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.     

Modeling activity-dependent synapse restructuring

The spread of electrical activity in a dendritic tree is shaped, in part, by its morphology. Conversely, experimental evidence is growing that electrical and chemical activity can slowly shape the morphology of the dendrite. In this theoretical study, the dendritic spines are dynamic elements, with biophysical properties that change in response to patterns of electrical activity. Recent experiments and diagrammatic models suggest that activity-dependent processes can regulate structural modifications in dendritic spines as well as their distribution along the dendrite. This study considers how local changes in spine structure (minutes to hours) can influence patterns of electrical activity along the dendrite; and how electrical activity due to synaptic events and excitable membrane dynamics can, over time, influence the morphology of the dendrite. The model presents a slow subsystem for structural synaptic plasticity associated with long-term potentiation. A perturbation problem evolves naturally when the spine stem shortens, since the ratio of spine stem resistance to input resistance is small. Hence, the difference between the spine head and dendritic potentials become negligible. This paper presents an asymptotic expansion of head potential in terms of dendritic potential. This leads to a reduced model for post-synaptic restructuring that captures the dynamics of the full model in a briefer computation period when the spines are well connected to the dendrite.     

Spatio-temporal filtering properties of a dendritic cable with active spines: A modeling study in the spike-diffuse-spike framework

The spike-diffuse-spike (SDS) model describes a passive dendritic tree with active dendritic spines. Spine-head dynamics is modeled with a simple integrate-and-fire process, whilst communication between spines is mediated by the cable equation. In this paper we develop a computational framework that allows the study of multiple spiking events in a network of such spines embedded on a simple one-dimensional cable. In the first instance this system is shown to support saltatory waves with the same qualitative features as those observed in a model with Hodgkin-Huxley kinetics in the spine-head. Moreover, there is excellent agreement with the analytically calculated speed for a solitary saltatory pulse. Upon driving the system with time-varying external input we find that the distribution of spines can play a crucial role in determining spatio-temporal filtering properties. In particular, the SDS model in response to periodic pulse train shows a positive correlation between spine density and low-pass temporal filtering that is consistent with the experimental results of Rose and Fortune [1999, ‘Mechanisms for generating temporal filters in the electrosensory system,’ The Journal of Experimental Biology 202: 1281–1289]. Further, we demonstrate the robustness of observed wave properties to natural sources of noise that arise both in the cable and the spine-head, and highlight the possibility of purely noise induced waves and coherent oscillations. Action Editor: Erik De Schutter     

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.     

Propagation of CaMKII translocation waves in heterogeneous spiny dendrites

CaMKII (Ca²⁺-calmodulin-dependent protein kinase II) is a key regulator of glutamatergic synapses and plays an essential role in many forms of synaptic plasticity. It has recently been observed experimentally that stimulating a local region of dendrite not only induces the local translocation of CaMKII from the dendritic shaft to synaptic targets within spines, but also initiates a wave of CaMKII translocation that spreads distally through the dendrite with an average speed of order 1μm/s. We have previously developed a simple reaction–diffusion model of CaMKII translocation waves that can account for the observed wavespeed and predicts wave propagation failure if the density of spines is too high. A major simplification of our previous model was to treat the distribution of spines as spatially uniform. However, there are at least two sources of heterogeneity in the spine distribution that occur on two different spatial scales. First, spines are discrete entities that are joined to a dendritic branch via a thin spine neck of submicron radius, resulting in spatial variations in spine density at the micron level. The second source of heterogeneity occurs on a much longer length scale and reflects the experimental observation that there is a slow proximal to distal variation in the density of spines. In this paper, we analyze how both sources of heterogeneity modulate the speed of CaMKII translocation waves along a spiny dendrite. We adapt methods from the study of the spread of biological invasions in heterogeneous environments, including homogenization theory of pulsating fronts and Hamilton–Jacobi dynamics of sharp interfaces.     

A Compartmental Model for Activity-Dependent Dendritic Spine Branching

Dendritic spines are small, mushroom-like protrusions from the arbor of a neuron in the central nervous system. Interdependent changes in the morphology, biochemistry, and activity of spines have been associated with learning and memory. Moreover, post-mortem cortices from patients with Alzheimer’s or Parkinson’s disease exhibit biochemical and physical alterations within their dendritic arbors and a reduction in the number of dendritic spines. For over a decade, experimentalists have observed perforations in postsynaptic densities on dendritic spines after induction of long-term potentiation, a sustained enhancement of response to a brief electrical or chemical stimulus, associated with learning and memory. In more recent work, some suggest that activity-dependent intraspine calcium may regulate the surface area of the spine head, and reorganization of postsynaptic densities on the surface. In this paper, we develop a model of a dendritic spine with the ability to partition its transmission and receptor zones, as well as the entire spine head. Simulations are initially performed with fixed parameters for morphology to study electrical properties and identify parameters that increase efficacy of the synaptic connection. Equations are then introduced to incorporate calcium as a second messenger in regulating continuous changes in morphology. In the model, activity affects compartmental calcium, which regulates spine head morphology. Conversely, spine head morphology affects the level of local activity, whether the spines are modeled with passive membrane properties, or excitable membrane using Hodgkin–Huxley kinetics. Results indicate that merely separating the postsynaptic receptors on the surface of the spine may add to the diversity of circuitry, but does not change the efficacy of the synapse. However, when the surface area of the spine is a dynamic variable, efficacy of the synapse may vary continuously over time.     

