Models describing nonlinear interactions in graded neuron synapses

 An intracellular recording made from the retinal luminosity horizontal cell (LHC) demonstrated that repetitive red flashes enhanced the cell's responsiveness to red stimulus and depressed its responsiveness to green stimulus and that repetitive green flashes suppressed the cell's red response but produced little change in its green response. Based on the idea that the spectral plasticity of LHCs may reflect some synaptic efficacy changes between the LHC and various cones, a self-organizing system is proposed to investigate the possible manner of information processing and storage within the synapses. The results of model analysis suggest that the stimulus-pattern-related spectral plasticity is attributable to the excitatory interaction within the same kinds of synapses and the inhibitory interaction between different kinds of synapses. This system is able to encode and store the history of signal transmission in a graded and cumulative fashion. Received: 7 December 2001 / Accepted: 26 November 2002 / Published online: 28 March 2003 Correspondence to: Pei-Ji Liang (email: pjliang@sjtu.edu.cn, Tel./Fax: (86) 2164070495) Acknowledgements. This research was supported by the National Basic Research Program (G1999054000) of China and the National Foundation of Natural Science of China (No. 30170263).     

A versatile ODE approximation to a network model for the spread of sexually transmitted diseases

 We develop a moment closure approximation (MCA) to a network model of sexually transmitted disease (STD) spread through a steady/casual partnership network. MCA has been used previously to approximate static, regular lattices, whereas application to dynamic, irregular networks is a new endeavour, and application to sociologically-motivated network models has not been attempted. Our goals are 1) to investigate issues relating to the application of moment closure approximations to dynamic and irregular networks, and 2) to understand the impact of concurrent casual partnerships on STD transmission through a population of predominantly steady monogamous partnerships. We are able to derive a moment closure approximation for a dynamic irregular network representing sexual partnership dynamics, however, we are forced to use a triple approximation due to the large error of the standard pair approximation. This example underscores the importance of doing error analysis for moment closure approximations. We also find that a small number of casual partnerships drastically increases the prevalence and rate of spread of the epidemic. Finally, although the approximation is derived for a specific network model, we can recover approximations to a broad range of network models simply by varying model parameters which control the structure of the dynamic network. Thus our moment closure approximation is very flexible in the kinds of network models it can approximate. Received: 26 August 2001 / Revised version: 15 March 2002 / Published online: 23 August 2002 C.T.B. was supported by the NSF. Key words or phrases: Moment closure approximation – Network model – Pair approximation – Sexually transmitted diseases – Steady/casual partnership network     

Neural computations in the tiger salamander and mudpuppy outer retinae and an analysis of GABA action from horizontal cells

 A neural network architecture based on the neural anatomy and function of retinal neurons in tiger salamander and mudpuppy retinae is proposed to study basic aspects of early visual information processing. The model predictions for the main response characteristics of retinal neurons are found to be in agreement with neurophysiological data, including the antagonistic role of horizontal cells in the outer plexiform layer. The examination of possible γ-aminobutyric acid (GABA) action from horizontal cells suggests that GABA_A alone, GABA_B alone, or their weighted combination can generate the response characteristics observed in bipolar cells. Received: 25 June 2002 / Accepted: 28 January 2003 / Published online: 20 May 2003 Acknowledgements. The authors would like to thank an anonymous reviewer for valuable comments. Correspondence to: S. X. Yang (e-mail: syang@uoguelph.ca)     

Object segmentation model: analytical results and biological implications

 A simple, biologically motivated neural network for segmentation of a moving object from a visual scene is presented. The model consists of two parts: an object selection model which employs a scaling approach for receptive field sizes, and a subsequent network implementing a spotlight by means of multiplicative synapses. The network selects one object out of several, segments the rough contour of the object, and encodes the winner object's position with high accuracy. Analytical equations for the performance level of the network, e.g., for the critical distance of two objects above which they are perceived as separate, are derived. The network preferentially chooses the object with the largest angular velocity and the largest angular width. An equation for the velocity and width preferences is presented. Additionally it is shown that for certain neurons of the model, flat receptive fields are more favourable than Gaussian ones. The network exhibits performances similar to those known from amphibians. Various electrophysiological and behavioral results – e.g., the distribution of the diameters of the receptive fields of tectal neurons, of the tongue-projecting salamander Hydromantes italicus and the range of optimal prey velocities for prey catching – can be understood on the basis of the model. Received: 7 December 2000 / Accepted: 13 February 2001     

