M. Drew LaMar

Laser ablation of Dbx1 neurons in the pre-Bötzinger complex stops inspiratory rhythm and impairs output in neonatal mice

To understand the neural origins of rhythmic behavior one must characterize the central pattern generator circuit and quantify the population size needed to sustain functionality. Breathing-related interneurons of the brainstem pre-Bötzinger complex (preBötC) that putatively comprise the core respiratory rhythm generator in mammals are derived from Dbx1-expressing precursors. Here, we show that selective photonic destruction of Dbx1 preBötC neurons in neonatal mouse slices impairs respiratory rhythm but surprisingly also the magnitude of motor output; respiratory hypoglossal nerve discharge decreased and its frequency steadily diminished until rhythm stopped irreversibly after 85±20 (mean ± SEM) cellular ablations, which corresponds to ∼15% of the estimated population. These results demonstrate that a single canonical interneuron class generates respiratory rhythm and contributes in a premotor capacity, whereas these functions are normally attributed to discrete populations. We also establish quantitative cellular parameters that govern network viability, which may have ramifications for respiratory pathology in disease states. DOI: http://dx.doi.org/10.7554/eLife.03427.001

Markov chain models of coupled calcium channels: Kronecker representations and iterative solution methods

Mathematical models of calcium release sites derived from Markov chain models of intracellular calcium channels exhibit collective gating reminiscent of the experimentally observed phenomenon of stochastic calcium excitability (i.e., calcium puffs and sparks). Calcium release site models are stochastic automata networks that involve many functional transitions, that is, the transition probabilities of each channel depend on the local calcium concentration and thus the state of the other channels. We present a Kronecker-structured representation for calcium release site models and perform benchmark stationary distribution calculations using both exact and approximate iterative numerical solution techniques that leverage this structure. When it is possible to obtain an exact solution, response measures such as the number of channels in a particular state converge more quickly using the iterative numerical methods than occupation measures calculated via Monte Carlo simulation. In particular, multi-level methods provide excellent convergence with modest additional memory requirements for the Kronecker representation of calcium release site models. When an exact solution is not feasible, iterative approximate methods based on the power method may be used, with performance similar to Monte Carlo estimates. This suggests approximate methods with multi-level iterative engines as a promising avenue of future research for large-scale calcium release site models.

Modeling vocal fold motion with a hydrodynamic semicontinuum model

Vocal fold (VF) motion is a fundamental process in voice production, and is also a challenging problem for numerical computation because the VF dynamics depend on nonlinear coupling of air flow with the response of elastic channels (VF), which undergo opening and closing, and induce internal flow separation. The traditional modeling approach makes use of quasisteady flow approximation or Bernoullis law which ignores air compressibility, and is known to be invalid during VF opening. A hydrodynamic semicontinuum system for VF motion is presented. The airflow is modeled by a modified quasi-one-dimensional Euler system with coupling to VF velocity. The VF is modeled by a lumped two mass system with a built-in geometric condition on flow separation. The modified Euler system contains the Bernoullis law as a special case, and is derivable from the two-dimensional compressible Navier-Stokes equations in the inviscid limit. The computational domain contains also solid walls next to VFs (flexible walls). It is shown numerically that several salient features of VFs are captured, especially transients such as the double peaks of the driving subglottal pressures at the opening and the closing stages of VF motion consistent with fully resolved two-dimensional direct simulations, and experimental data. The system is much simpler to compute than a VF model based on two-dimensional Navier-Stokes system.

Periodic Orbit Continuation in Multiple Time Scale Systems

Continuation methods utilizing boundary value solvers are an effective tool for computing unstable periodic orbits of dynamical systems. Auto [1] is the standard implementation of these procedures. Unfortunately, the collocation methods used in Auto often require very fine meshes for convergence on problems with multiple time scales. This inconvenience prompts the search for alternative methods for computing such periodic orbits; we introduce here new multiple-shooting algorithms based on geometric singular perturbation theory.

