# The Keith Worsley workshop on Computational Modeling of Brain Dynamics: from stochastic models to Neuroimages (09w5092)

Arriving in Banff, Alberta Sunday, June 14 and departing Friday June 19, 2009

## Organizers

Viktor Jirsa (Theoretical Neuroscience Group)

Pedro A. Valdés-Sosa (Cuban Neuroscience Center)

Keith Worsley (McGill University)

## Objectives

We are at a critical stage in understanding the neural underpinning of human cognitive processes with a sharp contradiction between the enormous amounts of data gathered by available neuroimaging technologies and the relatively low level of application of theoretical models to explain in detail the observed phenomena. The objective of this workshop is to bring together leading experts of the Human Brain Mapping Project together with computational neuroscientists, mathematicians and statisticians in order to assess the current state of database construction, neural modeling, numerical methods, computational technologies and statistical procedures that could bridge the gap between existing theory and the explanation of individually recorded brain images. We will emphasize the interpretation of data obtained from the human electroencephalogram (EEG) and functional magnetic resonance imaging (MRI), especially the understanding of oscillatory activity recorded in concurrent EEG-fMRI experiments. We hope to create links between several lines of work that have been developing in isolation and that are apparently ready to cross-fertilize. In fact if the workshop is approved the detailed questions enumerated below could be distributed and work advanced previous to the workshop. During the workshop the insights provided and collaborations that would be enhanced would have a positive impact in future research into normal brain function and its alterations in neuropsychiatric disorders as well as stimulating new areas of work in applied mathematics, statistics and biophysics. An additional objective would be to place an overview of the workshop in a high impact factor journal such as NeuroImage or Human Brain Mapping possibly accompanied by a selection of requested papers.

We propose to dedicate sessions successively to review and discuss the following areas:

1. Neural Mass Modeling (Day 1): The field of mesoscopic neural mass modeling based has been vigorously developed [1]. Currently local models formulated in terms of stochastic ordinary differential equations [2;2;3] are able to simulate normal and pathological EEG signals and have provided n insight into the nonlinear dynamics of the brain. More recently these models have been extended to global models based on stochastic partial differential equations [4-6]. The main difficulties to apply this modeling to actual data until recently has been: i) the lack of specific data for a given subject regarding cortical morphology, brain connectivity , and head volume conductor properties; ii) Limitations in numerical methods for the simulation and estimation of stochastic ordinary and partial differential equations, iii) Paucity of statistical methods that can adequately address the huge amounts of data provided by neuoroimages. More specifically in this session we will consider :

a. [7]An overview of the basic physiology and results of neural mass modeling: (Freeman[3], Lopes da Silva [2]

b. The current state and limitations of local models: Deco [8], Faugeras, Robinson [9], Suffcynzki and Wendling [10].

c. The current state and limitations of global models: Breakspear [11], Jirsa, Nunez [6], Wright [12].

d. Stochastic characterization of states and bifurcations of nonlinear local and global models: Faugeras , Harrison [13], Wennekers [14].

e. The basic information about brain morphology and connectivity, head volume conductor properties, population characteristics of hemodynamic and EEG characteristics that should be provided by the Human Brain Mapping initiative.

2. Structural constraints on brain dynamics provided by the Human Brain Mapping Project (Day 2): Over the past decade a wealth of information has been acquired and integrated into publicly available databases about the detailed cyto-architecture [15], gross brain morphology [16] and connectivity [17] of the human brain. This can provide the information that has been lacking for detailed neural modeling. New questions must be addressed and combined functional/morphological information gathered. More specifically in this session we will consider :

a. Current state of anatomical databases: Evans [18], ,Zilles [19],

b. General principles of brain connectivity, relevance for neural modeling:, Hilgetag [20;21] , Kotter [22], Sporns [21], Amunts [23].

c. In vivo determination of head volume conductor parameters and connectivity measures: Valdes-Hernandez [24].

3. Integration and estimation of neural stochastic models: In this session recent advances in the integration of stochastic differential equations will be examined. Particular emphasis will be places on the use of exponential integrators in order to preserve the qualitative dynamics of continuous neural mass models when discretized. Emphasis will also be placed on filtering and maximum likelihood estimation. More specifically in this session we will consider :

a. An Overview on bridging dynamical system and nonlinear time series theory: Ozaki .[25-27]

b. Simulation and estimation of ordinary stochastic differential equations, differential algebraic equations , and delay equations.: Jimenez [28;29], Burrage [30],Biscay [28;31], Roy [32].

c. Integration of stochastic partial differential equations: Berland [33].

