Rotating wave solutions of the FitzHugh–Nagumo equations

This paper will treat the bifurcation and numerical simulation of rotating wave (RW) solutions of the FitzHugh-Nagumo (FHN) equations. These equations are often used as a simple mathematical model of excitable media. The dependence of the solutions on a uniformly applied current, and also on the diffusion coefficients or domain size will be studied. Ranges of applied current and/or diffusion coefficients in which RW solutions are observed will be described using bifurcation theory and continuation methods. The bifurcation of time-periodic solutions of these FHN equations without diffusion is described first. Similar methods are then used to find RW solutions on a circular ring and numerical simulations are described. These results are then extended to investigate RW solutions on annular rings of finite cross-section. Scaling arguments are used to show how the existence of solutions depends on either the diffusion coefficient or on the size of the region.     

Weak periodic signal detection by sine-Wiener-noise-induced resonance in the FitzHugh–Nagumo neuron

Based on the FitzHugh–Nagumo (FHN) neuron model subjected to sine-Wiener (SW) noise, impacts of SW noise on weak periodic signal detection are investigated by calculating response measure Q for characterizing synchronization between the input signal and the output temporal activities of the neuron. It is numerically demonstrated that the response measure Q can achieve the optimal value under appropriate and moderate intensity or correlation time of SW noise, suggesting the occurrence of SW-noise-induced stochastic resonance. Furthermore, the optimal value of Q is sensitive to correlation time. Consequently, the correlation time of SW noise has a great influence on the performance of signal detection in the FHN neuron.     

Reconstructing parameters of the FitzHugh–Nagumo system from boundary potential measurements

We consider distributed parameter identification problems for the FitzHugh–Nagumo model of electrocardiology. The model describes the evolution of electrical potentials in heart tissues. The mathematical problem is to reconstruct physical parameters of the system through partial knowledge of its solutions on the boundary of the domain. We present a parallel algorithm of Newton–Krylov type that combines Newton’s method for numerical optimization with Krylov subspace solvers for the resulting Karush–Kuhn–Tucker system. We show by numerical simulations that parameter reconstruction can be performed from measurements taken on the boundary of the domain only. We discuss the effects of various model parameters on the quality of reconstructions. Action Editor: David Terman     

Instability in a generalized Keller–Segel model

We present a generalized Keller–Segel model where an arbitrary number of chemical compounds react, some of which are produced by a species, and one of which is a chemoattractant for the species. To investigate the stability of homogeneous stationary states of this generalized model, we consider the eigenvalues of a linearized system. We are able to reduce this infinite dimensional eigenproblem to a parametrized finite dimensional eigenproblem. By matrix theoretic tools, we then provide easily verifiable sufficient conditions for destabilizing the homogeneous stationary states. In particular, one of the sufficient conditions is that the chemotactic feedback is sufficiently strong. Although this mechanism was already known to exist in the original Keller–Segel model, here we show that it is more generally applicable by significantly enlarging the class of models exhibiting this instability phenomenon which may lead to pattern formation.     

Advection–diffusion equations for generalized tactic searching behaviors

 Many organisms search for limiting resources by using repeated responses to local cues, which cumulatively cause movement towards more favorable parts of their environment. This paper presents a general asymptotic expression, derived under the assumption of shallow environmental gradients, for the population-level flux of organisms moving at a constant speed and reorienting at rates determined by the environmental conditions experienced since the last reorientation. The expression takes the form of an advection-diffusion equation, in which the diffusivity and advection velocity are determined by statistics of the turning algorithm that are directly comparable to empirical observations. This work provides a mechanism with which to systematically evaluate a wide variety of tactic and kinetic strategies for determining turning behaviors. The model is applied to searchers on spatially-variable, random distributions of discrete resource patches. Such algorithms are functions of the integrated resource density encountered between turns. It is shown that behaviors in which the turning time distribution is a function of integrated density cannot result in taxis. In contrast, behaviors in which the turning rate is a function of integrated density can result in taxis. These two classes of search algorithm differ in that the latter requires the searcher to “learn” about its local environment, whereas the former requires no such assessment. This suggests neural or physiological mechanisms for remembering previous encounters may be a biological requirement for searchers on discrete resource distributions. Received: 21 September 1995/Revised version: 18 July 1996     

A generalized cholera model and epidemic–endemic analysis

The transmission of cholera involves both human-to-human and environment-to-human pathways that complicate its dynamics. In this paper, we present a new and unified deterministic model that incorporates a general incidence rate and a general formulation of the pathogen concentration to analyse the dynamics of cholera. Particularly, this work unifies many existing cholera models proposed by different authors. We conduct equilibrium analysis to carefully study the complex epidemic and endemic behaviour of the disease. Our results show that despite the incorporation of the environmental component, there exists a forward transcritical bifurcation at R ₀=1 for the combined human–environment epidemiological model under biologically reasonable conditions.     

