Análisis de la actividad neuronal y cerebral mediante técnicas de redes complejas

El cerebro humano es probablemente uno de los sistemas más complejos a los que nos enfrentamos en la actualidad, si bien es también uno de los más fascinantes. Sin embargo, la compresión de cómo el cerebro organiza su actividad para llevar a cabo tareas complejas es un problema plagado de restos y obstáculos. En sus inicios la neuroimagen y la electrofisiología tenían como objetivo la identificación de regiones asociadas a activaciones relacionadas con tareas especificas, o con patrones locales que variaban en el tiempo dada cierta actividad. Sin embargo, actualmente existe un consenso acerca de que la actividad cerebral tiene un carácter temporal multiescala y espacialmente extendido, lo que lleva a considerar el cerebro como una gran red de áreas cerebrales coordinadas, cuyas conexiones funcionales son continuamente creadas y destruidas. Hasta hace poco, el énfasis de los estudios de la actividad cerebral funcional se han centrado en la identidad de los nodos particulares que forman estas redes, y en la caracterización de métricas de conectividad entre ellos: la hipótesis subyacente es que cada nodo, que es una representación mas bien aproximada de una región cerebral dada, ofrece a una única contribución al total de la red. Por tanto, la neuroimagen funcional integra los dos ingredientes básicos de la neuropsicología: la localización de la función cognitiva en módulos cerebrales especializados y el rol de las fibras de conexión en la integración de dichos módulos. Sin embargo, recientemente, la estructura y la función cerebral han empezado a ser investigadas mediante la Ciencia de la Redes, una interpretación mecánico-estadística de una antigua rama de las matemáticas: La teoría de grafos. La Ciencia de las Redes permite dotar a las redes funcionales de una gran cantidad de propiedades cuantitativas (robustez, centralidad, eficiencia, ...), y así enriquecer el conjunto de elementos que describen objetivamente la estructura y la función cerebral a disposición de los neurocientíficos. La conexión entre la Ciencia de las Redes y la Neurociencia ha aportado nuevos puntos de vista en la comprensión de la intrincada anatomía del cerebro, y de cómo las patrones de actividad cerebral se pueden sincronizar para generar las denominadas redes funcionales cerebrales, el principal objeto de estudio de esta Tesis Doctoral. Dentro de este contexto, la complejidad emerge como el puente entre las propiedades topológicas y dinámicas de los sistemas biológicos y, específicamente, en la relación entre la organización y la dinámica de las redes funcionales cerebrales. Esta Tesis Doctoral es, en términos generales, un estudio de cómo la actividad cerebral puede ser entendida como el resultado de una red de un sistema dinámico íntimamente relacionado con los procesos que ocurren en el cerebro. Con este fin, he realizado cinco estudios que tienen en cuenta ambos aspectos de dichas redes funcionales: el topológico y el dinámico. De esta manera, la Tesis está dividida en tres grandes partes: Introducción, Resultados y Discusión. En la primera parte, que comprende los Capítulos 1, 2 y 3, se hace un resumen de los conceptos más importantes de la Ciencia de las Redes relacionados al análisis de imágenes cerebrales. Concretamente, el Capitulo 1 está dedicado a introducir al lector en el mundo de la complejidad, en especial, a la complejidad topológica y dinámica de sistemas acoplados en red. El Capítulo 2 tiene como objetivo desarrollar los fundamentos biológicos, estructurales y funcionales del cerebro, cuando éste es interpretado como una red compleja. En el Capítulo 3, se resumen los objetivos esenciales y tareas que serán desarrolladas a lo largo de la segunda parte de la Tesis. La segunda parte es el núcleo de la Tesis, ya que contiene los resultados obtenidos a lo largo de los últimos cuatro años. Esta parte está dividida en cinco Capítulos, que contienen una versión detallada de las publicaciones llevadas a cabo durante esta Tesis. El Capítulo 4 está relacionado con la topología de las redes funcionales y, específicamente, con la detección y cuantificación de los nodos mas importantes: aquellos denominados “hubs” de la red. En el Capítulo 5 se muestra como las redes funcionales cerebrales pueden ser vistas no como una única red, sino más bien como una red-de-redes donde sus componentes tienen que coexistir en una situación de balance funcional. De esta forma, se investiga cómo los hemisferios cerebrales compiten para adquirir centralidad en la red-de-redes, y cómo esta interacción se mantiene (o no) cuando se introducen fallos deliberadamente en la red funcional. El Capítulo 6 va un paso mas allá al considerar las redes funcionales como sistemas vivos. En este Capítulo se muestra cómo al analizar la evolución de la topología de las redes, en vez de tratarlas como si estas fueran un sistema estático, podemos caracterizar mejor su estructura. Este hecho es especialmente relevante cuando se quiere tratar de encontrar diferencias entre grupos que desempeñan una tarea de memoria, en la que las redes funcionales tienen fuertes fluctuaciones. En el Capítulo 7 defino cómo crear redes parenclíticas a partir de bases de datos de actividad cerebral. Este nuevo tipo de redes, recientemente introducido para estudiar las anormalidades entre grupos de control y grupos anómalos, no ha sido implementado nunca en datos cerebrales y, en este Capítulo explico cómo hacerlo cuando se quiere evaluar la consistencia de la dinámica cerebral. Para concluir esta parte de la Tesis, el Capítulo 8 se centra en la relación entre las propiedades topológicas de los nodos dentro de una red y sus características dinámicas. Como mostraré más adelante, existe una relación entre ellas que revela que la posición de un