4. Large scale simulations of neural networks (Day3): In this session we shall look at the computational issues of large and medium scale modeling of neural masses with emphasis on methods for producing simulations that can be checked against experimental results. This would be in themorning and the afternoon would be dedicated to group discussions and a walkabout the BIRS area.

a. Overview of large scale modeling, what exhaustive modeling can tell us aout neural masses: Markram [34].

b. Medium scale models: Deco [8], Horwitz [35].

5. Statistical inference on Brain images based on neural modeling (Day 4): In this session we

shall examine the current state and limitations of Statistical Parametric Mapping (SPM) of Neuroimages. The use of Bayesian state space models shall be examined in detail as well as procedures to deal with p much larger than n (p=numbr of variables, n=number of observations).

a. Overview of inference for statistical images: Worsley [36;37], Friston [38].

b. SPM of EEG/ERP models: David [38], Demiralp [39], Moran, Sotero [40].

c. SPM modeling functional images: Trujillo [40-42], Uludag [43].

d. SPM models for joint EEG/fMRI recordings: Babajani [44], Riera [45], Valdes[22;25;46]

6. Wrap up (Day 5): Overall discussion of issues, planning of collaborations and of publications.

Bibliography

[1] P. Suffczynski, F. Wendling, J. J. Bellanger, and F. H. L. da Silva, "Some insights into computational models of (patho)physiological brain activity," Proceedings of the Ieee, vol. 94, no. 4, pp. 784-804, 2006.

[2] F. H. Lopesdas, A. Hoeks, H. Smits, and L. H. Zetterbe, "Model of Brain Rhythmic Activity - Alpha-Rhythm of Thalamus," Kybernetik, vol. 15, no. 1, pp. 27-37, 1974.

[3] W. J. Freeman and G. Vitiello, "Nonlinear brain dynamics as macroscopic manifestation of underlying many-body field dynamics," Physics of Life Reviews, vol. 3, no. 2, pp. 93-118, 2006.

[4] J. J. Wright, C. J. Rennie, G. J. Lees, P. A. Robinson, P. D. Bourke, C. L. Chapman, E. Gordon, and D. L. Rowe, "Simulated electrocortical activity at microscopic, mesoscopic and global scales," International Journal of Bifurcation and Chaos, vol. 14, no. 2, pp. 853-872, 2004.

[5] V. K. Jirsa, "Connectivity and dynamics of neural information processing," Neuroinformatics, vol. 2, no. 2, pp. 183-204, 2004.

[6] P. L. Nunez, B. M. Wingeier, and R. B. Silberstein, "Spatial-temporal structures of human alpha rhythms: Theory, microcurrent sources, multiscale measurements, and global binding of local networks," Human Brain Mapping, vol. 13, no. 3, pp. 125-164, 2001.

[7] F. Grimbert and O. Faugeras, "Bifurcation analysis of Jansen's neural mass model," Neural Computation, vol. 18, no. 12, pp. 3052-3068, 2006.

[8] G. Deco and D. Marti, "Deterministic analysis of stochastic bifurcations in multi-stable neurodynamical systems," Biological Cybernetics, vol. 96, no. 5, pp. 487-496, 2007.

[9] P. A. Robinson, C. J. Rennie, D. L. Rowe, and S. C. O'Connor, "Estimation of multiscale neurophysiologic parameters by electroencephalographic means," Human Brain Mapping, vol. 23, no. 1, pp. 53-72, 2004.

[10] B. M. Bouwman, P. Suffczynski, F. H. L. da Silva, E. Maris, and C. M. van Rijn, "GABAergic mechanisms in absence epilepsy: a computational model of absence epilepsy simulating spike and wave discharges after vigabatrin in WAG/Rij rats," European Journal of Neuroscience, vol. 25, no. 9, pp. 2783-2790, 2007.

[11] M. Breakspear and J. R. Terry, "Nonlinear interdependence in neural systems: Motivation, theory, and relevance," International Journal of Neuroscience, vol. 112, no. 10, pp. 1263-1284, 2002.

[12] J. J. Wright, "Simulation of EEG: dynamic changes in synaptic efficacy, cerebral rhythms, and dissipative and generative activity in cortex," Biological Cybernetics, vol. 81, no. 2, pp. 131-147, 1999.

