Geometry and wound healing mechanisms
We will present work on the mechanisms used for establishing or restoring epithelial integrity which are motivated by experimental work on development and wound healing in Zebrafish and drosophila and on gap closure in monolayers of MDCK cells or keratinocytes. These works concern mathematical modeling of the dynamics of epithelial tissues pulled by lamellipodal crawling or the contraction of actomyosin cables at the gap boundary. We are particularly interested in the influence of the wound/gap geometry and of the adhesion to the substrate on the closure mechanism.
Cecile Appert-Rolland [slides]
Intracellular transport of cargos: tug-of-war, anomalous diffusion, and lattice deformation
Cells' life involves many processes that require the transport of cargos (vesicles, organelles...). As the interior of the cell is very crowded, this transport must be active. It relies mostly on two families of molecular motors that step in opposite directions along microtubules. There are still several features that were observed but not clearly understood.
In particular, it was evidenced that cargos can be pulled by teams of opposite motors. This phenomenon, called tug-of-war, seems quite inefficient as far as transport properties are concerned. Using an explicit stochastic model, we shall first show that such cargo-motors complexes can exhibit a quite complex dynamics in agreement with some observations in real cells. We will also propose some hypothesis based on numerical observations to explain why such transport mechanism can be beneficial in the frame of intracellular transport - or more generally in crowded heterogeneous media.
In a second part, we shall explore the interplay between transport and lattice dynamics. On the one hand, we have shown that lattice dynamics can facilitate or even allow transport when it is impeded by obstacles. On the other hand, motors can participate to the deformation of the cytoskeletal network. We shall present a model allowing to couple the stochastic dynamics of motors with filament deformation in simple geometries that could be reproduced in in vitro experiments.
Matthias Birkner [slides]
A conditional coalescent limit in fixed pedigrees
Kingman's coalescent, introduced in 1982, is a widely-used model for the genealogies of samples in mathematical genetics; it is usually derived by averaging out the demographic stochasticity in a population model. Wakeley, King, Bobbi and Ramachandran (2012) argued that the way random models of reproduction are used in coalescent theory is not justified for diploid biparental organisms, as the population pedigree for diploid organisms - that is, the set of all family relationships among members of the population - although unknown, should be treated as a fixed parameter, not as a random quantity. However, using simulations they observed that some distributional properties of samples often resemble those under Kingman's coalescent even if the population pedigree is kept fixed and randomness comes only from the Mendelian mechanism of inheritance. We prove that the conditional distribution of the gene genealogy of a finite sample given the pedigree converges for a relatively general class of diploid biparental population models to the law of Kingman's coalescent as the population size tends to infinity and in this way corroborate Wakeley et al's observations. This is based on joint work with Andrey Tyukin.
Luis Bonilla [slides]
Tumor induced angiogenesis
Angiogenesis is a multiscale process by which blood vessels grow from existing ones and carry oxygen to distant organs. Angiogenesis is essential for normal organ growth and wounded tissue repair but it may also be induced by tumors to amplify their own growth. Mathematical and computational models contribute to understanding angiogenesis and developing anti-angiogenic drugs, but most work only involves numerical simulations and analysis has lagged. A recent stochastic model of tumor induced angiogenesis including branching, elongation, and anastomosis (fusion) of blood vessels captures some of its intrinsic multiscale structures, yet allows one to extract a deterministic integropartial differential description of the vessel tip density . Vessel tips proliferate due to branching, elongate following Langevin dynamics and, when they meet other vessels, join them by anastomosis and stop moving. Stalk endothelial cells follow the tip cells, so that the trajectories thereof constitute the advancing blood vessel. Anastomosis keeps the number of actively moving vessel tips relatively small, so that we cannot use the law of large numbers to derive equations for their density. Nevertheless, we show that ensemble averages over many replicas of the stochastic process correspond to the solution of the deterministic equations with appropriate boundary conditions . Using asymptotic and numerical methods, we find that the density of active vessel tips advances towards the tumor as a stable soliton-like wave whose shape and velocity change slowly according to some differential equations. Analyzing these equations paves the way for controlling angiogenesis through the soliton, the engine that drives this process [3, 4]. This is joint work with M. Carretero and F. Terragni.
