Representing stimulus motion with waves in adaptive neural fields

Traveling waves of neural activity emerge in cortical networks both spontaneously and in response to stimuli. The spatiotemporal structure of waves can indicate the information they encode and the physiological processes that sustain them. Here, we investigate the stimulus-response relationships of traveling waves emerging in adaptive neural fields as a model of visual motion processing. Neural field equations model the activity of cortical tissue as a continuum excitable medium, and adaptive processes provide negative feedback, generating localized activity patterns. Synaptic connectivity in our model is described by an integral kernel that weakens dynamically due to activity-dependent synaptic depression, leading to marginally stable traveling fronts (with attenuated backs) or pulses of a fixed speed. Our analysis quantifies how weak stimuli shift the relative position of these waves over time, characterized by a wave response function we obtain perturbatively. Persistent and continuously visible stimuli model moving visual objects. Intermittent flashes that hop across visual space can produce the experience of smooth apparent visual motion. Entrainment of waves to both kinds of moving stimuli are well characterized by our theory and numerical simulations, providing a mechanistic description of the perception of visual motion.

Ultrasonic Traveling Waves for Near-Wall Positioning of Single Microbubbles in a Flowing Channel

Although microbubbles are used primarily in the medical industry as ultrasonic contrast agents, they can also be manipulated by acoustic waves for targeted drug delivery, sonothrombolysis and sonoporation. Acoustic waves can also potentially remove microbubbles from tubing systems (e.g., in hemodialysis) to prevent the negative effects associated with circulating microbubbles. A deeper understanding of the interactions between the acoustic radiation force, the microbubble and the channel wall could greatly benefit these applications. In this study, single air-filled microbubbles were injected into a flowing (polydimethylsiloxane) channel and monitored by a high-speed camera while passing through a pulsed ultrasonic wave zone (0.5 MHz). This study compared various bubble sizes, flow rates and acoustic pressure amplitudes to better understand the three physical regimes observed: free bubble translation (away from the wall); on-wall translation; and bubble-wall attachment. Comparison with a theoretical model revealed that the acoustic radiation force needs to exceed the combined repulsive forces (shear lift, wall lubrication and repulsive Van der Waal forces) for the dead state of bubble-wall attachment. The bubble dynamics revealed through this investigation provide an opportunity for efficient positioning of microbubbles in a channel flow, for either in vivo manipulation or removal in biological applications.

Mechanics and self-organization in tissue development

Self-organization is an all-important feature of living systems that provides the means to achieve specialization and functionality at distinct spatio-temporal scales. Herein, we review this concept by addressing the packing organization of cells, the sorting/compartmentalization phenomenon of cell populations, and the propagation of organizing cues at the tissue level through traveling waves. We elaborate on how different theoretical models and tools from Topology, Physics, and Dynamical Systems have improved the understanding of self-organization by shedding light on the role played by mechanics as a driver of morphogenesis. Altogether, by providing a historical perspective, we show how ideas and hypotheses in the field have been revisited, developed, and/or rejected and what are the open questions that need to be tackled by future research.

Mathematical modeling of atherogenesis: Atheroprotective role of HDL

Atherosclerosis is a chronic inflammatory cardiovascular disease in which arteries harden through the build-up of plaques. This work is devoted to the mathematical modeling and analysis of the inflammatory process of atherosclerosis. We propose a mathematical model formed by three coupled partial differential equations of reaction-diffusion type. We take into account three key-role players: the inflammatory immune cells, the inflammatory cytokines and the oxidized low density lipoproteins. A stability analysis of the kinetic system is performed. It leads to the presence of three stable fixed points relevant to appropriate biological states of atherogenesis; no inflammation, stabilized inflammation (stable plaque) and advanced inflammation (vulnerable plaque). The cases that may occur are subject to the variation of the parameters values. A detailed discussion showing how the model fits the biological phenomena is then established. We investigate as well the existence of solutions of traveling waves type along with numerical simulations that show the wave propagation in different cases. This shows that the inflammatory process propagates inside the intima as a traveling wave. Then, we consider the effect of high density lipoprotein (HDL) on the atherosclerotic plaque formation. To do that, we elaborate a map that determines the level of risk of plaque formation with respect to the prevalence of HDL in the blood. These results confirm but also generalize previous results published in the literature. They also give a deeper understanding to the propagation of the inflammation inside the artery in terms of the interplay among the different main players in the whole process.

