Neural fields with fast learning dynamic kernel

We introduce a modified-firing-rate model based on Hebbian-type changing synaptic connections. The existence and stability of solutions such as rest state, bumps, and traveling waves are shown for this type of model. Three types of kernels, namely exponential, Mexican hat, and periodic synaptic connections, are considered. In the former two cases, the existence of a rest state solution is proved and the conditions for their stability are found. Bump solutions are shown for two kinds of synaptic kernels, and their stability is investigated by constructing a corresponding Evans function that holds for a specific range of values of the kernel coefficient strength (KCS). Applying a similar method, we consider exponential synaptic connections, where traveling wave solutions are shown to exist. Simulation and numerical analysis are presented for all these cases to illustrate the resulting solutions and their stability.     

The numerical solution of stiff differential equations

This paper first discusses the conditions in which a set of differential equations should give stable solutions, starting with linear systems assuming that these do not differ greatly in this respect from non-linear systems. Methods of investigating the stability of particular systems are briefly discussed. Most real biochemical systems are known from observation to be stable, but little is known of the regions over which stability persists; moreover, models of biochemical systems may not be stable, because of inaccurate choice of parameter values.The separate problem of stability and accuracy in numerical methods of approximating the solution of systems of non-linear equations is then treated. Stress is laid on the consistently unsatisfactory results given by explicit methods for systems containing "stiff" equations, and implicit multistep methods are particularly recommended for this class of problem, which is likely to include many biochemical model systems. Finally, an iteration procedure likely to give convergence both in multistep methods and in the steady-state approach is recommended, and areas in which improvement in methods is likely to occur are outlined.     

Generation of the nervous impulse and periodic oscillations

Usually the models for the excitation and propagation of the nervous impulse are studied either in the space-clamp situation or on a model axon extended on both sides to infinity. Following Fitzhugh in the present paper the release of an impulse train at the axon hillock is studied within the scope of Fitzhugh's BVP model. The existence and stability of periodic oscillations are studied by direct methods, also the relation to Liénard's equation. The exact correspondence between the BVP model and the socalled Nagumo-equation is established. For typical examples the solutions are computed by numerical methods.     

Biomechanical modeling and optimal control of human posture

The present work describes the biomechanical modeling of human postural mechanics in the saggital plane and the use of optimal control to generate open-loop raising-up movements from a squatting position. The biomechanical model comprises 10 equivalent musculotendon actuators, based on a 40 muscles model, and three links (shank, thigh and HAT-Head, Arms and Trunk). Optimal control solutions are achieved through algorithms based on the Consistent Approximations Theory (Schwartz and Polak, 1996), where the continuous non-linear dynamics is represented in a discrete space by means of a Runge-Kutta integration and the control signals in a spline-coefficient functional space. This leads to non-linear programming problems solved by a sequential quadratic programming (SQP) method. Due to the highly non-linear and unstable nature of the posture dynamics, numerical convergence is difficult, and specific strategies must be implemented in order to allow convergence. Results for control (muscular excitations) and angular trajectories are shown using two final simulation times, as well as specific control strategies are discussed.     

Separable models in age-dependent population dynamics

A class of population models is considered in which the parameters such as fecundity, mortality and interaction coefficients are assumed to be age-dependent. Conditions for the existence, stability and global attractivity of steady-state and periodic solutions are derived. The dependence of these solutions on the maturation periods is analyzed. These results are applied to specific single and multiple population models. It is shown that periodic solutions cannot occur in a general class of single population age-dependent models. Conditions are derived that determine whether increasing the maturation period has a stabilizing effect. In specific cases, it is shown that any number of switches in stability can occur as the maturation period is increased. An example is given of predator-prey model where each one of these stability switches corresponds to a stable steady state losing its stability via a Hopf bifurcation to a periodic solution and regaining its stability upon further increase of the maturation period.     

Attractivity of coherent manifolds in metapopulation models

The likelihood that coupled dynamical systems will completely synchronize, or become “coherent”, is often of great applied interest. Previous work has established conditions for local stability of coherent solutions and global attractivity of coherent manifolds in a variety of spatially explicit models. We consider models of communities coupled by dispersal and explore intermediate regimes in which it can be shown that states in phase space regions of positive measure are attracted to coherent solutions. Our methods yield rigorous and practically useful coherence criteria that facilitate useful analyses of ecological and epidemiological problems.     

