Coupled stochastic model of M. xanthus swarming and experimental bacteria tracking will be used in this talk to demonstrate how the flexibility and adhesion between cells as well as cell reversals result in bacteria effectively colonizing surfaces. A connection will be described between a microscopic one-dimensional stochastic model of reversing non overlapping bacteria and a macroscopic nonlinear diffusion equation describing the dynamics of cellular density. Biologically-justified multi-scale model consisting of stochastic cell-based submodel representing individual bacteria coupled with the thin film Navier–Stokes equation will be used to suggest a mechanism of P. aeruginosa wave propagation as well as branched tendril formation at the edge of the population.

One of the major challenges in neuroscience is to determine how noise that is present at the molecular and cellular levels affects dynamics and information processing at the macroscopic level of synaptically coupled neuronal populations. Often noise is incorporated into deterministic network models using extrinsic noise sources. An alternative approach is to assume that noise arises intrinsically as a collective population effect, which has led to a master equation formulation of stochastic neural networks. In this talk we show how to extend the master equation formulation by introducing a stochastic model of neural population dynamics in the form of a velocity jump Markov process. The latter has the advantage of keeping track of synaptic processing as well as spiking activity, and reduces to the neural master equation in a particular limit. The population synaptic variables evolve according to piecewise deterministic dynamics, which depends on population spiking activity. The latter is characterised by a set of discrete stochastic variables evolving according to a jump Markov process, with transition rates that depend on the synaptic variables. We consider the particular problem of rare transitions between metastable states of a network operating in a bistable regime in the deterministic limit. Assuming that the synaptic dynamics is much slower than the transitions between discrete spiking states, we use a WKB approximation and singular perturbation theory to determine the mean first passage time to cross the separatrix between the two metastable states. Such an analysis can also be applied to other velocity jump Markov processes, including stochastic voltage-gated ion channels and stochastic gene networks.

The selection of dispersion is a classical problem in ecology and evolutionary biology. Deterministic dynamical models of two competing species differing only in their passive dispersal rates suggest that the lower mobility species has a competitive advantage in inhomogeneous environments, and that dispersion is a neutral characteristic in homogeneous environments. We consider models including local population fluctuations due to both individual movement and random birth and death events to investigate the effect of demographic stochasticity on the competition between species with different dispersal rates. For homogeneous environments where deterministic models predict degenerate dynamics in the sense that there are many (marginally) stable equilibria with the species' coexistence ratio depending only on initial data, demographic stochasticity breaks the degeneracy. A novel large carrying capacity asymptotic analysis, confirmed by direct numerical simulations, shows that a preference for faster dispersers emerges on a precisely defined time scale. While there is no evolutionarily stable rate for competitors to choose in these spatially homogeneous models, the stochastic selection mechanism is the essential counterbalance in spatially inhomogeneous models including demographic fluctuations which do display an evolutionarily stable dispersal rate. This is joint work with Yen Ting Lin and Hyejin Kim.

We first derive the equations for weak noise perturbations of exponentially stable limit cycles. With this perturbation theory, it become possible to compute quantities such as Liapunov exponents, di usion constants, and the effects of noise on frequency. We show that there are resonances between the frequency of the oscillations and the time scale of the noise. Next we apply this theory to synchronization of oscillators that receive partially correlated noise. We derive the invariant density and order parameters for the degree of synchronization. We show that some types of oscillators are better synchronizers than others. We conclude with some surprising eff ects of heterogeneity on the synchronization.

In this talk, I will present rigorous results on the dynamics of a piecewise affine system of pulse-coupled oscillators with global interaction, inspired by experiments on synchronization in colonies of bacteria-embedded genetic circuits. Due to global coupling, any cluster composed by a group of oscillators in sync is invariant in time. Hence the analysis essentially boils down to estimating possible asymptotic cluster distributions depending on the initial conditions. I will show that, as the coupling strength increases, the system exhibits a sharp transition between a regime of arbitrary asymptotic distributions, to a strongly clustered regime where every surviving distribution must contain a giant cluster. I will also report on manifestations of this phase transition in the dynamics of uniformly drawn random initial conditions. The most significant feature is that, when the coupling strength is sufficiently large, with positive probability, the number of asymptotic clusters remains bounded in the thermodynamic limit, while the maximum number linearly diverges.

Neural field models can describe spatially organized activity in large populations of neurons. These models are integrodifferential equations where the kernel of the integral term describes the strength of synaptic connections between neurons. Traveling waves (Pinto and Ermentrout 2001), stationary pulses (Amari 1977), and spiral waves (Laing 2005) have all been identified as solutions to various neural field models. These analyses often employ two assumptions -- (1) effects of noise are small enough to be ignored and (2) connections between neurons only depend upon the distance between them (spatially homogeneous). However, recent studies have shown noise can cause traveling front (Bressloff and Webber 2012) and stationary bump (Kilpatrick and Ermentrout 2013) solutions to wander purely diffusively about their mean position. These analyses employ a small noise expansion technique developed for fronts evolving in stochastic PDEs (Armero et al 1998). The pure diffusion of spatially structured solution relies on the translation symmetry of the system, which only occurs in neural fields if synaptic connections are spatially homogeneous.