 L.L. Bonilla, V. Capasso, M. Alvaro and M. Carretero, Hybrid modeling of tumor-induced angiogenesis, Phys. Rev. E 90, 062716 (2014)
 F. Terragni, M. Carretero, V. Capasso and L.L. Bonilla, Stochastic Model of Tumor-induced Angiogenesis : Ensemble Averages and Deterministic Equations, Phys. Rev. E 93, 022413 (2016)
 L.L. Bonilla, M. Carretero, F. Terragni and B. Birnir, Soliton driven angiogenesis, Sci.Rep. 6, 31296 (2016)
 L.L.Bonilla, M.Carretero and F.Terragni, Soliton like attractor for blood vessel tip density in angiogenesis. Phys.Rev.E 94, 062415 (2016)
The Fisher-KPP equation with noise, and branching processes
The Fisher-KPP equation is a deterministic equation with diffusion, growth and saturation. It can be used to model the spread of an infection or the growth of a population in the limit of very large densities. However, for populations of a finite size, one needs to add a noise term. I will present an overview of my work on the noisy Fisher-KPP equation and on its relation to branching processes where particles diffuse and duplicate randomly.
Leonid Bunimovich [slides]
Cross-immunoreactivity generates local immunodeficiency
Cross-immunoreactivity is a well known phenomenon which mean that antibodies induced by presence of some virus (antigen) can also neutralize some other viruses. Cross-immunoreactivity networks were always modeled as ("mean-field") complete graphs. It was experimentally found in CDC that for the Hepatitis C this network has a complex (scale free) structure (topology). Based on this fact a model of the Hepatitis C dynamics/evolution was built which explained clinical observations that were not explained before. A major general finding is the phenomenon of local immunodeficiency, i.e. some viruses remain hidden from human immune system being protected by other viruses thanks to their positions in the cross-immunoreactivity network.
Velocity-jump processes: large deviations and acceleration of fronts in transport-reaction equations
I will present WKB asymptotics for a simple velocity-jump process involving free transport and reorientation at a constant rate following a Gaussian velocity distribution. I will highlight the particular scaling of large deviations, as opposed to the one for the heat equation obtained in the diffusive limit. I will derive the corresponding nonlocal Hamilton-Jacobi. I will present applications of this work to the accelaration of transport-reaction fronts, as opposed to constant speed of propagation of reaction-diffusion traveling waves. This is joint work with Emeric Bouin (CEREMADE, Dauphine), Emmanuel Grenier (UMPA, ENSL) and Grégoire Nadin (LJLL, UPMC).
Francesca Collet [slides]
Rhythmic collective behavior in mean-field systems
An important problem in complex system is to understand how many interacting components organize to produce a coherent behavior at a macroscopic level. Basic examples include polarization (e.g. spin alignments in Ising-like models) and synchronization (e.g. phase locking in interacting rotators). We will consider a different, and less understood, phenomenon of self-organization: the emergence of periodic behavior in systems whose units have no tendency to evolve periodically. We will discuss how the interplay between interaction, noise and a reversibility breaking mechanism can be responsible for self-sustained oscillations. In particular, we will highlight the role of dissipation and interaction network topology in this respect. Our attention will be on mean-field stochastic models. This talk is based on joint works with Paolo Dai Pra, Marco Formentin and Daniele Tovazzi.
Bastien Fernandez [slides]
Landau damping in the Kuramoto model
The phenomenology of the Kuramoto model (continuum limit) strongly relies on the nonlinear stability of its stationary states. To understand and to rigorously assert stability in this infinite-dimensional setting have been long-standing challenges that show similar features of the Landau damping in the Vlasov equation. In this talk, I will present our results on stability conditions and asymptotic stability of various stationary states, that mathematically confirm the intuited phenomenology and its dependence on parameters. Joint works with H. Dieter, D. Gérard-Varet and G. Giacomin.
Amic Frouvelle [slides]
Nonlinear stability of aligned states for a alignment process on the sphere
We consider a model of alignment of unit vectors (the spatial homogeneous version of a model of interacting self-propelled particles who try to align their velocity, which is constrained to be of unit norm). We consider the kinetic equation satisfied by the limit law in the case of a large number of particles. When an angular noise is present we have convergence, at low noise intensity, towards a von Mises distribution at an exponential rate. In the noiseless situation, we are able to show convergence towards a Dirac mass when the initial condition is a measure without atom, but we lose the exponential rate.
