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ABSTRACT
Breathing in mammals depends on rhythms that originate from the preBötzinger complex (preBötC) of the ventral medulla and a network of brainstem and spinal premotor neurons. The rhythm-generating core of the preBötC, as well as some premotor circuits, consist of interneurons derived from Dbx1-expressing precursors (Dbx1 neurons), but the structure and function of these networks remain incompletely understood. We previously developed a cell-specific detection and laser ablation system to interrogate respiratory network structure and function in a slice model of breathing that retains the preBötC, the respiratory-related hypoglossal (XII) motor nucleus and XII premotor circuits. In spontaneously rhythmic slices, cumulative ablation of Dbx1 preBötC neurons decreased XII motor output by ∼50% after ∼15 cell deletions, and then decelerated and terminated rhythmic function altogether as the tally increased to ∼85 neurons. In contrast, cumulatively deleting Dbx1 XII premotor neurons decreased motor output monotonically but did not affect frequency nor stop XII output regardless of the ablation tally. Here, we couple an existing preBötC model with a premotor population in several topological configurations to investigate which one may replicate the laser ablation experiments best. If the XII premotor population is a "small-world" network (rich in local connections with sparse long-range connections among constituent premotor neurons) and connected with the preBötC such that the total number of incoming synapses remains fixed, then the in silico system successfully replicates the in vitro laser ablation experiments. This study proposes a feasible configuration for circuits consisting of Dbx1-derived interneurons that generate inspiratory rhythm and motor pattern.

KEYWORDS
central pattern generator, pre-Bötzinger complex, respiration

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ABSTRACT
We show that finding a subgraph realization with the minimum generalized Randić index for a given base graph and degree sequence is solvable in polynomial time by formulating the problem as the minimum weight perfect b-matching problem of Edmonds (J Res Natl Bur Stand 69B (1965), 125–130). However, the realization found via this reduction is not guaranteed to be connected. Approximating the minimum weight perfect b-matching problem subject to a connectivity constraint is shown to be NP-hard. For instances in which the optimal solution to the minimum Randić index problem is not connected, we describe a heuristic to connect the graph using pairwise edge exchanges that preserves the degree sequence. Although we focus on finding graph realizations with minimum Randić index, our results extend to finding graph realizations with maximum Randić index as well. Applications of the Randić index are provided to synchronization of neuronal networks controlling respiration in mammals and to normalizing cortical thickness networks in diagnosing individuals with dementia.

KEYWORDS
generalized Randić index, network realization, degree sequence, minimum weight perfect b-matching, connectivity constraint, synchronization, cortical networks

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ABSTRACT
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.

KEYWORDS
mathematics, biology, education, collaboration, professional development

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ABSTRACT
The mammalian breathing rhythm putatively originates from Dbx1-derived interneurons in the preBötzinger complex (preBötC) of the ventral medulla. Cumulative deletion of ∼15% of Dbx1 preBötC neurons in an in vitro breathing model stops rhythmic bursts of respiratory-related motor output. Here we assemble in silico models of preBötC networks using random graphs for structure, and ordinary differential equations for dynamics, to examine the mechanisms responsible for the loss of spontaneous respiratory rhythm and motor output measured experimentally in vitro. Model networks subjected to cellular ablations similarly discontinue functionality. However, our analyses indicate that model preBötC networks remain topologically intact even after rhythm cessation, suggesting that dynamics coupled with structural properties of the underlying network are responsible for rhythm cessation. Simulations show that cumulative cellular ablations diminish the number of neurons that can be recruited to spike per unit time. When the recruitment rate drops below 1 neuron/ms the network stops spontaneous rhythmic activity. Neurons that play pre-eminent roles in rhythmogenesis include those that commence spiking during the quiescent phase between respiratory bursts and those with a high number of incoming synapses, which both play key roles in recruitment, i.e., recurrent excitation leading to network bursts. Selectively ablating neurons with many incoming synapses impairs recurrent excitation and stops spontaneous rhythmic activity and motor output with lower ablation tallies compared with random deletions. This study provides a theoretical framework for the operating mechanism of mammalian central pattern generator networks and their susceptibility to loss-of-function in the case of disease or neurodegeneration.

KEYWORDS
cumulative ablation, Dbx1, network, recurrent excitation, rhythmogenesis, synaptic transmission

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ABSTRACT
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.

