On the use of correlation as a measure of network connectivity

Numerous studies have demonstrated that brain networks derived from neuroimaging data have nontrivial topological features, such as small-world organization, modular structure and highly connected hubs. In these studies, the extent of connectivity between pairs of brain regions has often been measured using some form of statistical correlation. This article demonstrates that correlation as a measure of connectivity in and of itself gives rise to networks with non-random topological features. In particular, networks in which connectivity is measured using correlation are inherently more clustered than random networks, and as such are more likely to be small-world networks. Partial correlation as a measure of connectivity also gives rise to networks with non-random topological features. Partial correlation networks are inherently less clustered than random networks. Network measures in correlation networks should be benchmarked against null networks that respect the topological structure induced by correlation measurements. Prevalently used random rewiring algorithms do not yield appropriate null networks for some network measures. Null networks are proposed to explicitly normalize for the inherent topological structure found in correlation networks, resulting in more conservative estimates of small-world organization. A number of steps may be needed to normalize each network measure individually and control for distinct features (e.g. degree distribution). The main conclusion of this article is that correlation can and should be used to measure connectivity, however appropriate null networks should be used to benchmark network measures in correlation networks.     

Anatomical insights into disrupted small-world networks in schizophrenia

Schizophrenia is characterized by lowered efficiency in distributed information processing, as indicated by research that identified a disrupted small-world functional network. However, whether the dysconnection manifested by the disrupted small-world functional network is reflected in underlying anatomical disruption in schizophrenia remains unresolved. This study examined the topological properties of human brain anatomical networks derived from diffusion tensor imaging in patients with schizophrenia and in healthy controls. We constructed the weighted brain anatomical network for each of 79 schizophrenia patients and for 96 age and gender matched healthy subjects using diffusion tensor tractography and calculated the topological properties of the networks using a graph theoretical method. The topological properties of the patients' anatomical networks were altered, in that global efficiency decreased but local efficiency remained unchanged. The deleterious effects of schizophrenia on network performance appear to be localized as reduced regional efficiency in hubs such as the frontal associative cortices, the paralimbic/limbic regions and a subcortical structure (the left putamen). Additionally, scores on the Positive and Negative Symptom Scale correlated negatively with efficient network properties in schizophrenia. These findings suggest that complex brain network analysis may potentially be used to detect an imaging biomarker for schizophrenia.     

Anatomically informed interpolation of fMRI data on the cortical surface

Analyzing functional magnetic resonance imaging (fMRI) data restricted to the cortical surface is of particular interest for two reasons: (1) to increase detection sensitivity using anatomical constraints and (2) to compare or use fMRI results in the context of source localization from magneto/electro-encephalography (MEEG) data, which requires data to be projected on the same spatial support. Designing an optimal scheme to interpolate fMRI raw data or resulting activation maps on the cortical surface relies on a trade-off between choosing large enough interpolation kernels, because of the distributed nature of the hemodynamic response, and avoiding mixing data issued from different anatomical structures. We propose an original method that automatically adjusts the level of such a trade-off, by defining interpolation kernels around each vertex of the cortical surface using a geodesic Vorono? diagram. This Vorono?-based interpolation method was evaluated using simulated fMRI activation maps, manually generated on an anatomical MRI, and compared with a more standard approach where interpolation kernels were defined as local spheres of radius r=3 or 5 mm. Several validation parameters were considered: the spatial resolution of the simulated activation map, the spatial resolution of the cortical mesh, the level of anatomical/functional data misregistration and the location of the vertices within the gray matter ribbon. Using an activation map at the spatial resolution of standard fMRI data, robustness to misregistration errors was observed for both methods, whereas only the Vorono?-based approach was insensitive to the position of the vertices within the gray matter ribbon.     

Small-world directed networks in the human brain: multivariate Granger causality analysis of resting-state fMRI

Small-world organization is known to be a robust and consistent network architecture, and is a hallmark of the structurally and functionally connected human brain. However, it remains unknown if the same organization is present in directed influence brain networks whose connectivity is inferred by the transfer of information from one node to another. Here, we aimed to reveal the network architecture of the directed influence brain network using multivariate Granger causality analysis and graph theory on resting-state fMRI recordings. We found that some regions acted as pivotal hubs, either being influenced by or influencing other regions, and thus could be considered as information convergence regions. In addition, we observed that an exponentially truncated power law fits the topological distribution for the degree of total incoming and outgoing connectivity. Furthermore, we also found that this directed network has a modular structure. More importantly, according to our data, we suggest that the human brain directed influence network could have a prominent small-world topological property.     

