Well posedness and Maximum Entropy Approximation for the Dynamics of Quantitative Traits

We study the Fokker-Planck equation derived in the large system limit of the Markovian process describing the dynamics of quantitative traits. The Fokker-Planck equation is posed on a bounded domain and its transport and diffusion coefficients vanish on the domain's boundary. We first argue that, despite this degeneracy, the standard no-flux boundary condition is valid. We derive the weak formulation of the problem and prove the existence and uniqueness of its solutions by constructing the corresponding contraction semigroup on a suitable function space. Then, we prove that for the parameter regime with high enough mutation rate the problem exhibits a positive spectral gap, which implies exponential convergence to equilibrium. Next, we provide a simple derivation of the so-called Dynamic Maximum Entropy (DynMaxEnt) method for approximation of moments of the Fokker-Planck solution, which can be interpreted as a nonlinear Galerkin approximation. The limited applicability of the DynMaxEnt method inspires us to introduce its modified version that is valid for the whole range of admissible parameters. Finally, we present several numerical experiments to demonstrate the performance of both the original and modified DynMaxEnt methods. We observe that in the parameter regimes where both methods are valid, the modified one exhibits slightly better approximation properties compared to the original one.Comment: 28 pages, 4 tables, 5 figure

Probabilistic models of individual and collective animal behavior

Recent developments in automated tracking allow uninterrupted, high-resolution recording of animal trajectories, sometimes coupled with the identification of stereotyped changes of body pose or other behaviors of interest. Analysis and interpretation of such data represents a challenge: the timing of animal behaviors may be stochastic and modulated by kinematic variables, by the interaction with the environment or with the conspecifics within the animal group, and dependent on internal cognitive or behavioral state of the individual. Existing models for collective motion typically fail to incorporate the discrete, stochastic, and internal-state-dependent aspects of behavior, while models focusing on individual animal behavior typically ignore the spatial aspects of the problem. Here we propose a probabilistic modeling framework to address this gap. Each animal can switch stochastically between different behavioral states, with each state resulting in a possibly different law of motion through space. Switching rates for behavioral transitions can depend in a very general way, which we seek to identify from data, on the effects of the environment as well as the interaction between the animals. We represent the switching dynamics as a Generalized Linear Model and show that: (i) forward simulation of multiple interacting animals is possible using a variant of the Gillespie's Stochastic Simulation Algorithm; (ii) formulated properly, the maximum likelihood inference of switching rate functions is tractably solvable by gradient descent; (iii) model selection can be used to identify factors that modulate behavioral state switching and to appropriately adjust model complexity to data. To illustrate our framework, we apply it to two synthetic models of animal motion and to real zebrafish tracking data.Comment: 26 pages, 11 figure

Topics in Applied Stochastic Dynamics.

Randomness in natural systems come from various sources, for example from the discrete nature of the underlying dynamical process when viewed on a small scale. In this thesis we study the effect of stochasticity on the dynamics in three applications, each with different sources and effects of randomness. In the first application we study the Hodgkin-Huxley model of the neuron with a random ion channel mechanism via numerical simulation. Randomness affects the nonlinear mechanism of a neuron’s firing behavior by spike induction as well as by spike suppression. The sensitivity to different types of channel noise is explored and robustness of the dynamical properties is studied using two distinct stochastic models. In the second application we compare and contrast the effectiveness of mixing of a passive scalar by stirring using different notions of mixing efficiency. We explore the non-commutativity of the limits of large Peclet numbers and large spatial scale separation between the flow and sources and sinks, and propose and examine a conceptual approach that captures some compat- ible features of the different models and measures of mixing. In the last application we design a stochastic dynamical system that mimics the properties of so-called ho- mogeneous Rayleigh-Benard convection and show that arbitrary small noise changes the dynamical properties of the model. The system’s properties are further exam- ined using the first exit time problem. The three applications show that randomness of small magnitude may play important and counterintuitive roles in determinig a system’s properties.Ph.D.Applied and Interdisciplinary MathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/64775/1/kbodova_1.pd

Statistical properties of noise-induced firing and quiescence in a Hodgkin-Huxley model

http://deepblue.lib.umich.edu/bitstream/2027.42/112533/1/12868_2009_Article_1225.pd

Resolution

Advection-Diffusion Equation for a Passive Scala

Pseudorapidity densities of charged particles with transverse momentum thresholds in pp collisions at √ s = 5.02 and 13 TeV

The pseudorapidity density of charged particles with minimum transverse momentum (pT) thresholds of 0.15, 0.5, 1, and 2 GeV/c is measured in pp collisions at the center of mass energies of √s=5.02 and 13 TeV with the ALICE detector. The study is carried out for inelastic collisions with at least one primary charged particle having a pseudorapidity (η) within 0.8pT larger than the corresponding threshold. In addition, measurements without pT-thresholds are performed for inelastic and nonsingle-diffractive events as well as for inelastic events with at least one charged particle having |η|2GeV/c), highlighting the importance of such measurements for tuning event generators. The new measurements agree within uncertainties with results from the ATLAS and CMS experiments obtained at √s=13TeV.