Inverse Stochastic Resonance with an example in Purkinje Neurons
Inverse Stochastic Resonance is a noise induced phenomenon where persistent repetitive firing is quenched by noise with a tuned strength. It is linked to bistability of fixed points and limit cycles. In neuronal models this is most generically connected to type II excitability where sustained oscillations appear through a subcritical Andronov-Hopf bifurcation. In this talk I will describe the basic phenomenon and dynamics of systems likely to show it. I will also give a concrete experimental example in cerebellar Purkinje neurons.
The balanced state: the standard model and beyond
Strong temporal irregularity and right-skewed, long-tailed distributions of firing rates are distinctive features of cortical spiking activity. Both features are quite puzzling upon consideration of the large number of synaptic inputs a cortical cell receives and the weak correlations among these inputs. A minimal theoretical framework accounting naturally for these features – the balance hypothesis – was proposed by van Vreeswijk and Sompolinsky in two seminal papers at the end of the 1990s. In fact, we showed recently that log-normal distributions of mean firing rates as reported in cortex in-vivo, emerge naturally in the "balanced regime". I will summarize the general phenomenology of the balanced regime as well as its functional consequences for the selectivity of neuronal responses in primary visual cortex. In the second part of my talk I will address two limitations of the balance hypothesis in its “standard” formulation: 1) it precludes multi-stability and thus the persistence of neurons activity (e.g., delay activity in working memory tasks) 2) it does not account for temporal fluctuations observed in the neuronal activity on time scales on the order of 100 ms-1s as observed in cortex. I will describe possible solutions to these limitations.
Inequalities for critical exponents in sandpiles
I will discuss results about critical exponents in the Abelian sandpile model. The first part of the talk will provide general background on sandpiles. In the second part I will present joint work with J. Hanson and S. Bhupatiraju where we derive rigorous inequalities for three exponents: the toppling probability, the radius and the avalanche size.
Quenching and Reviving Oscillations in Complex Networks
Coupled oscillators are shown to experience two structurally different oscillation quenching types: amplitude death (AD) and oscillation death (OD). We demonstrate that both AD and OD can occur in one system and find that the transition between them underlies a classical, Turing-type bifurcation, providing a clear classification of these significantly different dynamical regimes.
In certain circumstances, harmful oscillations are undesirable which should be suppressed. However, oscillatory behavior is an essential determinant for proper functioning of various physical and biological processes in a wide variety of natural systems. Here we propose a rather simple and generic scheme to revoke both AD and OD in coupled dynamical networks of nonlinear oscillators. Specifically, by introducing a simple feedback factor in the diffusive coupling, we show that it can destabilize stable (in)homogeneous steady states to restore dynamic behaviors of coupled systems.
By introducing a processing delay in the coupling, we find that it can effectively annihilate the quenching of oscillation, amplitude death (AD), in a network of coupled oscillators by switching the stability of AD. It revives the oscillation in the AD regime to retain sustained rhythmic functioning of the networks, which is in sharp contrast to the propagation delay with the tendency to induce AD. This processing delay-induced phenomenon occurs both with and without the propagation delay. We demonstrate the approach in experiments with an oscillatory chemical reaction.
 W. Zou, D.V. Senthilkumar, M. Zhan, and J. Kurths, Phys. Rev. Lett. 111, 014101 (2013)
 A. Koseska, E. Volkov, and J. Kurths, Phys. Rev. Lett. 111, 024103 (2013)
 W. Zou et al., Nature Commun. 6, 7709 (2015)
 R. Nagao et al., CHAOS 26, 094808 (2016)
Quantitative convergence towards a self similar profile in an age-structured renewal equation for subdiffusion
Continuous-time random walks are generalisations of random walks frequently used to account for the consistent observations that many molecules in living cells undergo anomalous diffusion, i.e. subdiffusion. Here, we describe the subdiffusive continuous-time random walk using age-structured partial differential equations with age renewal upon each walker jump, where the age of a walker is the time elapsed since its last jump. In the spatially-homogeneous (zero-dimensional) case, we follow the evolution in time of the age distribution. An approach inspired by relative entropy techniques allows us to obtain quantitative explicit rates for the convergence of the age distribution to a self-similar profile, which corresponds to convergence to a stationnary profile for the rescaled variables. An important difficulty arises from the fact that the equation in self-similar variables is not autonomous and we do not have a specific analyitcal solution. Therefore, in order to quantify the latter convergence, we estimate attraction to a time-dependent "pseudo-equilibrium", which in turn converges to the stationnary profile.
