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Simple models of excitable dynamics on graphs are an efficient framework for studying the interplay between network topology and dynamics. This topic is of practical relevance to diverse fields, ranging from neuroscience to engineering. Here we analyze how a single excitation propagates through a random network as a function of the excitation threshold, that is, the relative amount of activity in the neighborhood required for the excitation of a node. We observe that two sharp transitions delineate a region of sustained activity. Using analytical considerations and numerical simulation, we show that these transitions originate from the presence of barriers to propagation and the excitation of topological cycles, respectively, and can be predicted from the network topology. Our findings are interpreted in the context of network reverberations and self-sustained activity in neural systems, which is a question of long-standing interest in computational neuroscience.

Dynamical systems theory offers a different, geometrical view for describing the evolution of a system. It leads to the puzzling notions of deterministic chaos and universal routes to chaos.

In this chapter, I discuss the status of time when modeling natural systems. In such an operational perspective, time is defined and measured by a clock, i.e. the comparison with a given physical phenomenon. A central notion is that of time scales. I argue that the representation of time (e.g. discrete vs. continuous) is essentially a modeling choice, depending on the time scale of the observation or description compared to the characteristic times of the system. In integrated approaches developed for investigating systems presenting several levels of organization, it may be fruitful to consider several independent time variables describing evolution at different nested time scales (“multiple scale method”). This would solve some puzzling but recurrent statements, such as an evolution composed of a succession of equilibrium states, and leads to a practical understanding of diverse systems, from combustion and kinetic theory of gases in physics to enzymatic catalysis in biochemistry to adaptive dynamics in ecology. At a more fundamental level, the confrontation of descriptions performed at different scales and within different frameworks, from molecular dynamics to statistical mechanics to thermodynamics, is essential to delineate the limits of each of them, and overall reach a more informative understanding of reality. In particular, such a multi-scale confrontation offers a simple answer to the long-lasting opposition between microscopic reversibility and macroscopic irreversibility of isolated systems (the celebrated Second Principle of the thermodynamics). In biology, time scales are also essential in discussing the robustness of living systems and their evolutionary strategies.

In higher organisms, all cells share the same genome, but every cell expresses only a limited and specific set of genes that defines the cell type. During cell division, not only the genome, but also the cell type is inherited by the daughter cells. This intriguing phenomenon is achieved by a variety of processes that have been collectively termed epigenetics: the stable and inheritable changes in gene expression patterns. This article reviews the extremely rich and exquisitely multi-scale physical mechanisms that govern the biological processes behind the initiation, spreading and inheritance of epigenetic states. These include not only the changes in the molecular properties associated with the chemical modifications of DNA and histone proteins, such as methylation and acetylation, but also less conventional ones, such as the physics that governs the three-dimensional organization of the genome in cell nuclei. Strikingly, to achieve stability and heritability of epigenetic states, cells take advantage of many different physical principles, such as the universal behavior of polymers and copolymers, the general features of non-equilibrium dynamical systems, and the electrostatic and mechanical properties related to chemical modifications of DNA and histones. By putting the complex biological literature under this new light, the emerging picture is that a limited set of general physical rules play a key role in initiating, shaping and transmitting this crucial "epigenetic landscape". This new perspective not only allows to rationalize the normal cellular functions, but also helps to understand the emergence of pathological states, in which the epigenetic landscape becomes dysfunctional., 34 pages, 13 figures

Ionic currents across neuron and glial cells membranes lie at the origin of the entire brain electrophysiology. They are the common root of functional brain dynamics and mesoscopic or macroscopic phenomena such as extracellular fields. In particular, they provide the relevant basis to relate cellular electrophysiology and macroscopic dipole models. In order to derive robust features and to envision the multi-scale approaches required to connect the different levels of observation, an essential prerequisite is to have minimal model of elementary ionic motions. In this paper, we propose a general cellular automata framework allowing to investigate the distribution of ionic currents in heterogeneous media interspersed with membranes, from which follows the local electromagnetic field.

We investigate the evolution of a system composed of N non-interacting point particles of mass m in a container divided into two chambers by a movable adiabatic piston of mass M≫m. Using a two-time-scale perturbation approach in terms of the small parameter α=2m/(M+m), we show that the evolution towards thermal equilibrium proceeds in two stages. The first stage is a fast, deterministic, adiabatic relaxation towards mechanical equilibrium. The second stage, which takes place at times $$\mathcal{O}$$ (M), is a slow fluctuation-driven, diathermic relaxation towards thermal equilibrium. A very simple equation is derived which shows that in the second stage, the position of the piston is given by X M (t)= L[1/2−ξ(αt)] where the function ξ is independent of M. Numerical simulations support the assumptions underlying our analytical derivations and illustrate the large mass range in which the picture holds.