Long-range interaction effects on coupled excitable nodes: traveling waves and chimera state

In this paper, analytical and numerical studies of the influence of the long-range interaction parameter on the excitability threshold in a ring of FitzHugh-Nagumo (FHN) system are investigated. The long-range interaction is introduced to the network to model regulation of the Gap junctions or hemichannels activity at the connexins level, which provides links between pre-synaptic and post-synaptic neurons. Results show that the long-range coupling enhances the range of the threshold parameter. We also investigate the long-range effects on the network dynamics, which induces enlargement of the oscillatory zone before the excitable regime. When considering bidirectional coupling, the long-range interaction induces traveling patterns such as traveling waves, while when considering unidirectional coupling, the long-range interaction induces multi-chimera states. We also studied the difference between the dynamics of coupled oscillators and coupled excitable neurons. We found that, for the coupled system, the oscillation period decreases with the increasing of the coupling parameter. For the same values of the coupling parameter, the oscillation period of the Oscillatory dynamics is greater than the oscillation period of the excitable dynamics. The analytical approximation shows good agreement with the numerical results.

The formation of spreading front: the singular limit of three-component reaction-diffusion models

Understanding the invasion processes of biological species is a fundamental issue in ecology. Several mathematical models have been proposed to estimate the spreading speed of species. In recent decades, it was reported that some mathematical models of population dynamics have an explicit form of the evolution equations for the spreading front, which are represented by free boundary problems such as the Stefan-like problem (e.g., Mimura et al., Jpn J Appl Math 2:151-186, 1985; Du and Lin, SIAM J Math Anal 42:377-405, 2010). To understand the formation of the spreading front, in this paper, we will consider the singular limit of three-component reaction-diffusion models and give some interpretations for spreading front from the viewpoint of modeling. As an application, we revisit the issue of the spread of the grey squirrel in the UK and estimate the spreading speed of the grey squirrel based on our result. Also, we discuss the relation between some free boundary problems related to population dynamics and mathematical models describing Controlling Invasive Alien Species. Lastly, we numerically consider the traveling wave solutions, which give information on the spreading behavior of invasive species.

Traveling waves in non-local pulse-coupled networks

Traveling phase waves are commonly observed in recordings of the cerebral cortex and are believed to organize behavior across different areas of the brain. We use this as motivation to analyze a one-dimensional network of phase oscillators that are nonlocally coupled via the phase response curve (PRC) and the Dirac delta function. Existence of waves is proven and the dispersion relation is computed. Using the theory of distributions enables us to write and solve an associated stability problem. First and second order perturbation theory is applied to get analytic insight and we show that long waves are stable while short waves are unstable. We apply the results to PRCs that come from mitral neurons. We extend the results to smooth pulse-like coupling by reducing the nonlocal equation to a local one and solving the associated boundary value problem.

Deciphering the dynamics of lamellipodium in a fish keratocytes model

Motile cells depend on an intricate network of feedback loops that are essential in driving cell movement. Integrin-based focal adhesions (FAs) along with actin are the two key factors that mediate such motile behaviour. Together, they generate excitable dynamics that are essential for forming protrusions at the leading edge of the cell and, in certain cases, traveling waves along the membrane. A partial differential equation (PDE) model of a self-organizing lamellipodium in crawling keratocytes has been previously developed to understand how the three spatiotemporal patterns of activity observed in such cells, namely, stalling, waving and smooth motility, are produced. The model consisted of three key variables: the density of barbed actin filaments, newly formed FAs called nascent adhesions (NAs) and VASP, an anti-capping protein that gets sequestered by NAs during maturation. Using parameter sweeping techniques, the distinct regimes of behaviour associated with the three activity patterns were identified. In this study, we convert the PDE model into an ordinary differential equation (ODE) model to examine its excitability properties and determine all the patterns of activity exhibited by this system. Our results reveal that there are two additional regimes not previously identified, including bistability and oscillatory-like type IV excitability (generated by three steady states and their manifolds, rather than limit cycles). These regimes are also present in the PDE model. Applying slow-fast analysis on the ODE model shows that it exhibits a canard explosion through a folded-saddle and that rough motility seen in keratocytes is likely due to noise-dependent motility governed by dynamics near the interface of bistability and type IV excitability. The two parameter bifurcation suggests that the increase in the proportion of rough motion is due to a shift in activity towards the bistable and type IV excitable regimes induced by a decrease in NA maturation rate. Our results thus provide important insight into how microscopic mechanical effects are integrated to produce the observed modes of motility.

