Lateralization of Travelling Wave Response in the Hearing Organ of Bushcrickets

Travelling waves are the physical basis of frequency discrimination in many vertebrate and invertebrate taxa, including mammals, birds, and some insects. In bushcrickets (Tettigoniidae), the crista acustica is the hearing organ that has been shown to use sound-induced travelling waves. Up to now, data on mechanical characteristics of sound-induced travelling waves were only available along the longitudinal (proximal-distal) direction. In this study, we use laser Doppler vibrometry to investigate in-vivo radial (anterior-posterior) features of travelling waves in the tropical bushcricket Mecopoda elongata. Our results demonstrate that the maximum of sound-induced travelling wave amplitude response is always shifted towards the anterior part of the crista acustica. This lateralization of the travelling wave response induces a tilt in the motion of the crista acustica, which presumably optimizes sensory transduction by exerting a shear motion on the sensory cilia in this hearing organ.     

Landscape geometry and travelling waves in the larch budmoth

Travelling waves in cyclic populations refer to temporal shifts in peak densities moving across space in a wave‐like fashion. The epicentre hypothesis states that peak densities begin in specific geographic foci and then spread into adjoining areas. Travelling waves have been confirmed in a number of population systems, begging questions about their causes. Herein we apply a newly developed statistical technique, wavelet phase analysis, to historical data to document that the travelling waves in larch budmoth (LBM) outbreaks arise from two epicentres, both located in areas with high concentrations of favourable habitat. We propose that the spatial arrangement of the landscape mosaic is responsible for initiating the travelling waves. We use a tri‐trophic model of LBM dynamics to demonstrate that landscape heterogeneity (specifically gradients in density of favourable habitat) alone, is capable of inducing waves from epicentres. Our study provides unique evidence of how landscape features can mould travelling waves.     

Travelling wave solutions of diffusive Lotka-Volterra equations

We establish the existence of travelling wave solutions for two reaction diffusion systems based on the Lotka-Volterra model for predator and prey interactions. For simplicity, we consider only 1 space dimension. The waves are of transition front type, analogous to the travelling wave solutions discussed by Fisher and Kolmogorov et al. for a scalar reaction diffusion equation. The waves discussed here are not necessarily monotone. For any speed c there is a travelling wave solution of transition front type. For one of the systems discussed here, there is a distinguished speed c^* dividing the waves into two types, waves of speed c < c^* being one type, waves of speed c ? c^* being of the other type. We present numerical evidence that for this system the wave of speed c^* is stable, and that c^* is an asymptotic speed of propagation in some sense. For the other system, waves of all speeds are in some sense stable. The proof of existence uses a shooting argument and a Lyapunov function. We also discuss some possible biological implications of the existence of these waves.     

Discrete-time travelling waves: Ecological examples

Integrodifference equations are discrete-time models that possess many of the attributes of continuous-time reaction-diffusion equations. They arise naturally in population biology as models for organisms with discrete nonoverlapping generations and well-defined growth and dispersal stages. I examined the varied travelling waves that arise in some simple ecologically-interesting integrodifference equations. For a scalar equation with compensatory growth, I observed only simple travelling waves. For carefully chosen redistribution kernels, one may derive the speed and approximate the shape of the observed waveforms. A model with overcompensation exhibited flip bifurcations and travelling cycles in addition to simple travelling waves. Finally, a simple predator-prey system possessed periodic wave trains and a variety of travelling waves.     

Separation of arterial pressure into a nonlinear superposition of solitary waves and a windkessel flow

A simplified model of arterial blood pressure intended for use in model-based signal processing applications is presented. The main idea is to decompose the pressure into two components: a travelling wave which describes the fast propagation phenomena predominating during the systolic phase and a windkessel flow that represents the slow phenomena during the diastolic phase. Instead of decomposing the blood pressure pulse into a linear superposition of forward and backward harmonic waves, as in the linear wave theory, a nonlinear superposition of travelling waves matched to a reduced physical model of the pressure, is proposed. Very satisfactory experimental results are obtained by using forward waves, the N-soliton solutions of a Korteweg–de Vries equation in conjunction with a two-element windkessel model. The parameter identifiability in the practically important 3-soliton case is also studied. The proposed approach is briefly compared with the linear one and its possible clinical relevance is discussed.     

一类具有分布时滞的扩散种群模型行波解的存在性

研究了一类具有分布时滞的扩散种群模型行波解的存在性,证明了当平均时滞充分小时,方程具有连接两个平衡点的单调行波解.     

