The article is dedicated to the mathematical modeling of the regulatory mechanisms of the functioning of nerve cells and the dynamics of the transformation of stem cells into brain cells. Following modes have been revealed according to the results of the investigation: state of calm stem cells, stationary mode, activation of transformation into neuronal cells, self-oscillatory mode of neurogenesis, uncontrolled, chaotic process and a sharp loss of newly formed neurons due to apoptosis - the “black hole” effect. The revealed conditions of disturbances in the regulation of transformation of nerve cells, depending on external and internal factors, will help to develop new strategies for the treatment of diseases of the human central nervous system. Research data can be useful in creating strong artificial intelligence.
Neurobiology had dogmas about the nervous system, comprising nerve cells devoid of the ability of neurogenesis, about the immobility of the mature brain for a long time. The end of the 19th century was marked by the discovery of new knowledge about the process of transformation of nerve stem cells and the birth of new neurons [1-3]. At the same time, the regulatory mechanisms of neurogenesis are not fully understood. An increase in the rate of propagation of excitation in nerve structures may be evidence in favor of neurogenesis. The modern method of quantitatively describing nervous and cerebral processes originates from the investigations of Hodgkin-Huxley in 1952. They made a very successful attempt at a mathematical analysis of the processes of excitation of nervous tissue. The mathematical model of excitation of Hodgkin-Huxley determines the total current I through the cell membrane through conduction in relation to ions of potassium, sodium and others [5]: * Copyright c 2021 for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
( )n ;
n n n ( )m ;
m m m ( )h ,
h h h du dt g n4 (u u ) g m3h(u u ) g (u u );
K k Na Na e e
(
It should be noted that the main tasks of mathematical modeling of processes in the human nervous system are knowledge of the patterns of the structural and functional organization of the nervous system at various levels of the hierarchy, the investigation of the complexity of regulatory mechanisms both within the nervous system and in the interaction of the nervous system with the environment. Revealing the regulatory mechanisms of neurogenesis, the interconnected functioning of human neurons is an urgent task due to the fact that the issues of self-organization, self-regulation and adaptation of the body have not yet been studied in detail to achieve stable functioning in the process of processing external and internal information flows in the learning process. For a more realistic reflection of the regulatory mechanisms of neurogenesis, a complete understanding of intracellular regulation in health and in the case of anomalies, it is important to take into account the temporal relationships in the feedback system, the cooperative nature of the course of biological processes. and an inhibition effect by the final product.
The brilliant work of Hodgkin-Huxley, crowned with the Nobel Prize, inspired entire teams and led to the powerful development of quantitative research on cells of the nervous system. Here, the analysis of information processes in the body is carried out mainly with the help of a quantitative study of the mechanism of distribution and transfer of potentials or spikes in nerve cells. In this article, the modeling of the regulatory mechanisms of the interconnected functioning of neurons is based on the concepts of OR (Oscillator-Regulators) - elements of the regulatory system capable of perceiving and synthesizing signals of a certain nature, and ASTA (Active System with Time Average) - the signaling environment of the regulatory system, in which the interconnected the activity of the elements is carried out, on the basis of feedback, with some average time (the time elapsed from the moment of formation of signals until the moment of their (or their products) impact on the activity of the elements). OR together with ASTA constitute the ORASTA regulatory system. The geometry of such control systems is dynamic, in which the concept of a fixed point loses its meaning. The functioning of the regulatory mechanisms of such systems, for brevity, is designated by the term “regulation” [5]. As defined by B.N.Hidirov– regulation is the science that involves the study of interconnected activity of regulatory mechanisms [5].