Effects of noise on models of spiny dendrites

We study the effects of noise in two models of spiny dendrites. Through the introduction of different types of noise to both the Spike-diffuse-spike (SDS) and Baer–Rinzel (BR) models we investigate the change in behaviour of the travelling wave solution present in both deterministic systems, as noise intensity increases. We show that the speed of wave propagation in both the SDS and BR models respectively differs as the noise intensity in the spine heads increases. In contrast the cable is very robust to noise and as such the speed shows very little variation from the deterministic system. We introduce a space-dependent spine density, ρ(x), to the original Baer–Rinzel model and show how this modified model can mimic behaviour (under influence of noise) of both original systems, through variation of one parameter. We also show that the correlation time and length scales of the noise can enhance propagation of travelling wave solutions where the white noise dominates the underlying signal and produces noise induced phenomena.     

Subtle re-location of dendritic spine branches containing membrane with fast spiking mechanisms can alter synaptic efficacy

The potential physiological impact of morphological changes in the active dendritic spines, which are believed to be associated with altered synaptic efficacy, was investigated in a computer simulation study using the NEURON package [1]. A compartmental model of a simplified neuron was built, which included 30 complex spines (neck, head, and active zone) and accommodating AMPA-type synaptic inputs with alpha-function conductances. Hodgkin-Huxley type excitable membranes were inserted into the spine heads. It was shown that arranging spines in dense clusters, as opposed to a uniformly random spine distribution, has a negligible effect on the synaptic signal transfer (other model conditions, including synaptic input and spine density, remained unchanged). However, if a proportion (e.g., 3–20%) of the spines partly fuse with their neighbors forming branched spines, this could increase dramatically the cell response to the unchanged synaptic input. Results of this pilot study provide the basis for a more detailed investigation of the relationship between the spine arrangement and synaptic function, considering dual-component synaptic currents and mechanisms controlling ion fluxes in the dendritic compartments.     

A diffusion-activation model of CaMKII translocation waves in dendrites

Ca2+-calmodulin-dependent protein kinase II (CaMKII) is a key regulator of glutamatergic synapses and plays an essential role in many forms of synaptic plasticity. It has recently been observed that stimulating dendrites locally with a single glutamate/glycine puff induces a local translocation of CaMKII into spines that subsequently spreads in a wave-like manner towards the distal dendritic arbor. Here we present a mathematical model of the diffusion, activation and translocation of dendritic CaMKII. We show how the nonlinear dynamics of CaMKII diffusion-activation generates a propagating translocation wave, provided that the rate of activation is sufficiently fast. We also derive an explicit formula for the wave speed as a function of physiological parameters such as the diffusivity of CaMKII and the density of spines. Our model provides a quantitative framework for understanding the spread of CaMKII translocation and its possible role in heterosynaptic plasticity.     

Spatial asynchrony and periodic travelling waves in cyclic populations of field voles.

We demonstrate evidence for the presence of travelling waves in a cyclic population of field voles in northern Britain by fitting simple, empirical models to spatially referenced time series data. Population cycles were broadly synchronous at all sites, but use of Mantel correlations suggested a strong spatial pattern along one axis at a projection line 72 degrees from North. We then fitted a generalized additive model to log population density assuming a fixed-form travelling wave in one spatial dimension for which the density at each site was offset in time by a constant amount from a standard density-time curve. We assumed that the magnitude of this offset would be proportional to the spatial separation between any given site and the centroid of the sampling sites, where separation is the distance between sites in a fixed direction. After fitting this model, we estimated that the wave moved at an average speed of 19 km yr-1, heading from West to East at an angle of 78 degrees from North. Nomadic avian predators which could synchronize populations over large areas are scarce and the travelling wave may be caused by density-dependent dispersal by field voles and/or predation by weasels, both of which act at a suitably small spatial scale.     

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.     

Stability of periodic travelling wave solutions of a nerve conduction equation

Summary Nagumo's nerve conduction equation has travelling wave solutions of pulse type and periodic wave type. We consider the stability of the latter ones. We denote byL(c) the minimum spatial period of a periodic travelling wave solution whose propagation speed isc. It is shown that this travelling wave solution is unstable ifL′(c)<0.     

