The demographic effects of functional traits: an integral projection model approach reveals population-level consequences of reproduction-defence trade-offs.

Quantitatively linking individual variation in functional traits to demography is a necessary step to advance our understanding of trait-based ecological processes. We constructed a population model for Asclepias syriaca to identify how functional traits affect vital rates and population growth and whether trade-offs in chemical defence and demography alter population growth. Plants with higher foliar cardenolides had lower fibre, cellulose and lignin levels, as well as decreased sexual and clonal reproduction. Average cardenolide concentrations had the strongest effect on population growth. In both the sexual and clonal pathway, the trade-off between reproduction and defence affected population growth. We found that both increasing the mean of the distribution of individual plant values for cardenolides and herbivory decreased population growth. However, increasing the variance in both defence and herbivory increased population growth. Functional traits can impact population growth and quantifying individual-level variation in traits should be included in assessments of population-level processes.

Functional Interactions between Mammalian Respiratory Rhythmogenic and Premotor Circuitry.

UNLABELLED: 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. SIGNIFICANCE STATEMENT: To produce a breathing-related motor pattern, a brainstem core oscillator circuit projects to a population of premotor interneurons, but the assemblage of this network remains incompletely understood. Here we applied network modeling and numerical simulation to discover respiratory circuit configurations that successfully replicate photonic cell ablation experiments targeting either the core oscillator or premotor network, respectively. If premotor neurons are interconnected in a so-called "small-world" network with a fixed number of incoming synapses balanced between premotor and rhythmogenic neurons, then our simulations match their experimental benchmarks. These results provide a framework of experimentally testable predictions regarding the rudimentary structure and function of respiratory rhythm- and pattern-generating circuits in the brainstem of mammals.

Threshold Digraphs.

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.

Reduction of calcium release site models via moment fitting of phase-type distributions.

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.

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.

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.

Effect of node-degree correlation on synchronization of identical pulse-coupled oscillators.

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.

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.

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 Bernoulli's 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 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.