Travelling Waves in Hybrid Chemotaxis Models

Hybrid models of chemotaxis combine agent-based models of cells with partial differential equation models of extracellular chemical signals. In this paper, travelling wave properties of hybrid models of bacterial chemotaxis are investigated. Bacteria are modelled using an agent-based (individual-based) approach with internal dynamics describing signal transduction. In addition to the chemotactic behaviour of the bacteria, the individual-based model also includes cell proliferation and death. Cells consume the extracellular nutrient field (chemoattractant), which is modelled using a partial differential equation. Mesoscopic and macroscopic equations representing the behaviour of the hybrid model are derived and the existence of travelling wave solutions for these models is established. It is shown that cell proliferation is necessary for the existence of non-transient (stationary) travelling waves in hybrid models. Additionally, a numerical comparison between the wave speeds of the continuum models and the hybrid models shows good agreement in the case of weak chemotaxis and qualitative agreement for the strong chemotaxis case. In the case of slow cell adaptation, we detect oscillating behaviour of the wave, which cannot be explained by mean-field approximations.     

Arterial pulsation-driven cerebrospinal fluid flow in the perivascular space: a computational model

This study was conducted to determine whether local arterial pulsations are sufficient to cause cerebrospinal fluid (CSF) flow along perivascular spaces (PVS) within the spinal cord. A theoretical model of the perivascular space surrounding a "typical" small artery was analysed using computational fluid dynamics. Systolic pulsations were modelled as travelling waves on the arterial wall. The effects of wave geometry and variable pressure conditions on fluid flow were investigated. Arterial pulsations induce fluid movement in the PVS in the direction of arterial wave travel. Perivascular flow continues even in the presence of adverse pressure gradients of a few kilopascals. Flow rates are greater with increasing pulse wave velocities and arterial deformation, as both an absolute amplitude and as a proportion of the PVS. The model suggests that arterial pulsations are sufficient to cause fluid flow in the perivascular space even against modest adverse pressure gradients. Local increases in flow in this perivascular pumping mechanism or reduction in outflow may be important in the etiology of syringomyelia.     

Periodic travelling waves in cyclic predator–prey systems

Predation is an established cause of cycling in prey species. Here, the ability of predation to explain periodic travelling waves in prey populations, which have recently been found in a number of spatiotemporal field studies, is examined. The nature of periodic waves in these systems, and the way in which they can be generated by the invasion of predators into a prey population is discussed. A theoretical calculation that predicts, as a function of two parameter ratios, whether such an invasion will lead to a stable periodic travelling wave that would be observed in practice is presented ‐ the alternative outcome is spatiotemporal chaos. The calculation also predicts quantitative details of the periodic waves, such as speed and amplitude. The results give new insights into the types of predator‐prey systems in which one would expect to see periodic travelling waves following an invasion by predators.     

Travelling waves of a delayed SIR epidemic model with nonlinear incidence rate and spatial diffusion

This paper is concerned with the existence of travelling waves to a SIR epidemic model with nonlinear incidence rate, spatial diffusion and time delay. By analyzing the corresponding characteristic equations, the local stability of a disease-free steady state and an endemic steady state to this system under homogeneous Neumann boundary conditions is discussed. By using the cross iteration method and the Schauder's fixed point theorem, we reduce the existence of travelling waves to the existence of a pair of upper-lower solutions. By constructing a pair of upper-lower solutions, we derive the existence of a travelling wave connecting the disease-free steady state and the endemic steady state. Numerical simulations are carried out to illustrate the main results.     

Travelling waves in a tissue interaction model for skin pattern formation

Tissue interaction plays a major role in many morphogenetic processes, particularly those associated with skin organ primordia. We examine travelling wave solutions in a tissue interaction model for skin pattern formation which is firmly based on the known biology. From a phase space analysis we conjecture the existence of travelling waves with specific wave speeds. Subsequently, analytical approximations to the wave profiles are derived using perturbation methods. We then show numerically that such travelling wave solutions do exist and that they are in good agreement with our analytical results. Finally, the biological implications of our analysis are discussed.     

Arterial Pulsation-driven Cerebrospinal Fluid Flow in the Perivascular Space: A Computational Model

This study was conducted to determine whether local arterial pulsations are sufficient to cause cerebrospinal fluid (CSF) flow along perivascular spaces (PVS) within the spinal cord. A theoretical model of the perivascular space surrounding a "typical" small artery was analysed using computational fluid dynamics. Systolic pulsations were modelled as travelling waves on the arterial wall. The effects of wave geometry and variable pressure conditions on fluid flow were investigated. Arterial pulsations induce fluid movement in the PVS in the direction of arterial wave travel. Perivascular flow continues even in the presence of adverse pressure gradients of a few kilopascals. Flow rates are greater with increasing pulse wave velocities and arterial deformation, as both an absolute amplitude and as a proportion of the PVS. The model suggests that arterial pulsations are sufficient to cause fluid flow in the perivascular space even against modest adverse pressure gradients. Local increases in flow in this perivascular pumping mechanism or reduction in outflow may be important in the etiology of syringomyelia.     

Calcium waves with fast buffers and mechanical effects

In the paper we consider the existence of calcium travelling waves for systems with fast buffers. We prove the convergence of the travelling waves to an asymptotic limit as the kinetic coefficients characterizing the interaction between calcium and buffers tend to infinity. To be more precise, we prove the convergence of the speeds as well as the calcium component concentration profile to the profile of the travelling wave of the reduced equation. Additionally, we take into account the effect of coupling between the mechanical and chemical processes and show the existence as well the monotonicity of the profiles of concentrations. This property guarantees their positivity.     

