Simple models for excitable and oscillatory neural networks

 Chains of coupled oscillators of simple “rotator” type have been used to model the central pattern generator (CPG) for locomotion in lamprey, among numerous applications in biology and elsewhere. In this paper, motivated by experiments on lamprey CPG with brainstem attached, we investigate a simple oscillator model with internal structure which captures both excitable and bursting dynamics. This model, and that for the coupling functions, is inspired by the Hodgkin–Huxley equations and two-variable simplifications thereof. We analyse pairs of coupled oscillators with both excitatory and inhibitory coupling. We also study traveling wave patterns arising from chains of oscillators, including simulations of “body shapes” generated by a double chain of oscillators providing input to a kinematic musculature model of lamprey.. Received: 25 November 1996 / Revised version: 9 December 1997     

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     

An improved algorithm for the detection of dynamical interdependence in bivariate time-series

 We describe a new algorithm for the detection of dynamical interdependence in bivariate time-series data sets. By using geometrical and dynamical arguments, we produce a method that can detect dynamical interdependence in weakly coupled systems where previous techniques have failed. We illustrate this by comparison of our algorithm with another commonly used technique when applied to a system of coupled Hénon maps. In addition, an improvement of ∼20% in the detection rate is observed when the technique is applied to human scalp EEG data, as compared with existing techniques. Such an improvement may assist an understanding of the role of large-scale nonlinear processes in normal brain function. Received: 28 December 2001 / Accepted in revised form: 2 October 2002 Correspondence to: J.R. Terry (e-mail: J.R.Terry@lboro.ac.uk) Acknowledgements. JRT acknowledges the support of the Royal Society and the London Mathematical Society for jointly funding a trip to the School of Physics at the University of Sydney and The Brain Dynamics Centre at Westmead Hospital, Sydney. The authors thank Professor Peter Robinson for useful conversations and the provision of local facilities within the School of Physics during the visit of JRT. The support of the EPSRC via Grant GR/N00340 is also acknowledged. MJB is a recipient of a University Postgraduate Award from the University of Sydney.     

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 theory of hippocampal memory based on theta phase precession

 The neural dynamics of the hippocampal network for memory encoding of novel temporal sequences is proposed based on the theta rhythm modulated firing of place cells called theta phase precession. It is hypothesized that theta phase precession is generated at the entorhinal cortex by phase locking between local field theta oscillation and neural oscillators and that the hippocampal closed network with feedforward and backward projections employ theta phase precession to create selectivity in the associative connections needed for temporal sequence storage. Our analyses and computer experiments reveal that the phase precession generated by phase locking instantaneously endows stable phase relations among neural activities in the successively changing neural population. It is concluded that theta phase precession provides biologically plausible dynamics for the memory encoding of novel temporal sequences as episodic events. Received: 18 December 2002 / Accepted: 18 March 2003 / Published online: 20 May 2003 Correspondence to: Y. Yamaguchi (e-mail: yokoy@brain.riken.go.jp, Fax: +81-48-4676938) Acknowledgements. The author would like to express acknowledgement to Drs. McNaughton and Skaggs for their discussion and comment and to Dr. Amari for his continuous encouragement. Further thanks are given to Mr. Haga and Dr. Wu for their discussion and cooperation.     

On the dynamics of discrete food chains: low- and high-frequency behavior and optimality of chaos

 In this paper we derive and analyze a discrete version of Rosenzweig's (Am. Nat. 1973) food-chain model. We provide substantial analytical and numerical evidence for the general dynamical patterns of food chains predicted by De Feo and Rinaldi (Am. Nat. 1997) remaining largely unaffected by this discretization. Our theoretical analysis gives rise to a classification of the parameter space into various regions describing distinct governing dynamical behaviors. Predator abundance has a local optimum at the edge of chaos. Received: 13 August 1999 / Revised version: 12 March 2002 / Published online: 17 October 2002 Mathematics Subject Classification (1991): 92D40 Keywords or phrases: Discrete food-chain – Discrete Hopf (Neimark-Sacker) bifurcation – Pulsewise birth processes – Mean yield maximization – Nicholson-Bailey model     

An in silico central pattern generator: silicon oscillator,coupling, entrainment,and physical computation