We show that a variety of rich behaviors arise when one considers more realistic spatially heterogeneous synaptic connections. Namely, spatial heterogeneity establishes a multistable potential landscape in space that can then be traversed by solutions with the aid of noise. The dynamics then evolves as a particle kicked between multiple wells. Traveling wave solutions become pinned to stationary attractors in the deterministic system, but noise leads to rare events where the wave position moves between attractors, recovering wave propagation. Aperiodic spatial heterogeneity leads to a spatial bias in the rate of these transitions which must then be calculated for each well. Analytic and numerical results are discussed.

In this expository discussion, we explore the use of 'mass transport,' the classical Monge mass transfer problem and the contemporary development of its kinetic or evolutionary counterpart, to describe intracellular transport mechanisms. At its core, this theory offers a gradient flow for entropy or free energy and thus represents some way of understanding randomness in a given system whose elements undergo conformational change and dissipation. We also describe how qualitative conclusions may be drawn from the equations we find.

We focus on the biological problem of tracking organelles as they move through cells. In the past, intracellular movements were mostly recorded manually, however, the results are insufficient to capture the full complexity of organelle motions. An automated tracking algorithm promises to provide a complete analysis of noisy microscope’s data. In our work, we adopt statistical techniques from a Bayesian random set point of view. Instead of considering each individual organelle, we examine a random set whose members are the states of organelles and/or newborn organelles and we establish a Bayesian filtering algorithm involving such set-states. The propagated multi-object densities are approximated using a Gaussian mixture scheme. Our algorithm is then successfully implemented using synthetic and experimental data. This work is joint with Andreas Nebenfuhr.

The theory of structured populations is a mathematical framework for developing and analyzing ecological models that can take account of relatively realistic detail at the level of individual organisms. This framework in turn has given rise to the theory of adaptive dynamics, a versatile framework for dealing with the evolution of the adaptable traits of individuals through repeated mutant substitutions directed by ecologically driven selection. The step from the former to the latter theory is possible thanks to effective procedures for calculating the expected rate of invasion of mutants with altered trait values into a community the dynamics of which has relaxed to an attractor. The mathematical underpinning is through a sequence of limit theorems starting from individual-based stochastic processes and culminating in (i) a differential equation for long-term trait evolution and (ii) various geometrical tools for classifying the evolutionary singular points such as Evolutionarily Steady Strategies, where evolution gets trapped, and branching points, where an initially quasi-monomorphic population starts to diversify. Traits that have been studied using adaptive dynamics tools are i.a. the virulence of infectious diseases and various other sorts of life-history parameters such as age at maturation. As one example, adaptive dynamics models of respiratory diseases tell that such diseases will evolve towards the upper air passages and hence towards lesser virulence, while at the same time diversifying as a result of limited cross-immunity. Since the upper airways offer the largest scope for disease persistence, they also allow for the largest disease diversification. Moreover, the upward evolution brings with it a tendency for vacating the lower reaches, which leads to the prediction that emerging respiratory diseases will tend to act low and therefore be both unusually virulent and not overly infective.

Markov pure jump processes are used to model chemical reactions at molecular level, dynamics of wireless communication networks and the spread of epidemic diseases in small populations, among many other phenomena. There exist algorithms like the SSA by Gillespie or the Modified Next Reaction Method by Anderson, that simulates a single trajectory with the exact distribution of the process, but it can be time consuming when many reactions take place during a short time interval. The approximated Gillespie's tau-leap method, on the other hand, can be used to reduce computational time, but it may lead to non physical values due to a positive one-step exit probability, and it also introduces a time discretization error. This work presents a novel hybrid algorithm for simulating individual trajectories which adaptively switches between the SSA and the tau-leap method. The switching strategy is based on the comparison of the expected inter-arrival time of the SSA and an adaptive time step derived from a Chernoff-type bound for the one-step exit probability. Since this bound is non-asymptotic we do not need to make any distributional approximation for the tau-leap increments. This hybrid method allows: (i) to control the global exit probability of a simulated trajectory, (ii) to obtain accurate and computable estimates for the expected value of any smooth observable of the process with low computational work. We present numerical examples that illustrate the performance of the proposed method.

Kinetochores are nano-structures that mechanically couple chromosomes to dynamic microtubules to generate the forces necessary for proper chromosome segregation during mitosis. Recent studies reveal new details of the kinetochore’s molecular composition and structure, demonstrating the mechanically compliant nature of the kinetochore linkage to the microtubule. This finding stands in contrast to previous theoretical models of kinetochore motility (Hill, 1985; Molodtsov et al., 2005), which assumed an infinitely stiff linkage. Here, we present a compliant kinetochore-clutch model where an array of compliant linkers (“clutches”) interacts reversibly with a dynamic microtubule tip. We explore the behavior of a kinetochore-clutch system with various clutch parameters including: (1) clutch affinity for the MT-lattice, (2) clutch stiffness, (3) preference for tubulin intra-/interdimer association, (4) diffusion rate, and (5) tensional load force. We find that clutch stiffness is critical in governing the distribution of linkers over the MT tip, and thus in turn, modulating microtubule tip dynamics. Surprisingly, clutch affinity for the MT-lattice can be varied over a wide range with minimal effect on system behavior. We also find that clutch diffusion on the MT lattice is not critical for kinetochore coupling. Finally, we find that tensional load force shifts the distribution of linkers toward the MT tip to directly suppress net disassembly and prime MTs for rescue. Together, our theoretical studies predict a potentially important role for clutch mechanical compliance in kinetochore motility and control of microtubule assembly-disassembly.