nodo dentro una red está íntimamente correlacionada con sus propiedades dinámicas. Finalmente, la última parte de esta Tesis Doctoral está compuesta únicamente por el Capítulo 9, el cual contiene las conclusiones y perspectivas futuras que pueden surgir de los trabajos expuestos. En vista de todo lo anterior, espero que esta Tesis aporte una perspectiva complementaria sobre uno de los más extraordinarios sistemas complejos frente a los que nos encontramos: El cerebro humano. ABSTRACT The human brain is probably one of the most complex systems we are facing, thus being a timely and fascinating object of study. Characterizing how the brain organizes its activity to carry out complex tasks is highly non-trivial. While early neuroimaging and electrophysiological studies typically aimed at identifying patches of task-specific activations or local time-varying patterns of activity, there has now been consensus that task-related brain activity has a temporally multiscale, spatially extended character, as networks of coordinated brain areas are continuously formed and destroyed. Up until recently, though, the emphasis of functional brain activity studies has been on the identity of the particular nodes forming these networks, and on the characterization of connectivity metrics between them, the underlying covert hypothesis being that each node, constituting a coarse-grained representation of a given brain region, provides a unique contribution to the whole. Thus, functional neuroimaging initially integrated the two basic ingredients of early neuropsychology: localization of cognitive function into specialized brain modules and the role of connection fibres in the integration of various modules. Lately, brain structure and function have started being investigated using Network Science, a statistical mechanics understanding of an old branch of pure mathematics: graph theory. Network Science allows endowing networks with a great number of quantitative properties, thus vastly enriching the set of objective descriptors of brain structure and function at neuroscientists’ disposal. The link between Network Science and Neuroscience has shed light about how the entangled anatomy of the brain is, and how cortical activations may synchronize to generate the so-called functional brain networks, the principal object under study along this PhD Thesis. Within this context, complexity appears to be the bridge between the topological and dynamical properties of biological systems and, more specifically, the interplay between the organization and dynamics of functional brain networks. This PhD Thesis is, in general terms, a study of how cortical activations can be understood as the output of a network of dynamical systems that are intimately related with the processes occurring in the brain. In order to do that, I performed five studies that encompass both the topological and the dynamical aspects of such functional brain networks. In this way, the Thesis is divided into three major parts: Introduction, Results and Discussion. In the first part, comprising Chapters 1, 2 and 3, I make an overview of the main concepts of Network Science related to the analysis of brain imaging. More specifically, Chapter 1 is devoted to introducing the reader to the world of complexity, specially to the topological and dynamical complexity of networked systems. Chapter 2 aims to develop the biological, topological and functional fundamentals of the brain when it is seen as a complex network. Next, Chapter 3 summarizes the main objectives and tasks that will be developed along the forthcoming Chapters. The second part of the Thesis is, in turn, its core, since it contains the results obtained along these last four years. This part is divided into five Chapters, containing a detailed version of the publications carried out during the Thesis. Chapter 4 is related to the topology of functional networks and, more specifically, to the detection and quantification of the leading nodes of the network: the hubs. In Chapter 5 I will show that functional brain networks can be viewed not as a single network, but as a network-of-networks, where its components have to co-exist in a trade-off situation. In this way, I investigate how the brain hemispheres compete for acquiring the centrality of the network-of-networks and how this interplay is maintained (or not) when failures are introduced in the functional network. Chapter 6 goes one step beyond by considering functional networks as living systems. In this Chapter I show how analyzing the evolution of the network topology instead of treating it as a static system allows to better characterize functional networks. This fact is especially relevant when trying to find differences between groups performing certain memory tasks, where functional networks have strong fluctuations. In Chapter 7 I define how to create parenclitic networks from brain imaging datasets. This new kind of networks, recently introduced to study abnormalities between control and anomalous groups, have not been implemented with brain datasets and I explain in this Chapter how to do it when evaluating the consistency of brain dynamics. To conclude with this part of the Thesis, Chapter 8 is devoted to the interplay between the topological properties of the nodes within a network and their dynamical features. As I will show, there is an interplay between them which reveals that the position of a node in a network is intimately related with its dynamical properties. Finally, the last part of this PhD Thesis is composed only by Chapter 9, which contains the conclusions and future perspectives that may arise from the exposed results. In view of all, I hope that reading this Thesis will give a complementary perspective of one of the most extraordinary complex systems: The human brain.