We consider the evolution of a system composed of N non-interacting point particles of mass m in a cylindrical container divided into two regions by a movable adiabatic wall (the adiabatic piston). We study the thermodynamic limit for the piston where the area A of the cross-section, the mass M of the piston, and the number N of particles go to infinity keeping A/M and N/M fixed. The length of the container is a fixed parameter which can be either finite or infinite. In this thermodynamic limit we show that the motion of the piston is deterministic and the evolution is adiabatic. Moreover if the length of the container is infinite, we show that the piston evolves toward a stationary state with velocity approximately proportional to the pressure difference. If the length of the container is finite, introducing a simplifying assumption we show that the system evolves with either weak or strong damping toward a well-defined state of mechanical equilibrium where the pressures are the same, but the temperatures different. Numerical simulations are presented to illustrate possible evolutions and to check the validity of the assumption.

International audience; Investigating average thermodynamic quantities is not sufficient to understand conformational transitions of a finite-size polymer. We propose that such transitions are better described in terms of the probability distribution of some finite-size order parameter, and the evolution of this distribution as a control parameter varies. We demonstrate this claim for the coil-globule transition of a linear polymer and its mapping onto a two-state model. In a biological context, polymer models delineate the physical constraints experienced by the genome at different levels of organization, from DNA to chromatin to chromosome. We apply our finite-size approach to the formation of plectonemes in a DNA segment submitted to an applied torque and the ensuing helix-coil transition that can be numerically observed, with a coexistence of the helix and coil states in a range of parameters. Polymer models are also essential to analyze recent in vivo experiments providing the frequency of pairwise contacts between genomic loci. The probability distribution of these contacts yields quantitative information on the conformational fluctuations of chromosome regions. The changes observed in the shape of the distribution when the cell type or the physiological conditions vary may reveal an epigenetic modulation of the conformational constraints experienced by the chromosomes.

This chapter is dedicated to the study of the superconductor–insulator transition in high temperature superconductors. We saw, in Sect.1.3.3, that metal superconductors were not so interesting from the point of view of critical phenomena. This is because their coherence length at T=0 is of the order of a micron, so the critical region is around 10−14K, which is obviously impossible to observe. However this is not true for high temperature superconductors, which exhibit a critical temperature of the order of 100 K. The most important high temperature superconductor family is that of the superconducting cuprates discovered in 1986 by Bednorz and Muller. Since then other families have been discovered for example C60, MgB2 and AsFe. However the most important family is still the one that was discovered first, the cuprate family, which is the only one with superconducting properties above 100K. This is also the most documented family both experimentally and theoretically. For these reasons, in this chapter we will focus only on superconducting cuprates. Their coherence length at T=0 being of the order of \(15\r{A}\), the critical region can reach a few tens of kelvins! So these materials represent, in principle, an ideal case in which to study critical phenomena, even more so given that the transition can be induced by a magnetic field (as for all superconductors) or by doping (a unique property of high temperature superconductors). In principle, these two parameters, can induce quantum transitions at T=0. This physical system certainly seems interesting to test new descriptions of critical phenomena.

After the wide panorama presented in this book, we can draw a few general conclusions on scaling approaches. We have seen that they apply in situations when the system no longer has a characteristic scale, as for example observed for a temperature T→T c in the case of phase transitions, for a number of monomers N→∞ in polymers, for a filling density p→p c in percolation, for a time span t→∞ in growth or diffusion, at the onset of chaos in the transition to chaos by period doubling or for a Reynolds number Re→∞ in turbulence.

It is a fact of life in our daily experience and was a real enigma for a century– pure matter dramatically changes at extremely precise temperatures. In the nineteenth century, pioneering scientists, from Gay-Lussac to Van der Waals, carried out meticulous measurements of the fluid state, paving the way for microscopic descriptions which underlie our natural sciences today. The study of properties of gases at low density enabled the introduction of absolute temperature as a measure of the kinetic energy of molecules. The striking generality of thermal behaviour and mechanical properties of gases with vastly varying chemical properties was thereby elucidated. Thermodynamics and its microscopic interpretations was born on the wave of this success. However the pioneers of fluid observations also tackled liquid–vapour transformations and discovered another elegant generality which was far from evident a priori: In a dilute gas, molecules are almost isolated and therefore one thinks their chemical properties are unimportant, but what about in a liquid where the molecules are in constant interaction?

