Classic models of biology include: Mathematical modeling of biological processes: Textbook

Despite the diversity of living systems, they all have the following specific features that must be taken into account when constructing models.

• 1. Complex systems. All biological systems are complex, multicomponent, spatially structured, and their elements have individuality. When modeling such systems, two approaches are possible. The first is aggregated, phenomenological. This approach identifies the defining characteristics of a system (for example, the total number of species) and considers quality properties behavior of these quantities over time (stability steady state, the presence of oscillations, the existence of spatial heterogeneity). This approach is historically the most ancient and characteristic dynamic theory populations. Another approach is detailed consideration elements of the system and their interactions, building a simulation model whose parameters have a clear physical and biological meaning. Such a model does not allow analytical research, but with good experimental study of the fragments of the system, it can give a quantitative prediction of its behavior under various external influences.
• 2. Reproducing systems (capable of autoreproduction). This most important property living systems determine their ability to process inorganic and organic matter for the biosynthesis of biological macromolecules, cells, and organisms. In phenomenological models, this property is expressed in the presence in the equations of autocatalytic terms that determine the possibility of growth (in non-limited conditions - exponential), the possibility of instability of the stationary state in local systems ( necessary condition the emergence of oscillatory and quasi-stochastic regimes) and instability of a homogeneous stationary state in spatially distributed systems (the condition of spatially inhomogeneous distributions and autowave regimes). Important role in the development of complex spatiotemporal regimes, processes of interaction of components (biochemical reactions) and processes of transfer, both chaotic (diffusion) and related to direction, play a role external forces(gravity, electromagnetic fields) or with adaptive functions of living organisms (for example, the movement of cytoplasm in cells under the influence of microfilamepts).
• 3. Open systems, constantly passing through themselves flows of matter and energy. Biological systems are far from thermodynamic equilibrium and are therefore described nonlinear equations. Linear Onsager relations connecting forces and flows are valid only near thermodynamic equilibrium.
• 4. Biological objects have complex multi-level regulation system. In biochemical kinetics, this is expressed in the presence of feedback loops in circuits, both positive and negative. In the equations of local interactions feedbacks are described by nonlinear functions, the nature of which determines the possibility of the occurrence and properties of complex kinetic regimes, including oscillatory and quasistochastic ones. This type of nonlinearity, when taking into account the spatial distribution and transport processes, determines the patterns of stationary structures (spots various shapes, periodic dissipative structures) and types of autowave behavior (moving fronts, traveling waves, leading centers, spiral waves, etc.).
• 5. Living systems have complex spatial structure. Living cell and the organelles it contains have membranes, any living organism contains great amount membranes, total area which amounts to tens of hectares. Naturally, the environment within living systems cannot be considered homogeneous. The very emergence of such spatial structure and the laws of its formation represent one of the problems of theoretical biology. One of the approaches to solving such a problem is the mathematical theory of morphogenesis.

Membranes not only separate different reaction volumes of living cells, but also separate living from non-living (environment). They play a key role in metabolism, selectively allowing flows of inorganic ions and organic molecules. In the membranes of chloroplasts, the primary processes of photosynthesis take place - storing light energy in the form of high-energy chemical compounds, used later for synthesis organic matter and other intracellular processes. The key stages of the respiration process are concentrated in the membranes of mitochondria, membranes nerve cells determine their ability for nerve conduction. Mathematical models of processes in biological membranes constitute an essential part of mathematical biophysics.

Existing models are mainly systems differential equations. However, it is obvious that continuous models are not able to describe in detail the processes occurring in such individual and structured complex systems what living systems are. In connection with the development of the computational, graphical and intellectual capabilities of computers, simulation models built on the basis of discrete mathematics, including models of cellular automata, play an increasingly important role in mathematical biophysics.

6. Simulation models of specific complex living systems, as a rule, take into account the available information about the object as much as possible. Simulation models are used to describe objects at various levels of organization of living matter - from biomacromolecules to models of biogeocenoses. IN the latter case models should include blocks describing both living and “inert” components. Classic example simulation models are models molecular dynamics, in which the coordinates and momenta of all atoms that make up the biomacromolecule and the laws of their interaction are specified. A computer-calculated picture of the “life” of the system allows you to trace how physical laws manifest themselves in the functioning of the simplest biological objects - biomacromolecules and their environment. Similar models, in which the elements (building blocks) are no longer atoms, but groups of atoms, are used in modern technology computer design of biotechnological catalysts and medicines, acting on certain active groups membranes of microorganisms, viruses or performing other targeted actions.

Simulation models are created to describe physiological processes, occurring in vital organs: nerve fiber, heart, brain, gastrointestinal tract, bloodstream. They play “scenarios” of processes occurring normally and in various pathologies, and study the influence of various processes on the processes. external influences, including medications. Simulation models are widely used to describe plant production process and are used to develop an optimal regime for growing plants in order to obtain maximum yield or obtain the most evenly distributed ripening of fruits over time. Such developments are especially important for expensive and energy-intensive greenhouse farming.

We will consider in this section analytical models. In analytical models, input. and exit. The parameters are related by explicit expressions: equations, inequalities, etc. If we solve systems of Kolmogorov-Erlang equations, this is analytical modeling, but if we focus on a graph model and conduct a statistical experiment, determining how the system serves the flow of requests, then this is simulation modeling. To solve an analytical model, you usually have to use numerical methods for solving problems, but some models also provide an analytical solution, because different methods are used to solve different mathematical problems; sometimes analytical models are divided by methods (integral, differential, linear, etc.), but usually by areas of application (physical, chemical, biological, pedagogical, technical). Let's look at some examples of analytical math. models that are the simplest and at the same time classic.

Mathematical models in physics and technology

In physics, modeling is primarily used to describe industrial processes involving the solution of differential equations and partial derivatives. All other models are usually a simplified version of these processes. The basis for constructing models is the following: laws and equations:

Some of the equations are written in one-dimensional form or using the radius vector

;

2. Model of an oscillatory system

Let's look at it from simple to complex. As an example, there are many objects around us where vibration is important (motors). Oscillations are also common in electrical systems. We will assume that we have one-dimensional oscillations (along one axis).

The position of the object is determined by one x coordinate, the equation will be
.

The solution to this differential. equation is well known, it is

Oscillations Harmonic with phase shift, undamped.

We complicate the model - introduce attenuation

(K- attenuation coefficient)

If K is small (K<<1), то решение не будет сильно отличаться. Решение системы приводит к возникновению
.

K=0.1 - attenuation is clearly visible (periodical). With increasing K (
) - aperiodic damping, when there is not a single period.