[13] L. M. Harrison, O. David, and K. J. Friston, "Stochastic models of neuronal dynamics," Philosophical Transactions of the Royal Society B-Biological Sciences, vol. 360, no. 1457, pp. 1075-1091, 2005.

[14] T. Wennekers, M. Garagnani, and F. Pulvermuller, "Language models based on Hebbian cell assemblies," Journal of Physiology-Paris, vol. 100, no. 1-3, pp. 16-30, 2006.

[15] S. B. Eickhoff, T. Paus, S. Caspers, M. H. Grosbras, A. C. Evans, K. Zilles, and K. Amunts, "Assignment of functional activations to probabilistic cytoarchitectonic areas revisited," Neuroimage, vol. 36, no. 3, pp. 511-521, 2007.

[16] J. Mazziotta, A. Toga, A. Evans, P. Fox, J. Lancaster, K. Zilles, R. Woods, T. Paus, G. Simpson, B. Pike, C. Holmes, L. Collins, P. Thompson, D. MacDonald, M. Iacoboni, T. Schormann, K. Amunts, N. Palomero-Gallagher, S. Geyer, L. Parsons, K. Narr, N. Kabani, G. Le Goualher, D. Boomsma, T. Cannon, R. Kawashima, and B. Mazoyer, "A probabilistic atlas and reference system for the human brain: International Consortium for Brain Mapping (ICBM)," Philosophical Transactions of the Royal Society of London Series B-Biological Sciences, vol. 356, no. 1412, pp. 1293-1322, 2001.

[17] O. Sporns, G. Tononi, and R. Kotter, "The human connectome: A structural description of the human brain," Plos Computational Biology, vol. 1, no. 4, pp. 245-251, 2005.

[18] P. Shaw, D. Greenstein, J. Lerch, L. Clasen, R. Lenroot, N. Gogtay, A. Evans, J. Rapoport, and J. Giedd, "Intellectual ability and cortical development in children and adolescents," Nature, vol. 440, no. 7084, pp. 676-679, 2006.

[19] A. W. Toga, P. M. Thompson, S. Mori, K. Amunts, and K. Zilles, "Towards multimodal atlases of the human brain," Nature Reviews Neuroscience, vol. 7, no. 12, pp. 952-966, 2006.

[20] C. C. Hilgetag, "Principles of brain connectivity organization," Behavioral and Brain Sciences, vol. 29, no. 1, p. 18-+, 2006.

[21] O. Sporns, D. R. Chialvo, M. Kaiser, and C. C. Hilgetag, "Organization, development and function of complex brain networks," Trends in Cognitive Sciences, vol. 8, no. 9, pp. 418-425, 2004.

[22] P. A. Valdes-Sosa, R. Kotter, and K. J. Friston, "Introduction: multimodal neuroimaging of brain connectivity," Philosophical Transactions of the Royal Society B-Biological Sciences, vol. 360, no. 1457, pp. 865-867, 2005.

[23] U. Burgel, K. Amunts, L. Hoemke, H. Mohlberg, J. M. Gilsbach, and K. Zilles, "White matter fiber tracts of the human brain: Three-dimensional mapping at microscopic resolution, topography and intersubject variability," Neuroimage, vol. 29, no. 4, pp. 1092-1105, 2006.

[24] Y. Iturria-Medina, E. J. Canales-Rodriguez, L. Melie-Garcia, P. A. Valdes-Hernandez, E. Martinez-Montes, Y. eman-Gomez, and J. M. Sanchez-Bornot, "Characterizing brain anatomical connections using diffusion weighted MRI and graph theory," Neuroimage, vol. 36, no. 3, pp. 645-660, 2007.

[25] P. A. Valdes, J. C. Jimenez, J. Riera, R. Biscay, and T. Ozaki, "Nonlinear EEG analysis based on a neural mass model," Biological Cybernetics, vol. 81, no. 5-6, pp. 415-424, 1999.

[26] A. Galka, O. Yamashita, T. Ozaki, R. Biscay, and P. Valdes-Sosa, "A solution to the dynamical inverse problem of EEG generation using spatiotemporal Kalman filtering," Neuroimage, vol. 23, no. 2, pp. 435-453, 2004.

