The simplest stochastic models of biochemical processes treat the system as a continuous time Markov chain with the state being the number of molecules of each species and with reactions modeled as possible transitions of the chain. I will develop the relevant mathematical representations for the processes and then show how different computational methods can be developed and analyzed by utilizing the stochastic representations. Topics discussed will be a subset of: model development, approximation techniques, and variance reduction for Monte Carlo methods.

Alcohol abuse is a major problem, especially among students on and around college campuses. We use the mathematical framework of Mubayi et al. (2013) and study the role of environmental factors in the dynamics of an alcohol drinking population. Sensitivity and uncertainty analyses are carried out on the relevant functions (for example, on the drinking reproduction number and the extinction time of moderate and heavy drinking in the presence of intervention) to understand the impact of interventions on the distributions of drinkers. The reproduction number helps determine whether or not the high-risk alcohol drinking behavior will spread and becomes persistent in the population whereas extinction time of high-risk drinking measures the effectiveness of control programs. We found that the reproduction number is most sensitive to social interactions, whereas the time to extinction of high-risk drinkers is significantly sensitive to the rate of prevention programs and the graduation rate. The results also suggest that in a population, higher effectiveness of intervention programs in low-risk environments, more than in high-risk environments, is needed to reduce heavy drinking in the population.

Pairwise maximum entropy distributions provide a good approximation of spike patterns produced in the retina under certain conditions but show significant departures under other conditions and in other neural circuits. We perform systematic modeling of retinal microcircuits to understand the mechanisms underlying the generation of higher-order correlations. By modulating filtering properties and recurrent connectivity, we find conditions in which, other factors being equal, the circuit generates higher-order interactions (HOIs). The most prominent is when stimulus statistics and stimulus filtering combine to produce bimodal current inputs to the cell. These findings offer a mechanism for experimental results in parasol retinal ganglion cells.

In several mammalian species, the durations of sleep episodes are exponentially distributed, while the duration of wake episodes is sub-exponentially distributed. Motivated by the experimental finding that mutually-inhibitory neuronal networks regulate switching between the behavioral states of sleep and wake, we have constructed a model in which to explore possible mechanisms by which the observed statistical properties of state durations may emerge from the neuronal interactions. The model consists of two excitatory neuronal networks coupled by inhibitory inter-connections; each network is a random graph, and each node fires at state-dependent Poisson rates. The full model alternates between states in which one network is highly active while the other exhibits diminished activity; these states can be regarded as "sleep" or "wake" depending upon which network is more active. The dynamics of the model will be discussed in the context of other recent stochastic neuronal network models.

People use their senses, most dominantly sight and hearing, to interpret geographical cues in order to navigate to a target location. How does the process work if no cues are present and the only information is the initial direction towards the target location? Experiments suggest that people are not capable of walking straight and form surprisingly small looped trajectories. The analysis of experimental data infers a parametric family of stochastic difference models with individual-based set of parameters reflecting the directional bias and other properties of individual's motion.

This is joint work with Michal Janosi.

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.

We will consider stochastic dynamical systems defined on networks that exhibit the phenomenon of collective metastability---by this we mean network dynamics where none of the individual nodes' dynamics are metastable, but the configuration is metastable in its collective behavior. In particular, we show how many biological oscillators --- both "pulse-coupled" and "phase-coupled" --- fall into this framework. We also examine a non-standard spectral problem that generically appears in these problems.

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.

Synaptic transmission is a central component of neural processing. Short term depression occurs when repeated driving of a synapse reduces its efficacy, a feature that is common across the nervous system. The mechanics of synaptic discharge are well characterized and involve both probabilistic release and uptake of neurotransmitter during activity. However, many studies which consider the impact of depression on information flow use a deterministic model of depression, based on trial averaged response. Using techniques from stochastic calculus we derive simplified expressions for the information flow across a synapse which account for fluctuations in neurotransmitter release. We show that the timescales of the stochastic synaptic dynamics impart high pass information filtering that is absent in deterministic models. We expand our work to consider how synaptic depression affects the flow of correlated activity in networks of neurons. This is joint work with Robert Rosenbaum and Jonathan Rubin.

The dynamical evolution of a tumor growth model, under immune surveillance and subject to asymmetric non-Gaussian alpha-stable Levy noise, is explored. The lifetime of a tumor staying in the range between the tumor-free state and the stable tumor state, and the likelihood of noise-inducing tumor extinction, are characterized by the mean exit time (also called mean residence time) and the escape probability, respectively. For various initial densities of tumor cells, the mean exit time and the escape probability are computed with different noise parameters. It is found that unlike the Gaussian noise and symmetric non-Gaussian noise, the asymmetric non-Gaussian noise plays a constructive role in the tumor evolution in this simple model. By adjusting the noise parameters, the mean exit time can be shortened and the escape probability can be increased, simultaneously. This suggests that a tumor may be mitigated with higher probability in a shorter time, under certain external environmental stimuli.