Natural frequency
, frequency within force R. When the frequencies are equal, we get a sharp increase in the amplitude of oscillations - resonance, . If resonance is produced during oscillation, the natural oscillations will die out, leaving forced oscillations at the frequency of the forced force.

TO<<1, W>>p.

Modulation. Inside there are natural oscillations, their amplitude is modeled with the frequency of natural oscillations (beats)

If K<0, м.б. (т.к. она только мешает) – параметрический резонанс.

Example: car springs (usually useful for swinging vibrations).

Resonance can be negative or positive in value. The emission of electromagnetic waves is based on resonances, both ordinary and parametric. The emission and reception of electromagnetic waves are resonant. Parametric resonance is advantageous because it is much more powerful than usual. This is a convenient means for generating, for example, microwave oscillations (tape recorder). For parametric resonance, a natural frequency is not needed, so you can pump in energy until the destruction of this resonator. But there may also be harm, destruction, which is unpleasant.

Modulation- the basis of radio communication. There is a carrier frequency that is modulated and then demodulated. Sound is low frequency (36 kHz), and radio waves travel at high frequencies, which means megahertz are needed. There is amplitude, phase and frequency modulation. The beating effect is usually harmful, disturbing - it is a source of noise. Sometimes special noise generators are made using beats.

Thin layer thermal conductivity model

glass (thin, long),
- the temperature will be uniform, therefore
.
border
Usually this equation is not solved explicitly, but using a cellular approximation. By solving this system of equations, we find the values ​​at the grid nodes. Other problems of thermal conductivity, electrostatics and electrodynamics are modeled in a similar way. The main problem is the complexity of the calculation, which is why powerful computers are required.

Another model is the movement of a body thrown at an angle to the horizon. To solve it, the so-called shooting method is used; it is already close to simulation modeling.

Also, a model of the rocket’s movement:

- Tsiolkovsky equation.

Kinetic and structural models in chemistry

In chemistry, models of chemical reactions and the structure of a chemical model are mainly common. connections. For chem. reactions, the most important thing is the kinetics, i.e. change in the course of reactions over time, i.e. The faster the reaction goes, the less reactant remains, and vice versa. At the beginning of the twentieth century, Adolf Lotka formulated a model of kinetic reactions, which was called the Volterra-Lotka model. Chain of transformations of substances:

A differential system has been obtained. equations. These equations are similar in meaning to the Kolmogorov-Erlang equations. This shows that these were also kinetic equations and all kinetic processes are similar to each other.

In chemistry, kinetic equations are complicated by the fact that the quantities are not constant, but depend on such quantities as,

chemical composition of substances (temperature obeys the law of heat capacity, R depends on diffusion, which is given by the equation
- Fick's law of diffusion. Darcy’s law of filtration transfer has a similar relationship). As a result, we have to solve these complex equations simultaneously with the kinetic equation.

In chemistry, the structural models of molecules are of great importance: H-O-H, especially convenient for organic substances (they have a very complex structure).

When studying a new chemical. substances make a new chemical. analysis - determine the proportions containing certain substances. Then you can determine which atoms the molecule consists of, but also how they are connected. A valence bond is introduced. Some atoms have a 1st valence bond, others have a 2nd, etc. Isomers of the substance were discovered with the same number of molecules, but with different properties.

2 tasks:

Determine the internal structure of a molecule and relate its structure and chemical properties. properties, i.e. study of isomers.

Designing isomers - learn how to create stable structures for molecules of various types and give them hypotheses. properties.

Both of these problems have become so popular in organic chemistry that special systems for modeling molecules have even been created.

Mathematical models in biology

Biology is extremely related to chemistry and biochemistry => structural modeling has moved from chemistry to biology. Biological structures are very complex chemical structures => the science of biochemistry has appeared, which studies the chemistry of biological structures. This is where structural modeling methods have proven to be very useful. The most famous problems associated with gene modeling.

Genes are molecules from which the so-called information components of living beings are formed - DNA, RNA. Basically, the genes have already been studied and known, but questions remain about which genes are included in this or that DNA and how they are connected to each other. Because even in the simplest DNA there are tens of thousands of genes, the world project “DNA model” arose, first in the simplest creatures, now in humans (completion). Structural modeling is leading in biochemistry.

Models of intraspecific struggle

Individuals of the same species compete with each other. At the beginning, when there are few individuals and the conditions are favorable and the population grows rapidly, restrictions occur due to the struggle between individuals of the same species. The very first simple model was the growth model - the unbridled growth model. In this model there is no intraspecific competition; it will be modernized.

The more a, the less growth, however, and this model could not describe some of the phenomena that occurred in real ecosystems. In some systems, there were fluctuations in numbers from year to year. We introduced one more parameter and complicated the model

Coefficient b determines the nonlinear dependence of the growth rate R from the number. Numerical study of this model revealed 4 characteristic situations:

Monotonous growth

Situation of damped oscillations

The situation of undamped oscillations

Situation of fluctuations (random changes)

These models are discrete, but it is also possible to construct a continuous, kinetic one, its equation:

. wherein r- a kind of analogue of speed. This two-parameter model is called the logistic kinetic model (Voltaire-Lottky model).

Models of interspecific competition

If two species coexist and actively influence each other, then processes of interspecific competition and struggle arise. The most famous model (kinetic) of Voltaire is Trays, competition of two types:

Coefficients determine the relationship between 2 types. If, then the increase in individuals of the second species leads to a decrease in individuals of the first species. The second type suppresses the first. If, then individuals of the second species do not influence. Obviously, the more wolves, the fewer hares. The model has 6 parameters - its study is very difficult, so usually some of the parameters are fixed. In general, studies of this parametric model have shown that predator and prey populations experience cyclical changes. In biology, simulation modeling is also very often used.

Simulation modeling in biology

Life model

It simulates the reproduction of the simplest creatures, sets some restrictions on reproduction, death, etc., and then runs an experiment and traces the dynamics over time. The simplest option (school). Take a table of empty and filled (live) cells. Rules are set, for example, if a living cell is surrounded by 4 or more living ones, then it dies from overpopulation, if there is only one near it or not, it dies from loneliness. If a dead person is adjacent to 3 living ones, she comes to life. Experiment:

The initial random configuration of living cells is set

Specifies the number of time points that will be tracked

In a cycle, the table is updated at points in time according to the given rules, and the picture changes are observed. Similar systems were studied, and it turned out that such a table could contain stable configurations that do not collapse.

Models in economics

Economic sciences are one of the most important areas of application of modeling; this is where models provide the greatest efficiency, for example, if you optimize the spending of the entire state in one model, the effect will be expressed in billions of dollars. The following types of models can be distinguished:

LP model (linear) – model of resources, reserves, etc.