[27] T. Ozaki, "A Bridge Between Nonlinear Time-Series Models and Nonlinear Stochastic Dynamic-Systems - A Local Linearization Approach," Statistica Sinica, vol. 2, no. 1, pp. 113-135, 1992.

[28] H. De la Cruz, R. J. Biscay, F. Carbonell, T. Ozaki, and J. C. Jimenez, "A higher order local linearization method for solving ordinary differential equations," Applied Mathematics and Computation, vol. 185, no. 1, pp. 197-212, 2007.

[29] J. C. Jimenez, L. M. Pedroso, F. Carbonell, and V. Hernandez, "Local linearization method for numerical integration of delay differential equations," Siam Journal on Numerical Analysis, vol. 44, no. 6, pp. 2584-2609, 2006.

[30] K. Burrage, I. Lenane, and G. Lythe, "Numerical methods for second-order stochastic differential equations," Siam Journal on Scientific Computing, vol. 29, no. 1, pp. 245-264, 2007.

[31] F. Carbonell, J. C. Jimenez, and R. J. Biscay, "Weak local linear discretizations for stochastic differential equations: Convergence and numerical schemes," Journal of Computational

and Applied Mathematics, vol. 197, no. 2, pp. 578-596, 2006.

[32] D. Roy and M. K. Dash, "A novel stochastic locally transversal linearization (LTL) technique for engineering dynamical systems: Strong solutions," Applied Mathematical Modelling, vol. 29, no. 10, pp. 913-937, 2005.

[33] H. Berland, B. Skaflestad, and W. M. Wright, "EXPINT - A MATLAB package for exponential integrators," Acm Transactions on Mathematical Software, vol. 33, no. 1 2007.

[34] H. Markram, "Blue Brain Project," 2007.

[35] B. Horwitz, K. J. Friston, and J. G. Taylor, "Neural modeling and functional brain imaging: an overview," Neural Networks, vol. 13, no. 8-9, pp. 829-846, 2000.

[36] J. E. Taylor, K. J. Worsley, and F. Gosselin, "Maxima of discretely sampled random fields, with an application to 'bubbles'," Biometrika, vol. 94, no. 1, pp. 1-18, 2007.

[37] K. J. Worsley, "An improved theoretical P value for SPMs based on discrete local maxima," Neuroimage, vol. 28, no. 4, pp. 1056-1062, 2005.

[38] O. David, J. M. Kilner, and K. J. Friston, "Mechanisms of evoked and induced responses in MEG/EEG," Neuroimage, vol. 31, no. 4, pp. 1580-1591, 2006.

[39] T. Demiralp, Z. Bayraktaroglu, D. Lenz, S. Junge, N. A. Busch, B. Maess, M. Ergen, and C. S. Herrmann, "Gamma amplitudes are coupled EEG during visual to theta phase in human perception," International Journal of Psychophysiology, vol. 64, no. 1, pp. 24-30, 2007.

[40] R. C. Sotero, N. J. Trujillo-Barreto, Y. Iturria-Medina, F. Carbonell, and J. C. Jimenez, "Realistically coupled neural mass models can generate EEG rhythms," Neural Computation, vol. 19, no. 2, pp. 478-512, 2007.

[41] R. C. Sotero and N. J. Trujillo-Barreto, "Modelling the role of excitatory and inhibitory neuronal activity in the generation of the BOLD signal," Neuroimage, vol. 35, no. 1, pp. 149-165, 2007.

[42] N. J. Trujillo-Barreto, E. ubert-Vazquez, and P. A. Valdes-Sosa, "Bayesian model averaging in EEG/MEG imaging," Neuroimage, vol. 21, no. 4, pp. 1300-1319, 2004.

[43] N. Harel, K. Ugurbil, K. Uludag, and E. Yacoub, "Frontiers of brain mapping using MRI," Journal of Magnetic Resonance Imaging, vol. 23, no. 6, pp. 945-957, 2006.

[44] A. Babajani and H. Soltanian-Zadeh, "Integrated MEG/EEG and fMRI model based on neural masses," Ieee Transactions on Biomedical Engineering, vol. 53, no. 9, pp. 1794-1801, 2006.

[45] J. J. Riera, X. H. Wan, J. C. Jimenez, and R. Kawashima, "Nonlinear local electrovascular coupling. I: A theoretical model," Human Brain Mapping, vol. 27, no. 11, pp. 896-914, 2006.