Exploring the origins of topological frustration: Design of a minimally frustrated model of fragment B of protein A

Topological frustration in an energetically unfrustrated off-lattice model of the helical protein fragment B of protein A from Staphylococcus aureus was investigated. This Gō-type model exhibited thermodynamic and kinetic signatures of a well-designed two-state folder with concurrent collapse and folding transitions and single exponential kinetics at the transition temperature. Topological frustration is determined in the absence of energetic frustration by the distribution of Fersht φ values. Topologically unfrustrated systems present a unimodal distribution sharply peaked at intermediate φ, whereas highly frustrated systems display a bimodal distribution peaked at low and high φ values. The distribution of φ values in protein A was determined both thermodynamically and kinetically. Both methods yielded a unimodal distribution centered at φ = 0.3 with tails extending to low and high φ values, indicating the presence of a small amount of topological frustration. The contacts with high φ values were located in the turn regions between helices I and II and II and III, intimating that these hairpins are in large part required in the transition state. Our results are in good agreement with all-atom simulations of protein A, as well as lattice simulations of a three- letter code 27-mer (which can be compared with a 60-residue helical protein). The relatively broad unimodal distribution of φ values obtained from the all-atom simulations and that from the minimalist model for the same native fold suggest that the structure of the transition state ensemble is determined mostly by the protein topology and not energetic frustration.

New lower bounds for the topological complexity of aspherical spaces

Date of Acceptance: 5/04/2015 15 pages, 4 figures

Equilibrium distributions of topological states in circular DNA: Interplay of supercoiling and knotting