Models built on the transport problem (distribution and transportation of goods)

Integer programming models (the result belongs to the domain of integers, number of people, number of factories, etc.) are models of the first type with integer parameters.

Dynamic programming models - mainly related to the development of any production, company, etc.

Game models associated with confrontation and competition.

Predictive models related to forecasting situations with a lack of information or random events.

Automatic control models (to make the control system optimal)

nonlinear models are solved only in selected cases.

34. Stochastic modeling. Monte Carlo method in simulation. Generating random and pseudo-random numbers. Generation methods and algorithms. Generating random numbers distributed according to exponential, normal and arbitrary distribution laws.

Stochastic programming– a section of mathematical programming, a set of methods for solving optimization problems of a probabilistic nature. This means that either the parameters of the constraints (conditions) of the problem, or the parameters of the objective function, or both are random variables (contain random components).

Optimization problem- an economic and mathematical problem, the goal of which is to find the best distribution of available resources. It is solved using an optimal model using mathematical programming methods, i.e., by searching for the maximum or minimum of some functions under given restrictions (conditional optimization) and without restrictions (unconditional optimization). The solution to an optimization problem is called an optimal solution, an optimal plan, or an optimal point.

Random variables are characterized by means, variance, correlation, regression, distribution function, etc.

Statistical Modeling– modeling using random processes and phenomena.

There are 2 options for using statistical modeling:

– in stochastic models there may be random parameters or interactions. The relationship between the parameters is random or very complex.

– even for deterministic models can be used statistical methods. Static modeling is almost always used in simulation models

Models, where there is a one-to-one relationship between the parameters and there are no random parameters are called deterministic.

Deterministic Processes– certain processes in which all processes are determined by laws.

Man considers all processes to be deterministic, but over time, random processes are discovered. Random process- this is a process the course of which can be different depending on the case, and the probability of one or another course is determined.

A study of the processes showed that they are of 2 types:

a) Processes that are random in nature;

b) Very complex deterministic processes;

A central theorem has been proven, according to which the addition of various processes increases the random nature. So, if you add completely different sequences that are not related to each other, then the result in the limit tends to a normal distribution. But it is known that the normal distribution is independent events, therefore, the combination of deterministic events in the limit leads to their randomness.

That. in nature there are no completely purely deterministic processes; there is always a mixture of deterministic and random processes. The effect of a random factor is called “noise”. Sources of noise are complex deterministic processes (Brownian motion of molecules).

In simulation modeling, complex processes are often replaced by random ones; therefore, in order to make a simulation model, you need to learn how to model random processes using static modeling methods. Random processes in quantum mechanics are represented by a sequence of random numbers, the value of which varies randomly.

In statistical modeling, the Monte Carlo statistical test method is very often used. Monte Carlo method is a numerical method for solving mathematical problems by modeling random variables.

The essence of the method: in order to determine a constant or deterministic characteristic of a process, you can use a static experiment, the parameters of which, in the limit, are related to the quantity being determined. Essence of the method Monte Carlo consists of the following: you need to find the value of a of some studied quantity. To do this, choose the following random variable
, the mathematical expectation of which is equal to :
. In practice, they do this: they produce tests, as a result of which they obtain possible values
; calculate their arithmetic mean
and accept as an estimate (approximate value) the required number :
.

Let's consider the essence of the method on examples its use.

Over the past decades, there has been significant progress in quantitative (mathematical) description functions of various biosystems at various levels of life organization: molecular, cellular, organ, organismal, population, biogeocenological (ecosystem). Life is determined by many different characteristics of these biosystems and processes occurring at the appropriate levels of system organization and integrated into a single whole during the functioning of the system. Models based on essential postulates about the principles of system functioning, which describe and explain a wide range of phenomena and express knowledge in a compact, formalized form, can be spoken of as biosystem theories. Construction of mathematical models(theories) of biological systems became possible thanks to the exceptionally intensive analytical work of experimenters: morphologists, biochemists, physiologists, specialists in molecular biology, etc. As a result of this work, the morphofunctional schemes of various cells were crystallized, within which various physical processes occur in an orderly manner in space and time. chemical and biochemical processes that form very complex interweavings.

The second very important circumstance, which contributes to the involvement of the mathematical apparatus in biology, is the careful experimental determination of the rate constants of numerous intracellular reactions that determine the functions of the cell and the corresponding biosystem. Without knowledge of such constants, a formal mathematical description of intracellular processes is impossible.

And finally, third condition What determined the success of mathematical modeling in biology was the development of powerful computing tools in the form of personal computers, supercomputers and information technologies. This is due to the fact that usually the processes that control a particular function of cells or organs are numerous, covered by feedforward and feedback loops and, therefore, described complex systems of nonlinear equations with a large number of unknowns. Such equations cannot be solved analytically, but can be solved numerically using a computer.

Numerical experiments on models capable of reproducing a wide class of phenomena in cells, organs and the body allow us to evaluate the correctness of the assumptions made when constructing the models. Although experimental facts are used as model postulates, the need for some assumptions and assumptions is an important theoretical component of modeling. These assumptions and assumptions are hypotheses, which can be subjected to experimental verification. Thus, models become sources of hypotheses, moreover, experimentally verifiable. An experiment aimed at testing a given hypothesis can refute or confirm it and thereby help refine the model.

This interaction between modeling and experiment occurs continuously, leading to an increasingly deeper and more accurate understanding of the phenomenon:

• the experiment refines the model,
• the new model puts forward new hypotheses,
• the experiment refines the new model, etc.

Currently field of mathematical modeling of living systems unites a number of different and already established traditional and more modern disciplines, the names of which sound quite general, so that it is difficult to strictly delimit the areas of their specific use. At present, specialized areas of application of mathematical modeling of living systems are developing particularly rapidly - mathematical physiology, mathematical immunology, mathematical epidemiology, aimed at developing mathematical theories and computer models of relevant systems and processes.

Like any scientific discipline, mathematical (theoretical) biology has its own subject, methods, methods and procedures of research. As subject of research mathematical (computer) models are used biological processes, simultaneously representing both an object of research and a tool for studying biological objects themselves. In connection with this dual essence of biomathematical models, they imply use of existing and development of new methods for analyzing mathematical systems(theories and methods of relevant branches of mathematics) in order to study the properties of the model itself as a mathematical object, as well as the use of the model to reproduce and analyze experimental data obtained in biological experiments. At the same time, one of the most important purposes of mathematical models (and theoretical biology in general) is the ability to predict biological phenomena and scenarios for the behavior of a biosystem under certain conditions and their theoretical justification before conducting the corresponding biological experiments.

The main research method and the use of complex models of biological systems is computational computer experiment, which requires the use of adequate calculation methods for the corresponding mathematical systems, calculation algorithms, technologies for the development and implementation of computer programs, storage and processing of computer modeling results.