A Resonance Approach to Cochlear Mechanics

Background

How does the cochlea analyse sound into its component frequencies? In the 1850s Helmholtz thought it occurred by resonance, whereas a century later Békésy''s work indicated a travelling wave. The latter answer seemed to settle the question, but with the discovery in 1978 that the cochlea emits sound, the mechanics of the cochlea was back on the drawing board. Recent studies have raised questions about whether the travelling wave, as currently understood, is adequate to explain observations.

Approach

Applying basic resonance principles, this paper revisits the question. A graded bank of harmonic oscillators with cochlear-like frequencies and quality factors is simultaneously excited, and it is found that resonance gives rise to similar frequency responses, group delays, and travelling wave velocities as observed by experiment. The overall effect of the group delay gradient is to produce a decelerating wave of peak displacement moving from base to apex at characteristic travelling wave speeds. The extensive literature on chains of coupled oscillators is considered, and the occurrence of travelling waves, pseudowaves, phase plateaus, and forced resonance in such systems is noted.

Conclusion and significance

This alternative approach to cochlear mechanics shows that a travelling wave can simply arise as an apparently moving amplitude peak which passes along a bank of resonators without carrying energy. This highlights the possible role of the fast pressure wave and indicates how phase delays and group delays of a set of driven harmonic oscillators can generate an apparent travelling wave. It is possible to view the cochlea as a chain of globally forced coupled oscillators, and this model incorporates fundamental aspects of both the resonance and travelling wave theories.     

The emergence of wave emitting centres in an excitable medium

The response of an excitable biological medium to a double local stimulus is considered within the context of a mathematical model for a layer of starving cells of Dictyostelium discoideum, with both spatially one- and two-dimensional (1D and 2D) system being investigated. In contrast to the response usually seen in excitable media, whereby each superthreshold stimulus delivered to the relaxed medium results in the initiation of just one travelling wave, a source emitting a sequence of waves can develop in the present excitable medium after the second stimulus. In a 1D system, only transient wave sources forming a limited number of waves are found. In 2D systems, a permanent wave sources consisting in a pair of spirals are observed as well as the transient wave sources forming circular wave patterns. The general features of the medium dynamics that underlie the observed responses to the double stimulus are discussed.     

From periodic travelling waves to travelling fronts in the spike-diffuse-spike model of dendritic waves

In the vertebrate brain excitatory synaptic contacts typically occur on the tiny evaginations of neuron dendritic surface known as dendritic spines. There is clear evidence that spine heads are endowed with voltage-dependent excitable channels and that action potentials invade spines. Computational models are being increasingly used to gain insight into the functional significance of a spine with an excitable membrane. The spike-diffuse-spike (SDS) model is one such model that admits to a relatively straightforward mathematical analysis. In this paper we demonstrate that not only can the SDS model support solitary travelling pulses, already observed numerically in more detailed biophysical models, but that it has periodic travelling wave solutions. The exact mathematical treatment of periodic travelling waves in the SDS model is used, within a kinematic framework, to predict the existence of connections between two periodic spike trains of different interspike interval. The associated wave front in the sequence of interspike intervals travels with a constant velocity without degradation of shape, and might therefore be used for the robust encoding of information.     

Slow and fast invasion waves in a model of acid-mediated tumour growth

This work is concerned with a reaction-diffusion system that has been proposed as a model to describe acid-mediated cancer invasion. More precisely, we consider the properties of travelling waves that can be supported by such a system, and show that a rich variety of wave propagation dynamics, both fast and slow, is compatible with the model. In particular, asymptotic formulae for admissible wave profiles and bounds on their wave speeds are provided.     

Modelling the spatio-temporal dynamics of multi-species host-parasitoid interactions: heterogeneous patterns and ecological implications

A mathematical model of the spatio-temporal dynamics of a two host, two parasitoid system is presented. There is a coupling of the four species through parasitism of both hosts by one of the parasitoids. The model comprises a system of four reaction-diffusion equations. The underlying system of ordinary differential equations, modelling the host-parasitoid population dynamics, has a unique positive steady state and is shown to be capable of undergoing Hopf bifurcations, leading to limit cycle kinetics which give rise to oscillatory temporal dynamics. The stability of the positive steady state has a fundamental impact on the spatio-temporal dynamics: stable travelling waves of parasitoid invasion exhibit increasingly irregular periodic travelling wave behaviour when key parameter values are increased beyond their Hopf bifurcation point. These irregular periodic travelling waves give rise to heterogeneous spatio-temporal patterns of host and parasitoid abundance. The generation of heterogeneous patterns has ecological implications and the concepts of temporary host refuge and niche formation are considered.     

Critical manifolds,travelling waves,and an example from population genetics

A generalized Morse index theory is used to study travelling waves in a natural selection-migration model for a diploid organism when the selective strength is weak.