[46] E. Martinez-Montes, P. A. Valdes-Sosa, F. Miwakeichi, R. I. Goldman, and M. S. Cohen, "Concurrent EEG/fMRI analysis by multiway Partial Least Squares," Neuroimage, vol. 22, no. 3, pp. 1023-1034, 2004.

We propose to dedicate sessions successively to review and discuss the following areas:

1. Neural Mass Modeling (Day 1): The field of mesoscopic neural mass modeling based has been vigorously developed [1]. Currently local models formulated in terms of stochastic ordinary differential equations [2;2;3] are able to simulate normal and pathological EEG signals and have provided n insight into the nonlinear dynamics of the brain. More recently these models have been extended to global models based on stochastic partial differential equations [4-6]. The main difficulties to apply this modeling to actual data until recently has been: i) the lack of specific data for a given subject regarding cortical morphology, brain connectivity , and head volume conductor properties; ii) Limitations in numerical methods for the simulation and estimation of stochastic ordinary and partial differential equations, iii) Paucity of statistical methods that can adequately address the huge amounts of data provided by neuoroimages. More specifically in this session we will consider :

a. [7]An overview of the basic physiology and results of neural mass modeling: (Freeman[3], Lopes da Silva [2]

b. The current state and limitations of local models: Deco [8], Faugeras, Robinson [9], Suffcynzki and Wendling [10].

c. The current state and limitations of global models: Breakspear [11], Jirsa, Nunez [6], Wright [12].

d. Stochastic characterization of states and bifurcations of nonlinear local and global models: Faugeras , Harrison [13], Wennekers [14].

e. The basic information about brain morphology and connectivity, head volume conductor properties, population characteristics of hemodynamic and EEG characteristics that should be provided by the Human Brain Mapping initiative.

2. Structural constraints on brain dynamics provided by the Human Brain Mapping Project (Day 2): Over the past decade a wealth of information has been acquired and integrated into publicly available databases about the detailed cyto-architecture [15], gross brain morphology [16] and connectivity [17] of the human brain. This can provide the information that has been lacking for detailed neural modeling. New questions must be addressed and combined functional/morphological information gathered. More specifically in this session we will consider :

a. Current state of anatomical databases: Evans [18], ,Zilles [19],

b. General principles of brain connectivity, relevance for neural modeling:, Hilgetag [20;21] , Kotter [22], Sporns [21], Amunts [23].

c. In vivo determination of head volume conductor parameters and connectivity measures: Valdes-Hernandez [24].

3. Integration and estimation of neural stochastic models: In this session recent advances in the integration of stochastic differential equations will be examined. Particular emphasis will be places on the use of exponential integrators in order to preserve the qualitative dynamics of continuous neural mass models when discretized. Emphasis will also be placed on filtering and maximum likelihood estimation. More specifically in this session we will consider :

a. An Overview on bridging dynamical system and nonlinear time series theory: Ozaki .[25-27]

b. Simulation and estimation of ordinary stochastic differential equations, differential algebraic equations , and delay equations.: Jimenez [28;29], Burrage [30],Biscay [28;31], Roy [32].

c. Integration of stochastic partial differential equations: Berland [33].

4. Large scale simulations of neural networks (Day3): In this session we shall look at the computational issues of large and medium scale modeling of neural masses with emphasis on methods for producing simulations that can be checked against experimental results. This would be in themorning and the afternoon would be dedicated to group discussions and a walkabout the BIRS area.

a. Overview of large scale modeling, what exhaustive modeling can tell us aout neural masses: Markram [34].

b. Medium scale models: Deco [8], Horwitz [35].

5. Statistical inference on Brain images based on neural modeling (Day 4): In this session we

shall examine the current state and limitations of Statistical Parametric Mapping (SPM) of Neuroimages. The use of Bayesian state space models shall be examined in detail as well as procedures to deal with p much larger than n (p=numbr of variables, n=number of observations).

a. Overview of inference for statistical images: Worsley [36;37], Friston [38].

b. SPM of EEG/ERP models: David [38], Demiralp [39], Moran, Sotero [40].

c. SPM modeling functional images: Trujillo [40-42], Uludag [43].

d. SPM models for joint EEG/fMRI recordings: Babajani [44], Riera [45], Valdes[22;25;46]

6. Wrap up (Day 5): Overall discussion of issues, planning of collaborations and of publications.

Bibliography

[1] P. Suffczynski, F. Wendling, J. J. Bellanger, and F. H. L. da Silva, "Some insights into computational models of (patho)physiological brain activity," Proceedings of the Ieee, vol. 94, no. 4, pp. 784-804, 2006.

