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ncISO-3D coalescent embedding of structural brain networks highlights the presence of brain lobes anatomical arrangement. The median connectivity matrix of 30 healthy controls (HC) of Dataset I has been mapped in the 3D hyperbolic space using the coalescent embedding ncISO technique. The figure shows, in a posterior view, the 3D geometry of the brain emerging from the embedding in the hyperbolic sphere. The colours-filled circles represent the nodes of the left hemisphere, whereas the white-filled ones represent the brain structures of the right hemisphere. Each node has been labelled according to its real anatomical localization in the different brain lobes. The red colour indicates the Frontal Lobe, the green colour the Parietal Lobe, the magenta colour the Temporal Lobe, the blue the Occipital Lobe, whereas the grey colour characterizes the nodes that have not been assigned to any lobe, since they represent grey matter structures placed in the deep white matter.
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The human brain displays a complex network topology, whose structural organization is widely studied using diffusion tensor imaging. The original geometry from which emerges the network topology is known, as well as the localization of the network nodes in respect to the brain morphology and anatomy. One of the most challenging problems of current...
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Network science and graph theory applications have recently spread widely to help in understanding how human cognitive functions are linked to neuronal network structure, thus providing a conceptual frame that can help in reducing the analytical brain complexity and underlining how network topology can be used to characterize and model vulnerabilit...
The structural network of the human brain has a rich topology which many have sought to characterise using standard network science measures and concepts. However, this characterisation remains incomplete and the non-obvious features of this topology have largely confounded attempts towards comprehensive constructive modelling. This calls for new p...
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... The possible implications and applications of our research on geometric separability of network community in various fields, such as biological networks, transportation systems, and more is straightforward. For instance, we could apply the multivariate community separability analysis (MCSA), to evaluate how different are the brain MRI structural connectome representations of control and unhealthy subjects considering the mesoscale lobular brain organization, which has been proven to be reflected in the brain connectome topology and geometry 40,41 . Or to evaluate the geometric separability of gene communities in co-expression networks across different tissues 42 . ...
Complexity science studies physical phenomena that cannot be explained by the mere analysis of the single units of a system but requires to account for their interactions. A feature of complexity in connected systems is the emergence of mesoscale patterns in a geometric space, such as groupings in bird flocks. These patterns are formed by groups of points that tend to separate from each other, creating mesoscale structures. When multidimensional data or complex networks are embedded in a geometric space, some mesoscale patterns can appear respectively as clusters or communities, and their geometric separability is a feature according to which the performance of an algorithm for network embedding can be evaluated. Here, we introduce a framework for the definition and measure of the geometric separability (linear and nonlinear) of mesoscale patterns by solving the travelling salesman problem (TSP), and we offer experimental evidence on embedding and visualization of multidimensional data or complex networks, which are generated artificially or are derived from real complex systems. For the first time in literature the TSP’s solution is used to define a criterion of nonlinear separability of points in a geometric space, hence redefining the separability problem in terms of the travelling salesman problem is an innovation which impacts both computer science and complexity theory.
... Finally, we show how, contrary to the greedy navigability measures that need nodes geometrical coordinates (patent geometrical space, known or inferred by network embedding), the proposed geometrical congruence measure can be applied as a marker to reveal differences in real networks with latent geometrical space such as brain connectomes, without the requirement of nodes geometrical coordinates, but only geometrical weights (presumably associated with the latent space). Indeed, brain connectomes topological weights are associated with a developmental neurobiological geometry which, according to some studies, might be approximated by hyperbolic distances [12][13][14] . However, as a matter of fact, the geometry is latent because both the type of geometry and the coordinates of the nodes in this developmental neurobiological space are unknown. ...
... This helps to discuss how the congruence of geodesics and pTSP varies according to a certain specific structural property which is adopted to shape the hyperbolic network according to the nPSO model. Three structural properties are discussed: if average degree d grows (m is the direct nPSO model parameter, and d is about 2*m for sparse networks; we consider m = [2, 6, 10] hence d about [4,12,20]), network density increases; if temperature T grows (we consider T = [0.1, 0.5, 0.9]), clustering decreases; if γ grows (we consider γ = [2, 2.5, 3]), super-hub structure of the network is mitigated. ...
... It regards whether navigability and congruence can be used to gain any evidence at support of the thesis that macroscale structural MRI brain connectomes might have a latent developmental neurobiological geometry which accounts for more information content than the 3D visible Euclidean geometry. Indeed, brain networks are associated with a developmental neurobiological geometry which also influenced by neuronal plasticity 22 and, according to some studies, might be approximated by hyperbolic distances [12][13][14] . However, as a matter of fact, it is latent because both the type of geometry and the coordinates of the nodes in this developmental neurobiological space are unknown. ...