 In biological systems, the task of computing a gait trajectory is shared between the biomechanical and nervous systems. We take the perspective that both of these seemingly different computations are examples of physical computation. Here we describe the progress that has been made toward building a minimal biped system that illustrates this idea. We embed a significant portion of the computation in physical devices, such as capacitors and transistors, to underline the potential power of emphasizing the understanding of physical computation. We describe results in the exploitation of physical computation by (1) using a passive knee to assist in dynamics computation, (2) using an oscillator to drive a monoped mechanism based on the passive knee, (3) using sensory entrainment to coordinate the mechanics with the neural oscillator, (4) coupling two such systems together mechanically at the hip and computationally via the resulting two oscillators to create a biped mechanism, and (5) demonstrating the resulting gait generation in the biped mechanism. Received: 31 October 2001 / Accepted in revised form: 17 September 2002 Correspondence to: M.A. Lewis     

Dynamics of annual influenza A epidemics with immuno-selection

 The persistence of Influenza A in the human population relies on continual changes in the viral surface antigens allowing the virus to reinfect the same hosts every few years. The epidemiology of such a drifting virus is modeled by a discrete season-to-season map. During the epidemic season only one strain is present and its transmission dynamics follows a standard epidemic model. After the season, cross-immunity to next year's virus is determined from the proportion of hosts that were infected during the season. A partial analysis of this map shows the existence of oscillations where epidemics occur at regular or irregular intervals. Received: 16 February 2001 / Revised version: 11 June 2002 / Published online: 28 February 2003 Key words or phrases: Infectious disease – Influenza drift – Cross-immunity – Seasonal epidemics – Iterated map     

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)     

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.     

A numerical method for renal models that represent tubules with abrupt changes in membrane properties

 The urine concentrating mechanism of mammals and birds depends on a counterflow configuration of thousands of nearly parallel tubules in the medulla of the kidney. Along the course of a renal tubule, cell type may change abruptly, resulting in abrupt changes in the physical characteristics and transmural transport properties of the tubule. A mathematical model that faithfully represents these abrupt changes will have jump discontinuities in model parameters. Without proper treatment, such discontinuities may cause unrealistic transmural fluxes and introduce suboptimal spatial convergence in the numerical solution to the model equations. In this study, we show how to treat discontinuous parameters in the context of a previously developed numerical method that is based on the semi-Lagrangian semi-implicit method and Newton's method. The numerical solutions have physically plausible fluxes at the discontinuities and the solutions converge at second order, as is appropriate for the method. Received: 13 November 2001 / Revised version: 28 June 2002 / Published online: 26 September 2002 This work was supported in part by the National Institutes of Health (National Institute of Diabetes and Digestive and Kidney Diseases, grant DK-42091.) Mathematics Subject Classification (2000): 65-04, 65M12, 65M25, 92-04, 92C35, 35-04, 35L45 Keywords or phrases: Mathematical models – Differential equations – Mathematical biology – Kidney – Renal medulla – Semi-Lagrangian semi-implicit     

Scleroderma laeve (Gasteromycetes, Sclerodermatales), new to Japan

 A Scleroderma species collected on sandy soil under trees of Lithocarpus edulis in Saitama Prefecture, central Japan, is identified as Scleroderma laeve, a new record for Japan. Macroscopic and microscopic features are given. Received: May 24, 2002 / Accepted: September 9, 2002 Acknowledgments We thank Ms. Ryoko Onuma, who offered some useful literature on Scleroderma. We are also grateful to Dr. Toshimitsu Fukiharu (Natural History Museum and Institute, Chiba) for his help with preserving the specimens. For collecting specimens, we are grateful to Ms. Ayano Kimura, Mr. Tomoya Matsuyama, and Mr. Takahiro Uchida. Correspondence to:T. Kasuya     