Exact sampling formulas for multi-type Galton-Watson processes

 Exact formulas for the mean and variance of the proportion of different types in a fixed generation of a multi-type Galton-Watson process are derived. The formulas are given in terms of iterates of the probability generating function of the offspring distribution. It is also shown that the sequence of types backwards from a randomly sampled particle in a fixed generation is a non-homogeneous Markov chain where the transition probabilities can be given explicitly, again in terms of probability generating functions. Two biological applications are considered: mutations in mitochondrial DNA and the polymerase chain reaction. Received: 10 June 2001 / Revised version: 21 November 2001 / Published online: 23 August 2002 Mathematics Subject Classification (2000): Primary 60J80, Secondary 92D10, 92D25 Key words or phrases: Multi-type Galton-Watson process – sampling formula – PCR – mitochondrial DNA     

Traveling-wave pattern generator controls movement and organization of sensory feedback in a spinal cord model

 A traveling wave in a two-dimensional spinal cord model constitutes a stable pattern generator for quadruped gaits. In the context of the somatotopic organization of the spinal cord, this pattern generator is sufficient to generate stable locomotive limb trajectories. The elastic properties of muscles alone, providing linear negative feedback, are sufficient to stabilize stance and locomotion in the presence of perturbative forces. We further show that such a pattern generator is capable of organizing sensory processing in the spinal cord. A single-layer perceptron was trained to associate the sensory feedback from the limb (coding force, length, and change of length for each muscle) with the two-dimensional activity profile of the traveling wave. This resulted in a well-defined spatial organization of the connections within the spinal network along a rostrocaudal axis. The spinal network driven by peripheral afferents alone supported autonomous locomotion in the positive feedback mode, whereas in the negative feedback mode stance was stabilized in response to perturbations. Systematic variation of a parameter representing the effect of gamma-motor neurons on muscle spindle activity in our model led to a corresponding shift of limb position during stance and locomotion, resulting in a systematic displacement alteration of foot positions. Received: 30 July 2001 / Accepted in revised form: 17 April 2002 Correspondence to: A. Kaske (e-mails: alexander.kaske@mtc.ki.se, alexander.kaske@vglab.com)     

A discrete,size-structured model of phytoplankton growth in the chemostat: introduction of inhomogeneous cell division size

 We introduce inhomogeneous, substrate dependent cell division in a time discrete, nonlinear matrix model of size-structured population growth in the chemostat, first introduced by Gage et al. [8] and later analysed by Smith [13]. We show that mass conservation is verified, and conclude that our system admits one non zero globally stable equilibrium, which we express explicitly. Then we run numerical simulations of the system, and compare the predictions of the model to data related to phytoplankton growth, whose obtention we discuss. We end with the identification of several parameters of the system. Received: 9 February 2000 / Revised version: 10 October 2001 / Published online: 23 August 2002 RID="*" ID="*" Present address: Department of Mathematics and Statistics, University of Victoria, B.C., Canada. e-mail: jarino@math.uvic.ca Key words or phrases: Chemostat – Structured population models – Discrete model – Inhomogeneous division size     

Year class coexistence or competitive exclusion for strict biennials?