A tutorial in connectome analysis: topological and spatial features of brain networks

High-throughput methods for yielding the set of connections in a neural system, the connectome, are now being developed. This tutorial describes ways to analyze the topological and spatial organizations of the connectome at the macroscopic level of connectivity between brain regions as well as the microscopic level of connectivity between neurons. We will describe topological features at three different levels: the local scale of individual nodes, the regional scale of sets of nodes, and the global scale of the complete set of nodes in a network. Such features can be used to characterize components of a network and to compare different networks, e.g. the connectome of patients and control subjects for clinical studies. At the global scale, different types of networks can be distinguished and we will describe Erd?s-Rényi random, scale-free, small-world, modular, and hierarchical archetypes of networks. Finally, the connectome also has a spatial organization and we describe methods for analyzing wiring lengths of neural systems. As an introduction for new researchers in the field of connectome analysis, we discuss the benefits and limitations of each analysis approach.     

Statistical parametric network analysis of functional connectivity dynamics during a working memory task

Network analysis has become a tool of choice for the study of functional and structural Magnetic Resonance Imaging (MRI) data. Little research, however, has investigated connectivity dynamics in relation to varying cognitive load. In fMRI, correlations among slow (<0.1 Hz) fluctuations of blood oxygen level dependent (BOLD) signal can be used to construct functional connectivity networks. Using an anatomical parcellation scheme, we produced undirected weighted graphs linking 90 regions of the brain representing major cortical gyri and subcortical nuclei, in a population of healthy adults (n=43). Topological changes in these networks were investigated under different conditions of a classical working memory task - the N-back paradigm. A mass-univariate approach was adopted to construct statistical parametric networks (SPNs) that reflect significant modifications in functional connectivity between N-back conditions. Our proposed method allowed the extraction of 'lost' and 'gained' functional networks, providing concise graphical summaries of whole-brain network topological changes. Robust estimates of functional networks are obtained by pooling information about edges and vertices over subjects. Graph thresholding is therefore here supplanted by inference. The analysis proceeds by firstly considering changes in weighted cost (i.e. mean between-region correlation) over the different N-back conditions and secondly comparing small-world topological measures integrated over network cost, thereby controlling for differences in mean correlation between conditions. The results are threefold: (i) functional networks in the four conditions were all found to satisfy the small-world property and cost-integrated global and local efficiency levels were approximately preserved across the different experimental conditions; (ii) weighted cost considerably decreased as working memory load increased; and (iii) subject-specific weighted costs significantly predicted behavioral performances on the N-back task (Wald F=13.39,df(1)=1,df(2)=83,p<0.001), and therefore conferred predictive validity to functional connectivity strength, as measured by weighted cost. The results were found to be highly sensitive to the frequency band used for the computation of the between-region correlations, with the relationship between weighted cost and behavioral performance being most salient at very low frequencies (0.01-0.03 Hz). These findings are discussed in relation to the integration/specialization functional dichotomy. The pruning of functional networks under increasing cognitive load may permit greater modular specialization, thereby enhancing performance.     

Small-world and scale-free organization of voxel-based resting-state functional connectivity in the human brain

The brain is a complex dynamic system of functionally connected regions. Graph theory has been successfully used to describe the organization of such dynamic systems. Recent resting-state fMRI studies have suggested that inter-regional functional connectivity shows a small-world topology, indicating an organization of the brain in highly clustered sub-networks, combined with a high level of global connectivity. In addition, a few studies have investigated a possible scale-free topology of the human brain, but the results of these studies have been inconclusive. These studies have mainly focused on inter-regional connectivity, representing the brain as a network of brain regions, requiring an arbitrary definition of such regions. However, using a voxel-wise approach allows for the model-free examination of both inter-regional as well as intra-regional connectivity and might reveal new information on network organization. Especially, a voxel-based study could give information about a possible scale-free organization of functional connectivity in the human brain. Resting-state 3 Tesla fMRI recordings of 28 healthy subjects were acquired and individual connectivity graphs were formed out of all cortical and sub-cortical voxels with connections reflecting inter-voxel functional connectivity. Graph characteristics from these connectivity networks were computed. The clustering-coefficient of these networks turned out to be much higher than the clustering-coefficient of comparable random graphs, together with a short average path length, indicating a small-world organization. Furthermore, the connectivity distribution of the number of inter-voxel connections followed a power-law scaling with an exponent close to 2, suggesting a scale-free network topology. Our findings suggest a combined small-world and scale-free organization of the functionally connected human brain. The results are interpreted as evidence for a highly efficient organization of the functionally connected brain, in which voxels are mostly connected with their direct neighbors forming clustered sub-networks, which are held together by a small number of highly connected hub-voxels that ensure a high level of overall connectivity.     