Angiogenesis and vessel co-option in a mathematical model of diffusive tumor growth: The role of chemotaxis

This work considers the propagation of a tumor from the stage of a small avascular sphere in a host tissue and the progressive onset of a tumor neovasculature stimulated by a pro-angiogenic factor secreted by hypoxic cells. The way new vessels are formed involves cell sprouting from pre-existing vessels and following a trail via a chemotactic mechanism (CM). Namely, it is first proposed a detailed general family of models of the CM, based on a statistical mechanics approach. The key hypothesis is that the CM is composed by two components: i) the well-known bias induced by the angiogenic factor gradient; ii) the presence of stochastic changes of the velocity direction, thus giving rise to a diffusive component. Then, some further assumptions and simplifications are applied in order to derive a specific model to be used in the simulations. The tumor progression is favored by its acidic aggression towards the healthy cells. The model includes the evolution of many biological and chemical species. Numerical simulations show the onset of a traveling wave eventually replacing the host tissue with a fully vascularized tumor. The results of simulations agree with experimental measures of the vasculature density in tumors, even in the case of particularly hypoxic tumors.

Traveling Waves and Estimation of Minimal Wave Speed for a Diffusive Influenza Model with Multiple Strains

Antiviral treatment remains one of the key pharmacological interventions against influenza pandemic. However, widespread use of antiviral drugs brings with it the danger of drug resistance evolution. To assess the risk of the emergence and diffusion of resistance, in this paper, we develop a diffusive influenza model where influenza infection involves both drug-sensitive and drug-resistant strains. We first analyze its corresponding reaction model, whose reproduction numbers and equilibria are derived. The results show that the sensitive strains can be eliminated by treatment. Then, we establish the existence of the three kinds of traveling waves starting from the disease-free equilibrium, i.e., semi-traveling waves, strong traveling waves and persistent traveling waves, from which we can get some useful information (such as whether influenza will spread, asymptotic speed of propagation, the final state of the wavefront). On the other hand, we discuss three situations in which semi-traveling waves do not exist. When the control reproduction number \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_{C}$$\end{document}RC is larger than 1, the conditions for the existence and nonexistence of traveling waves are determined completely by the reproduction numbers \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_{SC}$$\end{document}RSC, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_{RC}$$\end{document}RRC and the wave speed c. Meanwhile, we give an interval estimation of minimal wave speed for influenza transmission, which has important guiding significance for the control of influenza in reality. Our findings demonstrate that the control of influenza depends not only on the rates of resistance emergence and transmission during treatment, but also on the diffusion rates of influenza strains, which have been overlooked in previous modeling studies. This suggests that antiviral treatment should be implemented appropriately, and infected individuals (especially with the resistant strain) should be tested and controlled effectively. Finally, we outline some future directions that deserve further investigation.

A Non-local Cross-Diffusion Model of Population Dynamics II: Exact, Approximate, and Numerical Traveling Waves in Single- and Multi-species Populations

We study traveling waves in a non-local cross-diffusion-type model, where organisms move along gradients in population densities. Such models are valuable for understanding waves of migration and invasion and how directed motion can impact such scenarios. In this paper, we demonstrate the emergence of traveling wave solutions, studying properties of both planar and radial wave fronts in one- and two-species variants of the model. We compute exact traveling wave solutions in the purely diffusive case and then perturb these solutions to analytically capture the influence directed motion has on these exact solutions. Using linear stability analysis, we find that the minimum wavespeeds correspond to the purely diffusive case, but numerical simulations suggest that advection can in general increase or decrease the observed wavespeed substantially, which allows a single species to more rapidly move into unoccupied resource-rich spatial regions or modify the speed of an invasion for two populations. We also find interesting effects from the non-local interactions in the model, suggesting that single species invasions can be enhanced with stronger non-locality, but that invasion of a competitive species may be slowed due to this non-local effect. Finally, we simulate pattern formation behind waves of invasion, showing that directed motion can have substantial impacts not only on wavespeed but also on the existence and structure of emergent patterns, as predicted in the first part of our study (Taylor et al. in Bull Math Biol, 2020).

