# mathematical modeling

## Modeling of wave patterns at the combustion front

In experimental studies of the propagation of combustion waves in gaseous media, it was found that, under certain conditions, autowave – spiral or target – patterns appear at the wave front. The purpose of the present study is to propose a mathematical model that can explain this phenomenon based on the known chemical kinetics of hydrogen combustion. Model. The original detailed model was first reduced to four equations that adequately describe the propagation of the combustion wave.

## Calcium concentration in astrocytes: Emergence of complicated spontaneous oscillations and their cessation

The purpose of this work is to show the mechanisms of transitions between different dynamic modes of spontaneous astrocytic calcium activity. With this aim, dynamics of recently introduced Lavrentovich–Hemkin mathematical model was examined by both analytical and numerical techniques. Methods. In order to obtain the conditions for the oscillations cessation, the linear stability analysis for the equilibrium point was carried out. Complicated dynamics was studied numerically by calculations of time traces and bifurcation diagrams. Results.

## Control of network bursting discharges by local electrical stimulation in spiking neuron network

Goal. The paper is devoted to controlling the dynamics of spike neural networks by local periodic stimulation of various network sections. Methods. The simulation uses a network of synaptically connected spike neurons distributed in two-dimensional space. The dynamics of the transmembrane potential of neurons is described by the Izhikevich model, short-term synaptic plasticity is represented by the model Tsodyksa–Markram, the effects of changes in the efficiency of connections between neurons are modeled using spike-timing-dependent plasticity (STDP). Results.

## Development of the Russian state in the 20th and 21st centuries: Mathematical modeling based on the socio-energy approach

Purpose. The article is devoted to modeling the socio-political development of Russia in 1910–2009 based on the author’s socio-energy approach. In this paper, we briefly talk about the basics of the proposed approach, its principles and basic equations. Methods. The mathematical model is based on the Langevin diffusion equation. We also introduce the concepts of social energy, coefficients of the state of society and give them definitions. Results.

## Control of network bursting discharges by local electrical stimulation in spiking neuron network

Aim. Control of spiking neuron network dynamics represents a fundamental problem generally unsolved for both real brain networks and their simulating mathematical models. The purpose of the work is to study the possibility of controlling the dynamics of network spike bursts using periodic stimulation of various zones of the neural network. Methods. We consider very simplified relative to real brain circuits, but still very complicated dynamically, model of synaptically coupled spiking neurons distributed in two-dimensional space.

## Calcium concentration in astrocytes: emergence of complicated spontaneous oscillations and their cessation

The purpose of this work is to show the mechanisms of transitions between different dynamic modes of spontaneous astrocytic calcium activity. With this aim, dynamics of recently introduced Lavrentovich-Hemkinmathematical model was examined by both analytical and numerical techniques.

Methods. In order to obtainthe conditions for the oscillations cessation, the linear stability analysis for the equilibrium point was carried out. Complicated dynamics was studied numerically by calculations of time traces and bifurcation diagrams.

## Development of the Russian state in the 20th and 21st centuries: mathematical modeling based on the socio-energy approach

Purpose. The article is devoted to modeling the socio-political development of Russia in 1910-2009 based on the author's socio-energy approach. In this paper, we briefly talk about the basics of the proposed approach, its principles and basic equations.

Methods. The mathematical model is based on the Langevin diffusion equation. We also introduce the concepts of social energy, coefficients of the state of society and give them definitions.

## Mathematical models of the world-system evolution

We propose new mathematical models of the evolution of the human society based on the synergistic approach. They describe the dynamics of the indicators of the major integral development of the World-System such as the total population and the level of the technological development. Our models capture the basic laws of the space and temporal development of the society. They indicate the hyperbolic growth of the population that agrees with the demographical data and the cyclic dynamics.

## Nonlinear dynamics of synthetic gene regulatory circuits

Built in a cell synthetic gene regulatory elements may function rather independently on the original natural system. Experimental and theoretical studies of small synthetic networks allow for a better understanding of fundamental dynamical mechanisms of gene regulation. This paper gives an introduction to the modern mathematical approaches and methods in this field, primarily in the framework of nonlinear dynamics.

## Cognitive neurodynamics two strategies navigation behavior of organisms

The conceptual model and computer simulations results of path integration in freescalable nonlinear oscillator neural networks with even cyclic inhibition (ECI-networks) are discussed in this paper. To estimate the phase shifting under input impact the ECInetworks contain two subsystems namely reference and information ones. The population of reference (nonencoding) oscillatory units has significant role in generation and stabilization of numerous time scales despite it don’t assist directly in the phase pattern encoding of input signals.