[2] F. H. Lopesdas, A. Hoeks, H. Smits, and L. H. Zetterbe, "Model of Brain Rhythmic Activity - Alpha-Rhythm of Thalamus," Kybernetik, vol. 15, no. 1, pp. 27-37, 1974.

[3] W. J. Freeman and G. Vitiello, "Nonlinear brain dynamics as macroscopic manifestation of underlying many-body field dynamics," Physics of Life Reviews, vol. 3, no. 2, pp. 93-118, 2006.

[4] J. J. Wright, C. J. Rennie, G. J. Lees, P. A. Robinson, P. D. Bourke, C. L. Chapman, E. Gordon, and D. L. Rowe, "Simulated electrocortical activity at microscopic, mesoscopic and global scales," International Journal of Bifurcation and Chaos, vol. 14, no. 2, pp. 853-872, 2004.

[5] V. K. Jirsa, "Connectivity and dynamics of neural information processing," Neuroinformatics, vol. 2, no. 2, pp. 183-204, 2004.

[6] P. L. Nunez, B. M. Wingeier, and R. B. Silberstein, "Spatial-temporal structures of human alpha rhythms: Theory, microcurrent sources, multiscale measurements, and global binding of local networks," Human Brain Mapping, vol. 13, no. 3, pp. 125-164, 2001.

[7] F. Grimbert and O. Faugeras, "Bifurcation analysis of Jansen's neural mass model," Neural Computation, vol. 18, no. 12, pp. 3052-3068, 2006.

[8] G. Deco and D. Marti, "Deterministic analysis of stochastic bifurcations in multi-stable neurodynamical systems," Biological Cybernetics, vol. 96, no. 5, pp. 487-496, 2007.

[9] P. A. Robinson, C. J. Rennie, D. L. Rowe, and S. C. O'Connor, "Estimation of multiscale neurophysiologic parameters by electroencephalographic means," Human Brain Mapping, vol. 23, no. 1, pp. 53-72, 2004.

[10] B. M. Bouwman, P. Suffczynski, F. H. L. da Silva, E. Maris, and C. M. van Rijn, "GABAergic mechanisms in absence epilepsy: a computational model of absence epilepsy simulating spike and wave discharges after vigabatrin in WAG/Rij rats," European Journal of Neuroscience, vol. 25, no. 9, pp. 2783-2790, 2007.

[11] M. Breakspear and J. R. Terry, "Nonlinear interdependence in neural systems: Motivation, theory, and relevance," International Journal of Neuroscience, vol. 112, no. 10, pp. 1263-1284, 2002.

[12] J. J. Wright, "Simulation of EEG: dynamic changes in synaptic efficacy, cerebral rhythms, and dissipative and generative activity in cortex," Biological Cybernetics, vol. 81, no. 2, pp. 131-147, 1999.

[13] L. M. Harrison, O. David, and K. J. Friston, "Stochastic models of neuronal dynamics," Philosophical Transactions of the Royal Society B-Biological Sciences, vol. 360, no. 1457, pp. 1075-1091, 2005.

[14] T. Wennekers, M. Garagnani, and F. Pulvermuller, "Language models based on Hebbian cell assemblies," Journal of Physiology-Paris, vol. 100, no. 1-3, pp. 16-30, 2006.

[15] S. B. Eickhoff, T. Paus, S. Caspers, M. H. Grosbras, A. C. Evans, K. Zilles, and K. Amunts, "Assignment of functional activations to probabilistic cytoarchitectonic areas revisited," Neuroimage, vol. 36, no. 3, pp. 511-521, 2007.

[16] J. Mazziotta, A. Toga, A. Evans, P. Fox, J. Lancaster, K. Zilles, R. Woods, T. Paus, G. Simpson, B. Pike, C. Holmes, L. Collins, P. Thompson, D. MacDonald, M. Iacoboni, T. Schormann, K. Amunts, N. Palomero-Gallagher, S. Geyer, L. Parsons, K. Narr, N. Kabani, G. Le Goualher, D. Boomsma, T. Cannon, R. Kawashima, and B. Mazoyer, "A probabilistic atlas and reference system for the human brain: International Consortium for Brain Mapping (ICBM)," Philosophical Transactions of the Royal Society of London Series B-Biological Sciences, vol. 356, no. 1412, pp. 1293-1322, 2001.