Altered brain anatomical networks and disturbed connection density in brain tumor patients revealed by diffusion tensor tractography

Purpose

Brain tumor patients are usually accompanied by impairments in cognitive functions, and these dysfunctions arise from the altered diffusion tensor of water molecules and disrupted neuronal conduction in white matter. Diffusion tensor imaging (DTI) is a powerful noninvasive imaging technique that can reflect diffusion anisotropy of water and brain white matter neural connectivity in vivo. This study was aimed to analyze the topological properties and connection densities of the brain anatomical networks in brain tumor patients based on DTI and provide new insights into the investigation of the structural plasticity and compensatory mechanism of tumor patient’s brain.

Methods

In this study, the brain anatomical networks of tumor patients and healthy controls were constructed using the tracking of white matter fiber bundles based on DTI and the topological properties of these networks were described quantitatively. The statistical comparisons were performed between two groups with six DTI parameters: degree, regional efficiency, local efficiency, clustering coefficient, vulnerability, and betweenness centrality. In order to localize changes in structural connectivity to specific brain regions, a network-based statistic approach was utilized. By comparing the edge connection density of brain network between two groups, the edges with greater difference in connection density were associated with three functional systems.

Results

Compared with controls, tumor patients show a significant increase in small-world feature of cerebral structural network. Two-sample two-tailed t test indicates that the regional properties are altered in 17 regions (\(p<0.05\)). Study reveals that the positive and negative changes in vulnerability take place in the 14 brain areas. In addition, tumor patients lose 3 hub regions and add 2 new hubs when compared to normal controls. Eleven edges show much significantly greater connection density in the patients than in the controls. Most of the edges with greater connection density are linked to regions located in the limbic/subcortical and other systems. Besides, most of the edges connect the two hemispheres of the brains.

Conclusion

The stronger small-world property in the tumor patients proves the existence of compensatory mechanism. The changes in the regional properties, especially the betweenness centrality and vulnerability, aid in understanding the brain structural plasticity. The increased connection density in the tumor group suggests that tumors may induce reorganization in the structural network.

     

Modulations of functional connectivity in the healthy and schizophrenia groups during task and rest

Connectivity analysis using functional magnetic resonance imaging (fMRI) data is an important area, useful for the identification of biomarkers for various mental disorders, including schizophrenia. Most studies to date have focused on resting data, while the study of functional connectivity during task and the differences between task and rest are of great interest as well. In this work, we examine the graph-theoretical properties of the connectivity maps constructed using spatial components derived from independent component analysis (ICA) for healthy controls and patients with schizophrenia during an auditory oddball task (AOD) and at extended rest. We estimate functional connectivity using the higher-order statistical dependence, i.e., mutual information among the ICA spatial components, instead of the typically used temporal correlation. We also define three novel topological metrics based on the modules of brain networks obtained using a clustering approach. Our experimental results show that although the schizophrenia patients preserve the small-world property, they present a significantly lower small-worldness during both AOD task and rest when compared to the healthy controls, indicating a consistent tendency towards a more random organization of brain networks. In addition, the task-induced modulations to topological measures of several components involving motor, cerebellum and parietal regions are altered in patients relative to controls, providing further evidence for the aberrant connectivity in schizophrenia.     

Local landmark-based mapping of human auditory cortex

Mammalian sensory cortex is functionally partitioned into cortical fields that are specialized for different processing operations. In theory, averaging functional and anatomical images across subjects can reveal both the average anatomy and the mean functional organization of sensory regions. However, this averaging process must overcome at least two obstacles: (1) the relative locations and sizes of cortical sensory areas vary in different subjects so that across-subject averaging introduces spatial smearing; (2) the relative locations and sizes of cortical areas vary between hemispheres, making it difficult to compare activations between hemispheres or to combine activations across hemispheres. These difficulties are particularly acute for small cortical regions such as auditory cortex. In whole-brain averaging procedures, considerable intersubject variance in the location and orientation of auditory cortex is introduced by variance of the size and shape of structures outside auditory cortex. Here, we compared these global methods with local landmark-based methods (LLMs) that use warping based on local anatomical landmarks. In comparison to maps made with global methods, LLMs produced anatomical maps of auditory cortex with clearer gyral and sulcal structure, and produce functional maps with improved resolution. These results suggest that LLMs have significant advantages over global mapping procedures in studying the details of auditory cortex organization.     

