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Song Z, Zhen B, Hu D. Multiple bifurcations and coexistence in an inertial two-neuron system with multiple delays. Cogn Neurodyn 2020; 14:359-374. [PMID: 32399077 DOI: 10.1007/s11571-020-09575-9] [Citation(s) in RCA: 12] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/20/2019] [Revised: 02/04/2020] [Accepted: 02/19/2020] [Indexed: 11/29/2022] Open
Abstract
In this paper, we construct an inertial two-neuron system with multiple delays, which is described by three first-order delayed differential equations. The neural system presents dynamical coexistence with equilibria, periodic orbits, and even quasi-periodic behavior by employing multiple types of bifurcations. To this end, the pitchfork bifurcation of trivial equilibrium is analyzed firstly by using center manifold reduction and normal form method. The system presents different sequences of supercritical and subcritical pitchfork bifurcations. Further, the nontrivial equilibrium bifurcated from trivial equilibrium presents a secondary pitchfork bifurcation. The system exhibits stable coexistence of multiple equilibria. Using the pitchfork bifurcation curves, we divide the parameter plane into different regions, corresponding to different number of equilibria. To obtain the effect of time delays on system dynamical behaviors, we analyze equilibrium stability employing characteristic equation of the system. By the Hopf bifurcation, the system illustrates a periodic orbit near the trivial equilibrium. We give the stability regions in the delayed plane to illustrate stability switching. The neural system is illustrated to have Hopf-Hopf bifurcation points. The coexistence with two periodic orbits is presented near these bifurcation points. Finally, we present some mixed dynamical coexistence. The system has a stable coexistence with periodic orbit and equilibrium near the pitchfork-Hopf bifurcation point. Moreover, multiple frequencies of the system induce the presentation of quasi-periodic behavior. The system presents stable coexistence with two periodic orbits and one quasi-periodic behavior.
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Affiliation(s)
- Zigen Song
- 1College of Information Technology, Shanghai Ocean University, Shanghai, 201306 China
| | - Bin Zhen
- 2School of Environment and Architecture, University of Shanghai for Science and Technology, Shanghai, 200093 China
| | - Dongpo Hu
- 3School of Mathematical Sciences, Qufu Normal University, Qufu, 273165 China
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Verisokin AY, Verveyko DV, Kuryshovav EA, Postnov DE. Noise-sustained patterns in a model of volume-coupled neural tissue. CHAOS (WOODBURY, N.Y.) 2018; 28:106326. [PMID: 30384648 DOI: 10.1063/1.5039854] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/11/2018] [Accepted: 08/17/2018] [Indexed: 06/08/2023]
Abstract
Computational neuroscience operates on models based on several important paradigms. Among them is the assumption that coupling in neural ensembles is provided by chemical or electrical synapses. This assumption works well under normal conditions. However, there is a growing body of data that show the importance of other communication pathways caused by bi-directional transport of substances between the cells and the intercellular space. This type of interaction is called "volume transmission" and has not been rarely addressed in the model studies. The volume transmission pathway naturally appears in multidimensional quantitative models of cellular processes, but is not sufficiently represented at the level of lumped and computationally effective neural models. In this paper, we propose a simple model that allows one to study the features of volume transmission coupling at various spatial scales and taking into account various inhomogeneities. This model is obtained by the extension of the well-known FitzHugh-Nagumo system by the addition of the nonlinear terms and equations to describe, at a qualitative level, the release of potassium into the intercellular space, its diffusion, and the reverse effect on the neurons. The study of model dynamics in various spatial configurations has revealed a number of characteristic spatio-temporal types of behavior that include self-organizing bursting and phase-locked firing patterns, different scenarios of excitation spreading, noise-sustained target patterns, and long-living slow moving wave segments.
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Affiliation(s)
- A Yu Verisokin
- Department of Theoretical Physics, Kursk State University, Radishcheva st., 33, 305000 Kursk, Russia
| | - D V Verveyko
- Department of Theoretical Physics, Kursk State University, Radishcheva st., 33, 305000 Kursk, Russia
| | - E A Kuryshovav
- Saratov State University, Astrakhanskaya st., 83, 410012 Saratov, Russia
| | - D E Postnov
- Saratov State University, Astrakhanskaya st., 83, 410012 Saratov, Russia
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Feng P, Wu Y, Zhang J. A Route to Chaotic Behavior of Single Neuron Exposed to External Electromagnetic Radiation. Front Comput Neurosci 2017; 11:94. [PMID: 29089882 PMCID: PMC5650994 DOI: 10.3389/fncom.2017.00094] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/25/2017] [Accepted: 10/02/2017] [Indexed: 11/13/2022] Open
Abstract
Non-linear behaviors of a single neuron described by Fitzhugh-Nagumo (FHN) neuron model, with external electromagnetic radiation considered, is investigated. It is discovered that with external electromagnetic radiation in form of a cosine function, the mode selection of membrane potential occurs among periodic, quasi-periodic, and chaotic motions as increasing the frequency of external transmembrane current, which is selected as a sinusoidal function. When the frequency is small or large enough, periodic, and quasi-periodic motions are captured alternatively. Otherwise, when frequency is in interval 0.778 < ω < 2.208, chaotic motion characterizes the main behavior type. The mechanism of mode transition from quasi-periodic to chaotic motion is also observed when varying the amplitude of external electromagnetic radiation. The frequency apparently plays a more important role in determining the system behavior.
