THE ASYMPTOTIC STRUCTURE OF THE HODGKIN–HUXLEY EQUATIONS

2003 ◽  
Vol 13 (12) ◽  
pp. 3805-3825 ◽  
Author(s):  
REBECCA SUCKLEY ◽  
VADIM N. BIKTASHEV
Keyword(s):  
Differential Equations ◽  
Action Potential ◽  
Slow Manifold ◽  
System Of Equations ◽  
Asymptotic Structure ◽  
Asymptotics Of Solutions ◽  
Cusp Singularity ◽  
Systems Of Differential Equations ◽  
Characteristic Features

We analyze the asymptotic structure of the Hodgkin–Huxley system of equations, in terms of the concepts of slow manifold and fast foliation, based on Tikhonov's theorem on asymptotics of solutions of slow–fast systems of differential equations. We test Zeeman's conjecture that the jump onset–slow return structure of the action potential in realistic equations of biological excitability may be due to a cusp singularity of the slow manifold with respect to the fast foliation. We find that although the cusp singularity can appear in such equations, the characteristic features in question cannot be reproduced within the Tikhonov scheme and require development of different asymptotic approaches.

scholarly journals Establishing conditions for the existence of bounded solutions to the weakly nonlinear pulse systems

2021 ◽  
Vol 4 (4(112)) ◽  
pp. 6-12
Author(s):  
Farhod Asrorov ◽  
Oleh Perehuda ◽  
Valentyn Sobchuk ◽  
Anna Sukretna
Keyword(s):  
Mathematical Modeling ◽  
Differential Equations ◽  
Bounded Solution ◽  
Pulse Action ◽  
Pulse Generation ◽  
System Of Equations ◽  
Bounded Solutions ◽  
Weakly Nonlinear ◽  
Systems Of Differential Equations ◽  
Wide Range

Processes that involve jump-like changes are observed in mechanics (the movement of a spring under an impact; clockwork), in radio engineering (pulse generation), in biology (heart function, cell division). Therefore, high-quality research of pulse systems is a relevant task in the modern theory of mathematical modeling. This paper considers the issue related to the existence of bounded solutions along the entire real axis (semi-axis) of the weakly nonlinear systems of differential equations with pulse perturbation at fixed time moments. A concept of the regular and weakly regular system of equations for the class of the weakly nonlinear pulse systems of differential equations has been introduced. Sufficient conditions for the existence of a bounded solution to the heterogeneous system of differential equations have been established for the case of poorly regularity of the corresponding homogeneous system of equations. The conditions for the existence of singleness of the bounded solution along the entire axis have been defined for the weakly nonlinear pulse systems. The results were applied to study bounded solutions to the systems with pulse action of a more general form. The established conditions make it possible to use the classical methods of differential equations to obtain statements about solvability and the continuous dependence of solutions on the parameters of a pulse system. It has been shown that classical qualitative methods for studying differential equations are mainly naturally transferred to dynamic systems with discontinuous trajectories. However, the presence of a pulse action gives rise to a series of new specific problems. The theory of systems with pulse influence has a wide range of applications. Such systems arise when studying pulsed automatic control systems, in the mathematical modeling of various mechanical, physical, biological, and other processes.

scholarly journals Analysis of approaches to mass-transfer modeling n non-stationary mode

2019 ◽  
pp. 55-64
Author(s):  
Yaroslav Pyanylo ◽  
Galyna Pyanylo
Keyword(s):  
Differential Equation ◽  
Mass Transfer ◽  
Numerical Methods ◽  
Differential Equations ◽  
Stationary Mode ◽  
System Of Equations ◽  
Physical Processes ◽  
Partial Derivatives ◽  
Systems Of Differential Equations ◽  
Mass Transfer Modeling

A significant number of natural and physical processes are described by differential equations in partial derivatives or systems of differential equations in partial derivatives. Numerical methods have been found to find their solutions. Partial derivatives systems are solved mainly by reducing the order of the system of equations or reducing it to one differential equation. This procedure leads to an increase in the order of the differential equation. There are various restrictions and errors that can lead to additional solutions, boundary conditions for intermediate derivatives, and so on. The work is devoted to the analysis of such situations and ways of exit.

SOLUTION OF SOME SYSTEMS OF DIFFERENTIAL EQUATIONS IN MATHEMATICAL SYSTEM MAPLE

2013 ◽  
Vol 1 (05) ◽  
pp. 58-65
Author(s):  
Yunona Rinatovna Krakhmaleva ◽  
◽  
Gulzhan Kadyrkhanovna Dzhanabayeva ◽  
Keyword(s):  
Differential Equations ◽  
Systems Of Differential Equations ◽  
Mathematical System

On decomposability of countable systems of differential equations

1993 ◽  
Vol 45 (10) ◽  
pp. 1598-1608
Author(s):  
A. M. Samoilenko ◽  
Yu. V. Teplinskii
Keyword(s):  
Differential Equations ◽  
Systems Of Differential Equations

NUMERICAL COMPUTATION OF CANARDS

2000 ◽  
Vol 10 (12) ◽  
pp. 2669-2687 ◽  
Author(s):  
JOHN GUCKENHEIMER ◽  
KATHLEEN HOFFMAN ◽  
WARREN WECKESSER
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Electrical Potential ◽  
Slow Manifold ◽  
Singularly Perturbed ◽  
Singularly Perturbed Systems ◽  
Singularly Perturbed System ◽  
Chemical Systems ◽  
Perturbed System ◽  
Physical And Chemical

Singularly perturbed systems of ordinary differential equations arise in many biological, physical and chemical systems. We present an example of a singularly perturbed system of ordinary differential equations that arises as a model of the electrical potential across the cell membrane of a neuron. We describe two periodic solutions of this example that were numerically computed using continuation of solutions of boundary value problems. One of these periodic orbits contains canards, trajectory segments that follow unstable portions of a slow manifold. We identify several mechanisms that lead to the formation of these and other canards in this example.

scholarly journals Differential Games for an Infinite 2-Systems of Differential Equations

Mathematics ◽  
2021 ◽  
Vol 9 (13) ◽  
pp. 1467
Author(s):  
Muminjon Tukhtasinov ◽  
Gafurjan Ibragimov ◽  
Sarvinoz Kuchkarova ◽  
Risman Mat Hasim
Keyword(s):  
Hilbert Space ◽  
Differential Equations ◽  
Differential Games ◽  
Differential Game ◽  
Infinite System ◽  
The State ◽  
Geometric Constraints ◽  
Control Parameters ◽  
Systems Of Differential Equations ◽  
Pursuit And Evasion

A pursuit differential game described by an infinite system of 2-systems is studied in Hilbert space l2. Geometric constraints are imposed on control parameters of pursuer and evader. The purpose of pursuer is to bring the state of the system to the origin of the Hilbert space l2 and the evader tries to prevent this. Differential game is completed if the state of the system reaches the origin of l2. The problem is to find a guaranteed pursuit and evasion times. We give an equation for the guaranteed pursuit time and propose an explicit strategy for the pursuer. Additionally, a guaranteed evasion time is found.

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