Chemomechanical machines and the motion of pseudopods
The notion of pseudopods can be (partially) explained by in vitro experiments which have only three elements, «receptor» coated plastic balls, actin filaments and ATP. We propose a simplified chemomechanical model which can be seen as a competitor to the Brownian ratchet type models. This is work in collaboration with S. Heinze and A. Schlichting.
Some PDE models in neurosciences
We are going to present several PDE models that describe the evolution of a network of neurons that interact via their common statistical distribution. In these different models we will focus above all on qualitative and asymptotique properties of solutions describing synchronization phenomena. Approach and technical difficulties differ from model to model: we will also discuss this aspect. This talk is based on collaborations with J. A. Carrillo, K. Pakdaman, B. Perthame and D. Smets.
A stochastic model for the evolution of a virus population
We introduce a discrete time model for a virus-like evolving population with high mutation probability. Different genomes correspond to different points (or sites) in the interval [0,1]. Each site has one or more individual on it (corresponding to the number of individuals with that genome). When a birth with mutation occurs a new site is selected uniformly in [0,1] and we put one individual on it. When a birth is without mutation we select one existing site at random and increase its population (or size) by 1. When a death occurs we kill the smallest site and all its population. We prove that the number of individuals per site converges to a geometric distribution whose parameter can be computed exactly, and that the size of the population at a site is independent of the location of the site.
Phase equilibria of phase-separating self-propelled particles
When synthetic self-propelled particles or bacteria slow down at high density, either because of collisions or due to biochemical interactions, they may undergo a liquid-gas phase separation. This leads to the emergence of cohesive matter without the need of cohesive forces. For the past ten years, various theories have (unsuccessfully) been put forward to account for the corresponding phase equilibria. I will show how the equilibrium concepts of pressure and chemical potential can be generalized in this non-equilibrium setting to determine the phase equilibria of systems undergoing the so-called Motility-Induced Phase Separation.
Spatiotemporal Self-Organization of Fluctuating Bacterial Colonies
Because they are not bound by the standard laws of equilibrium thermodynamics, active materials such as bird flocks, motile bacteria, self-organizing bio-polymers, or man-made self-propelled particles have many more routes towards self-assembly and self-organization than systems whose dynamics satisfy detailed-balance. While much remains to be done to build a statistical mechanics theory of these non-equilibrium systems, some generic principles governing their behavior have started to emerge. Motility-induced phase separation (MIPS) is one example. MIPS arises naturally in systems of self-propelled particles whose locomotive speed decreases strongly at high density, through a feedback in which particles accumulate where they move slowly and vice-versa. In this talk, we build on these results and model an enclosed system of bacteria, whose MIPS is coupled to slow population dynamics. Without noise, the system shows both static phase separation and a limit cycle, in which a rising global population causes a dense bacterial colony to form, which then declines by local cell death, before dispersing to re-initiate the cycle. Adding fluctuations, we find that static colonies are now metastable, moving between spatial locations via rare and strongly nonequilibrium pathways, whereas the limit cycle becomes quasi-periodic such that after each redispersion event the next colony forms in a random location. These results, which resemble some aspects of the biofilm-planktonic life cycle, can be explained by combining tools from large deviation theory with a bifurcation analysis in which the global population density plays the role of control parameter. The analytical and numerical tools presented should also be useful in other non-equilibrium systems in which noise-driven self-organization occurs. See the reference arXiv 1703.06923.
Public lecture by S. Douady (in french), Wednesday May 17th [video]
Les plantes font-elles des Mathématiques (bien avant nous) ?
Les plantes ont à première vue des formes trop variées et complexes. Mais justement, elles peuvent présenter des formes fractales parfaites, bien plus mathématiquement pures (et simples) que les cours de la bourse ou les côtes de l'Esterel. Et cela peut s'expliquer par leur algoritmhe même de croissance.
Elles s'amusent aussi à présenter des arrangements de feuilles ou fleurs, avec des nombres de spirales qui sont exactement des nombres de Fibonacci. Comment l'expliquer ? Encore une fois, en regardant comment les plantes poussent, on voit qu'elles sont contraintes de suivre des règles d'additions qui mènent directement à Fibonacci.
Enfin, la forme des feuilles a l'air irrégulière et très variable. Pourtant, elles peuvent aussi souvent être seulement construites avec des règles géométriques simples comme les ribambelles de papier.
Bref ces exemples nous montrent qu'au lieu de regarder les plantes commes des compagnions bien immobiles et décoratifs (ou nourrissants), nous pourrions les regarder un peu mieux et nous en inspirer.