Two variables define the topological state of closed double-stranded DNA: the knot type, K, and ΔLk, the linking number difference from relaxed DNA. The equilibrium distribution of probabilities of these states, P(ΔLk, K), is related to two conditional distributions: P(ΔLk|K), the distribution of ΔLk for a particular K, and P(K|ΔLk) and also to two simple distributions: P(ΔLk), the distribution of ΔLk irrespective of K, and P(K). We explored the relationships between these distributions. P(ΔLk, K), P(ΔLk), and P(K|ΔLk) were calculated from the simulated distributions of P(ΔLk|K) and of P(K). The calculated distributions agreed with previous experimental and theoretical results and greatly advanced on them. Our major focus was on P(K|ΔLk), the distribution of knot types for a particular value of ΔLk, which had not been evaluated previously. We found that unknotted circular DNA is not the most probable state beyond small values of ΔLk. Highly chiral knotted DNA has a lower free energy because it has less torsional deformation. Surprisingly, even at |ΔLk| > 12, only one or two knot types dominate the P(K|ΔLk) distribution despite the huge number of knots of comparable complexity. A large fraction of the knots found belong to the small family of torus knots. The relationship between supercoiling and knotting in vivo is discussed.

Dissection of De Novo Membrane Insertion Activities of Internal Transmembrane Segments of ATP-Binding-Cassette Transporters: Toward Understanding Topological Rules for Membrane Assembly of Polytopic Membrane Proteins

The membrane assembly of polytopic membrane proteins is a complicated process. Using Chinese hamster P-glycoprotein (Pgp) as a model protein, we investigated this process previously and found that Pgp expresses more than one topology. One of the variations occurs at the transmembrane (TM) domain including TM3 and TM4: TM4 inserts into membranes in an Nin-Cout rather than the predicted Nout-Cin orientation, and TM3 is in cytoplasm rather than the predicted Nin-Cout orientation in the membrane. It is possible that TM4 has a strong activity to initiate the Nin-Cout membrane insertion, leaving TM3 out of the membrane. Here, we tested this hypothesis by expressing TM3 and TM4 in isolated conditions. Our results show that TM3 of Pgp does not have de novo Nin-Cout membrane insertion activity whereas TM4 initiates the Nin-Cout membrane insertion regardless of the presence of TM3. In contrast, TM3 and TM4 of another polytopic membrane protein, cystic fibrosis transmembrane conductance regulator (CFTR), have a similar level of de novo Nin-Cout membrane insertion activity and TM4 of CFTR functions only as a stop-transfer sequence in the presence of TM3. Based on these findings, we propose that 1) the membrane insertion of TM3 and TM4 of Pgp does not follow the sequential model, which predicts that TM3 initiates Nin-Cout membrane insertion whereas TM4 stops the insertion event; and 2) “leaving one TM segment out of the membrane” may be an important folding mechanism for polytopic membrane proteins, and it is regulated by the Nin-Cout membrane insertion activities of the TM segments.

Coupled Translocation Events Generate Topological Heterogeneity at the Endoplasmic Reticulum Membrane

Topogenic determinants that direct protein topology at the endoplasmic reticulum membrane usually function with high fidelity to establish a uniform topological orientation for any given polypeptide. Here we show, however, that through the coupling of sequential translocation events, native topogenic determinants are capable of generating two alternate transmembrane structures at the endoplasmic reticulum membrane. Using defined chimeric and epitope-tagged full-length proteins, we found that topogenic activities of two C-trans (type II) signal anchor sequences, encoded within the seventh and eighth transmembrane (TM) segments of human P-glycoprotein were directly coupled by an inefficient stop transfer (ST) sequence (TM7b) contained within the C-terminus half of TM7. Remarkably, these activities enabled TM7 to achieve both a single- and a double-spanning TM topology with nearly equal efficiency. In addition, ST and C-trans signal anchor activities encoded by TM8 were tightly linked to the weak ST activity, and hence topological fate, of TM7b. This interaction enabled TM8 to span the membrane in either a type I or a type II orientation. Pleiotropic structural features contributing to this unusual topogenic behavior included 1) a short, flexible peptide loop connecting TM7a and TM7b, 2) hydrophobic residues within TM7b, and 3) hydrophilic residues between TM7b and TM8.