Early development and quorum sensing in bacterial biofilms

 We develop mathematical models to examine the formation, growth and quorum sensing activity of bacterial biofilms. The growth aspects of the model are based on the assumption of a continuum of bacterial cells whose growth generates movement, within the developing biofilm, described by a velocity field. A model proposed in Ward et al. (2001) to describe quorum sensing, a process by which bacteria monitor their own population density by the use of quorum sensing molecules (QSMs), is coupled with the growth model. The resulting system of nonlinear partial differential equations is solved numerically, revealing results which are qualitatively consistent with experimental ones. Analytical solutions derived by assuming uniform initial conditions demonstrate that, for large time, a biofilm grows algebraically with time; criteria for linear growth of the biofilm biomass, consistent with experimental data, are established. The analysis reveals, for a biologically realistic limit, the existence of a bifurcation between non-active and active quorum sensing in the biofilm. The model also predicts that travelling waves of quorum sensing behaviour can occur within a certain time frame; while the travelling wave analysis reveals a range of possible travelling wave speeds, numerical solutions suggest that the minimum wave speed, determined by linearisation, is realised for a wide class of initial conditions. Received: 10 February 2002 / Revised version: 29 October 2002 / Published online: 19 March 2003 Key words or phrases: Bacterial biofilm – Quorum sensing – Mathematical modelling – Numerical solution – Asymptotic analysis – Travelling wave analysis     

Pleosporales in Japan (1): the genus <Emphasis Type="Italic">Lophiostoma</Emphasis>

 Seven species of the genus Lophiostoma were the subject of this study. Among these, Lophiostoma mucosum is described and illustrated as a new species. All other species, L. macrostomum, L. semiliberum, L. arundinis, L. caulium, L. caudatum, and L. winteri, are reported for the first time in Japan. A key to the species of Lophiostoma in Japan is given. Received: August 5, 2002 / Accepted: November 28, 2002 Acknowledgments We are grateful to Dr. Hideki Takahashi (curator of SAPA) for the loan of fungal material. Correspondence to:Y. Harada     

Migraine aura dynamics after reverse retinotopic mapping of weak excitation waves in the primary visual cortex

 A kinematical model for excitable wave propagation is analyzed to describe the dynamics of a typical neurological symptom of migraine. The kinematical model equation is solved analytically for a linear dependency between front curvature and velocity. The resulting wave starts from an initial excitation and moves in the medium that represents the primary visual cortex. Due to very weak excitability the wave propagates only across a confined area and eventually disappears. This cortical excitation pattern is projected onto a visual hemifield by reverse retinotopic mapping. Weak excitability explains the confined appearance of aura symptoms in time and sensory space. The affected area in the visual field matches in growth and form the one reported by migraine sufferers. The results can be extended from visual to tactile and to other sensory symptoms. If the spatiotemporal pattern from our model can be matched in future investigations with those from introspectives, it would allow one to draw conclusions on topographic mapping of sensory input in human cortex. Received: 25 April 2002 / Accepted: 20 February 2003 / Published online: 20 May 2003 RID="*" ID="*" Present address: M. A. Dahlem Leibniz-Institut für Neurobiologie, Brenneckestr. 6, 39118 Magdeburg, Germany Acknowledgements. We would like to thank V. Zykov for useful discussions on wave Propagation, and one of us (MAD) would like to thank Ed Chronicle for useful discussions on functional excitability. This project was supported by a scholarship Landesstipendium Sachsen-Anhalt to MAD. Correspondence to: M. A. Dahlem (e-mail: dahlem@ifn-magdeburg.de)     

A temperature-sensitive DNA adenine methyltransferase mutant of Salmonella typhimurium

A temperature-sensitive mutant of Salmonella typhimurium was isolated earlier after transposon mutagenesis with Tn10d Tet. The mutant D220 grows well at 28 °C but has a lower growth rate and forms filaments at 37 °C. Transposon-flanking fragments of mutant D220 DNA were cloned and sequenced. The transposon was inserted in the dam gene between positions 803 and 804 (assigned allele number: dam-231 : : Tn10d Tet) and resulted in a predicted ten-amino-acid-shorter Dam protein. The insertion created a stop codon that led to a truncated Dam protein with a temperature-sensitive phenotype. The insertion dam-231 : : Tn10d Tet resulted in a dam“leaky” phenotype since methylated and unmethylated adenines in GATC sequences were present. In addition, the dam-231 : : Tn10d Tet insertion rendered dam mutants temperature-sensitive for growth depending upon the genetic background of the S. typhimurium strain. The wild-type dam gene of S. typhimurium exhibited 82% identity with the Escherichia coli dam gene.     