KEYWORDS
pre-Bötzinger complex, Dbx1, neurons, respiration, networks, breathing, central pattern generator

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ABSTRACT
A digraph whose degree sequence has a unique vertex labeled realization is called threshold. In this paper we present several characterizations of threshold digraphs and their degree sequences, and show these characterizations to be equivalent. Using this result, we obtain a new, short proof of the Fulkerson-Chen theorem on degree sequences of general digraphs.

KEYWORDS
directed graph, degree sequence, realization, split graph, split digraph, threshold graph

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ABSTRACT
Diamondback terrapins are a species of turtle found along the coast of the United States from Massachusetts to Texas. Many of the states in this range list the terrapins as endangered, threatened, or a species of concern. However, little is known about their actual population sizes across the range. Mitro (2003) created a linear age class model for a terrapin population in Rhode Island. Here we examine the effects on the same population of human-related threats such as crab pots and road traffic using a nonlinear, stage-based model. This model shows that crab potting has a larger negative effect on the population (which causes a decline occurring at 6.6% mortality of affected stages). We conclude that, in areas of existing crab potting, conservation efforts should focus on reducing terrapin mortality through the use of bycatch reduction devices.

KEYWORDS
matrix model, diamondback terrapin, crab potting, marriage function

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ABSTRACT
Zooplankton serve as an important link in marine ecosystem food webs and accurate yet mathematically and computationally tractable models of zooplankton dynamics should serve as an important component of large system models. Current models of zooplankton dynamics may have unnecessarily high dimensions resulting from tracking all stage classes or may lose accuracy due to neglected system characteristics such as non-predatory mortality due to disease or starvation. Here we construct a six stage class model and compare it to the thirteen stage class model in Elliot and Tang (2011) of Acartia tonsa zooplankton in the Chesapeake Bay. We also incorporate both predatory and non-predatory zooplankton mortality in order to further study the mortality term. We use sensitivity, elasticity, and interval analysis to show that the six dimensional model retains the essential features of the thirteen dimensional model and that both models are most sensitive to error in the mortality term. Given the model sensitivity to error in the mortality terms, a nonlinear approach to zooplankton modeling that separates non-predatory mortality from predation by fish and intra-guild predation is warranted to further improve current zooplankton models.

KEYWORDS
linear model, zooplankton, Acartia tonsa, non-predatory mortality, dimension reduction, sensitivity analysis

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ABSTRACT
We generalize the class of split graphs to the directed case and show that these split digraphs can be identified from their degree sequences. The first degree sequence characterization is an extension of the concept of splittance to directed graphs, while the second characterization says a digraph is split if and only if its degree sequence satisfies one of the Fulkerson inequalities (which determine when an integer-pair sequence is digraphic) with equality.

KEYWORDS
directed graph, degree sequence, split graph

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ABSTRACT
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.

KEYWORDS
digraphs, degree sequence, uniform sampling, directed 3-cycle anchored

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ABSTRACT
Models of calcium (Ca 2+) release sites derived from continuous-time Markov chain (CTMC) models of intracellular Ca 2+ channels exhibit collective gating reminiscent of the experimentally observed phenomenon of Ca 2 + puffs and sparks. In order to overcome the state-space explosion that occurs in compositionally defined Ca 2+ release site models, we have implemented an automated procedure for model reduction that replaces aggregated states of the full release site model with much simpler CTMCs that have similar within-group phase-type sojourn times and inter-group transitions. Error analysis based on comparison of full and reduced models validates the method when applied to release site models composed of 20 three-state channels that are both activated and inactivated by Ca 2+ . Although inspired by existing techniques for fitting moments of phase-type distributions, the automated reduction method for compositional Ca 2+ release site models is unique in several respects and novel in this biophysical context.

KEYWORDS
calcium release site, calcium puff, calcium spark, Markov chain, phase-type distribution

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ABSTRACT
We explore the effect of correlations between the in- and out-degrees of random directed networks on the synchronization of identical pulse-coupled oscillators. Numerical experiments demonstrate that the proportion of initial conditions resulting in a globally synchronous state (prior to a large but finite time) is an increasing function of node-degree correlation. For those networks observed to globally synchronize, both the mean and standard deviation of time to synchronization are decreasing functions of node-degree correlation. Pulse-coupled oscillator networks with negatively correlated node degree often exhibit multiple coherent attracting states, with trajectories performing fast transitions between them. These effects of node-degree correlation on dynamics of pulse-coupled oscillators are consistent with aspects of network topology (e.g., the effect of node-degree correlation on the eigenvalues of the Laplacian matrix) that have been shown to affect synchronization in other contexts.