 We consider a discrete time model of semelparous biennial population dynamics. Interactions between individuals are modelled with the aid of an ``environmental' variable I. The impact on and the sensitivity to the environmental condition is age specific. The main result is that competitive exclusion between the year classes is possible as is their coexistence. For moderate values of the basic reproduction ratio R ₀ there is a strict dichotomy: depending on the other parameters we either find competitive exclusion or coexistence. We characterize rather precisely the patterns of age specific impact and sensitivity that lead to either of these outcomes. Received: 13 July 2001 / Revised version: 26 June 2002 / Published online: 19 November 2002 Key words or phrases: Competitive exclusion – Semelparous species – Periodical insects     

Cattaneo models for chemosensitive movement

 We derive models for chemosensitive movement based on Cattaneo's law of heat propagation with finite speed. We apply the model to pattern formation as observed in experiments with Dictyostelium discoideum, with Salmonella typhimurium and with Escherichia coli. For Salmonella typhimurium we make predictions on pattern formation which can be tested in experiments. We discuss the relations of the Cattaneo models to classical models and we develop an effective numerical scheme. Received: 8 October 2001 / Revised version: 2 August 2002 / Published online: 19 November 2002 Key words or phrases: Chemotaxis – Aggregation – Cattaneo model – Numerical schemes Acknowledgements. We are very grateful for comments of S. Noelle concerning the numerical scheme. We thank K.P. Hadeler and C. Schmeiser for helpful remarks. The research was supported by the Deutsche Forschungsgemeinschaft, research project ANumE and the Austrian Science Foundation, grant no. W008.     

Monotone travelling fronts in an age-structured reaction-diffusion model of a single species

 We consider a partially coupled diffusive population model in which the state variables represent the densities of the immature and mature population of a single species. The equation for the mature population can be considered on its own, and is a delay differential equation with a delay-dependent coefficient. For the case when the immatures are immobile, we prove that travelling wavefront solutions exist connecting the zero solution of the equation for the matures with the delay-dependent positive equilibrium state. As a perturbation of this case we then consider the case of low immature diffusivity showing that the travelling front solutions continue to persist. Our findings are contrasted with recent studies of the delayed Fisher equation. Travelling fronts of the latter are known to lose monotonicity for sufficiently large delays. In contrast, travelling fronts of our equation appear to remain monotone for all values of the delay. Received: 1 November 2001 / Revised version: 10 May 2002 / Published online: 23 August 2002 Mathematics Subject Classification (2000): 35K57, 92D25 Key words or phrases: Age-structure – Time-delay – Travelling Fronts – Reaction-diffusion     

Sensor fusion by neural networks using spatially represented information

 A neural network model based on a lateral-inhibition-type feedback layer is analyzed with regard to its capabilities to fuse signals from two different sensors reporting the same event (“multisensory convergence”). The model consists of two processing stages. The input stage holds spatial representations of the sensor signals and transmits them to the second stage where they are fused. If the input signals differ, the model exhibits two different processing modes: with small differences it produces a weighted average of the input signals, whereas with large differences it enters a decision mode where one of the two signals is suppressed. The dynamics of the network can be described by a series of two first-order low-pass filters, whose bandwidth depends nonlinearly on the level of concordance of the input signals. The network reduces sensor noise by means of both its averaging and filtering properties. Hence noise suppression, too, depends on the level of concordance of the inputs. When the network's neurons have internal noise, sensor noise suppression is reduced but still effective as long as the input signals do not differ strongly. The possibility of extending the scheme to three and more inputs is discussed. Received: 2 August 2000 / Accepted in revised form: 3 May 2001     

Modeling species richness controlled by community-intrinsic and community-extrinsic processes: coastal fish communities as an example

 This article focuses on the analysis of coastal fish communities along the Norwegian Skagerrak coast. Species numbers are estimated based on annual samples of the fish communities within 12 fjords from 1953 to 1994. On this basis, a community dynamics model (incorporating both community-intrinsic and community-extrinsic processes) was developed and analyzed. This model is then discussed on the basis of other community models available through the literature, both phenomenologically oriented and process-oriented models. Received: January 17, 2002 / Accepted: May 13, 2002 Acknowledgments We thank Dr. Masakado Kawata for the invi-tation to present this paper at the 19th Symposium of the Society of Population Ecology held in Yamagata, Japan, October 26–28, 2001: “Evolution of Biodiversity: Theories and Facts.” Valuable input was provided after the presentation at this meeting, which we greatly appreciated. The reformulation of the model in terms of ΔS was kindly suggested to us by Prof. Joan Roughgarden. Thanks to Dr. Hildegunn Viljugrein for advice on the BUGS analyses and to two anonymous reviewers for constructive comments. This work has been supported by grants from the Norwegian Science Council (NFR). Correspondence to:N.C. Stenseth     