The detuning factor in the dynamics of interlimb rhythmic coordination

 Dynamical models of two coupled biological oscillators interpret the detuning term as an arithmetic difference between the uncoupled frequencies, Δω= (ω₁−ω₂). This Δω interpretation of detuning was addressed in four experiments in which human subjects oscillated pendulums in their right and left hands in 1 : 1 frequency locking in antiphase (Experiments 1–3) or inphase (Experiment 4). Differences between the uncoupled frequencies were manipulated through differences in the equivalent simple pendulum lengths, and the effects of this manipulation on the detuning of relative phase from π or 0 and the standard deviation of relative phase SDφ were measured. In Experiment 1, the same values of ω _i were satisfied by several different physical configurations. The experiment confirmed that the detuning term is related strictly to the uncoupled frequencies rather than to other physical characteristics of the oscillators. Experiments 2, 3 and 4 showed, however, that the particular dependency of fixed point drift and SDφ on Δω depends on the particulars of ω₁ and ω₂. With variations in Δω brought about by different ω₁ and ω₂ that always formed a constant ratio, fixed point drift related inversely to Δω, and SDφ varied with Δω in ways that depended on the magnitude of the constant ratio. These outcomes do not conform to expectations from models of coordination dynamics that interpret detuning as (ω₁−ω₂). Received: 18 October 1993/Accepted in revised form: 2 December 1994     

Computational model of the cockroach escape behavior: winner and losers in a population code

 I present a comprehensive biologically oriented computational model to account for the escape response of the cockroach on the ground. This model is an expansion of previous work that accounted only for discriminating left from right wind directions [Ezrachi et al. (1999) Biol Cybern 81: 89–99]. The model is composed of computational elements describing the biological processes taking place in the various neurons and includes input which emulates empirical data. With this model it is possible to obtain escape behavior that resembles natural behavior. The model is used to address an ongoing debate as to whether the cockroach's turn direction is determined by computations carried out by the entire neuronal population (PC) or rather by a “winner-take-all” (WTA) mechanism. I suggest that the computation mechanism that underlies the cockroach escape response is composed of both PC and WTA principles. Based on the properties of the suggested new mechanism I denote it a “Darwinian population code.” Received: 26 March 2002 / Accepted in revised form: 24 June 2002 Acknowledgements. I thank H. Parnas for her advice and assistance, J. M. Camhi for helpful comments, and D. Lipson for developing the simulation tools. Correspondence to: E. A. Ezrachi (e-mail: erez@piano.ls.huji.ac.il, Tel.: +972-2-6585818, Fax: +972-2-6585569)     

Solution phase synthesis of the 14-residue peptaibol antibiotic trichovirin I

Summary.  The 14-residue peptaibol antibiotic trichovirin I 4A of the structure Ac-Aib-L-Asn-L-Leu-Aib-L-Pro-L-Ala-L-Val-Aib-L-Pro-Aib-L-Leu-Aib-L-Pro-L-Leuol (Aib = α-aminoisobutyric acid, Leuol = leucinol) was synthesized by stepwise conventional solution phase synthesis using the Z/OtBu(OMe) strategy and HOBt/EDC as coupling reagents. Intermediates were fully characterized and the identity of the synthetic peptide with the component 4A of the natural, microheterogeneous peptide mixture was proven by electrospray mass spectrometry, HPLC, and bioassay. Received March 25, 2002 Accepted June 14, 2002 Published online December 18, 2002 RID="*" ID="*"  Dedicated to Prof. Dr. Günther Jung. Tübingen University, on the occasion of his 65^th anniversary. Authors' address: Prof. Dr. Hans Brückner, Interdisciplinary Research Center, Institute of Nutritional Science, Department of Food Sciences, Justus-Liebig-University of Giessen, Heinrich-Buff-Ring 26, D-35392 Giessen, Germany, Fax: +49-641-99-39149, E-mail: hans.brueckner@ernaehrung.uni-giessen.de Abbreviations: Amino acids are abbreviated according to three-letter-nomenclature; Aib, α-aminoisobutyric acid (2-methylalanine); Iva (isovaline, 2-ethylalanine); Leuol, L-leucinol [(S)-2-amino-4-methyl-1-pentanol]; AAA, amino acid analysis; EI-MS, electron impact mass spectrometry; ESI-MS, electrospray ionization mass spectrometry; HPLC, high performance liquid chromatography; Z, benzyloxycarbonyl; Fmoc, 9-fluorenylmethyoxycarbonyl; OtBu, tertiary butoxy (tert-butylester); OMe, methoxy (methyl ester); OBzl, benzyloxy (benzyl ester); TDM, N,N,N′,N′-tetramethyl-4,4′-diamino-diphenylmethane (Arnold's base); for other abbreviations see Experimental.