Investigating the role of gap junctions in seizure wave propagation

The effect of gap junctions as well as the biological mechanisms behind seizure wave propagation is not completely understood. In this work, we use a simple neural field model to study the possible influence of gap junctions specifically on cortical wave propagation that has been observed in vivo preceding seizure termination. We consider a voltage-based neural field model consisting of an excitatory and an inhibitory population as well as both chemical and gap junction-like synapses. We are able to approximate important properties of cortical wave propagation previously observed in vivo before seizure termination. This model adds support to existing evidence from models and clinical data suggesting a key role of gap junctions in seizure wave propagation. In particular, we found that in this model gap junction-like connectivity determines the propagation of one-bump or two-bump traveling wave solutions with features consistent with the clinical data. For sufficiently increased gap junction connectivity, wave solutions cease to exist. Moreover, gap junction connectivity needs to be sufficiently low or moderate to permit the existence of linearly stable solutions of interest.

Can chemotaxis speed up or slow down the spatial spreading in parabolic-elliptic Keller-Segel systems with logistic source?

The current paper is concerned with the spatial spreading speed and minimal wave speed of the following Keller-Segel chemoattraction system, [Formula: see text]where [Formula: see text], a, b, [Formula: see text], and [Formula: see text] are positive constants. Assume [Formula: see text] . Then if in addition [Formula: see text] holds, it is proved that [Formula: see text] is the spreading speed of the solutions of (0.1) with nonnegative continuous initial function [Formula: see text] with nonempty compact support, that is, [Formula: see text]and [Formula: see text]where [Formula: see text] is the unique global classical solution of (0.1) with [Formula: see text]. It is also proved that, if [Formula: see text] and [Formula: see text] holds, then [Formula: see text] is the minimal speed of the traveling wave solutions of (0.1) connecting (0, 0) and [Formula: see text], that is, for any [Formula: see text], (0.1) has a traveling wave solution connecting (0, 0) and [Formula: see text] with speed c, and (0.1) has no such traveling wave solutions with speed less than [Formula: see text]. Note that [Formula: see text] is the spatial spreading speed as well as the minimal wave speed of the following Fisher-KPP equation, [Formula: see text]Hence, if [Formula: see text] and [Formula: see text], or [Formula: see text] and [Formula: see text], then the chemotaxis neither speeds up nor slows down the spatial spreading in (0.1).

Dynamics of Pivoting Electrical Waves in a Cardiac Tissue Model

Through a detailed mathematical analysis we seek to advance our understanding of how cardiac tissue conductances govern pivoting (spiral, scroll, rotor, functional reentry) wave dynamics. This is an important problem in cardiology since pivoting waves likely underlie most reentrant tachycardias. The problem is complex, and to advance our methods of analysis we introduce two new tools: a ray tracing method and a moving-interface model. When used in combination with an ionic model, they permit us to elucidate the role played by tissue conductances on pivoting wave dynamics. Specifically we simulate traveling electrical waves with an ionic model that can reproduce the characteristics of plane and pivoting waves in small patches of cardiac tissue. Then ray tracing is applied to the simulated pivoting waves in a manner to expose their real displacement. In this exercise we find loci with special characteristics, as well as zones where a part of a pivoting wave quickly transitions from a regenerative to a non-regenerative propagation mode. The loci themselves and the monitoring of the ionic model state variables in this zone permit to elucidate several aspects of pivoting wave dynamics. We then formulate the moving-interface model based on the information gathered with the above-mentioned analysis. Equipped with a velocity profile v(s), s: distance along of the pivoting wave contour and the steady- state action potential duration (APD) of a plane wave during entrainment, APDss(T), at period T, this simple model can predict: shape, orbit of revolution, rotation period, whether a pivoting wave will break up or not, and whether the tissue will admit pivoting waves or not. Because v(s) and APDss(T) are linked to the ionic model, dynamical analysis with the moving-interface model conveys information on the role played by tissue conductances on pivoting wave dynamics. The analysis conducted here enables us to better understand previous results on the termination of pivoting waves. We surmise the method put forth here could become a means to discover how to alter tissue conductances in a manner to terminate pivoting waves at the origin of reentrant tachycardias.