Polarity establishment, the formation of a front and back, is fundamental to many cellular processes. Two examples where cell polarization plays a central role are during migration and the budding process of Saccharomyces cerevisiae (yeast). Key signaling proteins involved polarity establishment are the Rho GTPases, which act as molecular switches. We combine mathematical modeling with experimental studies to investigate the role of Rho GTPases in both budding and cell migration. During the budding process, the initial clustering of the GTPase Cdc42 is shown to be oscillatory, revealing the presence of a negative feedback loop that disperses polarity factors. We use mathematical modeling to demonstrate that negative feedback confers robustness to the polarity circuit and makes the kinetics of competition between polarity factor clusters relatively insensitive to polarity factor concentration. These predictions are confirmed experimentally. Next we combine stochastic modeling with biosenors for monitoring the spatiotemporal dynamics of Rho GTPase activity to investigate the role of RhoG in cell migration. Our analysis reveals that cells can exist in two distinct states of migration characterized by differences in speed and persistence and that RhoG plays a role in cell’s ability reorient their direction of motion.

This is joint work with:

Audrey Howell, Meng Jin, Chi-Fang Wu, Daniel Lew (yeast polarity establishment)

Richard Allen, Chris Welch, Klaus Hahn (cell migration)

Methods for spatio-temporal modelling in molecular, cellular and population biology will be presented. Application areas include intracellular calcium dynamics, actin dynamics, gene regulatory networks, and collective behaviour of cells and animals.

Three classes of models will be considered: (1) microscopic (molecular-based, individual-based) models which are based on the simulation of trajectories of individual molecules (or individuals) and their localized interactions (for example, reactions); (2) mesoscopic (lattice-based) models which divide the computational domain into a finite number of compartments and simulate the time evolution of the numbers of molecules in each compartment; and (3) macroscopic (deterministic) models which are written in terms of reaction-diffusion-advection PDEs for spatially varying concentrations.

I will discuss connections between the modelling frameworks (1)-(3). I will consider chemical reactions both at a surface and in the bulk. I will also present and analyse hybrid (multiscale) algorithms which use models with a different level of detail in different parts of the computational domain. The main goal of this multiscale methodology is to use a detailed modelling approach in localized regions of particular interest (in which accuracy and microscopic detail is important) and a less detailed model in other regions in which accuracy may be traded for simulation efficiency. I will also discuss hybrid modelling of chemotaxis where an individual-based model of cells is coupled with PDEs for extracellular chemical signals.

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.

We can understand much about a simple neuron firing pattern from a phase plane plot of our simulations of the stochastic Morris Lecar model, where firings correspond to circuits around the stable limit cycle and sub-threshold oscillations correspond to circuits around the fixed point inside the unstable limit cycle.

The phase-plane plot suggests the question: How many sub-threshold circuits of the fixed point occur before the sub-threshold stochastic process is in its stationary distribution? In various ways our figures show the answer to be that only one or two circuits are necessary!

Story 1:

The construction of flagellar motors in motile bacteria such as Salmonella is a carefully regulated genetic process. Among the structures that are built are the basal body, the hook and the filament. Each of these is made only after the previous one has been completed. Furthermore the length of the hook is tightly regulated.

The question that will be addressed in this story is how Salmonella detects and regulates the length of its hooks. The model for hook length regulation is based on the hypothesis that the hook length is determined by the rate of secretion of the length regulatory molecule FliK and a cleavage reaction with the gatekeeper molecule FlhB. A stochastic model for this interaction is built and analyzed, showing excellent agreement with hook length data. We will also describe how mathematical modeling led to predictions that were subsequently tested experimentally, an example of a Math Biology success story.

Story 2: To fire or not to fire:

The firing of an action potential by a nerve occurs when the input stimulus exceeds a threshold. However, since ion channels are discrete and they open and close randomly, ion channel noise can cause an action potential to fire even when the stimulus threshold is not exceeded. This short story will explore the mechanism underlying these spontaneous events. Specifically, we will show that the classical fast-slow analysis of deterministic models can be misleading in understanding the stochastic process.

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 a wider variety of spatially structured 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. In addition, considering long-range connections between multi-layered networks. Each layer supports a stationary bump solution due to its recurrent architecture. Connections between layers reenforce the initial position of those bumps in the presence of noise. Perturbative analysis of the effect of noise on the bumps allows one to derive a multidimensional mean reverting stochastic process for the position of the bump in each layer. 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.