Reaction and diffusion processes are used to model chemical and biological processes over a wide range of spatial and temporal scales. Several routes to the diffusion process at various levels of description in time and space are discussed and the master equation for spatially-discretized systems involving reaction and diffusion is developed. We discuss an estimator for the appropriate compartment size for simulating reaction-diffusion systems and introduce a measure of fluctuations in a discretized system. We then describe a new computational algorithm for implementing a modified Gillespie method for compartmental systems in which reactions are aggregated into equivalence classes and computational cells are searched via an optimized tree structure. Finally, we discuss several examples that illustrate the issues that have to be addressed in general systems.

Actin polymerization in vivo is regulated spatially and temporally by a web of signaling proteins. We developed detailed physico-chemical, stochastic models of lamellipodia and filopodia, which are projected by eukaryotic cells during cell migration, and contain dynamically remodeling actin meshes and bundles. In particular, we investigated how molecular motors regulate growth dynamics of elongated organelles of living cells. Our simulations show that some processes, such as binding and unbinding of capping proteins, may be dominated by rare events, where stochastic treatment of filament growth dynamics is obligatory. We also studied mechanical regulation of the growth dynamics of lamellipodia-like branched actin networks. In such networks, the treadmilling process leads to a concentration gradient of G-actin, thus G-actin transport is essential to effective actin network assembly. We shed light on how actin transport due to diffusion and facilitated transport such as advective flow and active transport, tunes the growth dynamics of the branched actin network. Our work demonstrates the role of molecular transport in determining the shapes of the commonly observed force-velocity curves.

Individual-based population dynamics articulates stochastic behavior of individuals and considers deterministic equations at the population level as an emergent phenomenon. Using chemical species inside a small aqueous volume (a cell) as an example, we introduce Delbrück-Gillespie birth-and-death process for chemical reactions dynamics.  Using this formalism, we (1) illustrate the relation between nonlinear saddle-node bifurcation and first- and second-order phase transition; (2) introduce a thermodynamic theory for entropy and entropy production and prove 1st and 2nd Laws-like theorems; and (3) show a completely consistency between dynamics and the newly developed thermodynamics. To physics: we discuss the fundamental issue of "what is dissipation" and its relation to time reversibility in subsystems. To biology: we suggest the inter-basin-of-attraction stochastic dynamics as a possible mechanism for epigenetic variations at the cellular level.

In this talk I will introduce the main mathematical questions arising in the modeling of large-scale neuronal networks involved at functional scales in the brain. Such networks are composed of multiple populations (different neuronal types), in which each neuron has a stochastic dynamics and operate in a random environment. Understanding the collective dynamics of such neuronal assemblies involves mathematical tools developed in statistical physics, and most cortical activity regimes are out-of-equilibrium, related to periodic or chaotic solutions in law. I will specifically present two recent works on the subject. The first one deals with mesoscopic limits of spatially extended neural fields and the resulting spatio-temporal pattern formation, in particular the presence of transitions from stationary to synchronized periodic activity induced by noise or heterogeneity, and the second one will analyze in depth a phase transition between stationary and dynamical chaotic activity in relationship with the topological complexity of the network.

I will report on recent work which proposes that the network dynamics of the mammalian visual cortex are neither homogeneous nor synchronous but highly structured and strongly shaped by temporally localized barrages of excitatory and inhibitory firing we call `multiple-firing events' (MFEs). Our proposal is based on careful study of a network of spiking neurons built to reflect the coarse physiology of a small patch of layer 2/3 of V1. When appropriately benchmarked this network is capable of reproducing the qualitative features of a range of phenomena observed in the real visual cortex, including orientation tuning, spontaneous background patterns, surround suppression and gamma-band oscillations. Detailed investigation into the relevant regimes reveals causal relationships among dynamical events driven by a strong competition between the excitatory and inhibitory populations. Testable predictions are proposed; challenges for mathematical neuroscience will also be discussed. This is joint work with Aaditya Rangan.

We investigate the formation and movement of self-organizing collectives of individuals in homogeneous environments. We review a hyperbolic system of conservation laws based on the assumption that the interactions governing movement depend not only on distance between individuals, but also on whether neighbours move towards or away from the reference individual. The inclusion of direction-dependent communication mechanisms significantly enriches the model behavior; the model exhibits classical patterns such as stationary pulses and traveling trains, but also novel patterns such as zigzag pulses, breathers, and feathers. The same enrichment of model behavior is observed when we include direction-dependent communication mechanisms in individual-based models.