Tuesday, May 14, 2013 - 3:00pm - 5:00pm
- Chernoff-based Hybrid Tau-Leap
Alvaro Moraes (King Abdullah University of Science & Technology)
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.
- Modeling Sperm Trajectories Accounting for Signaling Noise
Sarah Olson (Worcester Polytechnic Institute)
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.
- Why People Walk in Circles?
Katarina Bodova (Comenius University in Bratislava)
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.
- Diffusive Stability of Turing Patterns via Normal Forms
Qiliang Wu (University of Minnesota, Twin Cities)
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.
- Influence of Environmental Factors and Intervention Programs
on Control of College Alcohol Drinking Patterns
Ridouan Bani (Northeastern Illinois University)
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.
- Evolution Meets Thermodynamics: Fitness and Entropy Production in a Cell Population Dynamics with Epigenetic Phenotype Switching
Hong Qian (University of Washington)
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 defined 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.
- Tracking Intracellular Rapid movements: a Bayesian Random Set Approach
Vasileios Maroulas (University of Tennessee)
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.
- Periodic Oscillations in a Genetic Circuit with Delayed Negative Feedback
David Lipshutz (University of California)
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.
- The Sub-Threshold Oscellation is Stationary of Subthreshold Dynamics in Stochastic Morris Lecar Model and in an Entorhinal Cortex Stellate Cell.
Priscilla Greenwood (University of British Columbia)
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!
- Capturing Effective Neuronal Dynamics in Random Networks with Complex Topologies
Duane Nykamp (University of Minnesota, Twin Cities)
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
- The Impact of Short Term Depression and Stochastic Vesicle Dynamics on Neural Variability, Information Transmission and Correlations
Robert Rosenbaum (University of Pittsburgh)
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.
- Near Critical Catalyst Reactant Branching Processes with Controlled
Dominik Reinhold (Clark University)
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.
- Analysis of the Stochastic Shielding Approximation for Markovian Ion Channel Models via Random Graphs
Peter Thomas (Case Western Reserve University)
Schmandt and Galán  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
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].
 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.
 Knowles, A. and Yin, J. Eigenvector Distribution of Wigner Matrices.
Probability Theory and Related Fields 155:543-582, 2013.
 Terence Tao and Van Vu. Random matrices: Universal properties of
eigenvectors. Random Matrices: Theroy and Applications, 1(1):1150001,
Supported by NSF grant EF-1038677.
Joint work with Deena R. Schmidt
- Stochastic and Multiscale Modelling in Molecular, Cell and Population Biology
Radek Erban (University of Oxford)
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
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
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.
- Filtering and Recurrent Connectivity Shape Higher-Order Correlations in Retinal Circuits
Andrea Barreiro (Southern Methodist University)
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.
- Adaptive Dynamics: Some Basic Theory and an Application
Johan Metz (Rijksuniversiteit te Leiden)
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.
- Non-Gaussian Effects in a Tumor Growth Model with Immunization
Jinqiao Duan (Institute for Pure and Applied Mathematics (IPAM))
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