Limits of a multi-patch SIS epidemic model

 We start from a stochastic SIS model for the spread of epidemics among a population partitioned into M sites, each containing N individuals; epidemic spread occurs through within-site (`local') contacts and global contacts. We analyse the limit behaviour of the system as M and N increase to ∞. Two limit procedures are considered, according to the order in which M and N go to ∞; independently of the order, the limiting distribution of infected individuals across sites is a probability measure, whose evolution in time is governed by the weak form of a PDE. Existence and uniqueness of the solutions to this problem is shown. Finally, it is shown that the infected distribution converges, as time goes to infinity, to a Dirac measure at the value x ^* , the equilibrium of a single-patch SIS model with contact rate equal to the sum of local and global contacts. Received: 18 July 2001 / Revised version: 16 March 2002 / Published online: 26 September 2002 Mathematics Subject Classification (2000): 92D30, 60F99 Key words or phrases: SIS epidemic – Metapopulation – Markov population processes – Weak convergence of measures     

Existence of traveling wave solutions in a diffusive predator-prey model

 We establish the existence of traveling front solutions and small amplitude traveling wave train solutions for a reaction-diffusion system based on a predator-prey model with Holling type-II functional response. The traveling front solutions are equivalent to heteroclinic orbits in R ⁴ and the small amplitude traveling wave train solutions are equivalent to small amplitude periodic orbits in R ⁴ . The methods used to prove the results are the shooting argument and the Hopf bifurcation theorem. Received: 25 May 2001 / Revised version: 5 August 2002 / Published online: 19 November 2002 RID="*" ID="*" Research was supported by the National Natural Science Foundations (NNSF) of China. RID="*" ID="*" Research was partially supported by the Natural Sciences and Engineering Research Council (NSERC) of Canada. On leave from the Department of Mathematics and Statistics, Dalhousie University, Halifax, Nova Scotia B3H 3J5, Canada. Mathematics Subject Classification (2000): 34C35, 35K57 Key words or phrases: Traveling wave solution – Wazewski set – Shooting argument – Hopf bifurcation Acknowledgements. We would like to thank the two referees for their careful reading and helpful comments.     

Neuronal network modelling of the effects of anaesthetic agents on somatosensory pathways

 The whole question of consciousness, awareness and depth of anaesthesia is both timely, little understood and deeply challenging. Models of the underlying neural pathway mechanisms/dynamics are necessary for understanding the interactions involved and their structure and function. A neuronal network of the somatosensory pathways is proposed in this paper based on experimental information and physiological investigation into anaesthesia. Existing mathematical neuronal models from the literature have been modified and then employed to describe the dynamics of the proposed pathway network. Effects of anaesthetic agents on the cortex were simulated in the model which describes the evoked cortical responses. By comparison with responses from anaesthetised rats, the model's responses are able to describe the dynamics of typical responses. Thus, the proposed model promises to be valuable for investigating the mechanisms of anaesthesia on the cortex and the effects of brain lesions. Received: 4 March 2002 / Accepted in revised form: 8 July 2002 Correspondence to: D. A. Linkens (e-mail: d.linkens@sheffield.ac.uk, Tel.: +44-114-2225133, Fax: +44-114-2731729) Acknowledgements. C.H. Ting was supported by a postgraduate scholarship from the University of Sheffield.     

