Mathematical Modeling of Regulatory Mechanisms for the Distribution of Excitation in the Central Nervous System

II.Main part

The use of methods of mathematical modeling and computational experimental means in the development of scientifically based methods of management and control of the functional activity of living systems under normal and abnormal conditions allows for a quantitative analysis of the laws of living systems. To study the regulation of the interaction of the central nervous system and the main vital organs of a person (activity of regulatory mechanisms), we can create a biological and mathematical model based on the concept of "OrAsta" proposed by B.N. Khidirov [7]. Self-management of living systems is often called self-management, self-regulation, regulatory mechanisms, and the science that studies the general laws of their functioning is called regulatory. Since a common feature of the models of regulatory systems is a quantitative analysis of the actions of a set of elements acting in a certain environment and capable of responding to external influences, they are OR (operator-regulator) - for the detection and synthesis of signals, a certain character can be considered through the concept of elements of a capable control system. And ASTA (Active System with Time Average), which is the signal medium of the regulator system, through which the elements are interconnected on the basis of feedback, is carried out with an average time h. Here, h is the time elapsed from the appearance of signals (or their products) to the effect on the activity of elements [5]. Together with OR and ASTA, the ORASTA regulatory system will be formed.
In this paper, based on the “ORASTA” method, we give an example of constructing a biological model for regulating the distribution of excitation in interrelated activities between the central nervous system and the main vital organs. In this case, the nervous regulation of the following major vital organs is studied: brain, heart, lungs, liver, spleen, kidneys, and skin (see Fig. 1).

Fig. 1. The concept of the regulation of the interaction of the brain and the main vital organs based on the "OrAsta" method
By using the method of modeling the regulation of living systems [5], we develop the following system of functional differential equations to regulate the interaction between the central nervous system and the main vital organs:

(1)

initial conditions are
here , , , , , , are respectively presented values, which are the main vital organs: brain, heart, lungs, liver, spleen, kidneys, and skin; {a} and {b} coefficients respectively represent an increase and decrease in the activity of the main vital organs; h is the time. The system of equations has positive values of all coefficients, which provide biologically non-negative solutions of the developed system of nonlinear functional differential equations of delayed type.
Determination of the exact characteristic solutions of the system of equations, their basic modes and properties is very difficult due to the nonlinearity of the equations under consideration and the large number of parameters and variables in them. In such cases (1), a qualitative analysis of the appearance of functional differential equations, the development and implementation of methods for obtaining numerical solutions on a computer is relevant. Therefore (1) the system of equations can be simplified by the reduction and scaling method. [8]. At the same time, if the state of equilibrium of the activity of the main vital organs is distinguished as follows if , then the expression can be written as follows:







here are the equilibrium points of (1) system of equations. Equilibrium points can be found from the above expression as follows:

We put each found value in the first equation of the system of equations (1) and get the following expression:

Consequently, Using the method of reduction and scaling operations according to the system of equations (1), which represents the regulatory mechanisms of the interaction of the central nervous system and the main vital organs, we have the following simplified equation in the form of a functional differential equation:

by putting definitions into the view and , we make equation (2) dimensionless. As a result, the equation of the mechanisms that regulate the interaction of the central nervous system and the main vital organs is as follows:

here is the value of the activity of the interrelated activity of the central nervous system and the main vital organs; coefficients represents an increase and decrease in the activity of the central nervous system respectively; are the non-negative parameters. Since the reduction method is used to derive the model equation, all biological properties of the interrelated activity of the central nervous system and the main vital organs are preserved.
A software has been developed for computer analysis of the mechanisms that regulate the interaction of the central nervous system and the main vital organs. Analysis of the solutions to Equation (3) shows that the following modes of operation are available: stationary state, periodic oscillations, irregular oscillations - deterministic chaos, abrupt changes - the “black hole” effect (see figures 2 to 5).

Fig. 2. Stationary solution of Equation
(a1=6, b1=0.84, A=2.4, B=0.12, C=1.3, D=1.5, E=2.3, F=0.8, G=0.3)



Fig. 3. Periodic oscillation solution of equation
(a1=4, b1=1.4, A=1.4, B=2.2, C=1.3, D=1.5, E=2.3, F=0.8, G=3)

Fig. 4. Solution of unstable oscillation (chaos) of equation
(a1=30, b1=4, A=2.4, B=1.2, C=1.3, D=1.5, E=23, F=0.8, G=30)

Fig. 5. The “black hole” solution of Equation
(a1=16.5, b1=8.2, A=4, B=3.4, C=1.3, D=1.5, E=2.3, F=2, G=3)