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ABSTRACT
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.

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ABSTRACT
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.

KEYWORDS
Computational Biology, Computer Simulation, Calcium Channels, Calcium Signaling, Ion Channel Gating, Algorithms, Models: Biological, Markov Chains, Kinetics

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ABSTRACT
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.

KEYWORDS
Time domain, Nonlinear filtering, Basilar membrane, cochlea, Auditory signal processing

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ABSTRACT
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 Bernoulli's law which ignores air compressibility, and is known to be invalid during VF opening. Ahydrodynamic 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 Bernoulli's 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. [ABSTRACT FROM AUTHOR]

KEYWORDS
AIR flow, VOCAL cords, NUMERICAL analysis, FLUID dynamics, HYDRODYNAMICS

NOTES
Accession Number: 20665966; LaMar, M. Drew 1 Yingyong Qi 2 Xin, Jack 3; Email Address: jxinmath.utexas.edu; Affiliation: 1: Department of Mathematics, University of Texas at Austin, Austin, Texas 78712 2: Qualcomm Inc., 5775 Morehouse Drive, San Diego, California 92121 3: Department of Mathematics and TICAM, University of Texas at Austin, Austin, Texas 78712; Source Info: Jul2003, Vol. 114 Issue 1, p455; Subject Term: VOCAL cords; Subject Term: AIR flow; Subject Term: NUMERICAL analysis; Subject Term: HYDRODYNAMICS; Subject Term: FLUID dynamics; Number of Pages: 10p; Illustrations: 1 chart, 2 diagrams, 6 graphs; Document Type: Article

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ABSTRACT
Networks appear throughout the sciences, forming a common thread linking research activities in many fields, such as sociology, biology, chemistry, engineering, marketing, and mathematics. For example, they are used in ecology to represent food webs and in engineering and computer science to design high quality internet router connections. Depending on the application, one network structural property may be more important than another. The structural properties of networks (e.g. degree distribution, clustering coefficient, assortativity) are usually characterized in terms of invariants [8], which are functions on networks that do not depend on the labeling of the nodes. In this chapter we focus on network invariants that are quantitative, that is, they can be characterized as network measures. An increasingly important application area is how network invariants affect the dynamics of a process on the network (e.g. respiration, current, traffic) [36]. In order to study the potential effect of incremental changes in network invariants on network dynamics, one or more network invariants are held constant, thereby creating a family of networks. In this chapter the degree distribution of a network is held constant whilst other network invariants are examined. In particular, we examine the effects of assortativity, the Randić index and eigenvalues of the Laplacian on network dynamics.

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Continuation methods utilizing boundary value solvers are an effective tool for computing unstable periodic orbits of dynamical systems. Auto [7] 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.

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ABSTRACT
Choosing a uniformly sampled simple directed graph realization of a degree sequence has many applications, in particular in social networks where self-loops are commonly not allowed. It has been shown in the past that one can perform a Markov chain arc-switching algorithm to sample a simple directed graph uniformly by performing two types of switches: a 2-switch and a directed 3-cycle reorientation. This paper discusses under what circumstances a directed 3-cycle reorientation is required. In particular, the class of degree sequences where this is required is a subclass of the directed 3-cycle anchored degree sequences. An important implication of this result is a reduced Markov chain algorithm that uses only 2-switches.

KEYWORDS
directed graph, degree sequence, realization, uniform distribution, random walk

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ABSTRACT
The Havel-Hakimi algorithm for constructing realizations of degree sequences for undirected graphs has been used extensively in the literature. A result by Kleitman and Wang extends the Havel-Hakimi algorithm to degree sequences for directed graphs. In this paper we go a step further and describe a modification of Kleitman and Wang's algorithm that is a more natural extension of Havel-Hakimi's algorithm, in the sense that our extension can be made equivalent to Havel-Hakimi's algorithm when the degree sequence has equal in and out degrees and an even degree sum. We identify special degree sequences, called directed 3-cycle anchored, that are ill-defined for the algorithm and force a particular local structure on all directed graph realizations. We give structural characterizations of these realizations, as well as characterizations of the ill-defined degree sequences, leading to a well-defined algorithm.

KEYWORDS
directed graphs, degree sequence, realization, degree sequence characterization, M-partition, forcibly P-digraphic