Isolated periodic solutions of the Hodgkin-Huxley equations

Periodic solutions of the current clamped Hodgkin-Huxley equations (Hodgkin & Huxley, 1952 J. Physiol. 117, 500) that arise by degenerate Hopf bifurcation were studied recently by Labouriau (1985 SIAM J. Math. Anal. 16, 1121, 1987 Degenerate Hopf Bifurcation and Nerve Impulse (Part II), in press). Two parameters, temperature T and sodium conductance gNa were varied from the original values obtained by Hodgkin & Huxley. Labouriau's work proved the existence of small amplitude periodic solution branches that do not connect locally to the stationary solution branch, and had not been previously computed. In this paper we compute these solution branches globally. We find families of isolas of periodic solutions (i.e. branches not connected to the stationary branch). For values of gNa in the range measured by Hodgkin & Huxley, and for physically reasonable temperatures, there are isolas containing orbitally asymptotically stable solutions. The presence of isolas of periodic solutions suggests that in certain current space clamped membrane experiments, action potentials could be observed even though the stationary state is stable for all current stimuli. Once produced, such action potentials will disappear suddenly if the current stimulus is either increased or decreased past certain values. Under some conditions, "jumping" between action potentials of different amplitudes might be observed.     

Localized states in an unbounded neural field equation with smooth firing rate function: a multi-parameter analysis

The existence of spatially localized solutions in neural networks is an important topic in neuroscience as these solutions are considered to characterize working (short-term) memory. We work with an unbounded neural network represented by the neural field equation with smooth firing rate function and a wizard hat spatial connectivity. Noting that stationary solutions of our neural field equation are equivalent to homoclinic orbits in a related fourth order ordinary differential equation, we apply normal form theory for a reversible Hopf bifurcation to prove the existence of localized solutions; further, we present results concerning their stability. Numerical continuation is used to compute branches of localized solution that exhibit snaking-type behaviour. We describe in terms of three parameters the exact regions for which localized solutions persist.     

Compensation type algorithms for neural nets: stability and convergence

Plasticity of synaptic connections plays an important role in the temporal development of neural networks which are the basis of memory and behavior. The conditions for successful functional performance of these nerve nets have to be either guaranteed genetically or developed during ontogenesis. In the latter case, a general law of this development may be the successive compensation of disturbances. A compensation type algorithm is analyzed here that changes the connectivity of a given network such that deviations from each neuron's equilibrium state are reduced. The existence of compensated networks is proven, the convergence and stability of simulations are investigated, and implications for cognitive systems are discussed.     

Spatial arrangement and macrodomain organization of bacterial chromosomes

Recent developments in fluorescence microscopy have shown that bacterial chromosomes have a defined spatial arrangement that preserves the linear order of genes on the genetic map. These approaches also revealed that large portions of the chromosome in Escherichia coli or Bacillus subtilis are concentrated in the same cellular space, suggesting an organization as large regions defined as macrodomains. In E. coli, two macrodomains of 1 Mb containing the replication origin (Ori) and the replication terminus (Ter) have been shown to relocalize at specific steps of the cell cycle. A genetic analysis of the collision probability between distant DNA sites in E. coli has confirmed the presence of macrodomains by revealing the existence of large regions that do not collide with each other. Two macrodomains defined by the genetic approach coincide with the Ori and Ter macrodomains, and two new macrodomains flanking the Ter macrodomain have been identified. Altogether, these results indicate that the E. coli chromosome has a ring organization with four structured and two less-structured regions. Implications for chromosome dynamics during the cell cycle and future prospects for the characterization and understanding of macrodomain organization are discussed.     

Existence and stability of travelling wave solutions of a nonlinear integral operator

In this paper, we establish the existence and stability property of travelling wave solutions of a nonlinear integral operator in the inferior case.     

一类具年龄结构和接种的非终生免疫型传染病模型

研究一类具年龄结构和接种的非终生免疫SIRS传染病模型平衡解的稳定性.首先利用特征线法讨论了模型平衡解的存在性,然后利用比较定理和逐次迭代法得到无病平衡解与地方病平衡解全局稳定性的充分条件.     

Convergence in almost periodic Fisher and Kolmogorov models

 We study convergence of positive solutions for almost periodic reaction diffusion equations of Fisher or Kolmogorov type. It is proved that under suitable conditions every positive solution is asymptotically almost periodic. Moreover, all positive almost periodic solutions are harmonic and uniformly stable, and if one of them is spatially homogeneous, then so are others. The existence of an almost periodic global attractor is also discussed. Received: 11 November 1996 / Revised version: 8 January 1998     

Steady-state stability of substrate inhibited enzyme catalysis in open reaction systems

Substrate inhibited enzyme reactions occurring in systems open to mass transfer may display multiple steady-state behavior. For a simple one-substrate case it is shown theoretically that the unstable steady-state region always lies within the conversion range of 50–100%. A criterion for the stability of a steady-state point is given. Numerical solutions of the appropriate transient equation show how the system approaches stable steady state in instances where there are three possible steady-state points. The consequences of the existence of an unstable region on the systems response to changes in its parameters is discussed.     