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Affiliation(s)
- Peihua Feng
- Department of Mechanics, School of Aerospace, Xi'an Jiaotong University, Xi'an, China
| | - Ying Wu
- State Key Laboratory for Strength and Vibration of Mechanical Structures, Shaanxi Engineering Laboratory for Vibration Control of Aerospace Structures, School of Aerospace, Xi'an Jiaotong University, Xi'an, China.,Key Laboratory for NeuroInformation of Ministry of Education, University of Electronic Science and Technology of China, Chengdu, China
| | - Jiazhong Zhang
- Department of Fluid Machinery and Engineering, School of Power and Energy Engineering, Xi'an Jiaotong University, Xi'an, China
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Jiancheng S, Min L, Chusheng H. Cooperative effect of random and time-periodic coupling strength on synchronization transitions in one-way coupled neural system: mean field approach. Cogn Neurodyn 2017; 11:383-390. [PMID: 28761557 DOI: 10.1007/s11571-017-9437-1] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/02/2016] [Revised: 03/26/2017] [Accepted: 04/10/2017] [Indexed: 10/19/2022] Open
Abstract
The cooperative effect of random coupling strength and time-periodic coupling strengh on synchronization transitions in one-way coupled neural system has been investigated by mean field approach. Results show that cooperative coupling strength (CCS) plays an active role for the enhancement of synchronization transitions. There exist an optimal frequency of CCS which makes the system display the best CCS-induced synchronization transitions, a critical frequency of CCS which can not further affect the CCS-induced synchronization transitions, and a critical amplitude of CCS which can not occur the CCS-induced synchronization transitions. Meanwhile, noise intensity plays a negative role for the CCS-induced synchronization transitions. Furthermore, it is found that the novel CCS amplitude-induced synchronization transitions and CCS frequency-induced synchronization transitions are found.
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Affiliation(s)
- Shi Jiancheng
- College of Chemistry and Material Sciences, Guangxi Teachers Education University, Nanning, 530001 China
| | - Luo Min
- College of Chemistry and Material Sciences, Guangxi Teachers Education University, Nanning, 530001 China
| | - Huang Chusheng
- College of Chemistry and Material Sciences, Guangxi Teachers Education University, Nanning, 530001 China
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Li J, Tang J, Ma J, Du M, Wang R, Wu Y. Dynamic transition of neuronal firing induced by abnormal astrocytic glutamate oscillation. Sci Rep 2016; 6:32343. [PMID: 27573570 PMCID: PMC5004107 DOI: 10.1038/srep32343] [Citation(s) in RCA: 36] [Impact Index Per Article: 4.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/28/2016] [Accepted: 08/05/2016] [Indexed: 02/01/2023] Open
Abstract
The gliotransmitter glutamate released from astrocytes can modulate neuronal firing by activating neuronal N-methyl-D-aspartic acid (NMDA) receptors. This enables astrocytic glutamate(AG) to be involved in neuronal physiological and pathological functions. Based on empirical results and classical neuron-glial "tripartite synapse" model, we propose a practical model to describe extracellular AG oscillation, in which the fluctuation of AG depends on the threshold of calcium concentration, and the effect of AG degradation is considered as well. We predict the seizure-like discharges under the dysfunction of AG degradation duration. Consistent with our prediction, the suppression of AG uptake by astrocytic transporters, which operates by modulating the AG degradation process, can account for the emergence of epilepsy.
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Affiliation(s)
- Jiajia Li
- State Key Laboratory for Strength and Vibration of Mechanical Structures, School of Aerospace, Xi’an Jiaotong University, Xi’an 710049, China
| | - Jun Tang
- College of Science, China University of Mining and Technology, Xuzhou 221116, China
| | - Jun Ma
- Department of Physics, Lanzhou University of Technology, Lanzhou 730050, China
| | - Mengmeng Du
- State Key Laboratory for Strength and Vibration of Mechanical Structures, School of Aerospace, Xi’an Jiaotong University, Xi’an 710049, China
| | - Rong Wang
- State Key Laboratory for Strength and Vibration of Mechanical Structures, School of Aerospace, Xi’an Jiaotong University, Xi’an 710049, China
| | - Ying Wu
- State Key Laboratory for Strength and Vibration of Mechanical Structures, School of Aerospace, Xi’an Jiaotong University, Xi’an 710049, China
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Zhang H, Zhang Y, Xu D, Liu X. Deterministic convergence of chaos injection-based gradient method for training feedforward neural networks. Cogn Neurodyn 2015; 9:331-40. [PMID: 25972981 DOI: 10.1007/s11571-014-9323-z] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/13/2014] [Revised: 11/26/2014] [Accepted: 12/10/2014] [Indexed: 02/03/2023] Open
Abstract
It has been shown that, by adding a chaotic sequence to the weight update during the training of neural networks, the chaos injection-based gradient method (CIBGM) is superior to the standard backpropagation algorithm. This paper presents the theoretical convergence analysis of CIBGM for training feedforward neural networks. We consider both the case of batch learning as well as the case of online learning. Under mild conditions, we prove the weak convergence, i.e., the training error tends to a constant and the gradient of the error function tends to zero. Moreover, the strong convergence of CIBGM is also obtained with the help of an extra condition. The theoretical results are substantiated by a simulation example.
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Affiliation(s)
- Huisheng Zhang
- Department of Mathematics, Dalian Maritime University, Dalian, 116026 People's Republic of China ; Research Center of Information and Control, Dalian University of Technology, Dalian, 116024 People's Republic of China
| | - Ying Zhang
- Department of Mathematics, Dalian Maritime University, Dalian, 116026 People's Republic of China
| | - Dongpo Xu
- College of Science, Harbin Engineering University, Harbin, 150001 People's Republic of China
| | - Xiaodong Liu
- Research Center of Information and Control, Dalian University of Technology, Dalian, 116024 People's Republic of China
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