We introduce in network geometry a measure of geometrical congruence (GC) to evaluate the extent a network topology follows an underlying geometry. This requires finding all topological shortest-paths for each nonadjacent node pair in the network: a nontrivial computational task. Hence, we propose an optimized algorithm that reduces 26 years of worst scenario computation to one week parallel computing. Analysing artificial networks with patent geometry we discover that, different from current belief, hyperbolic networks do not show in general high GC and efficient greedy navigability (GN) with respect to the geodesics. The myopic transfer which rules GN works best only when degree-distribution power-law exponent is strictly close to two. Analysing real networks—whose geometry is often latent—GC overcomes GN as marker to differentiate phenotypical states in macroscale structural-MRI brain connectomes, suggesting connectomes might have a latent neurobiological geometry accounting for more information than the visible tridimensional Euclidean.
... While novel, this brain graph embedding technique is minimally related to neuroanatomy. Later on, [34] proposed an alternative to brain graph embedding in the geometric space by learning the coalescent embedding of brain connectivities in the hyperbolic space. The proposed method successfully segregated the brain graph into spatially distinct subgraphs representing the brain lobes. ...
... Furthermore, it is necessary to develop GNNs that are trained with a frugal setting where a few brain graphs will be used in the learning step of the model as in [111]- [113]. Finally, there are not currently available studies that motivate and explain the advantages and disadvantages of GNN in comparison to the other ML-based techniques such as [33], [34]. Future studies might investigate the reason to adopt GNN in contrast to these other popular ML-based methods. ...
Noninvasive medical neuroimaging has yielded many discoveries about the brain connectivity. Several substantial techniques mapping morphological, structural and functional brain connectivities were developed to create a comprehensive road map of neuronal activities in the human brain –namely brain graph. Relying on its non-Euclidean data type, graph neural network (GNN) provides a clever way of learning the deep graph structure and it is rapidly becoming the state-of-the-art leading to enhanced performance in various network neuroscience tasks. Here we review current GNN-based methods, highlighting the ways that they have been used in several applications related to brain graphs such as missing brain graph synthesis and disease classification. We conclude by charting a path toward a better application of GNN models in network neuroscience field for neurological disorder diagnosis and population graph integration. The list of papers cited in our work is available at
https://github.com/basiralab/GNNs-in-Network-Neuroscience
.
... ese models can not only simulate the growth of networks but also explain the dynamic process of the classic BA network model in complex networks. Related studies are thus becoming increasingly popular, and network models with geometric properties have been used successfully in many fields in network science and other disciplines, including brain science [7,8], international trade [9], route transfer [10][11][12], and protein formation mechanisms [13,14]. ...
Network embedding is a frontier topic in current network science. The scale-free property of complex networks can emerge as a consequence of the exponential expansion of hyperbolic space. Some embedding models have recently been developed to explore hyperbolic geometric properties of complex networks—in particular, symmetric networks. Here, we propose a model for embedding directed networks into hyperbolic space. In accordance with the bipartite structure of directed networks and multiplex node information, the method replays the generation law of asymmetric networks in hyperbolic space, estimating the hyperbolic coordinates of each node in a directed network by the asymmetric popularity-similarity optimization method in the model. Additionally, the experiments in several real networks show that our embedding algorithm has stability and that the model enlarges the application scope of existing methods.
... Another hidden geometrical property of the connectomes is their hyperbolicity in the graph-metric space, as we show in the following. A previous study 68 refers to the brain network embedded in a hyperbolic space, i.e., the volume of the skull. Theoretically, the hyperbolicity of a path-connected geodesic metric space was proved 69,70 to be equivalent to the hyperbolicity of the graph associated with it. ...
Mapping the brain imaging data to networks, where nodes represent anatomical brain regions and edges indicate the occurrence of fiber tracts between them, has enabled an objective graph-theoretic analysis of human connectomes. However, the latent structure on higher-order interactions remains unexplored, where many brain regions act in synergy to perform complex functions. Here we use the simplicial complexes description of human connectome, where the shared simplexes encode higher-order relationships between groups of nodes. We study consensus connectome of 100 female (F-connectome) and of 100 male (M-connectome) subjects that we generated from the Budapest Reference Connectome Server v3.0 based on data from the Human Connectome Project. Our analysis reveals that the functional geometry of the common F&M-connectome coincides with the M-connectome and is characterized by a complex architecture of simplexes to the 14th order, which is built in six anatomical communities, and linked by short cycles. The F-connectome has additional edges that involve different brain regions, thereby increasing the size of simplexes and introducing new cycles. Both connectomes contain characteristic subjacent graphs that make them 3/2-hyperbolic. These results shed new light on the functional architecture of the brain, suggesting that insightful differences among connectomes are hidden in their higher-order connectivity.