Studying the human brain anatomical network via diffusion-weighted MRI and Graph Theory

Our goal is to study the human brain anatomical network. For this, the anatomical connection probabilities (ACP) between 90 cortical and subcortical brain gray matter areas are estimated from diffusion-weighted Magnetic Resonance Imaging (DW-MRI) techniques. The ACP between any two areas gives the probability that those areas are connected at least by a single nervous fiber. Then, the brain is modeled as a non-directed weighted graph with continuous arc weights given by the ACP matrix. Based on this approach, complex networks properties such as small-world attributes, efficiency, degree distribution, vulnerability, betweenness centrality and motifs composition are studied. The analysis was carried out for 20 right-handed healthy subjects (mean age: 31.10, S.D.: 7.43). According to the results, all networks have small-world and broad-scale characteristics. Additionally, human brain anatomical networks present bigger local efficiency and smaller global efficiency than the corresponding random networks. In a vulnerability and betweenness centrality analysis, the most indispensable and critical anatomical areas were identified: putamens, precuneus, insulas, superior parietals and superior frontals. Interestingly, some areas have a negative vulnerability (e.g. superior temporal poles, pallidums, supramarginals and hechls), which suggest that even at the cost of losing in global anatomical efficiency, these structures were maintained through the evolutionary processes due to their important functions. Finally, symmetrical characteristic building blocks (motifs) of size 3 and 4 were calculated, obtaining that motifs of size 4 are the expanded version of motif of size 3. These results are in agreement with previous anatomical studies in the cat and macaque cerebral cortex.     

Graph properties of synchronized cortical networks during visual working memory maintenance

Oscillatory synchronization facilitates communication in neuronal networks and is intimately associated with human cognition. Neuronal activity in the human brain can be non-invasively imaged with magneto- (MEG) and electroencephalography (EEG), but the large-scale structure of synchronized cortical networks supporting cognitive processing has remained uncharacterized. We combined simultaneous MEG and EEG (MEEG) recordings with minimum-norm-estimate-based inverse modeling to investigate the structure of oscillatory phase synchronized networks that were active during visual working memory (VWM) maintenance. Inter-areal phase-synchrony was quantified as a function of time and frequency by single-trial phase-difference estimates of cortical patches covering the entire cortical surfaces. The resulting networks were characterized with a number of network metrics that were then compared between delta/theta- (3–6 Hz), alpha- (7–13 Hz), beta- (16–25 Hz), and gamma- (30–80 Hz) frequency bands. We found several salient differences between frequency bands. Alpha- and beta-band networks were more clustered and small-world like but had smaller global efficiency than the networks in the delta/theta and gamma bands. Alpha- and beta-band networks also had truncated-power-law degree distributions and high k-core numbers. The data converge on showing that during the VWM-retention period, human cortical alpha- and beta-band networks have a memory-load dependent, scale-free small-world structure with densely connected core-like structures. These data further show that synchronized dynamic networks underlying a specific cognitive state can exhibit distinct frequency-dependent network structures that could support distinct functional roles.     

A wavelet-based method for measuring the oscillatory dynamics of resting-state functional connectivity in MEG

Determining the dynamics of functional connectivity is critical for understanding the brain. Recent functional magnetic resonance imaging (fMRI) studies demonstrate that measuring correlations between brain regions in resting-state activity can be used to reveal intrinsic neural networks. To study the oscillatory dynamics that underlie intrinsic functional connectivity between regions requires high temporal resolution measures of electrophysiological brain activity, such as magnetoencephalography (MEG). However, there is a lack of consensus as to the best method for examining connectivity in resting-state MEG data. Here we adapted a wavelet-based method for measuring phase-locking with respect to the frequency of neural oscillations. This method employs anatomical MRI information combined with MEG data using the minimum norm estimate inverse solution to produce functional connectivity maps from a "seed" region to all other locations on the cortical surface at any and all frequencies of interest. We test this method by simulating phase-locked oscillations at various points on the cortical surface, which illustrates a substantial artifact that results from imperfections in the inverse solution. We demonstrate that normalizing resting-state MEG data using phase-locking values computed on empty room data reduces much of the effects of this artifact. We then use this method with eight subjects to reveal intrinsic interhemispheric connectivity in the auditory network in the alpha frequency band in a silent environment. This spectral resting-state functional connectivity imaging method may allow us to better understand the oscillatory dynamics underlying intrinsic functional connectivity in the human brain.     