[17] O. Sporns, G. Tononi, and R. Kotter, "The human connectome: A structural description of the human brain," Plos Computational Biology, vol. 1, no. 4, pp. 245-251, 2005.

[18] P. Shaw, D. Greenstein, J. Lerch, L. Clasen, R. Lenroot, N. Gogtay, A. Evans, J. Rapoport, and J. Giedd, "Intellectual ability and cortical development in children and adolescents," Nature, vol. 440, no. 7084, pp. 676-679, 2006.

[19] A. W. Toga, P. M. Thompson, S. Mori, K. Amunts, and K. Zilles, "Towards multimodal atlases of the human brain," Nature Reviews Neuroscience, vol. 7, no. 12, pp. 952-966, 2006.

[20] C. C. Hilgetag, "Principles of brain connectivity organization," Behavioral and Brain Sciences, vol. 29, no. 1, p. 18-+, 2006.

[21] O. Sporns, D. R. Chialvo, M. Kaiser, and C. C. Hilgetag, "Organization, development and function of complex brain networks," Trends in Cognitive Sciences, vol. 8, no. 9, pp. 418-425, 2004.

[22] P. A. Valdes-Sosa, R. Kotter, and K. J. Friston, "Introduction: multimodal neuroimaging of brain connectivity," Philosophical Transactions of the Royal Society B-Biological Sciences, vol. 360, no. 1457, pp. 865-867, 2005.

[23] U. Burgel, K. Amunts, L. Hoemke, H. Mohlberg, J. M. Gilsbach, and K. Zilles, "White matter fiber tracts of the human brain: Three-dimensional mapping at microscopic resolution, topography and intersubject variability," Neuroimage, vol. 29, no. 4, pp. 1092-1105, 2006.

[24] Y. Iturria-Medina, E. J. Canales-Rodriguez, L. Melie-Garcia, P. A. Valdes-Hernandez, E. Martinez-Montes, Y. eman-Gomez, and J. M. Sanchez-Bornot, "Characterizing brain anatomical connections using diffusion weighted MRI and graph theory," Neuroimage, vol. 36, no. 3, pp. 645-660, 2007.

[25] P. A. Valdes, J. C. Jimenez, J. Riera, R. Biscay, and T. Ozaki, "Nonlinear EEG analysis based on a neural mass model," Biological Cybernetics, vol. 81, no. 5-6, pp. 415-424, 1999.

[26] A. Galka, O. Yamashita, T. Ozaki, R. Biscay, and P. Valdes-Sosa, "A solution to the dynamical inverse problem of EEG generation using spatiotemporal Kalman filtering," Neuroimage, vol. 23, no. 2, pp. 435-453, 2004.

[27] T. Ozaki, "A Bridge Between Nonlinear Time-Series Models and Nonlinear Stochastic Dynamic-Systems - A Local Linearization Approach," Statistica Sinica, vol. 2, no. 1, pp. 113-135, 1992.

[28] H. De la Cruz, R. J. Biscay, F. Carbonell, T. Ozaki, and J. C. Jimenez, "A higher order local linearization method for solving ordinary differential equations," Applied Mathematics and Computation, vol. 185, no. 1, pp. 197-212, 2007.

[29] J. C. Jimenez, L. M. Pedroso, F. Carbonell, and V. Hernandez, "Local linearization method for numerical integration of delay differential equations," Siam Journal on Numerical Analysis, vol. 44, no. 6, pp. 2584-2609, 2006.

[30] K. Burrage, I. Lenane, and G. Lythe, "Numerical methods for second-order stochastic differential equations," Siam Journal on Scientific Computing, vol. 29, no. 1, pp. 245-264, 2007.

[31] F. Carbonell, J. C. Jimenez, and R. J. Biscay, "Weak local linear discretizations for stochastic differential equations: Convergence and numerical schemes," Journal of Computational

and Applied Mathematics, vol. 197, no. 2, pp. 578-596, 2006.

[32] D. Roy and M. K. Dash, "A novel stochastic locally transversal linearization (LTL) technique for engineering dynamical systems: Strong solutions," Applied Mathematical Modelling, vol. 29, no. 10, pp. 913-937, 2005.

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