... Indeed, it was shown that the nPSO is better suited than the PSO as realistic benchmark for the evaluation of inference techniques [20]. On the other side, also algorithms for embedding real topologies in the hyperbolic space have been designed [1], [7], [8], [21]- [27], which can be adopted for example in community detection and link prediction applications [1], [22]- [24], to study the navigability of complex networks [10], [28], or in pioneering attempts to develop latent geometry network markers in order to detect the separation between two groups in different conditions [29]. ...
Analysis of 'big data' characterized by high-dimensionality such as word vectors and complex networks requires often their representation in a geometrical space by embedding. Recent developments in machine learning and network geometry have pointed out the hyperbolic space as a useful framework for the representation of this data derived by real complex physical systems. In the hyperbolic space, the radial coordinate of the nodes characterizes their hierarchy, whereas the angular distance between them represents their similarity. Several studies have highlighted the relationship between the angular coordinates of the nodes embedded in the hyperbolic space and the community metadata available. However, such analyses have been often limited to a visual or qualitative assessment. Here, we introduce the angular separation index (ASI), to quantitatively evaluate the separation of node network communities or data clusters over the angular coordinates of a geometrical space. ASI is particularly useful in the hyperbolic space - where it is extensively tested along this study - but can be used in general for any assessment of angular separation regardless of the adopted geometry. ASI is proposed together with an exact test statistic based on a uniformly random null model to assess the statistical significance of the separation. We show that ASI allows to discover two significant phenomena in network geometry. The first is that the increase of temperature in 2D hyperbolic network generative models, not only reduces the network clustering but also induces a 'dimensionality jump' of the network to dimensions higher than two. The second is that ASI can be successfully applied to detect the intrinsic dimensionality of network structures that grow in a hidden geometrical space.
... As with NeuroCave, a goal of the connectogram is to more effectively represent densely connected networks, as is the case for the human connectome. Cacciola et al. (2017) demonstrate that the intrinsic geometry of a structural brain (left), taken from the Circos tutorial website (http://circos.ca/tutorials/), versus a 3D platonic solid representation of a connectome and its modularity using NeuroCave (right). ...
... These topologies are automatically identified by the application, and ongoing development aims to enable the transformation of anatomical datasets into a range of topological spaces on-demand. Currently, we have applied a range of dimensionality reduction techniques to connectome datasets, including Isomap (Tenenbaum, De Silva, & Langford, 2000) and t-SNE (Maaten & Hinton, 2008), and we make use of these methods to help identify patterns in the "intrinsic geometry" (i.e., the geometry as determined by the brain connectivity itself, either structural or functional) of a connectome dataset (Cacciola et al., 2017;Ye, Ajilore, et al., 2015). Once these intrinsic datasets are loaded, users can switch between anatomical and these abstract topological spaces as needed to support particular analyses, making it possible to see the same data transformed in various ways in order to investigate the connectome from a range of different perspectives. ...
... Put together, these two intrinsic geometries suggest strong functional coupling between the left and right hemispheres, likely mediated by the callosal connections, during the resting state. This finding is consistent with recent results reported by Cacciola et al. (2017), who indicate a clear matching between intrinsic geometry and neuroanatomy. ...
We introduce NeuroCave, a novel immersive visualization system that facilitates the visual inspection of structural and functional connectome datasets. The representation of the human connectome as a graph enables neuroscientists to apply network-theoretic approaches in order to explore its complex characteristics. With NeuroCave, brain researchers can interact with the connectome—either in a standard desktop environment or while wearing portable virtual reality headsets (such as Oculus Rift, Samsung Gear, or Google Daydream VR platforms)—in any coordinate system or topological space, as well as cluster brain regions into different modules on-demand. Furthermore, a default side-by-side layout enables simultaneous, synchronized manipulation in 3D, utilizing modern GPU hardware architecture, and facilitates comparison tasks across different subjects or diagnostic groups or longitudinally within the same subject. Visual clutter is mitigated using a state-of-the-art edge bundling technique and through an interactive layout strategy, while modular structure is optimally positioned in 3D exploiting mathematical properties of platonic solids. NeuroCave provides new functionality to support a range of analysis tasks not available in other visualization software platforms.
