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## Sequential dynamics in motif of excitatory coupled elements

Regular and Chaotic Dynamics. 2015. Vol. 20. No. 6. P. 701-715.

In this article a new model of motif (small ensemble) of neuron-like elements is proposed. It is built with the use of generalized Lotka-Volterra model with excitatory couplings. The main motivation for this work comes from the problems of neuroscience where excitatory couplings are proved to be the predominant type of interaction between neurons of the brain. In this paper it is shown that there are two modes depending on the type of coupling between the elements: the mode with a stable heteroclinic cycle and the mode with a stable limit cycle. Our second goal is to examine the chaotic dynamics of generalized three-dimensional Lotka-Volterra model.

Keywords: strange attractormotif of excitatory coupled elementsLotka-Volterra modelheteroclinic cycle

Publication based on the results of:

Gonchenko S., Gonchenko M., Sinitsky I. O., Izvestiya. Mathematics 2020 Vol. 84 No. 1 P. 23-51

We considerone-parameter families (general unfoldings)of two- dimensional reversible diffeomorphisms that contain a diffeomorphism with a symmetric non-transversal heteroclinic cycle. We show that in such families there exist Newhouse intervals of parameters such that the values corresponding to the co-existence of infinitely many stable, completely unstable, saddle and symmetric elliptic periodic orbits are generic (that is, ...

Added: March 9, 2021

Kazakov A., Леванова Т. А., Коротков А. Г. et al., Известия высших учебных заведений. Прикладная нелинейная динамика 2018 Т. 26 № 5 С. 101-112

The phenomenological model of an ensemble of three neurons which are coupled by
chemical (synaptic) and electrical couplings is studies. A neuron is modeled by the oscillator of van
der Pol. The aim of work is a study of the influence of coupling’s strength and frequency detuning
between elements at regime of sequential activity that is observed in ...

Added: October 26, 2018

Kazakov A., Borisov A. V., Sataev I. R., Regular and Chaotic Dynamics 2014 Vol. 19 No. 6 P. 718-733

In this paper we consider the motion of a dynamically asymmetric unbalanced ball
on a plane in a gravitational field. The point of contact of the ball with the plane is subject
to a nonholonomic constraint which forbids slipping. The motion of the ball is governed by the
nonholonomic reversible system of 6 differential equations. In the case ...

Added: March 29, 2015

Gonchenko S. V., Gonchenko A. S., Kazakov A. et al., International Journal of Bifurcation and Chaos in Applied Sciences and Engineering 2018 Vol. 28 No. 11 P. 1830036-1-1830036-29

The paper is devoted to topical issues of modern mathematical theory of dynamical chaos and its applications. At present, it is customary to assume that dynamical chaos in finite-dimensional smooth systems can exist in three different forms. This is dissipative chaos, the mathematical image of which is a strange attractor; conservative chaos, for which the ...

Added: October 26, 2018

Kazakov A., Гонченко А. С., Гонченко С. В. et al., Известия высших учебных заведений. Радиофизика 2018 Т. 61 № 10 С. 867-882

We study dynamical properties of a Celtic stone moving along the plane. Both one- and two-parameter families of the corresponding nonholonomic models are considered, in which bifurcations are studied that lead to changing types of stable motions of the stone as well as to the onset of chaotic dynamics. It is shown that multistability phenomena ...

Added: October 26, 2018

Kazakov A., Gonchenko A. S., Gonchenko S. V. et al., International Journal of Bifurcation and Chaos in Applied Sciences and Engineering 2014 Vol. 24 No. 8 P. 1440005-1440030

We give a qualitative description of two main routes to chaos in three-dimensional maps. We discuss Shilnikov scenario of transition to spiral chaos and a scenario of transition to discrete Lorenz-like and figure-eight strange attractors. The theory is illustrated by numerical analysis of three-dimensional Henon-like maps and Poincar´ e maps in models of nonholonomic mechanics ...