Neural Signatures: Multiple Coding in Spiking–bursting Cells

Recent experiments have revealed the existence of neural signatures in the activity of individual cells of the pyloric central pattern generator (CPG) of crustacean. The neural signatures consist of cell-specific spike timings in the bursting activity of the neurons. The role of these intraburst neural fingerprints is still unclear. It has been reported previously that some muscles can reflect small changes in the spike timings of the neurons that innervate them. However, it is unclear to what extent neural signatures contribute to the command message that the muscles receive from the motoneurons. It is also unknown whether the signatures have any functional meaning for the neurons that belong to the same CPG or to other interconnected CPGs. In this paper, we use realistic neural models to study the ability of single cells and small circuits to recognize individual neural signatures. We show that model cells and circuits can respond distinctly to the incoming neural fingerprints in addition to the properties of the slow depolarizing waves. Our results suggest that neural signatures can be a general mechanism of spiking–bursting cells to implement multicoding. An erratum to this article can be found at     

Melatonin ameliorates sleep–wake disturbances and autism-like behaviors in the Ctnnd2 knock out mouse model of autism spectrum disorders

Autism spectrum disorder (ASD) is a prevalent neurodevelopmental disorder characterized by atypical patterns of social interaction and communication, as well as restrictive and repetitive behaviors. In addition, patients with ASD often presents with sleep disturbances. Delta (δ) catenin protein 2 (CTNND2) encodes δ-catenin protein, a neuron-specific catenin implicated in many complex neuropsychiatric diseases. Our previous study demonstrated that the deletion of Ctnnd2 in mice led to autism-like behaviors. However, to our knowledge, no study has investigated the effects of Ctnnd2 deletion on sleep in mice. In this study, we investigated whether the knockout (KO) of exon 2 of the Ctnnd2 gene could induce sleep–wake disorders in mice and identified the effects of oral melatonin (MT) supplementation on Ctnnd2 KO mice. Our results demonstrated that the Ctnnd2 KO mice exhibited ASD-like behaviors and sleep–wake disorders that were partially attenuated by MT supplementation. Overall, our current study is the first to identify that knockdown of Ctnnd2 gene could induce sleep–wake disorders in mice and suggests that treatment of sleep–wake disturbances by MT may benefit to autism-like behaviors causing by Ctnnd2 gene deletion.     

Characteristics of Symmetric Surface Plasmon Polariton Mode in Glass–Metal–Glass Waveguide

The propagation characteristics of symmetric surface plasmon polariton mode in a glass–metal–glass waveguide are presented. Gallium lanthanum sulfide has been taken as the glass and silver (Ag) has been used as the metal. The analysis has been done both numerically and analytically. A two-dimensional finite-difference time-domain-based simulation model has been developed in order to analyze the propagation characteristics numerically. The obtained results using numerical and analytical methods have been compared and a very good agreement has been found.     

Using the NZ–DNDC model to estimate agricultural N2O emissions in the Manawatu–Wanganui region

Nitrous oxide (N₂O) emissions are difficult to quantify at regional and national scales. There is considerable spatial and temporal variability in N₂O emissions from soil, partly because of variability in the underlying biogenic processes responsible for soil N₂O production. The process-based NZ–DNDC (New Zealand Denitrification-Decomposition) model was used, with georeferenced input data on soils, climate and land use, to map and predict net N₂O emissions from farming in the Manawatu–Wanganui region. The Manawatu–Wanganui region has a temperate, maritime climate and the major agricultural land use is pastoral grazing. We created databases of regional soil, climate and farm management information from various available data sources including national databases of climate, soil type and land use, and national agricultural production statistics. The error introduced by upscaling the model was assessed by comparing results using measured site data with the corresponding predictions using the regional approximations. We also examined the effect of climate conditions by rerunning the 2003 simulation using the climate data for the years ended June 1990 and 2004. The modelled net N₂O emissions for this region for the year ended June 2003 were 4.6?±?1.5 Gg N₂O–N per year. The total fertiliser and excretal N inputs for the region were approximately 224,140 tonnes, so the percentage emitted as N₂O was 2.0?±?0.7%. The modelled net N₂O emissions for the region for the year ended June 1990 were 3.8?±?2.1 Gg N₂O–N per year, indicating annual net N₂O emissions in the Manawatu–Wanganui region between 1990 and 2003 had increased by 0.8?±?0.6 Gg N₂O–N (an increase of about 20%). This change can be attributed to both changes in weather conditions and land use and farm management between 1990 and 2003.     