Intracellular transport is driven primarily by molecular motor proteins, such as kinesin and dynein, which convert chemical energy (stored in ATP) to directed motion along microtubules. Mathematical models of this process typically involve stochastic elements to describe the progress of binding and other chemical steps in the molecular motor stepping cycle, as well as for the interaction of the spatial dynamics with thermal fluctuations in the environment. Experiments and models have been pursued for the last 20 years to explore the functioning and characteristics of single molecular motors attached to a cargo load. Attention has turned more recently to understand how intracellular transport operates in the presence of multiple motors of various types, which is evidently relevant in vivo. Of particular interest is understanding mechanistically at a systems level how the various molecular motors, microtubule architecture, and regulatory factors interact to achieve the functional goals of intracellular transport. Achieving this goal appears to require the integration of experimental data and theoretical models over at least three scales of description, and the development of quantitative relationships between in vitro and in vivo experimental results. I will sketch some of the emerging conceptual challenges and opportunities in the science of intracellular transport, and several ways in which stochastic models and methods have been employed by collaborators and other groups to contribute to our understanding of this complex system. My recent and ongoing research in this area is pursued jointly with John Fricks, Scott McKinley, Will Hancock, and Avanti Athreya.

Dynamical system models with delayed feedback, state constraints and small noise arise in a variety of applications in biology. Under certain conditions oscillatory behavior has been observed. Here we consider a prototypical fluid model approximation for such a system --- a one-dimensional delay differential equation with reflection. We provide sufficient conditions for the existence, stability and uniqueness of slowly oscillating periodic solutions of such equations. We illustrate our findings with a simple genetic circuit model.

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.

We derive effective equations for the activity of recurrent spiking neuron models coupled via networks in which different motifs (patterns of connections) are overrepresented. We present an analysis of network dynamics that shed lights on how network structure influences mean behavior as well as leads to the initiation and propagation of variability and covariability. One key result is that network topology can increase the dimension of the effective dynamics. We demonstrate this behavior through simulations of spiking neuronal networks.

Joint work with Patrick Campbell and Michael Buice

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.

Invertebrate sperm navigate marine environments in order to reach and fertilize the egg via chemotaxis. The chemoattractant is an egg protein that binds to the flagellum and causes an increase in calcium within the flagellum. This increase in calcium then causes the flagellum to beat differently, changing the trajectory. As sperm navigate in a gradient of chemoattractant, they will have several periods of calcium influx and subsequent calcium efflux to return to a steady internal calcium concentration. This corresponds to changes in path curvature and looping to search for the egg. We highlight results from two different models to understand trajectories, changes in motility patterns, and the role of randomness in these models. The first model is a fluid structure interaction model accounting for a noisy calcium input to investigate trajectories. The second model is a random biased walk, not accounting for the fluid, to investigate strategies that lead the sperm to the egg in a given concentration of chemoattractant.

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.

We study a simple cell population dynamics in which subpopulations grow with different rates and individual cells can switch between different epigenetic phenotypes. The population dynamics and thermodynamics of Markov processes, separately defined two important concepts in mathematical terms: the fittness in the former and the (non-adiabatic) entropy production in the latter. Both appear in the cell population dynamics. The switching sustains the variations among the subpopulation growth thus continuous natural selection. As a form of Price's equation, the fitness increases with (i) natural selection, e.g., variations and (ii) positive covariance between the per capita growth and switching, which represents a Lamarchian-like behavior. A negative covariancebalances the natural selection in a steady state. The growth keeps the proportions of subpopulations away from their switching equilibrium, thus leads to a continous entropy production. Covariance between the per capita growth rate and the "free energy" of subpopulation, counteracts the entropy production.

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.

Near critical catalyst-reactant branching processes with controlled immigration are studied. The reactant population evolves according to a branching process whose branching rate is proportional to the total mass of the catalyst. The bulk catalyst evolution is that of a classical continuous time branching process; in addition there is a specific form of immigration. Immigration takes place exactly when the catalyst population falls below a certain threshold, in which case the population is instantaneously replenished to the threshold. Such models are motivated by problems in chemical kinetics where one wants to keep the level of a catalyst above a certain threshold in order to maintain a desired level of reaction activity. A diffusion limit theorem for the scaled processes is presented, in which the catalyst limit is described through a reflected diffusion, while the reactant limit is a diffusion with coefficients that are functions of both the reactant and the catalyst. Stochastic averaging principles under fast catalyst dynamics are established. In the case where the catalyst evolves ``much faster" than the reactant, a scaling limit, in which the reactant is described through a one dimensional SDE with coefficients depending on the invariant distribution of the reflected diffusion, is obtained. This is joint work with Amarjit Budhiraja, UNC-Chapel Hill.