Added: March 29, 2015

Kazakov A., Козлов А. Д., Журнал Средневолжского математического общества 2018 Т. 20 № 2 С. 187-198

In the paper a new method of constructing of three-dimensional flow systems with different chaotic attractors is presented. Using this method, an example of three-dimensional system possessing an asymmetric Lorenz attractor is obtained. Unlike the classical Lorenz attractor, the observed attractor does not have symmetry. However, the discovered asymmetric attractor, as well as classical one, ...

Added: October 26, 2018

Korotkov A., Kazakov A., Леванова Т. А. et al., Communications in Nonlinear Science and Numerical Simulation 2019 Vol. 71 P. 38-49

We investigated the phenomenological model of ensemble of two FitzHugh–Nagumo neuron-like elements with symmetric excitatory couplings. The main advantage of proposed model is the new approach to model the coupling which is implemented by smooth function that approximates rectangular function and reflects main important properties of biological synaptic coupling. The proposed coupling depends on three ...

Added: October 18, 2019

Kazakov A., Известия высших учебных заведений. Радиофизика 2018 Т. 61 № 8-9 С. 729-738

In this paper, a new scenario of the appearance of mixed dynamics in two-dimensional reversible diffeomorphisms is proposed. The key point of the scenario is a sharp increase of the sizes of both strange attractor and strange repeller which appears due to heteroclinic bifurcations of the invariant manifolds of saddle fixed points belonging to these ...

Added: October 26, 2018

Kazakov A., Гонченко С. В., Гонченко А. С. et al., Известия высших учебных заведений. Прикладная нелинейная динамика 2017 Т. 25 № 2 С. 4-36

We consider important problems of modern theory of dynamical chaos and its applications. At present, it is customary to assume that in the finite-dimensional smooth dynamical systems three fundamentally different forms of chaos can be observed. This is the dissipative chaos, whose mathematical image is a strange attractor; the conservative chaos, for which the whole ...

Added: October 13, 2017

Strange Attractors and Mixed Dynamics in the Problem of an Unbalanced Rubber Ball Rolling on a Plane

Kazakov A., Regular and Chaotic Dynamics 2013 Vol. 18 No. 5 P. 508-520

We consider the dynamics of an unbalanced rubber ball rolling on a rough plane. The termrubbermeans that the vertical spinning of the ball is impossible. The roughness of the plane means that the ball moves without slipping. The motions of the ball are described by a nonholonomic system reversible with respect to several involutions whose ...

Added: March 29, 2015

Kazakov A., Баханова Ю. В., Коротков А. Г., Журнал Средневолжского математического общества 2017 Т. 19 № 2 С. 13-24

Investigations of spiral chaos in generalized Lotka-Volterra systems and Rosenzweig-MacArthur systems that describe the interaction of three species are made in this work. It is shown that in systems under study the spiral chaos appears in agreement with Shilnikov's scenario, that is when changing a parameter in system a stable limit cycle and a saddle-focus ...

Added: October 13, 2017

Bizyaev I. A., Borisov A. V., Kazakov A., Regular and Chaotic Dynamics 2015 Vol. 20 No. 5 P. 605-626

In this paper, we present some results on chaotic dynamics in the Suslov problem which describe the motion of a heavy rigid body with a fixed point, subject to a nonholonomic constraint, which is expressed by the condition that the projection of angular velocity onto the body-fixed axis is equal to zero. Depending on the ...

Added: October 22, 2015

Lerman L., Trifonov K., Chaos 2021 Vol. 31 No. 2 Article 023113

An analytic reversible Hamiltonian system with two degrees of freedom is studied in a neighborhood of its symmetric heteroclinic connection made up of a symmetric saddle-center, a symmetric orientable saddle periodic orbit lying in the same level of a Hamiltonian, and two non-symmetric heteroclinic orbits permuted by the involution. This is a codimension one structure; therefore, ...

Added: February 6, 2021

Kuryzhov E., Karatetskaia E., Mints D., Russian Journal of Nonlinear Dynamics 2021 Vol. 17 No. 2 P. 165-174

We consider the system of two coupled one-dimensional parabola maps. It is well known that the parabola map is the simplest map that can exhibit chaotic dynamics, chaos in this map appears through an infinite cascade of period-doubling bifurcations. For two coupled parabola maps we focus on studying attractors of two types: those which resemble ...