Structure of the UreD–UreF–UreG–UreE complex in Helicobacter pylori: a model study

The molecular details of the protein complex formed by UreD, UreF, UreG, and UreE, accessory proteins for urease activation in the carcinogenic bacterium Helicobacter pylori, have been elucidated using computational modeling. The calculated structure of the complex supports the hypothesis of UreF acting as a GTPase activation protein that facilitates GTP hydrolysis by UreG during urease maturation, and provides a rationale for the design of new drugs against infections by ureolytic bacterial pathogens.     

A reaction–diffusion malaria model with incubation period in the vector population

Malaria is one of the most important parasitic infections in humans and more than two billion people are at risk every year. To understand how the spatial heterogeneity and extrinsic incubation period (EIP) of the parasite within the mosquito affect the dynamics of malaria epidemiology, we propose a nonlocal and time-delayed reaction–diffusion model. We then define the basic reproduction ratio R₀{\mathcal{R}_0} and show that R₀{\mathcal{R}_0} serves as a threshold parameter that predicts whether malaria will spread. Furthermore, a sufficient condition is obtained to guarantee that the disease will stabilize at a positive steady state eventually in the case where all the parameters are spatially independent. Numerically, we show that the use of the spatially averaged system may highly underestimate the malaria risk. The spatially heterogeneous framework in this paper can be used to design the spatial allocation of control resources.     

On the effect of sharp rises in blood pressure in the Shah–Humphrey model for intracranial saccular aneurysms

We consider the model originally proposed by Shah and Humphrey (J Biomech 32:593–599, 1999) for a class of intracranial saccular aneurysms and show that for constant pressure the addition of the viscoelastic term corresponding to the presence of cerebral spinal fluid outside the membrane, no matter how small, does ensure convergence to an equilibrium. Our arguments apply to a general equation of this type, and thus also hold for variations of this model such as that proposed by David and Humphrey (J Biomech 36:1143–1150, 2003). On the other hand, it is known that the presence of damping may destabilize periodic orbits of periodically forced systems or even prevent them from existing altogether. We present numerical simulations showing that for some forcing terms the high-frequency oscillations do not die out with time, and a much more complex behaviour may emerge as a discontinuous forcing term is approached. The key point for this situation to occur is related to the derivative of the forcing term, supporting the hypothesis that sharper rises (or falls) in blood pressure may increase the risk of aneurysm rupture.     

White-noise stimulation of the Hodgkin–Huxley model

 We studied the combined influence of noise and constant current stimulations on the Hodgkin–Huxley neuron model through time and frequency analysis of the membrane-potential dynamics. We observed that, in agreement with experimental data (Guttman et al. 1974), at low noise and low constant current stimulation the behavior of the model is well approximated by that of the linearized Hodgkin–Huxley system. Conversely, nonlinearities due to firing dominate at large noise or current stimulations. The transition between the two regimes is abrupt, and takes place in the same range of noise and current intensities as the noise-induced transition characterized by the qualitative change in the stationary distribution of the membrane potential (Tanabe and Pakdaman 2001a). The implications of these results are discussed. Received: 27 July 2001 / Accepted in revised form: 18 December 2001     

Does Actaea asiatica Have the Most Symmetric and Primitive Karyotype in the Ranunculaceae?

Actaea asiatica was previously reported to have the most symmetric and primitive karyotype, consisting of 10 m- and 6 sm-chromosomes, which is quite different from those of the remaining species in the genus Actaea, consisting of 10 m-, 4 sm- and two T-chromo somes. In this paper, the chromosomes of this species were re-examined. The results show that Actaea asiatica has the same karyotype as the other species in the genus. Compared with the species in other genera in the tribe Cimicifugeae, i.e. Beesia, Anemonopsis, Souliea and Cimicifuga, Actaea asiatica, together with the remaining species of the genus, has the most asymmetric and thus probably the most advanced karyotype in this tribe because of the presence of two T- chromosomes in their chromosome complements. The two T- chromosomes may serve as one of the most important cytological markers, by which the species in Actaea are clearly distinguishable cytologically from those in Beesia, Anemonop-sis, Souliea and Cimicifuga.     