... We hope that this article will contribute to establish a new bridge at the interface between physics of complex networks and computational machine learning theory, and that future extended studies will dig into real applications revealing the impact of coalescent network embedding in social, natural, and life sciences. Interestingly, a first important instance of impact on network medicine is given in the article of Cacciola et al. 42 , in which coalescent embedding in the hyperbolic space enhances our understanding of the hidden geometry of brain connectomes, introducing the idea of network latent geometry marker characterization of brain diseases with application in de novo drug naive Parkinson's disease patients. ...
Physicists recently observed that realistic complex networks emerge as discrete samples from a continuous hyperbolic geometry enclosed in a circle: the radius represents the node centrality and the angular displacement between two nodes resembles their topological proximity. The hyperbolic circle aims to become a universal space of representation and analysis of many real networks. Yet, inferring the angular coordinates to map a real network back to its latent geometry remains a challenging inverse problem. Here, we show that intelligent machines for unsupervised recognition and visualization of similarities in big data can also infer the network angular coordinates of the hyperbolic model according to a geometrical organization that we term “angular coalescence.” Based on this phenomenon, we propose a class of algorithms that offers fast and accurate “coalescent embedding” in the hyperbolic circle even for large networks. This computational solution to an inverse problem in physics of complex systems favors the application of network latent geometry techniques in disciplines dealing with big network data analysis including biology, medicine, and social science.
The general ideas of \blockquote{TGD and EEG} are discussed in the first part of the book so that it is enough to explain the organization of \blockquote{TGD and EEG: Part II}. The book contains parts.
\begin{enumerate}
\item The first part represents chapters devoted to TGD inspired models for hearing, music experience, and language.
Also included are two chapters about the notion of bioharmony which started as a model of musical harmony based on the properties of icosahedron and tetrahedron. Surprisingly, the model led to successful models for the genetic code. Gene would be analog of a music piece consisting of N codons represented as 3-chords of light. This led to an identification of the bioharmony for the "music of light" as a correlate for emotions. Bio-communications between genes can be seen as music based on 3N-cyclotron resonances for dark genes with N codons of dark gene generating dark 3N-photon.
The latest discovery was that the bioharmony corresponds to the so-called icosa-tetrahedral tessellation of hyperbolic 3-space playing a central role in TGD. In many respects this tessellation is completely unique, and this could make s genetic code a universal way to represent information using 6-bit sequence as a basic unit.
\item The second part contains topics related to the arrow of time in neuroscience. A vision that sensory perception and motor action represent time reversals of each other in zero energy ontology (ZEO) is introduced. Also the notion of quantum statistical brain is discussed.
\end{enumerate}
\item The second part contains topics related to the arrow of time in neuroscience. A vision that sensory perception and motor action represent time reversals of each other in zero energy ontology (ZEO) is introduced. Also the notion of quantum statistical brain is discussed.
\end{enumerate}
The basic them of this book is the notion of\index{magnetic body} magnetic body which is one of the most radical new notions of TGD inspired theory of\index{consciousness} consciousness and\index{quantum biology} quantum biology.
\begin{enumerate}
\item The concept derives from the topological quantization of fields implying also the notion of\index{topological light ray} topological light ray (\blockquote{massless extremal}, ME) and quantization of electric flux. The notion means that, in contrast to Maxwell's ED, TGD allows allows to assign to a given material system also field identity. Magnetic body as the intentional agent controlling biological body thus comes the basic hypothesis of TGD inspired quantum theory of living systems.
\item TGD Universe is fractal containing fractal copies of standard model physics at various\index{space-time sheet} space-time sheets and labeled by the collection of\index{p-adic prime} p-adic primes assignable to elementary particles and by the level of dark matter hierarchy characterized partially by the rational value of Planck constant labeling the pages of the book like structure formed by singular\index{covering space} covering spaces of the embedding space $M^4\times CP_2$ glued together along a four-dimensional back. Particles at different pages are dark relative to each other since purely local interactions defined in terms of the vertices of a scattering diagram involve only particles at the same page if the the number of particles is larger than two. p-Adic length scale hypothesis and the assignment of dark matter with macroscopic quantum phases characterized by a hierarchy of Planck constants allows to quantify the notion of magnetic body. One can identify dark magnetic\index{flux quanta} flux quanta relevant to biology as 4-surfaces at pages of the book for which Planck constant is large.