MARKOV CHAIN MODELS OF COUPLED INTRACELLULAR CALCIUM CHANNELS: KRONECKER STRUCTURED REPRESENTATIONS AND BENCHMARK STATIONARY DISTRIBUTION CALCULATIONS

Mathematical models of calcium release sites derived from Markov chain models of intracellular calcium channels exhibit collective gating reminiscent of the experimentally observed phenomenon of stochastic calcium excitability (i.e., calcium puffs and sparks). We present a Kronecker structured representation for calcium release site models and perform benchmark stationary distribution calculations using numerical iterative solution techniques that leverage this structure. In this context we find multi-level methods and certain preconditioned projection methods superior to simple Gauss-Seidel type iterations. Response measures such as the number of channels in a particular state converge more quickly using these numerical iterative methods than occupation measures calculated via Monte Carlo simulation.

Signal processing of acoustic signals in the time domain with an active nonlinear nonlocal cochlear model

A two-space dimensional active nonlinear nonlocal cochlear model is formulated in the time domain to capture nonlinear hearing effects such as compression, multi-tone suppression and difference tones. The micromechanics of the basilar membrane (BM) are incorporated to model active cochlear properties. An active gain parameter is constructed in the form of a nonlinear nonlocal functional of BM displacement. The model is discretized with a boundary integral method and numerically solved using an iterative second-order accurate finite difference scheme. A block matrix structure of the discrete system is exploited to simplify the numerics with no loss of accuracy. Model responses to multiple frequency stimuli are shown in agreement with hearing experiments. A nonlinear spectrum is computed from the model, and compared with FFT spectrum for noisy tonal inputs. The discretized model is efficient and accurate, and can serve as a useful auditory signal processing tool.

Directed 3-cycle anchored digraphs and their application in the uniform sampling of realizations from a fixed degree sequence

In this paper, we give structural and degree sequence characterizations for a new class of digraphs called directed 3-cycle anchored. A digraph in this class has the property that, for every realization of its degree sequence, there is a directed 3-cycle through each vertex of a labeled vertex set. We end by illustrating their use in the uniform sampling of simple directed graph realizations from a fixed degree sequence.

Transmission of Infectious Diseases: Data, Models, and Simulations

Abstract This chapter introduces students to various aspects of mathematical modeling in epidemiology, including data collection, development of models, and deriving and interpreting predictions of models. The agent-based models of this chapter are explored by using a teaching tool, IONTW (Infections On NeTWorks), which was developed by the authors using the NetLogo programming language. The chapter discusses in detail the process of building mathematical models of disease transmission. Students will frequently be challenged to critically examine the simplifying assumptions that are involved in this process. It also provides an intuitive introduction to thinking about simulation models in terms of pseudocode, intended to empower students who have no prior knowledge of computer programming to analyze simulation models at a semiformal level. The chapter includes numerous conceptual and simulation exercises aimed at developing deeper understanding of the concepts.

QUBES: a community focused on supporting teaching and learning in quantitative biology

This letter provides an overview of the Quantitative Undergraduate Biology Education and Synthesis (QUBES) Project funded through the National Science Foundation. The project has five distinct, but interdependent, initiatives which work together to support faculty and students in the teaching and learning of quantitative biology (QB). QUBES has adopted an integrated strategy to improving the frequency and effectiveness of QB instruction that includes coordinating a broad consortium of professional stakeholders, supporting faculty development and the implementation of new teaching practices, providing an infrastructure for collaboration and access to high quality materials, establishing new metrics for faculty teaching scholarship and documenting the project outcomes.

Disease Transmission Dynamics on Networks: Network Structure Versus Disease Dynamics

Abstract The first part of this chapter covers compartment-based models that are constructed under the uniform mixing assumption. The role of the basic reproductive ratio in these models is illustrated both by theoretical results and simulations with IONTW (Infections On NeTWorks), a teaching tool that was developed by the authors and uses the NetLogo language. The second part introduces students to network-based models of disease transmission. In these models it is assumed that the infection can be transmitted only during a direct contact between two hosts who are adjacent in the underlying contact network. Models of this kind allow for incorporation of more realistic details than do compartment-based models that assume uniform mixing. Students are guided through sequences of conceptual and simulation exercises to discover how the network structure influences the predictions of the model, including predictions about the effectiveness of possible control measures.