Multistability and stochastic dynamics of Rulkov neurons coupled via a chemical synapse

We study complex dynamics of two Rulkov neurons unidirectionally connected via a chemical synapse with respect to three control parameters: (i) a parameter responsible for the type of dynamical behavior of a solitary neuron, (ii) coupling strength, and (iii) noise intensity. The coupled system exhibits various scenarios on the route from a stable equilibrium to chaos with respect to the coupling strength. We observe a variety of dynamical regimes, including mono-, bi- and tri-stability, order-chaos transitions and vice versa, as well as the coexistence of in-phase and anti-phase synchronization. We also study transitions between in-phase and out-of-phase synchronization with statistics on the duration of synchronization intervals and transitions from order to chaos. In addition to numerical simulations, we demonstrate the effectiveness of the analytical confidence ellipses method based on stochastic sensitivity approach. © 2023 The Author(s)Russian Science Foundation, RSF: 21-11-00062The work was supported by the Russian Science Foundation (project No. 21-11-00062)

Noise-induced excitement and mixed-mode oscillatory regimes in the Chialvo model of neural activity

Map-based nonlinear neuron model proposed by Chialvo is considered under the influence of random disturbances. In the deterministic case, the model possesses both mono- and bistability regimes with the coexistence of equilibrium and oscillatory modes in a form of quasiperiodic closed invariant curves. Variability of noise-induced transformations in dynamics of this model is demonstrated. Phenomena of noise-induced excitement, stochastic generation of mixed-mode oscillations, and transitions between order and chaos are discussed. © 2022 Author(s).Russian Science Foundation, RSF, (21-11-00062)The work was supported by Russian Science Foundation, Russia (no. 21-11-00062)

Noise-induced complex dynamics and synchronization in the map-based Chialvo neuron model

The paper considers a stochastic version of the conceptual map-based Chialvo model of neural activity. Firstly, we focus on the parametric zone where this model exhibits mono- and bistability with coexistence of equilibria and oscillatory spiking attractors forming closed invariant curves. Stochastic effects of excitement and generation of bursting are studied both numerically and analytically by confidence ellipses. A phenomenon of the noise-induced transition to chaos in a localized two-parametric zone is discussed. Besides, we also study the phenomenon of synchronization between neurons by using a two-neuron network with a small coupling. In this scenario, we have found critical values of noise for which we obtain a good performance for the synchronization between the neurons of the network. © 2022 Elsevier B.V.Russian Science Foundation, RSF: 21-11-00062; European Regional Development Fund, ERDF: PID2019-105554GB-I00; Agencia Estatal de Investigación, AEIThe work of IB and LR on the analysis of stochastic excitement, generation of mixed-mode oscillations, order-chaos-order transitions, and stochastic sensitivity presented in Sections 2 , 3 has been financially supported by Russian Science Foundation ( 21-11-00062 ).The work of JU, JMS and MAFS on the analysis of synchronization presented in Sections 4 , 5 has been financially supported by the Spanish State Research Agency (AEI) and the European Regional Development Fund (ERDF) under Project No. PID2019-105554GB-I00

Modelos matemáticos para lesões em redes neurais com padrões complexos de conectividade

Orientador: Prof. Dr. Ricardo Luiz VianaTese (Doutorado) - Universidade Federal do Paraná, Setor de Ciências Exatas, Programa de Pós-Graduação em Física. Defesa: Curitiba, 18/09/2015Inclui referências : f. 139-146Resumo: O cérebro contem cerca de cem bilhões de neurônios que se conectam através de um padrão complexo de conectividade, que opera no sentido de otimizar o processo de transmissão de informação. Os neurônios, ao estabelecerem uma conexão entre si podem muitas vezes exibir sincronização. A presença de traumas e doenças degenerativas em regiões específicas do cérebro podem, através de efeitos locais, danificar o funcionamento cerebral como um todo. O propósito deste trabalho é tentar responder a questões do tipo: em uma rede neural, quais são as formas de lesões que causam maior impacto na dinâmica da rede? É possível identificar um tipo de lesão a partir do seu efeito? Existem topologias de rede que são mais robustas à lesões? Neste sentido, analisamos a sincronização de fase para uma rede de neurônios com diferentes topologias de rede, comparamos os resultados com um modelo de osciladores e analisamos diferentes tipos de lesões. Nossos resultados apontam que para o estudo de sincronização de fase, os neurônios podem ser considerados como osciladores, porém, o comportamento das frequências no estado sincronizado em redes neurais, em geral, não é similar ao comportamento de osciladores. No estudo das lesões, do ponto de vista dinâmico, para cada tipo de rede existe um comportamento distinto aos diferentes tipos de lesões. Entre neurônios globalmente acoplados, é possível distinguir a partir da dinâmica global se a lesão destrói apenas as conexões ou destrói os neurônios. Em redes complexas, o efeito das lesões é maior quando a lesão afeta os neurônios mais conectados ou com maior centralidade de intermediação. Em redes de pequeno mundo, a diferença entre os tipos de lesão é perceptível, porém, mais sutil do que para redes aleatórias e sem escala.Abstract: The brain is composed of around one hundread billion of neurons connected through synapses forming a complex pattern of connectivity. This complex connectivity is responsible to optimize the information process. When neurons are connected among themselves they can exhibit synchronization. The presence of traumas and neurodegenerative diseases in some brain areas causes not only local effects, but in the whole brain. The purpose of this work is to answer questions like: which are the type of lesions with bigger dynamical effects in the neural network? Is it possible to identify a type of lesion just looking at its dynamical effects in the network? Are there topologies against lesions which are more robust than others? In this sense, we analyse phase synchronization in a neural network with different network topologies. We compare the obtained results with a model of phase oscillators and we analysed different types of lesions. Our results show that neuronal phase synchronization is similar to phase synchronization in oscillators, however, frequency synchronization usually is different in both models. Related to lesions, from the dynamical point of view, for each type of network there is a distinct behavior for each type of lesion. Among globally coupled neurons, it is possible to dynamically distinguish when the lesion either disrupt or destroy the neurons. For complex networks, the most effective lesions are those that affects the most connected neurons or those with the largest betweenness. For small-world networks, the difference among types of lesions are distinguishable, though, they are subtle in comparison with random and scale-free networks

