We consider a system of two non-linear differential-difference equations that models a behaviour of two coupled neural oscillators with a synaptic connection of a threshold type. To find a solution ...of the system means to find normalized membrane potentials of neurons. We prove that there exists an antiphase solution. It is a special relaxation periodic regime such that one membrane potential is a half-period shift of the other membrane potential. We prove that the antiphase solution is stable and construct its asymptotic. We show that there exists a solution with a bursting-effect by choosing a specific connection chain delay. That is, for any natural n there is a connection chain delay such that an antiphase solution has exactly n asymptotically high spikes on a period after a refractive segment.
This paper considers a mathematical model of a ring neural network with an even number of synaptically interacting elements. The model is a system of scalar nonlinear differential-difference ...equations, the right-hand sides of which depend on a large parameter. The unknown functions being contained in the system characterize membrane potentials of neurons. It is of interest to search within the system for special periodic solutions, so-called impulse-refractory modes. Functions with odd numbers of the impulse-refraction cycle have an asymptotically large burst and the functions with even numbers are asymptotically small. For this purpose, two substitutions are made sequentially, making it possible to study of a two-dimensional system of nonlinear differential-difference singularly perturbed equations with two delays instead of the initial system. Further, as the large parameter tends to infinity, the limiting object is defined, which is a relay system of equations with two delays. Using the step-by-step method, we prove that the solution of a relay system with an initial function from a suitable class is a periodic function with required properties. Then, using the Poincaré operator and the Schauder principle, the existence of a relaxation periodic solution of a two-dimensional singularly perturbed system is proven. To do this, we construct asymptotics of this solution, and then prove its closeness to the solution of the relay system. The exponential estimate of the Frechet derivative of the Poincaré operator implies uniqueness of the solution of a two-dimensional differential-difference system of equations with two delays in the constructed class of functions and is used to justify the exponential orbital stability of this solution. Furthermore, with the help of reverse replacement, the proven result is transferred to the original system.
We study the mathematical model of a circular neural network with synaptic interaction between the elements. The model is a system of scalar nonlinear differential-difference equations, the right ...parts of which depend on large parameters. The unknown functions included in the system characterize the membrane potentials of the neurons. The search for relaxation cycles within the system of equations is of interest. Thus, we postulate the problem of finding its solution in the form of discrete travelling waves. This allows us to study a scalar nonlinear differential-difference equation with two delays instead of the original system. We define a limit object which represents a relay equation with two delays by passing the large parameter to infinity. Using this construction and the step-by-step method, we show that there are six cases for restrictions on the parameters. In each case there exists a unique periodic solution to the relay equation with the initial function from a suitable function class. Using the Poincaré operator and the Schauder principle, we prove the existence of relaxation periodic solutions of a singularly perturbed equation with two delays. We find the asymptotics of this solution and prove that the solution is close to the solution of the relay equation. The uniqueness and stability of the solutions of the differential-difference equation with two delays follow from the exponential bound on the Fréchet derivative of the Poincaré operator.
In the paper, a mathematical model of a neural network with an even number of ring synaptic interaction elements is considered. The model is a system of scalar nonlinear differentialdifference ...equations, the right parts of which depend on large parameters. The unknown functions included in the system characterize the membrane potentials of the neurons. The search of special impulse-refraction cycles within the system of equations is of interest. The functions with odd numbers of the impulse-refraction cycle have an asymptotically high pulses and the functions with even numbers are asymptotically small. Two changes allow to study a two-dimension nonlinear differential-difference system with two delays instead of the system. Further, a limit object that represents a relay system with two delays is defined by a large parameter tending to infinity. There exists the only periodic solution of the relay system with the initial function from a suitable function class. This is structurally proved, by using the step method. Next, the existence of relaxation periodic solutions of the two-dimension singularly perturbed system is proved by using the Poincare operator and the Schauder principle. The asymptotics of this solution is constructed, and it is proved that the solution is close to the decision of the relay system. Because of the exponential estimate of the Frechet derivative of the Poincare operator it implies the uniqueness and stability of solutions of the two-dimension differential-difference equation with two delays. Furthermore, with the help of reverse replacement the proved result is transferred to the original system.
In this paper the mathematical model of a neural network with a ring synaptic interaction elements is considered. The model is a system of scalar nonlinear differential-difference equations, the right ...parts of which depend on a large parameter. The unknown functions included in the system characterize the membrane potentials of the neurons. The search of relaxation cycles within the system of equations is interested. To this end solutions of the task are finded in the form of discrete traveling waves. It allows to research a scalar nonlinear differential-difference equations with two delays instead of system. Further, a limit a object that represents a relay equation with two delays is defined by large parameter tends to infinity. There are six cases of restrictions on the parameters. In every case exist alone periodic solution of relay equation started from initial function from suitable function class. It is structurally proved by using the step method. Next, the existence of a relaxation periodic solutions of a singularly perturbed equation with two delays is proved by using Poincare operator and Schauder principle. The asymptotics of this solution is constructed, and then it is proved that the solution is close to decision of the relay equation. Because of the exponential estimate Frechet derivative of the Poincare operator implies the uniqueness and stability of solutions of differential-difference equation with two delays.
We consider the scalar nonlinear differential-difference equation with two delays, which models electrical activity of a neuron. Under some additional suppositions for this equation well known method ...of quasi-normal forms can be applied. Its essence lies in the formal normalization of the Poincare - Dulac obtaining quasi-normal form and the subsequent application of the theorems of conformity. In this case, the result of the application of quasi-normal forms is a countable system of differential-difference equations, which can be turned into a boundary value problem of the Korteweg - de Vries equation. The investigation of this boundary value problem allows us to draw a conclusion about the behaviour of the original equation. Namely, for a suitable choice of parameters in the framework of this equation is implemented buffer phenomenon consisting in the presence of the bifurcation mechanism for the birth of an arbitrarily large number of stable cycles.
We consider a scalar nonlinear differential-difference equation with two delays, which models the behavior of a single neuron. Under some additional suppositions for this equation it is applied a ...well-known method of quasi-normal forms. Its essence lies in the formal normalization of the Poincare – Dulac, the production of a quasi-normal form and the subsequent application of the conformity theorems. In this case, the result of the application of quasi-normal forms is a countable system of differential-difference equations, which manages to turn into a boundary value problem of the Korteweg – de Vries equation. The investigation of this boundary value problem allows to make the conclusion about the behavior of the original equation. Namely, for a suitable choice of parameters in the framework of this equation it is implemented the buffer phenomenon consisting in the presence of the bifurcation mechanism for the birth of an arbitrarily large number of stable cycles.
The article examines the state of the antral mucosa of the stomach in 43 young patients with duodeno-gastric reflux without any clinical manifestations. In 69.9% of investigated revealed changes in ...the gastric mucosa, characteristic of reflux gastritis. The comparative characteristic of stereometric indicators of cellular elements, infiltrated own plate of a mucous membrane of a stomach is spent at a reflux-gastritis and the mixed gastritis (at presence Helicobacter pylori). Article is illustrated by 2 tables.
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