Finally, in connection with the main goal of using biomathematical models to understand the laws of functioning of biological systems, all stages of the development and use of mathematical models require mandatory reliance on theory and practice of biological science, and primarily on the results of natural experiments.

MINISTRY OF EDUCATION AND SCIENCE OF THE RUSSIAN FEDERATION

FEDERAL STATE BUDGET EDUCATIONAL

INSTITUTION OF HIGHER PROFESSIONAL EDUCATION

"UDMURT STATE UNIVERSITY"

Faculty of Biology and Chemistry

TRAINING AND METODOLOGY COMPLEX

BY DISCIPLINE

MATH MODELING

BIOLOGICAL PROCESSES

Direction of training

Direction of training 020400 Biology

Name of master's program

"Biology" (Botany) 020421 m

"Biology" (Immunobiotechnology) 020422 m

"Biology" (Cell Biology) 020423 m

The place of the discipline in the structure of the master's program. Student competencies formed as a result of mastering the discipline. The purpose of mastering the discipline. The structure of the discipline by type of academic work, the relationship between topics and developed competencies. Contents of the discipline.

5.1 Topics of lecture sessions and their annotations

5.2. Practical lesson plans.

5.3. Plans laboratory workshop.

5.4. Program independent work students.

Educational technologies. Assessment tools for ongoing monitoring of progress, intermediate certification. Educational, methodological and information support of the discipline. Material and technical support of discipline.

PROCEDURE FOR APPROVAL OF THE WORK PROGRAM

Developer work program disciplines

Examination of the work program

Approval of the work program of the discipline

Other documents on assessing the quality of the discipline’s work program

(if available - FEPO, reviews from employers, undergraduates, etc.)

1 . PLACE OF DISCIPLINE IN THE STRUCTURE OF PLO MASTER'S PROGRAM

Discipline enters the cycle the basic part of the mathematical and natural science cycle of the OOP master's program.

The discipline is addressed to 020400 Biology (qualification (degree) "Master"), first year of study.

The course is preceded by the following disciplines: computer science, natural science disciplines.

To successfully master the discipline, the following competencies must be developed:

capable of adapting and improving one’s scientific and cultural level (OK-3);

Successful completion of the course allows you to move on to studying the following disciplines: theoretical biology, synergetics, With Contemporary problems of biology, other disciplines of the mathematical and natural science cycle of the Master's program, the implementation of a master's thesis.

The course program is built according to the block-modular principle, in it sections highlighted:

The concept of differential and integral calculus. Modeling goals. Basic concepts. Models described by an autonomous differential equation Discrete models Models described by systems of two autonomous differential equations Stability of stationary states of nonlinear systems. Trigger systems. Oscillatory systems.

2 . STUDENT COMPETENCIES FORMED

AS A RESULT OF MASTERING DISCIPLINE

· independently analyzes available information, identifies fundamental problems, poses a problem and performs field and laboratory biological research when solving specific problems in specialization using modern equipment and computing tools, demonstrates responsibility for the quality of work and the scientific reliability of the results (PC-3);

· creatively applies modern computer technologies in the collection, storage, processing, analysis and transmission of biological information (PC-6);

· independently uses modern computer technologies to solve research, production and technological problems of professional activity, to collect and analyze biological information (PC-13);

As a result of mastering the discipline, the student must:

know:

· about methods of modeling biological systems with their subsequent analysis using differential and integral calculus.

be able to:

· be able to apply the acquired knowledge in practical work;

· correctly present the results of model calculations performed.

Own:

· skills of integral and differential calculus;

· skills in working with a personal computer when using available software products for numerical modeling of biological systems.

3 . PURPOSE AND OBJECTIVES OF MASTERING THE DISCIPLINE

The purpose of mastering the discipline MATHEMATICAL MODELING OF BIOLOGICAL PROCESSES

is:

give some basic knowledge and ideas about the possibilities of practicing numerical methods of mathematical analysis, mathematical modeling, classification of mathematical models and the scope of their applicability, show what fundamental qualitative questions can be answered by a mathematical model, in the form of which knowledge about a biological object is formalized. This is achieved by including in the course basic issues of integral and differential calculus, the foundations of the mathematical apparatus of the qualitative theory of differential equations. Based on this knowledge, the main types of temporal and spatial dynamic behavior inherent in biological systems of different levels are considered. The possibilities of mathematical modeling are illustrated by examples of specific models that can be considered classic.

Objectives of mastering the discipline:

to form ideas about the applicability of numerical methods of mathematical analysis in relation to mathematical modeling of biological systems;

introduce specific mathematical models that a research biologist can apply (adapt) to his research;

expand knowledge on the use of software in modeling biological processes.

4. STRUCTURE OF DISCIPLINE BY TYPE OF STUDY WORK,

RELATIONSHIP OF TOPICS AND FORMED COMPETENCIES

Topic 1.2.

test work on the material of the previous lesson, theoretical introduction to the topic of the lesson, implementation of practical tasks.

Topic 1.3.(2 hours) Theoretical part.

List of assignments and tasks submitted for laboratory work:

Topic 1.4.(2 hours) Theoretical part.

List of assignments and tasks submitted for laboratory work: test work on the material of the previous lesson, theoretical introduction to the topic of the lesson, implementation of practical tasks.

Topic 1.5.(3 hours) Theoretical part.

List of assignments and tasks submitted for laboratory work: test work on the material of the previous lesson, theoretical introduction to the topic of the lesson, implementation of practical tasks.

Topic 1 hour) Theoretical part.

List of assignments and tasks submitted for laboratory work: test work on the material of the previous lesson, theoretical introduction to the topic of the lesson, implementation of practical tasks.

Topic 1.7.(2 hours) Theoretical part.

List of assignments and tasks submitted for laboratory work: test work on the material of the previous lesson, theoretical introduction to the topic of the lesson, implementation of practical tasks.

Topic 1.8.(2 hours) Theoretical part.

List of assignments and tasks submitted for laboratory work: test work on the material of the previous lesson, theoretical introduction to the topic of the lesson, implementation of practical tasks.

Topic 1.9.(2 hours) Theoretical part.

List of assignments and tasks submitted for laboratory work: test work on the material of the previous lesson, theoretical introduction to the topic of the lesson, implementation of practical tasks.

Topic 1.10.(2 hours) Theoretical part. Study of the stability of stationary states of second order nonlinear systems. Classical system of V. Volterra. Analytical research (determination of stationary states and their stability) and construction of phase and kinetic portraits. Using the Maxima analytical calculation package.