Topological testing of the mechanism of homology search promoted by RecA protein

To initiate homologous recombination, sequence similarity between two DNA molecules must be searched for and homology recognized. How the search for and recognition of homology occurs remains unproven. We have examined the influences of DNA topology and the polarity of RecA–single-stranded (ss)DNA filaments on the formation of synaptic complexes promoted by RecA. Using two complementary methods and various ssDNA and duplex DNA molecules as substrates, we demonstrate that topological constraints on a small circular RecA–ssDNA filament prevent it from interwinding with its duplex DNA target at the homologous region. We were unable to detect homologous pairing between a circular RecA–ssDNA filament and its relaxed or supercoiled circular duplex DNA targets. However, the formation of synaptic complexes between an invading linear RecA–ssDNA filament and covalently closed circular duplex DNAs is promoted by supercoiling of the duplex DNA. The results imply that a triplex structure formed by non-Watson–Crick hydrogen bonding is unlikely to be an intermediate in homology searching promoted by RecA. Rather, a model in which RecA-mediated homology searching requires unwinding of the duplex DNA coupled with local strand exchange is the likely mechanism. Furthermore, we show that polarity of the invading RecA–ssDNA does not affect its ability to pair and interwind with its circular target duplex DNA.

Capabilities of liposomes for topological transformation

Dynamic behaviors of liposomes caused by interactions between liposomal membranes and surfactant were studied by direct real-time observation by using high-intensity dark-field microscopy. Solubilization of liposomes by surfactants is thought to be a catastrophic event akin to the explosion of soap bubbles in the air; however, the actual process has not been clarified. We studied this process experimentally and found that liposomes exposed to various surfactants exhibited unusual behavior, namely continuous shrinkage accompanied by intermittent quakes, release of encapsulated liposomes, opening up, and inside–out topological inversion.

Distinct Biochemical and Topological Properties of the 31- and 27-Kilodalton Plasma Membrane Intrinsic Protein Subgroups from Red Beet1

Plasma membrane vesicles from red beet (Beta vulgaris L.) storage tissue contain two prominent major intrinsic protein species of 31 and 27 kD (X. Qi, C.Y Tai, B.P. Wasserman [1995] Plant Physiol 108: 387–392). In this study affinity-purified antibodies were used to investigate their localization and biochemical properties. Both plasma membrane intrinsic protein (PMIP) subgroups partitioned identically in sucrose gradients; however, each exhibited distinct properties when probed for multimer formation, and by limited proteolysis. The tendency of each PMIP species to form disulfide-linked aggregates was studied by inclusion of various sulfhydryl agents during tissue homogenization and vesicle isolation. In the absence of dithiothreitol and sulfhydryl reagents, PMIP27 yielded a mixture of monomeric and aggregated species. In contrast, generation of a monomeric species of PMIP31 required the addition of dithiothreitol, iodoacetic acid, or N-ethylmaleimide. Mixed disulfide-linked heterodimers between the PMIP31 and PMIP27 subgroups were not detected. Based on vectorial proteolysis of right-side-out vesicles with trypsin and hydropathy analysis of the predicted amino acid sequence derived from the gene encoding PMIP27, a topological model for a PMIP27 was established. Two exposed tryptic cleavage sites were identified from proteolysis of PMIP27, and each was distinct from the single exposed site previously identified in surface loop C of a PMIP31. Although the PMIP31 and PMIP27 species both contain integral proteins that appear to occur within a single vesicle population, these results demonstrate that each PMIP subgroup responds differently to perturbations of the membrane.

Topological challenges to DNA replication: Conformations at the fork

The unwinding of the parental DNA duplex during replication causes a positive linking number difference, or superhelical strain, to build up around the elongating replication fork. The branching at the fork and this strain bring about different conformations from that of (−) supercoiled DNA that is not being replicated. The replicating DNA can form (+) precatenanes, in which the daughter DNAs are intertwined, and (+) supercoils. Topoisomerases have the essential role of relieving the superhelical strain by removing these structures. Stalled replication forks of molecules with a (+) superhelical strain have the additional option of regressing, forming a four-way junction at the replication fork. This four-way junction can be acted on by recombination enzymes to restart replication. Replication and chromosome folding are made easier by topological domain barriers, which sequester the substrates for topoisomerases into defined and concentrated regions. Domain barriers also allow replicated DNA to be (−) supercoiled. We discuss the importance of replicating DNA conformations and the roles of topoisomerases, focusing on recent work from our laboratory.

