Phases and phase transitions in living matter

Resumen

Standing on the shoulders of giants such as Landau, Anderson, Mandelbrot or Bak, emergent phenomena have meant a major step towards the comprehension of macro-structures and patterns in Nature. In particular, the criticality hypothesis, which proposes that –under some circumstances– living systems can lie in the vicinity of a phase transition, i.e. at the borderline between their ordered and disordered phases, has shed much light on the comprehension of several natural phenomena that, until recently, were poorly understood. This celebrated and provocative idea conjectures that living close to a critical point may confer a large number of benefits such as maximal dynamical range, maximal sensitivity to environmental changes, as well as an excellent trade-off between stability and flexibility. Based on this assumption, the aim of this thesis is to look into the criticality hypothesis, extending its horizons through the analysis of phases and phase transitions in Nature, developing a better understanding of certain empirical findings and behaviors of biological systems. Thus, the development of models trying to shed light –through numerical simulations and theoretical calculations– on the emergent behavior of particular biological systems constitute the common theme of this thesis. In chapter 1 a basic introduction and a schematic review on the criticality hypothesis, as well as some particular examples in living systems such as neuronal dynamics and gene regulation are outlined. Also, a brief, but necessary introduction to phases and phases transition and the Landau equilibrium theory of critical phenomena is presented. It covers the two principal phase transitions, first order and second order, certain theoretical notions of non-equilibrium systems, together with the introduction of the main self-organizing mechanisms to such phase transitions, as well as a brief summary on the effects of non-homogeneous underlying structures in the dynamical evolution, and phases, of most systems. Above all, one of the principal aims of this chapter is to be intended to allow for a self-contained book. In chapter 2 we try to shed light in the origin, nature and functional significance of complex patterns of neural activity in the human cortex, which operates in a state of sempiternal irregular activity, whose meaning and functionality are still not well understood. Such patterns include collective oscillations, emerging out of neural synchronization, as well as highly-heterogeneous outbursts of activity interspersed by periods of quiescence, called “neuronal avalanches”. A fascinating though still controversial hypothesis, to some extent backed by empirical evidence, suggests that the cortex might work at the edge of a phase transition, from which important functional advantages stem. However, the nature of such a phase transition is still not fully understood. Here, we adopt ideas from the physics of phase transitions, to construct a general (Landau-Ginzburg) theory of cortical networks, allowing us to analyze their possible collective phases and phase transitions. We conclude that the empirically reported scale-invariant avalanches can possibly come about if the cortex operated at the edge of a synchronization phase transition, at which neuronal avalanches and incipient oscillations coexist. Chapter 3 tackles the problem of neuronal synchronization in a more complex and realistic underlying structure (i.e. coupling scheme) given by the actual human-brain connectome network employing a parsimonious (mesoscopic) approach, the Kuramoto model, in order to preserve the essence of a minimal design, with the purpose of studying analytically and computationally the synchronization dynamics and to scrutinize the spontaneous emergence of coherent behavior in neural function. We elucidate the existence of a so-far-uncovered intermediate phase, placed between the standard synchronous and asynchronous phases, i.e. between order and disorder. This novel phase stems from the hierarchical modular organization of the connectome. Where one would expect a hierarchical synchronization process, we show that the interplay between structural bottlenecks and quenched intrinsic frequency heterogeneities at many different scales, gives rise to frustrated synchronization, metastability, and chimera-like states, resulting in a very rich and complex phenomenology. We uncover the origin of the dynamic freezing behind these features by using spectral graph theory and discuss how the emerging complex synchronization patterns relate to the need for the brain to access –in a robust though flexible way– a large variety of functional attractors and dynamical repertoires without ad hoc fine-tuning to a critical point. Also, we explore the role of noise, as an effective description of external perturbations, and we discuss how its presence accounts for the ability of the system to escape intermittently from such attractors and explore complex dynamic repertoires of locally coherent states, in analogy with experimentally