List of assignments and tasks submitted for laboratory work: test work on the material of the previous lesson, theoretical introduction to the topic of the lesson, implementation of practical tasks.

List of assignments and tasks submitted for laboratory work: test work on the material of the previous lesson, theoretical introduction to the topic of the lesson, implementation of practical tasks.

Topic 1.hour) Theoretical part.

List of assignments and tasks submitted for laboratory work: test work on the material of the previous lesson, theoretical introduction to the topic of the lesson, implementation of practical tasks.

5.4. Self-study program for master's students

SRS structure

Preparation for laboratory work – 12 works - 48 hours

The results of all types of SRS are assessed in points and are the basis of SRS.

When performing SRS, the educational and methodological materials specified in the corresponding section are used (see table SRS structure)

SRS control schedule

Legend: cr – test , To - colloquium, R - abstract, d – report, di – business game, rz – problem solving, chickens - course work , lr - laboratory work, dz - homework

6. EDUCATIONAL TECHNOLOGIES

When conducting classes and organizing independent work of undergraduates, traditional technologies of informative learning are used, which involve the transfer of information in ready-made form, the formation of educational skills according to the model: the theoretical part of the laboratory work is structured as: lecture-exposition, lecture-explanation.

The use of traditional technologies ensures the formation of the cognitive (knowledge) component of the professional competencies of a research biologist.

In the process of studying the theoretical sections of the discipline and performing practical tasks, new educational teaching technologies are used: lecture-visualization.

When conducting laboratory classes the following are used:

The concept of a model. Objects, goals and modeling methods. Models in different sciences. Physical and mathematical models. History of the first models in biology. Modern classification of models of biological processes: regression, simulation, qualitative models. Examples of various models used in your area of ​​scientific interest. Principles of simulation modeling and examples of models. Specifics of modeling living systems.

The concept of the derivative and how to find it (differentiation rules). Integral and methods for finding integrals. Solving problems on this topic.

Drawing up (derivation) of a differential equation. Some techniques for solving homogeneous and inhomogeneous differential equations. Solution by the method of separable variables. Solution of a general linear differential equation by the Lagrange method. Solving problems on this topic.

Methods for studying dynamic systems. Stationary state. Taylor's formula. Stability of stationary states (case of one equation): concept of stability, analytical method for determining the type of stability (Lyapunov method), graphical method for determining the type of stability. Solving problems on this topic.

Analysis of some population growth models. Malthus' models. Logistic model of Verhulst. Model of a flow cultivator. Solving problems on this topic.

Difference models of population growth. Analysis of the Malthus difference model (finding stationary states and analyzing them for stability). The discrete logistic Verhulst equation and its limitations for biological systems. Analysis of the discrete logistic Ricker equation (finding stationary states and analyzing them for stability). Qualitative analysis of difference models of population growth using the Lamerey diagram (ladder). Solving problems on this topic.

Analysis of models described by a system of two autonomous ordinary linear differential equations. Solution of a system of two linear autonomous ODEs. Analysis of the stability of the behavior of these models near singular points. Types of singular points. Solving problems on this topic.

A qualitative method for analyzing models described by a system of two autonomous ordinary linear differential equations. Phase plane. Isoclins. Construction of phase portraits. Kinetic curves. Solving problems on this topic.

Analysis of some models described by a system of two autonomous ordinary linear differential equations. Analysis of the kinetic model of a system of linear chemical reactions.

Study of the stability of stationary states of second-order nonlinear systems. Lyapunov's method of linearization of systems in the vicinity of a stationary state. Examples of studying the stability of stationary states of models of biological systems. Analysis of the Lotka kinetic equation (chemical reaction). Classical system of V. Volterra. Analytical research (determination of stationary states and their stability) and construction of phase and kinetic portraits.

Trigger systems. Competition. Analytical research (determination of stationary states and their stability) and construction of phase and kinetic portraits.

Oscillatory systems. Local model of Brusselator.

The main technology for assessing the level of development of competence(s) is: a point-rating system for assessing student performance (Order /01-04 "On the introduction of the Procedure for implementing a point-rating system for assessing the educational work of students at the Federal State Budgetary Educational Institution of Higher Professional Education "UdSU").

Total points = 100 points.

Attendance at classes and the student’s work during the class itself is assessed up to 15 points.

The test at the beginning of the lesson is worth up to 30 points.

Homework is graded up to 15 points.

Number of points allocated for credit up to 40 points

The discipline is considered mastered if at the stage of intermediate certification the student scored more than 14 points and the student’s final rating in the discipline for the semester is at least 61 points.

Scheme for converting points to traditional assessment

Table for converting final BRS points into the traditional grading system

Examples of test tasks, issued at the beginning of the lesson for 10-12 minutes.

Test task 1

Option 1

1) Find the derivative based on the definition of the derivative: y = (1+3x)2

2) The population size is described by the equation: https://pandia.ru/text/78/041/images/image004_19.gif" width="88" height="41">

Option 3

1) Find the derivative based on the definition of the concept of derivative: y = (1+x)2

2) The population size is described by the equation: https://pandia.ru/text/78/041/images/image006_13.gif" width="90" height="45">

Option 2

Option 3

Solve the following differential equation.

Find a solution to the Cauchy problem if x(0)=1

Test task 3

Poll 3. Option 2

Solve the following differential equation.

Poll 3. Option 3

Solve the following differential equation.

Poll 3. Option 4

Solve the following differential equation.

Approximate test tasks for home use(specific texts of the assignment are issued to undergraduates through the IIAS system and on paper):

Homework 1

Recommendations.

1) Prepare a speech and attach a handwritten text with the report about an example of a physical model

2) Prepare a speech and attach a handwritten text with the report about the example regression model in your specialty (I can ask anyone) – 3-4 minutes – one per group. Should not be the same as another group's example.

3) Prepare a speech and attach a handwritten text with the report about an example of a simulation model in your specialty (I can ask anyone) – 3-4 minutes – one per group. Should not be the same as another group's example.

4) Using the definition of derivative, find the derivative for the expression:

y= 1+ x+ x 2

5) Find derivatives:

https://pandia.ru/text/78/041/images/image014_10.gif" width="84" height="41 src=">

https://pandia.ru/text/78/041/images/image017_9.gif" width="108" height="27 src=">.gif" width="105" height="41 src=">, Where u And A constant..gif" width="153" height="28 src=">

8) The bacterial population grows from an initial size of 1000 individuals to p(t) in the moment t(in days) according to the equation https://pandia.ru/text/78/041/images/image023_6.gif" width="106" height="41 src=">. Find p(t) for all moments t>0 if p(0)=0. How many years will it take for the proportion of those who have recovered to reach 90%?