Added: September 8, 2021

Kazakov A., Борисов А. В., Кузнецов С. П., Успехи физических наук 2014 Т. 184 № 5 С. 493-500

Based on the results of numerical simulations we discuss and illustrate dynamical phenomena characteristic for the rattleback, a solid body of convex surface moving on a rough horizontal plane, which are associated with the lack of conservation for the phase volume in the nonholonomic mechanical system. Due to local compression of the phase volume, behaviors ...

Added: October 22, 2015

Kazakov A., Борисов А. В., Пивоварова Е. Н., Нелинейная динамика 2017 Т. 13 № 2 С. 277-297

This paper is concerned with the rolling motion of a dynamically asymmetric unbalanced ball (Chaplygin top) in a gravitational field on a plane under the assumption that there is no slipping and spinning at the point of contact. We give a description of strange attractors existing in the system and discuss in detail the scenario ...

Added: October 13, 2017

Kazakov A., Korotkov A., Levanova T. et al., IFAC-PapersOnLine 2018

We study the peculiarities of chaotic dynamics in the phenomenological model of the ensemble of two FitzHugh-Nagumo elements with weak excitatory couplings. This model was recently proposed as a suitable model for describing the behaviour of two coupled neurons. A rich diversity of different types of neuron-like behaviour, including regular in-phase, anti-phase, sequential spiking activities ...

Added: October 26, 2018

Kazakov A., Борисов А. В., Пивоварова Е. Н., Regular and Chaotic Dynamics 2016 Vol. 21 No. 7-8 P. 885-901

This paper is concerned with the rolling motion of a dynamically asymmetric unbalanced ball (Chaplygin top) in a gravitational field on a plane under the assumption that there is no slipping and spinning at the point of contact. We give a description of strange attractors existing in the system and discuss in detail the scenario ...

Added: January 30, 2017

Kotelnikova M. V., Aistov A., Вестник Нижегородского университета им. Н.И. Лобачевского. Серия: Социальные науки 2019 Т. 55 № 3 С. 183-189

The article describes a method that allows to improve the content of disciplines of the mathematical cycle by dividing them into invariant (general) and variable parts. The invariants were identified for such disciplines as «Linear algebra», «Mathematical analysis», «Probability theory and mathematical statistics» delivered to Bachelors program students of economics at several universities. Based on ...

Added: January 28, 2020

Borzykh D., ЛЕНАНД, 2021

Книга представляет собой экспресс-курс по теории вероятностей в контексте начального курса эконометрики. В курсе в максимально доступной форме изложен тот минимум, который необходим для осознанного изучения начального курса эконометрики. Данная книга может не только помочь ликвидировать пробелы в знаниях по теории вероятностей, но и позволить в первом приближении выучить предмет «с нуля». При этом, благодаря доступности изложения и небольшому объему книги, ...

Added: February 20, 2021

В. Л. Попов, Математические заметки 2017 Т. 102 № 1 С. 72-80

Мы доказываем, что аффинно-треугольные подгруппы являются борелевскими подгруппами групп Кремоны. ...

Added: May 3, 2017

Красноярск : ИВМ СО РАН, 2013

Труды Пятой Международной конференции «Системный анализ и информационные технологии» САИТ-2013 (19–25 сентября 2013 г., г.Красноярск, Россия): ...

Added: November 18, 2013

Min Namkung, Younghun K., Scientific Reports 2018 Vol. 8 No. 1 P. 16915-1-16915-18

Sequential state discrimination is a strategy for quantum state discrimination of a sender’s quantum
states when N receivers are separately located. In this report, we propose optical designs that can
perform sequential state discrimination of two coherent states. For this purpose, we consider not
only binary phase-shifting-key (BPSK) signals but also general coherent states, with arbitrary prior
probabilities. Since ...

Added: November 16, 2020