Analysis of an excitable dendritic spine with an activity-dependent stem conductance

 Dendritic spines are the major target for excitatory synaptic inputs in the vertebrate brain. They are tiny evaginations of the dendritic surface consisting of a bulbous head and a tenuous stem. Spines are considered to be an important locus for plastic changes underlying memory and learning processes. The findings that synaptic morphology may be activity-dependent and that spine head membrane may be endowed with voltage-dependent (excitable) channels is the motivation for this study. We first explore the dynamics, when an excitable, yet morphologically fixed spine receives a constant current input. Two parameter Andronov–Hopf bifurcation diagrams are constructed showing stability boundaries between oscillations and steady-states. We show how these boundaries can change as a function of both the spine stem conductance and the conductance load of the attached dendrite. Building on this reference case an idealized model for an activity-dependent spine is formulated and analyzed. Specifically we examine the possibility that the spine stem resistance, the tunable “synaptic weight” parameter identified by Rall and Rinzel, is activity-dependent. In the model the spine stem conductance depends (slowly) on the local electrical interactions between the spine head and the dendritic cable; parameter regimes are found for bursting, steady states, continuous spiking, and more complex oscillatory behavior. We find that conductance load of the dendrite strongly influences the burst pattern as well as other dynamics. When the spine head membrane potential exhibits relaxation oscillations a simple model is formulated that captures the dynamical features of the full model. Received: 10 January 1997/Revised version: 25 March 1997     

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.     

Peristaltic transport as the travelling deformation waves

Geometry approach to the theoretical and experimental investigations of peristaltic waves based on the travelling deformation waves and wave mass transfer theory (Dobrolyubov, 1991) is presented. The theory of travelling deformation waves is employed to determine uniformed expressions for mass transfer capability parameters of peristalsis. Slow (quasi-static) wave motion is considered which permits not to take into account dynamic phenomena.     

Integrodifference equations in the presence of climate change: persistence criterion,travelling waves and inside dynamics

To understand the effects that the climate change has on the evolution of species as well as the genetic consequences, we analyze an integrodifference equation (IDE) models for a reproducing and dispersing population in a spatio-temporal heterogeneous environment described by a shifting climate envelope. Our analysis on the IDE focuses on the persistence criterion, travelling wave solutions, and the inside dynamics. First, the persistence criterion, characterizing the global dynamics of the IDE, is established in terms of the basic reproduction number. In the case of persistence, a unique travelling wave is found to govern the global dynamics. The effects of the size and the shifting speed of the climate envelope on the basic reproduction number, and hence, on the persistence criterion, are also investigated. In particular, the critical domain size and the critical shifting speed are found in certain cases. Numerical simulations are performed to complement the theoretical results. In the case of persistence, we separate the travelling wave and general solutions into spatially distinct neutral fractions to study the inside dynamics. It is shown that each neutral genetic fraction rearranges itself spatially so as to asymptotically achieve the profile of the travelling wave. To measure the genetic diversity of the population density we calculate the Shannon diversity index and related indices, and use these to illustrate how diversity changes with underlying parameters.

     

Existence and stability of periodic travelling wave solutions to Nagumo's nerve equation

Summary Nagumo's nerve conduction equation has a one-parameter family of spatially periodic travelling wave solutions. First, we prove the existence of these solutions by using a topological method. (There are some exceptional cases in which this method cannot be applied in showing the existence.) A periodic travelling wave solution corresponds to a closed orbit of a third-order dynamical system. The Poincaré index of the closed orbit is determined as a direct consequence of the proof of the existence. Second, we prove that the periodic travelling wave solution is unstable if the Poincaré index of the corresponding closed orbit is + 1. By using this result, together with the result of the author's previous paper, it is concluded that the slow periodic travelling wave solutions are always unstable. Third, we consider the stability of the fast periodic travelling wave solutions. We denote by L(c) the spatial period of the travelling wave solution with the propagation speed c. It is shown that the fast solution is unstable if its period is close to L_min, the minimum of L(c).     

A condition for setting off ectopic waves in computational models of excitable cells

The purpose of this paper is to study the stability of steady state solutions of the Monodomain model equipped with Luo-Rudy I kinetics. It is well established that re-entrant arrhythmias can be created in computational models of excitable cells. Such arrhythmias can be initiated by applying an external stimulus that interacts with a partially refractory region, and spawn breaking waves that can eventually generate extremely complex wave patterns commonly referred to as fibrillation. An ectopic wave is one possible stimulus that may initiate fibrillation. Physiologically, it is well known that ectopic waves exist, but the mechanism for initiating ectopic waves in a large collection of cells is poorly understood. In the present paper we consider computational models of collections of excitable cells in one and two spatial dimensions. The cells are modeled by Luo-Rudy I kinetics, and we assume that the spatial dynamics is governed by the Monodomain model. The mathematical analysis is carried out for a reduced model that is known to provide good approximations of the initial phase of solutions of the Luo-Rudy I model. A further simplification is also introduced to motivate and explain the results for the more complicated models. In the analysis the cells are divided into two regions; one region (N) consists of normal cells as model by the standard Luo-Rudy I model, and another region (A) where the cells are automatic in the sense that they would act as pacemaker cells if they where isolated from their surroundings. We let delta denote the spatial diffusion and a denote a characteristic length of the automatic region. It has previously been shown that reducing diffusion or increasing the automatic region enhances ectopic activity. Here we derive a condition for the transition from stable resting state to ectopic wave spread. Under suitable assumptions on the model we provide mathematical and computational arguments indicating that there is a constant eta such that a steady state solution of this system is stable whenever delta approximately > etaa(2), and unstable whenever delta approximately < etaa(2).