Effects of the local dynamics in the synchronization of neural models

Orientador: Prof. Dr. Sergio Roberto LopesTese (doutorado) - Universidade Federal do Paraná, Setor de Ciências Exatas, Programa de Pós-Graduação em Física. Defesa : Curitiba, 24/02/2022Inclui referênciasResumo: O comportamento cooperativo de neurônios e áreas neuronais associadas ao comportamento de sincronização se apresenta como mecanismo fundamental para o funcionamento cerebral. Além disso, níveis anormais de sincronização têm sido relacionados a estados patológicos. Ao longo desta tese, abordam-se diferentes fenômenos de sincronização que surgem por meio da dinâmica coletiva de modelos de neurônios acoplados em um a rede. Primeiramente, mostra-se um a forte correlação entre a dinâmica individual do neurônio com o comportamento global da sincronização da rede, em que a periodicidade observada no neurônio isolado é refletida em um a sincronização de fase ao considerar um acoplamento fraco. Em segundo lugar, estuda-se o papel da biestabilidade na sincronização de um a rede de neurônios idênticos, acoplados através de um esquema de campo médio. Mostra-se que a simples existência de dois estados estáveis distintos pode levar a rede a diferentes estados de sincronização, dependendo da inicialização do sistema. Por fim, é investigado o mecanismo de sincronização explosiva de um a rede neural complexa composta por neurônios não-idênticos. A presença deste regime é acompanhada por um loop de histerese na dinâmica da rede, à medida que o parâmetro de acoplamento é adiabaticamente aumentado e reduzido. Demonstra-se que as transições de sincronização abruptas estão associadas a rotas para o caos e que os mecanismos dinâmicos para a região de biestabilidade são dados em termos de um a bifurcação de sela-nó e um a crise de fronteira. Portanto, os resultados desta tese mostram uma riqueza de comportamentos de sincronização associados a pequenas mudanças na dinâmica neuronal, trazendo novos insights para o estudo teórico das redes neurais.Abstract: The cooperative behavior of neurons and neuronal areas associated with the synchronization behavior proves to be a fundamental neural mechanism. In addition, abnormal levels of synchronization have been related to unhealthy neural behaviors. Throughout this thesis, it is explored different synchronization phenomena which emerge through the collective dynamics of models of neurons coupled in a network. Firstly, it is shown a strong correlation between the individual dynamics of the neuron with the global behavior of the synchronization, in which the periodicity seen in the isolated neuron is reflected in a phase synchronization in the weak coupling region. Secondly, it is studied the role of bistability in the synchronization of a network of identical neurons coupled through a mean-field scheme. It is shown that the simple existence of two distinct stable states can lead the network to different states of synchronization, depending on the initialization of the system. Lastly, it is investigated the mechanism for explosive synchronization of a complex neural network composed of non-identical neurons. The presence of this regime is accompanied by a hysteresis loop on the network dynamics as the coupling parameter is adiabatically increased and decreased. It is shown that the abrupt synchronization transitions are associated with routes to chaos. The dynamical mechanisms for the bistability region, are given in terms of a saddle-node bifurcation and a boundary crisis. Therefore, the results of this thesis show a richness of synchronization behaviors associated with small changes of the neuronal dynamics bringing new insights to the theoretical study of neural networks

Transition to High Dimensional Dynamics in Systems of Coupled Oscillators

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15th Conference on Dynamical Systems Theory and Applications DSTA 2019 ABSTRACTS

From Preface: This is the fifteen time when the conference „Dynamical Systems – Theory and Applications” gathers a numerous group of outstanding scientists and engineers, who deal with widely understood problems of theoretical and applied dynamics. Organization of the conference would not have been possible without a great effort of the staff of the Department of Automation, Biomechanics and Mechatronics. The patronage over the conference has been taken by the Committee of Mechanics of the Polish Academy of Sciences and the Ministry of Science and Higher Education. It is a great pleasure that our invitation has been accepted by so many people, including good colleagues and friends as well as a large group of researchers and scientists, who decided to participate in the conference for the first time. With proud and satisfaction we welcome nearly 255 persons from 47 countries all over the world. They decided to share the results of their research and many years experiences in the discipline of dynamical systems by submitting many very interesting papers. This booklet contains a collection of 338 abstracts, which have gained the acceptance of referees and have been qualified for publication in the conference edited books.Technical editor and cover design: Kaźmierczak, MarekCover design: Ogińska, Ewelina; Kaźmierczak, Mare