3) Find common decision for the following first order equations and solve the Cauchy problem for the specified conditions:

If x(0)=2

, if x(0)=1

Homework 3

Recommendations. The assignment report is provided only in handwritten form indicating all intermediate calculations (an electronic version is not needed). All calculations must be transparent (write what you are calculating, indicate the original calculation formula, then a formula with substituted numbers, then the answer).

1) Population growth is described by the Verhulst equation. The capacity of the ecological niche for it is 1000. Construct a graph of the dynamics of the population size if it is known that the initial number is equal to: a) 10; b) 700; c) 1200. The growth rate r is 0.5. Specify the coordinates of the inflection point.

2) Expand the function f (x) into a Taylor series in the vicinity of the point 0 x up to 4th order:

f (x) = x 3 +1, x 0 = 1;

https://pandia.ru/text/78/041/images/image028_5.gif" width="114" height="46 src=">

https://pandia.ru/text/78/041/images/image030_5.gif" width="71" height="41 src=">. Find stationary states of the equation and determine their type of stability analytically (Lyapunov method) and with using a function graph f (x) :

f (x) = x 3 + 8x – 6x 2

f (x) = x 4 + 2x 3 − 15x 2

Homework 4

Recommendations. The assignment report is provided only in handwritten form indicating all intermediate calculations (an electronic version is not needed). All calculations must be transparent (write what you are calculating, indicate the original calculation formula, then the formula with substituted numbers, then the answer).

1) (1.0 points) Using the Lamerey diagram, construct a graph of population dynamics if the dependence Nt+1 = f (Nt) has the form and draw a conclusion about the sustainability of the population development.

2) (2.5 points) Construct a phase portrait for each of the systems in the vicinity of the stationary state according to the plan:

2.1) Find the coordinates of the singular (stationary) point

2.3) Using the isocline method (isocline: 0o, +45o, –45o, 90o, angles of intersection with the X and Y axes) construct a phase portrait of the system

2.4) using isoclines and based on point 2.2, draw a sketch of the phase portrait

2.5) Determine the direction of movement of the trial (figurative) point along the integral curves obtained in 2.4.

2.6) Select arbitrary point on one of the integral curves obtained in paragraph 2.4 and construct a kinetic portrait of the system.

3) (1.5 points) In the process of studying a certain population, the following dependence of population size on time was revealed (see data below).

1) Does the development of this population obey the Malthus equation or the Verhulst equation? Prove it.

2) If the development of a population obeys the Malthus equation, determine:

r

2.2) doubling period T.

2) If the development of the population obeys the logistic equation, determine:

2.1) the value of the Malthusian parameter r (specific speed reproduction);

2.2) resource parameter value TO

2.3) using values r And TO estimate the time after which population growth will begin to slow down.

This control and evaluation technology provides assessment of the level of mastery of professional competencies.

8 EDUCATIONAL, METHODOLOGICAL AND INFORMATION SUPPORT

DISCIPLINES

Main literature

1. Riznichenko, on mathematical models in biology. Part 1. Description of processes in living systems over time. - M.; Izhevsk: RHD, 2002

Lectures. Methods of lecturing

Lectures are one of the main teaching methods in the discipline, which should solve the following problems:

· state essential material course syllabus covering the main points;

· to develop among undergraduates the need for independent work on educational and scientific literature.

The main task each lecture is to reveal the essence of the topic and analyze its main provisions. It is recommended to bring to the attention of undergraduates the structure of the course and its sections at the first lecture, and then indicate the beginning of each section, the essence and its objectives, and, having finished the presentation, summarize this section in order to link it with the next one.

Methodology for conducting laboratory classes

The purposes of the laboratory work are:

· establishing connections between theory and practice in the form experimental confirmation provisions of the theory;

· training master's students in the ability to analyze the results obtained;

· control of independent work of undergraduates in mastering the course;

· training in professional skills

The goals of the laboratory workshop are being achieved the best way in the event that the experiment is preceded by certain preparatory extracurricular work. Therefore, the teacher is obliged to inform all undergraduates about the laboratory work schedule so that they can engage in targeted home preparation.

Before the start of the next lesson, the teacher must make sure that the master's students are ready to perform laboratory work through a short interview and checking that the master's students have prepared protocols for the work.

Successful mastery of the discipline presupposes the active, creative participation of the master's student through systematic, daily work.

The study of the discipline should begin with the development of a work program, Special attention, paying attention to the goals and objectives, structure and content of the course.

Review the notes immediately after class, marking any material in the lecture notes that is difficult to understand. Try to find answers to difficult questions using the recommended literature. If you are unable to understand the material on your own, formulate questions and seek help from your teacher at a consultation or the next lecture.

Regularly set aside time to review the material you have covered, testing your knowledge, skills and abilities using test questions.

Performing laboratory work

During class, obtain a lab schedule from your teacher. Get all the necessary methodological support.

Before visiting the laboratory, study the theory of the question proposed for research, read the manual for the relevant work and prepare a protocol for the work, in which you include:

· job title;

· preparing tables for filling with experimental observational data;

· equations of chemical reactions of transformations that will be carried out during the experiment;

· calculation formulas.

The preparation of reports should be carried out after completion of work in the laboratory or in another place designated for classes.

To prepare to protect the report, you should analyze experimental results, compare them with known theoretical principles or reference data, summarize the research results in the form of conclusions on the work, prepare answers to the questions given in methodological guidelines to perform laboratory work.

9. MATERIAL AND TECHNICAL SUPPORT OF DISCIPLINE

To conduct a computer workshop, you need a computer class that allows you to provide a separate workplace for each student. Computers must have parameters sufficient for the functioning of the programs being studied. If you use insufficiently powerful computers, you can recommend using older versions of programs or replacing some of the programs you are studying with less resource-intensive ones. Computers must have access to the Internet. Computers must have Windows XP (or older) installed, as well as a set of programs being studied (see the relevant section of paragraph 8 EDUCATIONAL, METHODOLOGICAL AND INFORMATION SUPPORT OF DISCIPLINE).

The computer class should have a large blackboard, chalk, and a cloth.

We have already said that the mathematical approach to the study of certain phenomena of the real world usually begins with the creation of corresponding general concepts, i.e., from the construction of mathematical models that have essential properties for us of the systems and processes that we study. We also mentioned the difficulties associated with the construction of such models in biology, difficulties caused by the extreme complexity of biological systems. However, despite these difficulties, the “model” approach to biological problems is now successfully developing and has already brought certain results. We will look at some models related to various biological processes and systems.