The impact of aging on gray matter structural covariance networks

Previous anatomical volumetric studies have shown that healthy aging is associated with gray matter tissue loss in specific cerebral regions. However, these studies may have potentially missed critical elements of age-related brain changes, which largely exist within interrelationships among brain regions. This magnetic resonance imaging research aims to assess the effects of aging on the organization of gray matter structural covariance networks. Here, we used voxel-based morphometry on high-definition brain scans to compare the patterns of gray matter structural covariance networks that sustain different sensorimotor and high-order cognitive functions among young (n=88, mean age=23.5±3.1years, female/male=55/33) and older (n=88, mean age=67.3±5.9years, female/male=55/33) participants. This approach relies on the assumption that functionally correlated brain regions show correlations in gray matter volume as a result of mutually trophic influences or common experience-related plasticity. We found reduced structural association in older adults compared with younger adults, specifically in high-order cognitive networks. Major differences were observed in the structural covariance networks that subserve the following: a) the language-related semantic network, b) the executive control network, and c) the default-mode network. Moreover, these cognitive functions are typically altered in the older population. Our results indicate that healthy aging alters the structural organization of cognitive networks, shifting from a more distributed (in young adulthood) to a more localized topological organization in older individuals.     

Uncovering Intrinsic Modular Organization of Spontaneous Brain Activity in Humans

The characterization of topological architecture of complex brain networks is one of the most challenging issues in neuroscience. Slow (<0.1 Hz), spontaneous fluctuations of the blood oxygen level dependent (BOLD) signal in functional magnetic resonance imaging are thought to be potentially important for the reflection of spontaneous neuronal activity. Many studies have shown that these fluctuations are highly coherent within anatomically or functionally linked areas of the brain. However, the underlying topological mechanisms responsible for these coherent intrinsic or spontaneous fluctuations are still poorly understood. Here, we apply modern network analysis techniques to investigate how spontaneous neuronal activities in the human brain derived from the resting-state BOLD signals are topologically organized at both the temporal and spatial scales. We first show that the spontaneous brain functional networks have an intrinsically cohesive modular structure in which the connections between regions are much denser within modules than between them. These identified modules are found to be closely associated with several well known functionally interconnected subsystems such as the somatosensory/motor, auditory, attention, visual, subcortical, and the “default” system. Specifically, we demonstrate that the module-specific topological features can not be captured by means of computing the corresponding global network parameters, suggesting a unique organization within each module. Finally, we identify several pivotal network connectors and paths (predominantly associated with the association and limbic/paralimbic cortex regions) that are vital for the global coordination of information flow over the whole network, and we find that their lesions (deletions) critically affect the stability and robustness of the brain functional system. Together, our results demonstrate the highly organized modular architecture and associated topological properties in the temporal and spatial brain functional networks of the human brain that underlie spontaneous neuronal dynamics, which provides important implications for our understanding of how intrinsically coherent spontaneous brain activity has evolved into an optimal neuronal architecture to support global computation and information integration in the absence of specific stimuli or behaviors.     

局灶性脑损害患者视觉注意环路的小世界网络属性分析

摘要 目的：探讨局灶性脑损害患者视觉注意网络的连通性及其小世界特征。 方法：对2例局灶性后顶叶损害、2例背外侧前额叶损害、2例颞叶损害（其中1例为手术前后）和1例锥体束损害患者进行静息态fMRI检测,然后进行功能网络建立和小世界属性分析。 结果：所有局灶性脑损害患者的脑功能网络在给定阈值范围（0.05—0.5）内都具有小世界属性。局灶性后顶叶、背外侧前额叶、颞叶和锥体束损害患者之间,Gamma系数差异（P<0.05）和Sigma系数差异（P<0.01）均具有显著性意义。 结论：小世界网络分析为视觉注意网络的连通性研究提供了一个非常有价值的方法。我们推测,小世界网络检测方法应该可以作为局灶性脑损害神经功能损害的影像学生物标记。     

