlations (Smith et al., 2010). Granger causality and DBN rest on the theory of Martingales (i.e. Marko... |

9 |
Estimating the transfer function from neuronal activity to BOLD using simultaneous EEG-fMRI.
- MJ, Kilner, et al.
- 2010
(Show Context)
Citation Context ...he most prescient for fMRI responses. This is because it is generally assumed that fMRI signals scale with the predominant frequency of neuronal activity (Kilner et al., 2005; Laufs and Duncan, 2007; =-=Rosa et al., 2010-=-). We now turn to how different nodes are coupled and see how a separation of fast and slow dynamics in a distributed network of nodes provides a model for network dynamics. We will see that only the ... |

9 |
Endogenous multifractal brain dynamics are modulated by age, cholinergic blockade and cognitive performance.
- Suckling, AM, et al.
- 2010
(Show Context)
Citation Context ...el of neuronal dynamics. In this paper, we take a closer 75 look at what this linear approximation means, when considering 76 endogenous fluctuations that arise from self-organised dynamics (e.g., 77 =-=Suckling et al., 2008-=-; Honey et al, 2009). Having established the basic 78 form of our model, we then turn to model inversion and consider briefly 79 the distinction between deterministic and stochastic schemes. This 80 d... |

8 |
Discrete dynamic Bayesian network analysis of fMRI data
- Burge, Lane, et al.
(Show Context)
Citation Context ... approaches that do not (see Valdés-Sosa et al., 2010 for a full discussion). Other schemes that use dynamic graphs include Granger causality (Granger, 1969) and Dynamic Bayesian Networks (DBN: e.g., =-=Burge et al., 2009-=-; Rajapakse and Zhou, 2007). However, there is a growing appreciation that Granger causality may not be appropriate for fMRI time-series (e.g., Nalatore et al, 2007) and performs poorly in comparison ... |

8 | and backward connections in the brain: a DCM study of functional asymmetries - Chen, Henson, et al. - 2009 |

8 |
Low-dimensional manifolds in reaction-diffusion equations. 1. Fundamental aspects
- Davis
(Show Context)
Citation Context ...e manifold theorem and synergetic treatments of high-dimensional, self-organising systems (Ginzburg and Landau, 1950; Carr, 1981; Haken, 1983); see De Monte et al (2003), Melnik and Roberts, 2004 and =-=Davis, 2006-=-, for interesting examples and applications. Namely, we make the assumption that the eigenvalues λk = U− k IUk associated − with each mode ζk=U k ξ are distributed sparsely; λ1 N λ2 N λ3…∈R. That is, ... |

7 |
What’ and ‘where’ in the human
- Ungerleider, Haxby
- 1994
(Show Context)
Citation Context ...., hubs in graph theory; Bullmore and Sporns, 2009). There is an interesting structure to the anti-edges that speaks to the well known segregation of dorsal and ventral pathways in the visual system (=-=Ungerleider and Haxby, 1994-=-): The missing connections are between (i) the superior temporal sulcus and the early visual system, and (ii) the (ventral) superior temporal sulcus/angular gyrus and (dorsal) posterior parietal corte... |

4 | Combining structural connectivity and response latencies to model the structure of the visual system. - Capalbo, Postma, et al. - 2008 |

3 | Bubbling of attractors and synchronization of chaotic attractors - Ashwin, Buescu, et al. - 1994 |

3 | A Opt. Image Sci - Am |

3 | Computational models for multi-scale coupled dynamic problems,” Future Generation
- Melnik, Roberts
- 2004
(Show Context)
Citation Context ... (heuristically) to the centre manifold theorem and synergetic treatments of high-dimensional, self-organising systems (Ginzburg and Landau, 1950; Carr, 1981; Haken, 1983); see De Monte et al (2003), =-=Melnik and Roberts, 2004-=- and Davis, 2006, for interesting examples and applications. Namely, we make the assumption that the eigenvalues λk = U− k IUk associated − with each mode ζk=U k ξ are distributed sparsely; λ1 N λ2 N ... |

1 |
Post hoc model selection — under review
- Friston, Penny
- 2011
(Show Context)
Citation Context ...density over reduced parameters that is far from a prior of zero suggests the reduced parameters are needed to explain the data. Eq. (8) contains the Savage–Dickey density ratio (Dickey, 1971; seealso=-=Friston and Penny, 2011-=-) thatisusedfornestedmodel comparison, and indeed all classical inference using the extra sum of squares principle (such as F-tests or analysis of variance, ANOVA). We can approximate the marginal pos... |

1 |
Psychophysio1363 logical and modulatory interactions in neuroimaging
- Friston, Büchel, et al.
- 1997
(Show Context)
Citation Context ...d previously to illustrate psychophysiological interactions, structural equation modelling, multivariate autoregressive models, Kalman filtering, variational filtering, DEM and Generalised Filtering (=-=Friston et al., 1997-=-; Büchel and Friston, 1997, 1998;Fristonetal., 2003, 2008, 2010; Harrison et al., 2003; Stephan et al., 2008; Li et al., 2010). Data were acquired from a normal subject at two Tesla using a Magnetom V... |

1 | Variational 1367 free energy and the Laplace approximation - Friston, Mattout, et al. - 2007 |

1 |
DEM: a variational treatment of 1369 dynamic systems
- Friston, Trujillo-Barreto, et al.
- 2008
(Show Context)
Citation Context ...thermore, the 1262 attentional data set used in this paper can be downloaded from the 1263 above website, for people who want to reproduce the analyses 1264 described in this paper. Uncited reference =-=Friston et al., 2008-=- 1267 Acknowledgments 1268 This work was funded by the Wellcome Trust and supported by the 1269 China Scholarship Council (CSC), the NEUROCHOICE project by 1270 SystemsX.ch (JD, KES) and the Universit... |