Speaking about the role of models in biological research, it is important to note the following. Although we understand the term “model” in an abstract sense - as a certain system logical concepts, and not as a real physical device, yet a model is something significantly more than a simple description of a phenomenon or a purely qualitative hypothesis, in which there is still enough room for various kinds ambiguities and subjective opinions. Let us remind you next example, dating back to a fairly distant past. At one time, Helmholtz, while studying hearing, put forward the so-called resonance theory, which looked plausible from a purely qualitative point of view. However, carried out later quantitative calculations, taking into account the real values ​​of the masses, elasticity and viscosity of the components auditory system components showed the inconsistency of this hypothesis. In other words, an attempt to turn a purely qualitative hypothesis into an accurate model that allows its investigation mathematical methods, immediately revealed the inconsistency of the original principles. Of course, if we have built a certain model and even obtained good agreement between this model and the results of the corresponding biological experiment, this does not yet prove the correctness of our model. Now, if, based on studying our model, we can make some predictions about that biological system, which we simulate and then confirm these predictions real experiment, then this will be much more valuable evidence in favor of the correctness of the model.

But let's move on to specific examples.

2.Blood circulation

One of the first, if not the very first, work on mathematical modeling of biological processes should be considered the work of Leonhard Euler, in which he developed mathematical theory blood circulation, looking at the first approximation of the entire circulatory system as consisting of a reservoir with elastic walls, peripheral resistance and a pump. These ideas of Euler (as well as some of his other works) were first thoroughly forgotten, and then revived in more recent times. later works other authors.

3. Mendel's laws

A fairly old and well-known, but nevertheless very remarkable model in biology is the Mendelian theory of heredity. This model, based on probability theoretical concepts, is that the chromosomes of the parent cells contain certain sets of characteristics, which during fertilization are combined with each other independently and randomly. Subsequently, this basic idea underwent very significant clarifications; for example, it was discovered that different signs are not always independent of each other; if they are associated with the same chromosome, then they can only be transmitted in a certain combination. Further, it was discovered that different chromosomes do not combine independently, but there is a property called chromosome affinity that violates this independence, etc. At present, probability-theoretical and statistical methods have penetrated very widely into genetic research and even the term “mathematical genetics” received full citizenship rights. Currently, intensive work is being carried out in this area, many results have been obtained that are interesting both from biological and purely mathematical point vision. However, the very basis of these studies is the model that was created by Mendel more than 100 years ago.

4. Muscle models

One of the most interesting objects physiological research is a muscle. This object is very accessible, and the experimenter can carry out many studies simply on himself, having only relatively simple equipment. The functions that a muscle performs in a living organism are also quite clear and definite. Despite all this, numerous attempts to build a satisfactory model of muscle function have not yielded definitive results. It is clear that although a muscle can stretch and contract like a spring, their properties are completely different, and even to the very first approximation, a spring cannot be considered as a semblance of a muscle. For a spring, there is a strict relationship between its elongation and the load applied to it. This is not the case for a muscle: a muscle can change its length while maintaining tension, and vice versa, change the traction force without changing its length. Simply put, at the same length, a muscle can be relaxed or tense.

Among the various modes of operation possible for a muscle, the most significant are the so-called isotonic contraction (i.e., a contraction in which the muscle tension remains constant) and isometric tension, in which the length of the muscle does not change (both ends are fixed). Studying a muscle in these modes is important for understanding the principles of its operation, although under natural conditions muscle activity is neither purely isotonic nor purely isometric.

To describe the relationship between the speed of isotonic muscle contraction and the magnitude of the load, various mathematical formulas. The most famous of them is the so-called characteristic equation Hilla. It looks like

(P+a)V=b(P 0 -P),

Others are good famous formulas to describe the same connection is the Ober equation

P = P 0 e- V⁄P ±F

and the Polissar equation

V=const (A 1-P/P 0 - B 1-P/P 0).

Hill's equation has become widespread in physiology; it gives a fairly good agreement with experiment for the muscles of a wide variety of animals, although in fact it represents the result of a "fit" rather than an inference from some model. Two other equations, which give approximately the same dependence over a fairly wide range of loads as the Hill equation, were obtained by their authors from certain ideas about the physicochemical mechanism muscle contraction. There are a number of attempts to construct a model of muscle work, considering the latter as some combination of elastic and viscous elements. However, there is still no sufficiently satisfactory model that reflects all the main features of muscle work in various modes.

5. Neuron models, neural networks

Nerve cells, or neurons, are the “working units” that make up the nervous system and to which the animal or human body owes all its abilities to perceive external signals and control various parts bodies. A characteristic feature of nerve cells is that such a cell can be in two states - rest and excitation. In this, nerve cells are similar to elements such as radio tubes or semiconductor triggers, from which the logical circuits of computers are assembled. Over the past 15-20 years, many attempts have been made to model activities nervous system, based on the same principles on which the work of universal computers is based. Back in the 40s, American researchers McCulloch and Pitts introduced the concept of a “formal neuron,” defining it as an element (the physical nature of which does not matter) equipped with a certain number of “excitatory” and a certain number of “inhibitory” inputs. This element itself can be in two states - “rest” or “excitement”. An excited state occurs if the neuron receives a sufficient number of excitatory signals and there are no inhibitory signals. McCulloch and Pitts showed that with the help of circuits composed of such elements, it is possible, in principle, to implement any of the types of information processing that occur in a living organism. This, however, does not mean at all that we have thereby learned the actual principles of the nervous system. First of all, although nerve cells are characterized by the “all or nothing” principle, i.e. the presence of two clearly defined states - rest and excitation, it does not at all follow from this that our nervous system, like a universal computer, uses binary digital code consisting of zeros and ones. For example, in the nervous system, frequency modulation apparently plays a significant role, that is, the transmission of information using the length of time intervals between impulses. In general, in the nervous system there is apparently no such division of information encoding methods into “digital” discrete) and “analog” (continuous) as is available in modern computer technology.

In order for a system of neurons to work as a whole, it is necessary that there be certain connections between these neurons: impulses generated by one neuron must arrive at the inputs of other neurons. These connections can have a correct, regular structure, or they can be determined only by statistical patterns and be subject to certain random changes. In currently existing computing devices, no randomness in connections between elements is allowed, however, there are a number of theoretical studies on the possibility of constructing computing devices based on the principles of random connections between elements. There are quite serious arguments in favor of the fact that the connections between real neurons in the nervous system are also largely statistical, and not strictly regular. However, opinions on this matter differ.

In general, the following can be said about the problem of modeling the nervous system. We already know quite a lot about the peculiarities of the work of neurons, that is, those elements that make up the nervous system. Moreover, with the help of systems of formal neurons (understood in the sense of McCulloch and Pitts or in some other sense), simulating the basic properties of real nerve cells, it is possible to simulate, as already mentioned, very diverse ways of processing information. Nevertheless, we are still quite far from a clear understanding of the basic principles of the functioning of the nervous system and its individual parts, and, consequently, from creating its satisfactory model *.