The structural and functional unit of the nervous system is a neuron, which is a miniature nervous system. Each neuron consists of a body (inside which there is a nucleus, a molecular genetic system, and regulation and synthesis of proteins necessary for the formation and functioning of memory, thought processes (P) OR), an axon hillock (main generator of nerve impulses, trigger (T) or mini-OR) and dendritic processes diverging from the body in different directions (ASTA). The short branches are called dendrites (D) and the long branches are called axons (A). A neuron receives signals from other neurons through their branches, which form contacts-synapses on the body of the neuron or on its dendrites. The cell collects, integrates these signals and transmits them along the axon and its branches to other cells or executive organs. The equations of the model of regulatory mechanisms of neuron functioning, built taking into account cooperativity, temporal relationships in ORASTA, taking into account past events, predictive abilities and the possibility, in some cases, of signaling in ASTA without the participation of OR have the form: dT (t)
dt dA(t)
dt dD(t)
dt dP(t) dt a T (t )b A(t);
5 2 2 d A(t )b D(t); 1 3 3 p T 2(t ) 0 4 p T5(t ) 1 4 with the initial conditions:
a A2(t )P(t )D(t ) 1 1 1 1 a3 P(t 1 ) a4 P(t 1 ) b1T (t); 1a A(t )P(t )D(t ) 2 1 1 1 p2 D(t 4 ) p3 D(t 4 ) b4 P(t),
T (t) (t),
A(t)1(t), t[ ,0) t[ ,0) 2 (t ),t[ ,0) D(t ) 2* (t ),t(tk ,tk ] 3(t),t[ ,0) P(t) *
3(t),t(tk ,tk ]
max(1,..., 4)
(
The algorithm for solving problem (
Determination of the laws of solution behavior, analysis of the presence and
dynamics of critical points, determination of the nature of their stability allow in
advance, without starting quantitative studies, to determine the degree of suitability of
the equations for describing the dynamics of transformation. New cells, the nature of
the decrease in the number of stem cells due to apoptosis and determination of the
ranges of parameter values used in mathematical modeling [6-8]. The presence of a
trivial and functional attractor in the equation for the dynamics of neurogenesis makes
it possible to use it to describe the states of calm stem cells and activate
transformation into neuronal cells. Let us proceed to consider the nature of stability of the
equilibrium positions of the functional differential equation of the dynamics of
neurogenesis (
Linearization with respect to the equilibrium position X leads to a linear
differential-difference equation:
(
( 1)Z(t1)Z(t) h
e ( 1)e 1 Or:
h e ha ( 1)e 1 (
A detailed analysis of (
(Z )ez 0, where and are real numbers, have negative real parts if and only if: 1; 0; sin cos , where – root of the equation
atg 2 , if 0 . Consider first the trivial equilibrium 0 . Then h
and the first Hayes condition is satisfied. The second and third conditions
are as follows:
h (1 a) 0
; 0 , if 0
h
and
,
h a sin h cos
where – root of the equation
tg ,
0
(
Consequently, the right-hand side of the inequality of the third Hayes condition is positive (since sin * 0, cos * 0 )) and for the trivial equilibrium to be stable, a <1 must hold.
The emergence of a nontrivial equilibrium position will lead to the loss of stability of the trivial equilibrium position.
Let us proceed to consider the stability condition for a nontrivial equilibrium
position. For him, the first two conditions are satisfied, and the third condition is reduced
to:
ln a 1 ( h) * sin * cos *
(
Consequently, the parametric portrait of equation (
Domain A has a single stable trivial root. Upon transition to B, its stability is lost and the so-called "soft excitation" arises, at which a smooth change (in this case, the emergence of a nontrivial equilibrium position) of the rest position to a stable mode occurs. Upon transition to region C, the nontrivial equilibrium position loses its stability and oscillations arise around it (Figure 4).
The results of targeted computational experiments based on the developed program showed the possibility of analyzing the dynamics of the number of brain cells and revealed the presence of the following modes: state of calm stem cells, stationary mode, activation of transformation into neuronal cells (Figure 5), self-oscillatory mode of neurogenesis (Figure 6), loss self-oscillation mode (Figure 7), an uncontrolled, chaotic process (Figure 8) and a sharp loss of newly formed neurons due to apoptosis — the “black hole” effect (Figure 9).
One of the mechanisms for stabilizing the processes of neurogenesis is programmed cell death, that is, apoptosis. Programmed cell death is biologically reasonable as an effective way to remove non-viable and pathological nerve cells from the body. In the brain cell that has received the death signal, important processes of decision-making about apoptosis take place. This results in either continuation of normal operation or in the launch of the self-destruct program. Modeling the regulation of neurogenesis makes it possible to analyze the behavior of a model based on possible modes of normal behavior or transition to a "black hole" mode. 4
In connection with the successful implementation of the achievements of information technologies in various fields of science and the creation of strong artificial intelligence, the development and application of highly reliable mathematical methods for quantitative research of the abilities of perspective thinking, including brain research. plasticity is relevant [10-11]. The developed mathematical models of regulatory mechanisms of the functioning of neurons and neurogenesis of stem cells make it possible to quickly, in detail, in an environmentally safe way to analyze the functionality of the brain at the accepted level of modeling in various conditions of the external and internal environment, acting factors and biologically active substances. Effective mathematical modeling of the regulatory mechanisms of the interconnected functioning of human neurons under normal conditions and with anomalies requires the determination of the main stages, functioning factors and the most significant parameters of the functioning of the neural network. The results of targeted computational experiments based on the developed program showed the possibility of analyzing the interrelated activity of conjugated neurons in order to identify regulatory mechanisms for the precise organization of the implementation of information processes in the brain activity, to determine functional disorders and possible ways to prevent and treat anomalies. The developed equations of stem cell neurogenesis allow simulating the regulatory mechanisms of the main modes of functioning of the number of brain cells: rest, stationary mode, self-oscillations, irregular oscillations and breakdown of the oscillatory mode - the “black hole” effect. Identification of violations in the regulation of neural cell transformation will help the development of new strategies in the treatment of human nervous diseases. try in modern conditions. IOP Conference Series: Materials Science and Engineering, 734, 012051 (2020).