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Automatic search 1373 for fMRI connectivity mapping: an alternative to Granger causality testing using 1374 formal equivalences among SEM path modeling
- Gates, Molenaar, et al.
- 2010
(Show Context)
Citation Context ...s area has looked at the efficiency of various correlation schemes and Granger causality, when identifying the sparsity and connectivity structure of real and simulated data (e.g., Cole et al., 2010; =-=Gates et al., 2010-=-; Smith et al., 2010). Finally, discovery of causal network structure from neuroimaging data has also been pursued in the context of Bayesian networks. Ramsey et al. (2010) introduced an “independent ... |

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Multivariate autoregressive modeling of 1390 fMRI time series
- Harrison, Penny, et al.
- 2003
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Citation Context ...lling, multivariate autoregressive models, Kalman filtering, variational filtering, DEM and Generalised Filtering (Friston et al., 1997; Büchel and Friston, 1997, 1998;Fristonetal., 2003, 2008, 2010; =-=Harrison et al., 2003-=-; Stephan et al., 2008; Li et al., 2010). Data were acquired from a normal subject at two Tesla using a Magnetom VISION (Siemens, Erlangen) whole body MRI system, during a visual 852 853 854 855 856 8... |

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Dynamic Granger causality based 1392 on Kalman filter for evaluation of functional network connectivity in fMRI data
- Havlicek, Jan, et al.
- 2010
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Citation Context ...). One approach to large numbers of nodes (e.g., voxels) is to summarise distributed activity in terms of modes or patterns and then estimate the coupling among those patterns (cf, Chen et al., 2008; =-=Havlicek et al., 2010-=-). In terms of the increase in the size of model space with the number of nodes; as noted by one of our reviewers, one could employ a greedy search using the post hoc log-evidence. In our current impl... |

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Hierarchical organization of macaque 1395 and cat cortical sensory systems explored with a novel network processor
- Hilgetag, O'Neill, et al.
- 2000
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Citation Context ...graphs. However, here we have the 1010 unique opportunity to exploit asymmetries in reciprocal connections 1011 and revisit questions about hierarchical organisation (e.g., Capalbo et 1012 al., 2008; =-=Hilgetag et al., 2000-=-; Lee and Mumford, 2003; Reid et al., 1013 2009). There are many interesting analyses that one could consider, 1014 given a weighted (and signed) adjacency matrix. Here, we will 1015 illustrate a simp... |

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The existence of generalized synchronization of chaotic 1401 systems in complex networks
- Hu, Xu, et al.
- 2010
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Citation Context ...back to) the manifold. Please see the main text for a full description of the equations. ð2Þ approximates a hyper-diagonal subspace or a smoothly mapped 244 (synchronisation) manifold close by (e.g., =-=Hu et al., 2010-=-): The 245 presence of strong transverse flow towards this manifold and a 246 weakly stable or unstable flow on the manifold is exactly the sort of 247 behaviour described by Eq. (3) and has clear rel... |

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Hemodynamic correlates of EEG: a 1404 heuristic
- Kilner, Mattout, et al.
- 2005
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Citation Context ...ecause this summary of neuronal activity is the most prescient for fMRI responses. This is because it is generally assumed that fMRI signals scale with the predominant frequency of neuronal activity (=-=Kilner et al., 2005-=-; Laufs and Duncan, 2007; Rosa et al., 2010). We now turn to how different nodes are coupled and see how a separation of fast and slow dynamics in a distributed network of nodes provides a model for n... |

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Broadband criticality 1406 of human brain network synchronization
- Kitzbichler, Smith, et al.
- 2009
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Citation Context ... has additional plausibility for 191 neuronal systems, given their tendency to show self-organised 192 criticality and slowing (Stam and de Bruin, 2004; Shin and Kim, 193 2006; Suckling et al., 2008; =-=Kitzbichler et al., 2009-=-). Critical slowing 194 means that some modes decay slowly and show protracted correla- 195 tions over time. 196 Put simply, all this means is that the dynamics of any system 197 comprising many eleme... |

1 | Symmetry and phase-locking in chains of weakly 1408 coupled oscillators - Kopell, Ermentrout - 1986 |

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Stochastic DCM and 1416 generalised filtering. Under revision
- Li, Daunizeau, et al.
- 2010
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Citation Context ...0 of the DCM used in subsequent sections. This is exactly the same as the 71 conventional DCM for fMRI but includes endogenous fluctuations, 72 which are represented by random differential equations (=-=Li et al., 2010-=-). 73 These equations can be regarded as a [bi]linear approximation to any 74 nonlinear model of neuronal dynamics. In this paper, we take a closer 75 look at what this linear approximation means, whe... |

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Dynamic causal modelling for fMRI: a two1424 state model
- Marreiros, Kiebel, et al.
- 2008
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Citation Context ...pled reasons to suppose this 339 phase-variable (and its manifold) exist. In previous motivations, we 340 represented the macroscopic behaviour of each node with one 341 (Friston et al, 2003) or two (=-=Marreiros et al, 2008-=-) macroscopic 342 neuronal states; with no motivation for why this was appropriate or 343 sufficient. The current treatment provides that motivation and shows 344 that using a small number of macrosco... |

1 | Large-scale neural model validation of 1429 partial correlation analysis for effective connectivity investigation in functional 1430 - Marrelec, Kim, et al. - 2009 |

1 | problems for causal inference from fMRI - Six |