Modeling the Spatial Spread of Chagas Disease

The aim of this work is to understand the spatial spread of Chagas disease, which is primarily transmitted by triatomines. We propose a mathematical model using a system of partial differential reaction-diffusion equations to study and describe the spread of this disease in the human population. We consider the respective subclasses of infected and uninfected individuals within the human and triatomine populations. The dynamics of the infected human subpopulation considers two disease phases: acute and chronic. The human population is considered to be homogeneously distributed across a space to describe the local propagation of Chagas disease by triatomines during a short epidemic period. We determine the basic reproduction number that allows us to assess Chagas disease control measures, and we determine the speed of disease propagation by using traveling wave solutions for our model.

Analysis of heterogeneous cardiac pacemaker tissue models and traveling wave dynamics

The sinoatrial-node (SAN) is a complex heterogeneous tissue that generates a stable rhythm in healthy hearts, yet a general mechanistic explanation for when and how this tissue remains stable is lacking. Although computational and theoretical analyses could elucidate these phenomena, such methods have rarely been used in realistic (large-dimensional) gap-junction coupled heterogeneous pacemaker tissue models. In this study, we adapt a recent model of pacemaker cells (Severi et al., 2012), incorporating biophysical representations of ion channel and intracellular calcium dynamics, to capture physiological features of a heterogeneous population of pacemaker cells, in particular "center" and "peripheral" cells with distinct intrinsic frequencies and action potential morphology. Large-scale simulations of the SAN tissue, represented by a heterogeneous tissue structure of pacemaker cells, exhibit a rich repertoire of behaviors, including complete synchrony, traveling waves of activity originating from periphery to center, and transient traveling waves originating from the center. We use phase reduction methods that do not require fully simulating the large-scale model to capture these observations. Moreover, the phase reduced models accurately predict key properties of the tissue electrical dynamics, including wave frequencies when synchronization occurs, and wave propagation direction in a variety of tissue models. With the reduced phase models, we analyze the relationship between cell distributions and coupling strengths and the resulting transient dynamics. Further, the reduced phase model predicts parameter regimes of irregular electrical dynamics. Thus, we demonstrate that phase reduced oscillator models applied to realistic pacemaker tissue is a useful tool for investigating the spatial-temporal dynamics of cardiac pacemaker activity.

Counter-propagating wave patterns in a swarm model with memory

Hyperbolic transport-reaction equations are abundant in the description of movement of motile organisms. Here, we focus on a system of four coupled transport-reaction equations that arises from an age-structuring of a species of turning individuals. By modeling how the behavior depends on the time since the last reversal, we introduce a memory effect. The highlight consists of the explicit construction and characterization of counter-propagating traveling waves, patterns which have been observed in bacterial colonies. Stability analysis reveals conditions for the wave formation as well as pulsating-in-time spatially constant solutions.

Invasion waves and pinning in the Kirkpatrick-Barton model of evolutionary range dynamics

The Kirkpatrick-Barton model, well known to invasion biologists, is a pair of reaction-diffusion equations for the joint evolution of population density and the mean of a quantitative trait as functions of space and time. Here we prove the existence of two classes of coherent structures, namely "bounded trait mean differential" traveling waves and localized stationary solutions, using geometric singular perturbation theory. We also give numerical examples of these (when they appear to be stable) and of "unbounded trait mean differential" solutions. Further, we provide numerical evidence of bistability and hysteresis for this system, modeling an initially confined population that colonizes new territory when some biotic or abiotic conditions change, and remains in its enlarged range even when conditions change back. Our analytical and numerical results indicate that the dynamics of this system are more complicated than previously recognized, and help make sense of evolutionary range dynamics predicted by other models that build upon it and sometimes challenge its predictions.