Quantum simulation of a topological Mott insulator with Rydberg atoms in a Lieb lattice

We propose a realistic scheme to quantum simulate the so-far experimentally unobserved topological Mott insulator phase-an interaction-driven topological insulator-using cold atoms in an optical Lieb lattice. To this end, we study a system of spinless fermions in a Lieb lattice, exhibiting repulsive nearest-and next-to-nearest-neighbor interactions and derive the associated zero-temperature phase diagram within mean-field approximation. In particular, we analyze how the interactions can dynamically generate a charge density wave ordered, a nematic, and a topologically nontrivial quantum anomalous Hall phase. We characterize the topology of the different phases by the Chern number and discuss the possibility of phase coexistence. Based on the identified phases, we propose a realistic implementation of this model using cold Rydberg-dressed atoms in an optical lattice. The scheme, which allows one to access, in particular, the topological Mott insulator phase, robustly and independently of its exact position in parameter space, merely requires global, always-on off-resonant laser coupling to Rydberg states and is feasible with state-of-the-art experimental techniques that have already been demonstrated in the laboratory.

Topological SLAM using a graph-matching based method on omnidirectional images

Comunicación presentada en el XI Workshop of Physical Agents, Valencia, 9-10 septiembre 2010.

Colossal anisotropy in diluted magnetic topological insulators

We consider dilute magnetic doping in the surface of a three dimensional topological insulator where a two dimensional Dirac electron gas resides. We find that exchange coupling between magnetic atoms and the Dirac electrons has a strong and peculiar effect on both. First, the exchange-induced single ion magnetic anisotropy is very large and favors off-plane orientation. In the case of a ferromagnetically ordered phase, we find a colossal magnetic anisotropy energy, of the order of the critical temperature. Second, a persistent electronic current circulates around the magnetic atom and, in the case of a ferromagnetic phase, around the edges of the surface.

Topological structures of complex belief systems

The concepts of substantive beliefs and derived beliefs are defined, a set of substantive beliefs S like open set and the neighborhood of an element substantive belief. A semantic operation of conjunction is defined with a structure of an Abelian group. Mathematical structures exist such as poset beliefs and join-semilattttice beliefs. A metric space of beliefs and the distance of belief depending on the believer are defined. The concepts of closed and opened ball are defined. S′ is defined as subgroup of the metric space of beliefs Σ and S′ is a totally limited set. The term s is defined (substantive belief) in terms of closing of S′. It is deduced that Σ is paracompact due to Stone's Theorem. The pseudometric space of beliefs is defined to show how the metric of the nonbelieving subject has a topological space like a nonmaterial abstract ideal space formed in the mind of the believing subject, fulfilling the conditions of Kuratowski axioms of closure. To establish patterns of materialization of beliefs we are going to consider that these have defined mathematical structures. This will allow us to understand better cultural processes of text, architecture, norms, and education that are forms or the materialization of an ideology. This materialization is the conversion by means of certain mathematical correspondences, of an abstract set whose elements are beliefs or ideas, in an impure set whose elements are material or energetic. Text is a materialization of ideology.

Topological structures of complex belief systems (II): Textual materialization

Mythical and religious belief systems in a social context can be regarded as a conglomeration of sacrosanct rites, which revolve around substantive values that involve an element of faith. Moreover, we can conclude that ideologies, myths and beliefs can all be analyzed in terms of systems within a cultural context. The significance of being able to define ideologies, myths and beliefs as systems is that they can figure in cultural explanations. This, in turn, means that such systems can figure in logic-mathematical analyses.