The filtering properties of neural synapses are modulated by a form of short term depression arising from the depletion of neurotransmitter vesicles. The uptake and release of these vesicles is stochastic in nature, but a widely used model of synaptic depression does not take this stochasticity into account. While this deterministic model of synaptic depression accurately captures the trial-averaged synaptic response to a presynaptic spike train, it fails to capture variability introduced by stochastic vesicle dynamics. Our goal is to understand the impact of stochastic vesicle dynamics and short term depression on synaptic filtering, neural variability and neural correlations.

We derive compact, closed-form expressions for the synaptic filter induced by short term synaptic depression when stochastic vesicle dynamics are taken into account and when they are not. We find that stochasticity in vesicle uptake and release fundamentally alters the way in which a synapse filters presynaptic information. Predictably, the variability introduced by this stochasticity reduces the rate at which information is transmitted through a synapse and reduces correlations between two synaptic responses. Additionally, this variability introduces frequency-dependence to the transfer of information through a synapse: a model that ignores synaptic variability transmits slowly varying signals with the same fidelity as faster varying signals, but a model that takes this variability into account transmits faster varying signals with higher fidelity than slower signals. Differences between the models persist even when the presynaptic cell makes many contacts onto the postsynaptic cell. We extend our analysis to the population level and conclude that a slowly varying signal must be encoded by a large presynaptic population if it is to be reliably transmitted through depressing synapses, but faster varying signals can be reliably encoded by smaller populations. Our results provide useful analytical tools for understanding the filtering properties of depressing synapses and have important consequences for neural coding in the presence of short term synaptic depression.

Schmandt and Galán [1] introduced a stochastic shielding approximation as a fast, accurate method for generating sample paths from a finite state Markov process in which only a subset of states are observable. For example, in ion channel models, such as the Hodgkin-Huxley or other conductance based neural models, a nerve cell has a population of ion channels performing a random walk on a graph representing a finite set of states, only some of which allow a transmembrane current to pass. The stochastic shielding approximation consists of neglecting fluctuations associated with edges in the graph not directly affecting the observable states.

Here we consider the problem of finding the optimal complexity reducing mapping from a stochastic process on a graph to an approximate process on a smaller sample space, as determined by the choice of a particular linear measurement functional on the graph. The partitioning of ion channel states into conducting versus nonconducting states provides a case in point. In addition to establishing that Schmandt and Galán’s approximation is in fact optimal in a specific sense, we provide heuristic error estimates for the accuracy of the stochastic shielding approximation for an ensemble of Erdös-Rényi random graphs, using results from random matrix theory [2,3].

References:

[1] Nicolaus T. Schmandt and Roberto F. Galán. Stochastic-shielding approximation of markov chains and its application to efficiently simulate random ion-channel gating. Phys Rev Lett, 109(11):118101, 2012.

[2] Knowles, A. and Yin, J. Eigenvector Distribution of Wigner Matrices. Probability Theory and Related Fields 155:543-582, 2013.

[3] Terence Tao and Van Vu. Random matrices: Universal properties of eigenvectors. Random Matrices: Theroy and Applications, 1(1):1150001, 2012.

Acknowledgment:

Supported by NSF grant EF-1038677.

Joint work with Deena R. Schmidt

In deterministic dynamics, a stable limit cycle is a closed, isolated periodic orbit that attracts nearby trajectories. Points in its basin of attraction may be disambiguated by their asympototic phase. In stochastic systems with approximately periodic trajectories, asymptotic phase is no longer well defined, because all initial densities converge to the same stationary measure. We explore circumstances under which one may nevertheless define an analog of the "asymptotic phase". In particular, we consider jump Markov process models incorporating ion channel noise, and study a stochastic version of the classical Morris-Lecar system in this framework. We show that the stochastic asymptotic phase can be defined even for some systems in which no underlying deterministic limit cycle exists, such as an attracting heteroclinic cycle perturbed by additive noise. We also discuss an analysis of an efficient numerical approximation for simulating trajectories of randomly gated ion channels, called the stochastic shielding approximation, recently introduced by Schmandt and Galan (Physical Review Letters, 2012).

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.

We investigate dynamics near Turing patterns in reaction-diffusion systems posed on the real line. Linear analysis predicts diffusive decay of small perturbations. We construct a “normal form” coordinate system near such Turing patterns which exhibits an approximate discrete conservation law. The key ingredients to the normal form is a conjugation of the reaction-diffusion system on the real line to a lattice dynamical system. At each lattice site, we decompose perturbations into neutral phase shifts and normal decaying components. As an application of our normal form construction, we prove nonlinear stability of Turing patterns with respect to small localized perturbations, with sharp rates.

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.