Characterization of applied fields for ion mobility separations in traveling wave based structures for lossless ion manipulations (SLIM)

Ion mobility (IM) is rapidly gaining attention for the separation and analysis of biomolecules due to the ability to distinguish the shapes of ions. However, conventional constant electric field drift tube IM separations have limited resolving power, constrained by practical limitations on the path length and maximum applied voltage. The implementation of traveling waves (TW) in IM removes the latter limitation, allowing higher resolution to be achieved using extended path lengths. Both of these can be readily obtained in structures for lossless ion manipulations (SLIM), which are fabricated from arrays of electrodes patterned on two parallel surfaces where potentials are applied to generate appropriate electric fields between the surfaces. Here we have investigated the relationship between the primary SLIM variables, such as electrode dimensions, inter-surface gap, and the applied TW voltages, that directly impact the fields experienced by ions. Ion trajectory simulations and theoretical calculations have been utilized to understand the dependence of SLIM geometry and effective electric fields on IM resolution. The variables explored impact both ion confinement and the observed IM resolution using SLIM modules.

The effect of inhibition on the existence of traveling wave solutions for a neural field model of human seizure termination

In this paper we study the influence of inhibition on an activity-based neural field model consisting of an excitatory population with a linear adaptation term that directly regulates the activity of the excitatory population. Such a model has been used to replicate traveling wave data as observed in high density local field potential recordings (González-Ramírez et al. PLoS Computational Biology, 11(2), e1004065, 2015). In this work, we show that by adding an inhibitory population to this model we can still replicate wave properties as observed in human clinical data preceding seizure termination, but the parameter range over which such waves exist becomes more restricted. This restriction depends on the strength of the inhibition and the timescale at which the inhibition acts. In particular, if inhibition acts on a slower timescale relative to excitation then it is possible to still replicate traveling wave patterns as observed in the clinical data even with a relatively strong effect of inhibition. However, if inhibition acts on the same timescale as the excitation, or faster, then traveling wave patterns with the desired characteristics cease to exist when the inhibition becomes sufficiently strong.

Ecocultural range-expansion scenarios for the replacement or assimilation of Neanderthals by modern humans

Recent archaeological records no longer support a simple dichotomous characterization of the cultures/behaviors of Neanderthals and modern humans, but indicate much cultural/behavioral variability over time and space. Thus, in modeling the replacement or assimilation of Neanderthals by modern humans, it is of interest to consider cultural dynamics and their relation to demographic change. The ecocultural framework for the competition between hominid species allows their carrying capacities to depend on some measure of the levels of culture they possess. In the present study both population densities and the densities of skilled individuals in Neanderthals and modern humans are spatially distributed and subject to change by spatial diffusion, ecological competition, and cultural transmission within each species. We analyze the resulting range expansions in terms of the demographic, ecological and cultural parameters that determine how the carrying capacities relate to the local densities of skilled individuals in each species. Of special interest is the case of cognitive and intrinsic-demographic equivalence of the two species. The range expansion dynamics may consist of multiple wave fronts of different speeds, each of which originates from a traveling wave solution. Properties of these traveling wave solutions are mathematically derived. Depending on the parameters, these traveling waves can result in replacement of Neanderthals by modern humans, or assimilation of the former by the latter. In both the replacement and assimilation scenarios, the first wave of intrusive modern humans is characterized by a low population density and a low density of skilled individuals, with implications for archaeological visibility. The first invasion is due to weak interspecific competition. A second wave of invasion may be induced by cultural differences between moderns and Neanderthals. Spatially and temporally extended coexistence of the two species, which would have facilitated the transfer of genes from Neanderthal into modern humans and vice versa, is observed in the traveling waves, except when niche overlap between the two species is extremely high. Archaeological findings on the spatial and temporal distributions of the Initial Upper Palaeolithic and the Early Upper Palaeolithic and of the coexistence of Neanderthals and modern humans are discussed.