A nonlinear dynamical mechanism for bruit generation by an intracranial saccular aneurysm

 Intracranial saccular aneurysms have been clinically observed to emit a transient sound, a bruit, on each heartbeat. The mechanism causing the bruits has been a matter of contention. A qualitative analysis of the nonlinear dynamical properties of the Shah-Humphrey model for periodic pressure forcing of a thin-necked saccular aneurysm, using the Fung nonlinear constitutive model for the aneurysm material, shows that a small blood pressure jump on each beat, whether the pressure is weakly aperiodic or periodic, induces transients in the radial deformation response of the aneurysmal wall on each heartbeat. These transient vibrations, which have a component with frequency near the natural frequency of the system but are not resonant phenomena and which decay rapidly to a limit cycle during each distinct forcing pressure cycle, can generate the bruits. Received: 21 November 2000 / Revised version: 9 August 2001 / Published online: 23 August 2002 Mathematics Subject Classification (2000): 92B99, 70K40, 70K05 Key words or phrases: Intracranial saccular aneurysm – Bruit – Spectrum – Nonlinear dynamics – Transients – Vortex shedding – Fung model     

Development of a biologically-based controlled growth and differentiation model for developmental toxicology

 A mathematical model is developed with a highly controlled birth and death process for precursor cells. This model is both biologically- and statistically-based. The controlled growth and differentiation (CGD) model limits the number of replications allowed in the development of a tissue or organ and thus, more closely reflects the presence of a true stem cell population. Leroux et al. (1996) presented a biologically-based dose-response model for developmental toxicology that was derived from a partial differential equation for the generating function. This formulation limits further expansion into more realistic models of mammalian development. The same formulae for the probability of a defect (a system of ordinary differential equations) can be derived through the Kolmogorov forward equations due to the nature of this Markov process. This modified approach is easily amenable to the expansion of more complicated models of the developmental process such as the one presented here. Comparisons between the Leroux et al. (1996) model and the controlled growth and differentiation (CGD) model as developed in this paper are also discussed. Received: 8 June 2001 / Revised version: 15 June 2002 / Published online: 26 September 2002 Keywords or phrases: Teratology – Multistate process – Cellular kinetics – Numerical simulation     

Taeniolella rudis and Taeniolella longissima sp. nov. with secondary sympodioconidia from freshwater habitats

 Taeniolella rudis, with secondary sympodioconidia, is illustrated and described and its taxonomic assignment evaluated. A second new species, Taeniolella longissima, is described and illustrated. The ecological role of these fungi in the colonization of wood in freshwater habitats is discussed. Received: August 16, 2001 / Accepted: February 8, 2002     

The onset of oscillatory dynamics in models of multiple disease strains

 We examine a generalised SIR model for the infection dynamics of four competing disease strains. This model contains four previously-studied models as special cases. The different strains interact indirectly by the mechanism of cross-immunity; individuals in the host population may become immune to infection by a particular strain even if they have only been infected with different but closely related strains. Several different models of cross-immunity are compared in the limit where the death rate is much smaller than the rate of recovery from infection. In this limit an asymptotic analysis of the dynamics of the models is possible, and we are able to compute the location and nature of the Takens–Bogdanov bifurcation associated with the presence of oscillatory dynamics observed by previous authors. Received: 5 December 2001 / Revised version: 5 May 2002 / Published online: 17 October 2002 Keywords or phrases: Infection – Pathogen – Epidemiology – Multiple strains – Cross-immunity – Oscillations – Dynamics – Bifurcations     

On spreading speeds and traveling waves for growth and migration models in a periodic habitat

 It is shown that the methods previously used by the author [Wei82] and by R. Lui [Lui89] to obtain asymptotic spreading results and sometimes the existence of traveling waves for a discrete-time recursion with a translation invariant order preserving operator can be extended to a recursion with a periodic order preserving operator. The operator can be taken to be the time-one map of a continuous time reaction-diffusion model, or it can be a more general model of time evolution in population genetics or population ecology in a periodic habitat. Methods of estimating the speeds of spreading in various directions will also be presented. Received: 12 July 2001 / Revised version: 19 July 2002 / Published online: 17 October 2002 Mathematics Subject Classification (2000): 92D40, 92D25, 35K55, 35K57, 35B40 Keywords or phrases: Periodic – Spreading speed – Traveling wave