\item The question about the precise form of the hierarchy of Planck constants remained open for a long time. The recent view is that all integer multiples of basic value $\hbar_0$ are allowed. The first guess was $\hbar_0=\hbar$ but now it seems that $\hbar_0$ is considerably smaller than $\hbar$.
There are also arguments suggesting that hierarchies involving integer multiples of some integer multiple of $\hbar$ are realized and number theoretical vision could allow this kind of hierarchies. For instance, there are indications that Planck constants comings as $2^{11k_d}$- multiples of the standard Planck constant are in in a special role in biology (this might relate to proton electron mass ratio and to the fact that $2^{11}\simeq m_p/m_e$ could appear as a fundamental constant in TGD Universe, as well as to the fact that the phases $exp(i2\pi 2^{-k_d})$ are number theoretically simple).
\item The notion of\index{personal magnetic body} personal magnetic body (actually onion-like fractal hierarchy of them) is essential for the TGD inspired model of living matter and predicts a hierarchy of generalized\index{EEG} EEGs associated with the magnetic bodies and responsible for the communications from biological body or its part to the corresponding magnetic body. There is no reason to assume that only personal magnetic bodies of living systems are relevant. Rather, the view about entire magnetosphere as a conscious system controlling the behavior of biosphere emerges naturally. In this book this vision is developed. \end{enumerate}
Most of the material of this book has been written much before the dark matter revolution and formulation of the \index{zero energy ontology} zero energy ontology and that I have only later added comments to the existing text. I hope that I can later add new material in which the implications of the dark matter hierarchy are discussed in more detail.
\subsection{The organization of the book "Magnetospheric Consciousness}
The book is divided to 3 parts.
\begin{enumerate}
\item In the first part of the book the first chapter is devoted to the idea about\index{magnetosphere} a magnetosphere as a conscious system perhaps defining in some respects a fractally scaled up version of the biological body and brain. At the first look this idea sounds completely crazy but in TGD Universe p-adic fractality and the fractality associated with\index{dark matter} dark matter hierarchy make it look rather natural. Furthermore, magnetic body and electric body would be TGD counterparts for the Maxwellian fields and the explanatory power of these notions justifies their introduction. Second chapter represents a vision about the relationship between EEG and magnetosphere.
\item The second part of the book contains two chapters about the notion of semitrance. Semitrance is based on quantum\index{entanglement} entanglement of the sub-self of self, say the subsystem of the brain, with a remote system. The idea that sub-systems of two unentangled systems can entangle and in this manner give rise to a sharing and fusion of\index{mental image} mental images (stereo vision would be the basic example) makes sense only in many-sheeted space. A rigorous justification for the sharing of mental images comes from the notion of\index{finite measurement resolution} finite measurement resolution - one of the fundamental notions of quantum TGD. The proposal is that\index{semitrance} semitrance could have been a basic control and communication tool of collective levels of consciousness during the period of human consciousness which Jaynes calls bicamerality. Schizophrenics could be seen as modern bicamerals.
The idea that human consciousness might have had totally different character for only a few millennia ago, finds additional support from the notions of super- and hyper\index{genome} genome implicated naturally by the dark matter hierarchy and the notion of magnetic body.
Super genome could be seen as a book having magnetic\index{flux sheet} flux sheets as pages. Text lines would be defined by genomes for sequences of nuclei. This would make possible coherent\index{gene expression} gene expression at the level of organs.
The text lines of\index{hyper genome} hyper genome would consist of super genomes of different organisms, not necessarily of the same species. Hyper genome would make possible coherent gene expression at the level of social group and society and give rise also to social rules. The identification of \index{meme} memes as hyper genes looks rather attractive. The evolution of the hyper genome could be seen as the basic driver of the explosive evolution of human civilizations during the last two millennia and would also distinguish us from our cousins.
\item The two chapters of the third part of the book entitled \blockquote{Crazy Stuff} are devoted to a model of\index{crop circle} crop circles: it is left to the reader to decide whether the chapters should be taken as miserable crack-pottery, mental gymnastics with tongue in cheek, or as a fruit of a new brave vision about us and the Universe. In the first chapter it is proposed that crop circles are due to intentional action of magnetospheric higher level self or a higher level self using magnetosphere as a tool to build them.
In the second chapter two special crop circles, Chilbolton and\index{Crabwood} Crabwood crop circles, are discussed in detail and the proposal that they provide information about the genomes of the life forms responsible for the crop circles. Some candidates for these life forms are discussed: the most science-fictive identification allowed by TGD would be ourselves in distant\index{geometric future} geometric future using time mirror mechanism to affect\index{geometric past} geometric past.
\end{enumerate}