* (If we can create some kind of system that can solve the same problems as some other system, this does not mean that both systems work according to the same principles. For example, you can numerically solve a differential equation on a digital computer by giving it the appropriate program, or you can solve the same equation on an analog computer. We will get the same or almost the same results, but the principles of information processing in these two types of machines are completely different.)

6. Perception of visual images. Color vision

Vision is one of the main channels through which we receive information about outside world. Famous expression- it is better to see once than to hear a hundred times - this is also true, by the way, from a purely informational point of view: the amount of information that we perceive through vision is incomparably greater than that perceived by other senses. This importance visual system for a living organism, along with other considerations (specificity of functions, the possibility of conducting various studies without any damage to the system, etc.) stimulated its study and, in particular, attempts at a model approach to this problem.

The eye is an organ that serves as both an optical system and an information processing device. From both points of view, this system has a number of amazing properties. The eye's ability to adapt to a very wide range of light intensities and to correctly perceive all colors is remarkable. For example, a piece of chalk placed in a poorly lit room reflects less light than a piece of coal exposed to bright sunlight, nevertheless, in each of these cases we perceive the colors of the corresponding objects correctly. The eye conveys relative differences in illumination intensities well and even “exaggerates” them somewhat. Thus, a gray line on a bright white background seems darker to us than a solid field of the same gray. This ability of the eye to emphasize contrasts in illumination is due to the fact that visual neurons have an inhibitory effect on each other: if the first of two neighboring neurons receives a stronger signal than the second, then it has an intense inhibitory effect on the second, and the difference in the output of these neurons is the intensity is greater than the difference in the intensity of the input signals. Models consisting of formal neurons connected by both excitatory and inhibitory connections have attracted the attention of both physiologists and mathematicians. There are also interesting results and unresolved issues.

Of great interest is the mechanism of perception by the eye various colors. As you know, all shades of colors perceived by our eyes can be represented as combinations of three primary colors. Usually these primary colors are red, blue and yellow colors, corresponding to wavelengths 700, 540 and 450 Å, but this choice is not unambiguous.

The “three-color” nature of our vision is due to the fact that the human eye has three types of receptors, with maximum sensitivity in the yellow, blue and red zones, respectively. The question is how do we distinguish between these three receptors? a large number of color shades, is not very simple. For example, it is not yet clear enough what exactly this or that color is encoded in our eye: frequency nerve impulses, the localization of the neuron that preferentially responds to a given shade of color, or something else. There are some model ideas about this process of perception of shades, but they are still quite preliminary. There is no doubt, however, that here, too, systems of neurons connected to each other by both excitatory and inhibitory connections should play a significant role.

Finally, the eye is also very interesting as a kinematic system. A series of ingenious experiments (many of them were carried out in the laboratory of vision physiology of the Institute for Problems of Information Transmission in Moscow) established the following at first glance unexpected fact: if some image is motionless relative to the eye, then the eye does not perceive it. Our eye, when examining an object, literally “feels” it (these eye movements can be accurately recorded using appropriate equipment). Study of the motor apparatus of the eye and development of appropriate model representations are quite interesting both in themselves and in connection with other (optical, informational, etc.) properties of our visual system.

To summarize, we can say that we are still far from creating completely satisfactory models of the visual system that well describe all its basic properties. However, a number important aspects and (the principles of its operation are already quite clear and can be modeled in the form of computer programs for a digital computer or even in the form of technical devices.

7. Active medium model. Spread of excitation

One of the very characteristic properties many living tissues, primarily nerve tissue, this is their ability to excite and transfer excitation from one area to another adjacent to it. About once a second, a wave of excitement runs through our heart muscle, causing it to contract and drive blood throughout the body. Excitation along nerve fibers, spreading from the periphery (sensory organs) to the spinal cord and brain, informs us about the outside world, and in the opposite direction there are excitation commands that prescribe certain actions to the muscles.

Excitation in a nerve cell can occur on its own (as they say, “spontaneously”), under the influence of an excited neighboring cell, or under the influence of some external signal, say, electrical stimulation coming from some current source. Having passed into an excited state, the cell remains in it for some time, and then the excitement disappears, after which a certain period of cell immunity to new stimuli begins - the so-called refractory period. During this period, the cell does not respond to signals received by it. Then the cell returns to its original state, from which a transition to a state of excitation is possible. Thus, the excitation of nerve cells has a number of clearly defined properties, starting from which it is possible to construct an axiomatic model of this phenomenon. Further, purely mathematical methods can be used to study this model.

Ideas about such a model were developed several years ago in the works of I.M. Gelfand and M.L. Tsetlin, which were then continued by a number of other authors. Let's formulate axiomatic description the model in question.

By “excitable medium” we mean a certain set X elements (“cells”) with the following properties:

1.Each element can be in one of three states: rest, excitement and refractoriness;

2. From each excited element, excitation spreads through many elements at rest at a certain speed v;

3.If the item X hasn't been excited for some specific time T(x), then after this time it spontaneously goes into an excited state. Time T(x) called the period of spontaneous activity of the element X. This does not exclude the case when T(x)= ∞, i.e. when spontaneous activity is actually absent;

4. The state of excitement lasts for some time τ (which may depend on X), then the element moves for a while R(x) into a refractory state, after which a state of rest sets in.

Similar mathematical models arise in completely other areas, for example, in the theory of combustion, or in problems of the propagation of light in an inhomogeneous medium. However, the presence of a “refractory period” is characteristic feature namely biological processes.

The described model can be studied or analytical methods, or by implementing it on a computer. In the latter case, we are, of course, forced to assume that the set X(excitable medium) consists of a certain finite number of elements (in accordance with the capabilities of the existing computer technology- about several thousand). For analytical research it is natural to assume X some continuous variety (for example, consider that X- this is a piece of plane). The simplest case such a model is obtained if we take for X some segment (a prototype of a nerve fiber) and assume that the time during which each element is in an excited state is very short. Then the process of sequential propagation of impulses along such a “nerve fiber” can be described by a chain of ordinary first-order differential equations. Already in this simplified model, a number of features of the propagation process that are also found in real biological experiments are reproduced.

The question of the conditions for the emergence of such a model in such a model is very interesting. active environment so-called fibrillation. This phenomenon, observed experimentally, for example in the cardiac muscle, consists in the fact that instead of rhythmic coordinated contractions, random local excitations appear in the heart, devoid of periodicity and disrupting its functioning. First theoretical research This problem was tackled in the work of N. Wiener and A. Rosenbluth in the 50s. Currently, work in this direction is being intensively carried out in our country and has already yielded a number of interesting results.