recorded patterns of cerebral activity. In Chapter 4 we revisit the problem of deriving the mean-field values of avalanche exponents in systems with absorbing states. These are well known to coincide with those of an unbiased branching processes, as reported in neural avalanches, for at least, four different universality classes. We report on the emergence of non-universal continuously varying exponent values stemming from the presence of small external driving –that might induce avalanche merging– and that, to the best of our knowledge, has not been noticed in the past. Such active/quiescent transition is closely related with the “balanced amplification” theoretical approach recently proposed to explain the empirical neural avalanches of activity. Standing on the active phase with an excellent balance between excitation/inhibition, the weak stability of the basin of attraction of the system caused by a reactive dynamics is exploited, i.e. the dynamics is encoded in a “non-normal” matrix. Thus, the system exhibit large fluctuations reminiscent of “up” and “down” states and neural activity. We have progressed in a thorough understanding of such phenomenon as well as it has been extended to a wider scenario: a similar non-critical scale-invariance can be obtained by changing the regulatory mechanism that drives the dynamics, i.e. excluding inhibition and introducing synaptic plasticity. We believe that a simple and unified perspective as the one presented here can help to clarify the overall picture and underline the super-universality of the behavior giving rise to the unbiased branching processes exponents in active/quiescent phase transitions, as well as review, better understand and clarify certain processes with generic power laws but poised far away from criticality. Chapter 5 is the first to address other problems beyond neural dynamics such as gene regulation in complex biological systems. To this end, the well-founded Boolean approach to model gene regulatory networks is employed. A much discussed hypothesis proposed that such approach reproduces empirical findings the best if it is tuned to operate at criticality, exploiting its large number of functional advantages. Here, we study the effect of noise within the context of Boolean networks trained to learn complex tasks under supervision. We verify that quasi-critical networks are the ones learning in the fastest possible way –even for asynchronous updating rules– and that the larger the task complexity the smaller the distance to criticality. On the other hand, when additional sources of intrinsic noise in the network states and/or in its wiring pattern are introduced, the optimally performing networks become clearly subcritical. These results suggest that in order to compensate for inherent stochasticity, regulatory and other type of biological networks might become subcritical rather than being critical, all the most if the task to be performed has limited complexity. In chapter 6 we analyze the evolving modular structure of the network of dependencies between software packages in the different Debian GNU/Linux distributions released to date. Also, we explore the emergent properties and vulnerability of such networks and their role in the functionality of the system. In parallel, we show the interesting parallelisms between the architecture and emergent properties of software networks and that of regulatory interactions between genes. Indeed, such analogy allow us for an appealing explanation of recent empirical findings and enigmas of systems biology, the emergent cascading failures of “gene knockout” and possible functionalities of the, sometimes considered futile, non-coding DNA. Chapter 7 highlights key findings and conclusions derived from this thesis, seen in a global perspective, which allows the reader to appreciate its contribution to the understanding of the emergent (critical) properties and phases of living systems, as well as the open issues and the enormous amount of work that remains to be done. Although, in order to make this work available to the wider academic community, this thesis is written in English, a brief summary in Spanish (appendix [chap:Resumen-castellano]) is included in order to obtain the degree of Doctor of Philosophy in Physics with European level, fulfilling the requirements of the University of Granada. Furthermore, some chapters also contain annexes. In particular, chapter 1 (appendix [app:Introduction-to-critical]), chapter 2 (appendix [app:Landau-Ginzburg-theory-of]), chapter 3 (appendix [app:Kuramoto-model]) and chapter 4 (appendix [chap:Langevin-equations-in]), in order to clarify some calculations and remarks that are too specific (or exhaustive) and may be excluded without impairing a proper understanding of each chapter. Also, a list of publications of the author is reported. Of course, there has been more work beyond that explained here, as part of the learning process that gave birth to this thesis. Some of this additional work has been published and it has also been included in such list of publications.