Existence of Traveling Waves for the Generalized F-KPP Equation

Variation in genotypes may be responsible for differences in dispersal rates, directional biases, and growth rates of individuals. These traits may favor certain genotypes and enhance their spatiotemporal spreading into areas occupied by the less advantageous genotypes. We study how these factors influence the speed of spreading in the case of two competing genotypes under the assumption that spatial variation of the total population is small compared to the spatial variation of the frequencies of the genotypes in the population. In that case, the dynamics of the frequency of one of the genotypes is approximately described by a generalized Fisher-Kolmogorov-Petrovskii-Piskunov (F-KPP) equation. This generalized F-KPP equation with (nonlinear) frequency-dependent diffusion and advection terms admits traveling wave solutions that characterize the invasion of the dominant genotype. Our existence results generalize the classical theory for traveling waves for the F-KPP with constant coefficients. Moreover, in the particular case of the quadratic (monostable) nonlinear growth-decay rate in the generalized F-KPP we study in detail the influence of the variance in diffusion and mean displacement rates of the two genotypes on the minimal wave propagation speed.

An ALE-based finite element model of flagellar motion driven by beating waves: A parametric study

A computational model of flagellar motility is presented using the finite element method. Two-dimensional traveling waves of finite amplitude are propagated down the flagellum and the swimmer is propelled through a viscous fluid according to Newto's second law of motion. Incompressible Navier-Stokes equations are solved on a triangular moving mesh and arbitrary Lagrangian-Eulerian formulation is employed to accommodate the deforming boundaries. The results from the present study are validated against the data available in the literature and close agreement with previous works is found. The effects of wave parameters as well as head morphology on the swimming characteristics are studied for different swimming conditions. We have found that the swimming velocities are linear functions of finite amplitudes and that the rate of work is independent of the channel height for large amplitudes. Furthermore, we have also demonstrated that for the range of wave parameters that are often encountered in human sperm motility studies, the propulsive velocity versus the wavelength exhibits dissimilar trends for different channel heights. Various head configurations were analyzed and it is also observed that wall proximity amplifies the effects induced by different head shapes. By taking non-Newtonian fluids into account, we present new efficiency analyzes through which we have found that the model microorganism swims much more efficiently in shear-thinning fluids.

Donders is dead: cortical traveling waves and the limits of mental chronometry in cognitive neuroscience

An assumption nearly all researchers in cognitive neuroscience tacitly adhere to is that of space–time separability. Historically, it forms the basis of Donders’ difference method, and to date, it underwrites all difference imaging and trial-averaging of cortical activity, including the customary techniques for analyzing fMRI and EEG/MEG data. We describe the assumption and how it licenses common methods in cognitive neuroscience; in particular, we show how it plays out in signal differencing and averaging, and how it misleads us into seeing the brain as a set of static activity sources. In fact, rather than being static, the domains of cortical activity change from moment to moment: Recent research has suggested the importance of traveling waves of activation in the cortex. Traveling waves have been described at a range of different spatial scales in the cortex; they explain a large proportion of the variance in phase measurements of EEG, MEG and ECoG, and are important for understanding cortical function. Critically, traveling waves are not space–time separable. Their prominence suggests that the correct frame of reference for analyzing cortical activity is the dynamical trajectory of the system, rather than the time and space coordinates of measurements. We illustrate what the failure of space–time separability implies for cortical activation, and what consequences this should have for cognitive neuroscience.

Traveling Waves in Spatial SIRS Models

We study traveling wavefront solutions for two reaction-diffusion systems, which are derived respectively as diffusion approximations to two nonlocal spatial SIRS models. These solutions characterize the propagating progress and speed of the spatial spread of underlying epidemic waves. For the first diffusion system, we find a lower bound for wave speeds and prove that the traveling waves exist for all speeds bigger than this bound. For the second diffusion system, we find the minimal wave speed and show that the traveling waves exist for all speeds bigger than or equal to the minimal speed. We further prove the uniqueness (up to translation) of these solutions for sufficiently large wave speeds. The existence of these solutions are proved by a shooting argument combining with LaSalle's invariance principle, and their uniqueness by a geometric singular perturbation argument.