The minimal model of the hypothalamic–pituitary–adrenal axis

This paper concerns ODE modeling of the hypothalamic–pituitary– adrenal axis (HPA axis) using an analytical and numerical approach, combined with biological knowledge regarding physiological mechanisms and parameters. The three hormones, CRH, ACTH, and cortisol, which interact in the HPA axis are modeled as a system of three coupled, nonlinear differential equations. Experimental data shows the circadian as well as the ultradian rhythm. This paper focuses on the ultradian rhythm. The ultradian rhythm can mathematically be explained by oscillating solutions. Oscillating solutions to an ODE emerges from an unstable fixed point with complex eigenvalues with a positive real parts and a non-zero imaginary parts. The first part of the paper describes the general considerations to be obeyed for a mathematical model of the HPA axis. In this paper we only include the most widely accepted mechanisms that influence the dynamics of the HPA axis, i.e. a negative feedback from cortisol on CRH and ACTH. Therefore we term our model the minimal model. The minimal model, encompasses a wide class of different realizations, obeying only a few physiologically reasonable demands. The results include the existence of a trapping region guaranteeing that concentrations do not become negative or tend to infinity. Furthermore, this treatment guarantees the existence of a unique fixed point. A change in local stability of the fixed point, from stable to unstable, implies a Hopf bifurcation; thereby, oscillating solutions may emerge from the model. Sufficient criteria for local stability of the fixed point, and an easily applicable sufficient criteria guaranteeing global stability of the fixed point, is formulated. If the latter is fulfilled, ultradian rhythm is an impossible outcome of the minimal model and all realizations thereof. The second part of the paper concerns a specific realization of the minimal model in which feedback functions are built explicitly using receptor dynamics. Using physiologically reasonable parameter values, along with the results of the general case, it is demonstrated that un-physiological values of the parameters are needed in order to achieve local instability of the fixed point. Small changes in physiologically relevant parameters cause the system to be globally stable using the analytical criteria. All simulations show a globally stable fixed point, ruling out periodic solutions even when an investigation of the ‘worst case parameters’ is performed.     

An in vitro model quantifying the effect of calcification on the tissue–stent interaction in a stenosed aortic root

Transcatheter Aortic Valves rely on the tissue-stent interaction to ensure that the valve is secured within the aortic root. Aortic stenosis presents with heavily calcified leaflets and it has been proposed that this calcification also acts to secure the valve, but this has never been quantified. In this study, we developed an in vitro calcified aortic root model to quantify the role of calcification on the tissue-stent interaction. The in vitro model incorporated artificial calcifications affixed to the leaflets of porcine aortic heart valves. A self-expanding nitinol braided stent was deployed into non-calcified and artificially calcified porcine aortic roots and imaged by micro computed tomography. Mechanical tests were then conducted to dislodge the stent from the aortic root and it was found that, in the presence of calcification, there was a significant increase in pullout force (8.59 ± 3.68 N vs. 2.84 ± 1.55 N p = 0.045), stent eccentricity (0.05 ± 0.01 vs. 0.02 ± 0.01, p = 0.049), and coefficient of friction between the stent and aortic root (0.36 ± 0.12 vs. 0.09 ± 0.05, p = 0.018), when compared to non-calcified roots. This study quantifies for the first time the impact of calcification on the friction between the aortic tissue and transcatheter aortic valve stent, showing the role of calcification in anchoring the valve stent in the aortic root.     

How to model the local interaction in the predator–prey system at slow diffusion in a heterogeneous environment?

We examine the nonlinear reaction–diffusion–advection equations to modeling of the predator–prey system under heterogeneous carrying capacity of the prey, and Holling type II functional response. When advection and diffusion fluxes are absent or small, we detect the discrepancy between the resource (carrying capacity) and species distributions. The large diffusion eliminates this effect. We propose a modification of the functional response coefficients to provide the correlation between species distribution and resource in both cases. The numerical simulation of several models both under small and moderate advection–diffusion fluxes is carried out.