The use of anatomical constraints with MEG beamformers

Synthetic Aperture Magnetometry (SAM) is a beamformer approach for the localisation of neuronal activity from EEG/MEG data. SAM estimates the optimum orientation of each source in a predefined source space by a nonlinear search for the orientation that maximises the beamformer output. However, MEG is most sensitive to cortical sources and these sources are generally oriented perpendicular to the surface. The reconstructed neuronal activity can therefore reasonably be constrained to the cortical surface, orientated perpendicular to it, therefore removing the search for the optimum orientation for the computation of the beamformer weights. This paper sets out to compare the performance of a constrained and unconstrained beamformer (SAM), with respect to the localisation accuracy of the source reconstructions and the spatial resolution. Fifty sources were randomly placed on a cortical surface estimated from an MRI, and we simulated data over a range of different signal-to-noise ratios (SNRs) for each source. These datasets were analysed using both an unconstrained beamformer (SAM) and a constrained beamformer (with the sources orientated perpendicular to the cortical surface). The influence of errors in the estimation of the surface location and surface normals on the performance of the constrained beamformer, representing MEG/MRI coregistration and segmentation errors, were also examined. The spatial resolution of the beamformer improves, typically by a factor of four by applying anatomical constraints, and the localisation accuracy improves marginally. However, the advantage in spatial resolution disappears when errors are introduced into the orientation and location constraints, and, moreover, the localisation accuracy of the inaccurately constrained beamformer degrades rapidly. We conclude that the use of anatomical constraints is only advantageous if the MEG/MRI coregistration error is smaller than 2 mm and the error in the estimation of the cortical surface orientation is smaller than 10 degrees.     

Topographically specific functional connectivity between visual field maps in the human brain

Neural activity in mammalian brains exhibits large spontaneous fluctuations whose structure reveals the intrinsic functional connectivity of the brain on many spatial and temporal scales. Between remote brain regions, spontaneous activity is organized into large-scale functional networks. To date, it has remained unclear whether the intrinsic functional connectivity between brain regions scales down to the fine detail of anatomical connections, for example the fine-grained topographic connectivity structure in visual cortex. Here, we show that fMRI signal fluctuations reveal a detailed retinotopically organized functional connectivity structure between the visual field maps of remote areas of the human visual cortex. The structured coherent fluctuations were even preserved in complete darkness when all visual input was removed. While the topographic connectivity structure was clearly visible in within hemisphere connections, the between hemisphere connectivity structure differs for representations along the vertical and horizontal meridian respectively. These results suggest a tight link between spontaneous neural activity and the fine-grained topographic connectivity pattern of the human brain. Thus, intrinsic functional connectivity reflects the detailed connectivity structure of the cortex at a fine spatial scale. It might thus be a valuable tool to complement anatomical studies of the human connectome, which is one of the keys to understand the functioning of the human brain.     

Multi-manifold diffeomorphic metric mapping for aligning cortical hemispheric surfaces

Cortical surface-based analysis has been widely used in anatomical and functional studies because it is geometrically appropriate for the cortex. One of the main challenges in the cortical surface-based analysis is to optimize the alignment of the cortical hemispheric surfaces across individuals. In this paper, we introduce a multi-manifold large deformation diffeomorphic metric mapping (MM-LDDMM) algorithm that allows simultaneously carrying the cortical hemispheric surface and its sulcal curves from one to the other through a flow of diffeomorphisms. We present an algorithm based on recent derivation of a law of momentum conservation for the geodesics of diffeomorphic flow. Once a template is fixed, the space of initial momentum becomes an appropriate space for studying shape via geodesic flow since the flow at any point on curves and surfaces along the geodesic is completely determined by the momentum at the origin. We solve for trajectories (geodesics) of the kinetic energy by computing its variation with respect to the initial momentum and by applying a gradient descent scheme. The MM-LDDMM algorithm optimizes the initial momenta encoding the anatomical variation of each individual relative to a common coordinate system in a linear space, which provides a natural scheme for shape deformation average and template (or atlas) generation. We applied the MM-LDDMM algorithm for constructing the templates for the cortical surface and 14 sulcal curves of each hemisphere using a group of 40 subjects. The estimated template shape reflects regions which are highly variable across these subjects. Compared with existing single-manifold LDDMM algorithms, such as the LDDMM-curve mapping and the LDDMM-surface mapping, the MM-LDDMM mapping provides better results in terms of surface to surface distances in five predefined regions.