Mixed culture biooxidation of phenol. III. Existence of multiple steady states in continuous culture with wall growth

It is shown that two steady states exist in certain regions of operation of a 2-liter continuous stirred tank biological reactor. Transition was made from one steady state to another by applying shock loads of either phenol substrate which is inhibitory to the culture at high concentrations or by adding large additional amounts of concentrated organisms. The existence of the multiple steady states is ascribed to the existence of wall growth, and their position is determined by the amount of wall growth. Transient behavior of the system did not follow the predictions of the simple wall growth model but the culture appeared to undergo a lag period immediately after applying the shock load to the system. It is concluded that the stability of a continuous culture utilizing an inhibitory substrate is improved by increasing the degree of wall growth and decreasing the substrate feed concentration. It is also concluded that small scale experiments can usually not be interpreted correctly unless the effect of wall growth is taken into account.     

A novel approach for solving constrained nonlinear optimization problems using neurofuzzy systems

A neural network model for solving constrained nonlinear optimization problems with bounded variables is presented in this paper. More specifically, a modified Hopfield network is developed and its internal parameters are computed using the valid-subspace technique. These parameters guarantee the convergence of the network to the equilibrium points. The network is shown to be completely stable and globally convergent to the solutions of constrained nonlinear optimization problems. A fuzzy logic controller is incorporated in the network to minimize convergence time. Simulation results are presented to validate the proposed approach.     

Lagrange stability and ecological systems

This paper considers the ecological and mathematical relationships among various stability concepts. It is shown that diagonally dominant nonlinear systems have positive solutions for positive initial conditions. The boundedness of such positive solutions leads to a total systems concept of stability: Lagrange stability. This concept is discussed as a compatible method for determining the stability of ecosystems and their mathematical models. Stability in the sense of Lagrange is a total systems concept which does not directly address the problem of the stability of equilibria. A topological presentation is used to describe the Lagrange stability of nonlinear ecosystem compartment models. Finally, the concept of practical stability is developed and discussed with respect to maintaining the desired state of ecosystems in the presence of perturbations.     

Direct and inverse relationships between Riccati systems coupled with multiplicative terms

An analytical and computational framework for the derivation of solitary solutions to biological systems describing the cooperation and competition of species and expressed by the system of Riccati equations coupled with multiplicative terms is presented in this paper. It is demonstrated that relationships between these solitary solutions can be either direct or inverse. Thus, an infinitesimal perturbation of one population would lead to an infinitesimal change in the other population – if only both solitary solutions are coupled with the direct relationship. But, in general, that is not true if solitary solutions are coupled with the inverse relationship – an infinitesimal perturbation of one population may result into a non-infinitesimal change in the other population. Necessary and sufficient conditions for the existence of solitary solutions are derived in the space of the system's parameters and initial conditions.     

Diseases with chronic stage in a population with varying size

An epidemiological model of hepatitis C with a chronic infectious stage and variable population size is introduced. A non-structured baseline ODE model which supports exponential solutions is discussed. The normalized version where the unknown functions are the proportions of the susceptible, infected, and chronic individuals in the total population is analyzed. It is shown that sustained oscillations are not possible and the endemic proportions either approach the disease-free or an endemic equilibrium. The expanded model incorporates the chronic age of the individuals. Partial analysis of this age-structured model is carried out. The global asymptotic stability of the infection-free state is established as well as local asymptotic stability of the endemic non-uniform steady state distribution under some additional conditions. A numerical method for the chronic-age-structured model is introduced. It is shown that this numerical scheme is consistent and convergent of first order. Simulations based on the numerical method suggest that in the structured case the endemic equilibrium may be unstable and sustained oscillations are possible. Closer look at the reproduction number reveals that treatment strategies directed towards speeding up the transition from acute to chronic stage in effect contribute to the eradication of the disease.     

The distribution of orientation of optimal stimuli for cells of striate cortex

The mechanism mediating the adaptation of cortical Area 17 oriented line detector cells is modeled. A novel form of environment and convergence property (which places a requirement on distribution of optimal stimuli) is utilized to show that classical concepts of synaptic efficiency loss due only to disuse are inadequate under the convergence property. A reasonable alternative is presented: a form of synaptic control which reduces synaptic efficiency at non-active synapses as a function of cell firing rate. We show that adequate solutions to our convergence property exist for members of this class. Simulations of this adequate mechanism indicate that under the very disjoint environment defined, small perturbations of the environment's distribution of stimuli may lead to large perturbations of the distribution of stimuli to which Area 17 cells are optimally responsive. A simulation indicates that this effect may be accentuated by a lateral interaction which causeslike cell optimal stimuli to form near each other in cortex; and the effect may be reduced by a lateral interaction which causes like cell optimal stimuli to not form near each other. A form of neurophysiological experiment is suggested for verification.