In the introductory chapter (Chap.1) we showed several examples of critical behaviours near second order phase transitions. Let us remind ourselves of the key observations. When the temperature is near the critical temperature, most physical quantities obey a power law(T−T c ) x , where xis called a critical exponent. In this chapter we will see that this sort of behaviour is the signature of the scale invariance of the system close to the critical point.

What physical concept can unite the flow of liquid through ground coffee, electrical conduction in a conductor–insulator mixture, target collapse on shooting, the birth of a continent, polymer gelification and the spread of epidemics or forest fires? One question is common to all these processes: how is “something” produced at large scales by contributions at small scales? In all these situations, a quantity (liquid, electric charges, target fracture, dry land, molecular crosslinks, disease or fire) may or may not propagate from one element to its neighbour. As in the previous chapters, we are interested in asymptotic properties resulting at large scale, that is to say in a system that is large compared with the size of the individual elements. The analogue of temperature T here is the inverse of the relative population density p of elements (pores, conducting regions, impacts, etc) which varies between 0 and 1 for the maximum population density. p ∼ 0 corresponds to a large amount of disorder for a dilute population and p ∼ 1 corresponds to a large amount of order established. In these situations we can ask ourselves the same questions as for changes of states of matter in thermal systems.

During this book we have encountered many examples of critical phenomenon and have highlighted their common characteristics of divergence of the range of correlations, absence of characteristic scales, fluctuations of all sizes and anomalous response (at all scales in amplitude, spatial extent and duration) to even a tiny perturbation. These characteristics are reflected in many scaling laws, expressing quantitatively the scale invariance of the phenomena. Typically, criticality occurs for a particular value of the control parameter which is adjusted externally, for example the critical temperature in second order phase transitions, percolation threshold p c or bifurcation point in a dynamic system. However, it turns out that certain systems, maintained in a nonequilibrium state by a continuous supply of material or energy, can evolve spontaneously to a critical state, without external regulation. This is the concept of self-organised criticality, which we will present in this chapter, discussing various examples.

We investigate the stationary nonequilibrium states of a quasi one-dimensional system of heavy particles whose interaction is mediated by purely elastic collisions with light particles, in contact at the boundary with two heat baths with fixed temperatures $T^+$ and $T^-$. It is shown that Fourier law is satisfied with a thermal conductivity proportional to $\sqrt{T(x)}$ where $T(x)$ is the local temperature. Entropy flux and entropy production are also investigated., 12 pages, 4 figures

The piston problem is investigated in the case where the length of the cylinder is infinite (on both sides) and the ratio $m/M$ is a very small parameter, where $m$ is the mass of one particle of the gaz and $M$ is the mass of the piston. Introducing initial conditions such that the stochastic motion of the piston remains in the average at the origin (no drift), it is shown that the time evolution of the fluids, analytically derived from Liouville equation, agrees with the Second Law of thermodynamics. We thus have a non equilibrium microscopical model whose evolution can be explicitly shown to obey the two laws of thermodynamics., 29 pages, 9 figures submitted to Journal of Statistical Physics (2003)

This paper focuses on mechanical aspects of chromatin biological functioning. Within a basic geometric modeling of the chromatin assembly, we give for the first time the complete set of elastic constants (twist and bend persistence lengths, stretch modulus and twist-stretch coupling constant) of the so-called 30-nm chromatin fiber, in terms of DNA elastic properties and geometric properties of the fiber assembly. The computation naturally embeds the fiber within a current analytical model known as the ``extensible worm-like rope'', allowing a straightforward prediction of the force-extension curves. We show that these elastic constants are strongly sensitive to the linker length, up to 1 bp, or equivalently to its twist, and might locally reach very low values, yielding a highly flexible and extensible domain in the fiber. In particular, the twist-stretch coupling constant, reflecting the chirality of the chromatin fiber, exhibits steep variations and sign changes when the linker length is varied. We argue that this tunable elasticity might be a key feature for chromatin function, for instance in the initiation and regulation of transcription., Comment: 38 pages 15 figures