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Mondal A, Mondal A, Kumar Sharma S, Kumar Upadhyay R, Antonopoulos CG. Spatiotemporal characteristics in systems of diffusively coupled excitable slow-fast FitzHugh-Rinzel dynamical neurons. CHAOS (WOODBURY, N.Y.) 2021; 31:103122. [PMID: 34717324 DOI: 10.1063/5.0055389] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/28/2021] [Accepted: 09/27/2021] [Indexed: 06/13/2023]
Abstract
In this paper, we study an excitable, biophysical system that supports wave propagation of nerve impulses. We consider a slow-fast, FitzHugh-Rinzel neuron model where only the membrane voltage interacts diffusively, giving rise to the formation of spatiotemporal patterns. We focus on local, nonlinear excitations and diverse neural responses in an excitable one- and two-dimensional configuration of diffusively coupled FitzHugh-Rinzel neurons. The study of the emerging spatiotemporal patterns is essential in understanding the working mechanism in different brain areas. We derive analytically the coefficients of the amplitude equations in the vicinity of Hopf bifurcations and characterize various patterns, including spirals exhibiting complex geometric substructures. Furthermore, we derive analytically the condition for the development of antispirals in the neighborhood of the bifurcation point. The emergence of broken target waves can be observed to form spiral-like profiles. The spatial dynamics of the excitable system exhibits two- and multi-arm spirals for small diffusive couplings. Our results reveal a multitude of neural excitabilities and possible conditions for the emergence of spiral-wave formation. Finally, we show that the coupled excitable systems with different firing characteristics participate in a collective behavior that may contribute significantly to irregular neural dynamics.
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Affiliation(s)
- Arnab Mondal
- Department of Mathematics and Computing, Indian Institute of Technology (Indian School of Mines), Dhanbad 826004, India
| | - Argha Mondal
- School of Engineering, Amrita Vishwa Vidyapeetham, Amritapuri, Kollam 690525, India
| | - Sanjeev Kumar Sharma
- Department of Mathematics and Computing, Indian Institute of Technology (Indian School of Mines), Dhanbad 826004, India
| | - Ranjit Kumar Upadhyay
- Department of Mathematics and Computing, Indian Institute of Technology (Indian School of Mines), Dhanbad 826004, India
| | - Chris G Antonopoulos
- Department of Mathematical Sciences, University of Essex, Wivenhoe Park, Colchester CO4 3SQ, United Kingdom
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Rajagopal K, Parastesh F, Azarnoush H, Hatef B, Jafari S, Berec V. Spiral waves in externally excited neuronal network: Solvable model with a monotonically differentiable magnetic flux. CHAOS (WOODBURY, N.Y.) 2019; 29:043109. [PMID: 31042930 DOI: 10.1063/1.5088654] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/12/2019] [Accepted: 03/21/2019] [Indexed: 06/09/2023]
Abstract
Spiral waves are particular spatiotemporal patterns connected to specific phase singularities representing topological wave dislocations or nodes of zero amplitude, witnessed in a wide range of complex systems such as neuronal networks. The appearance of these waves is linked to the network structure as well as the diffusion dynamics of its blocks. We report a novel form of the Hindmarsh-Rose neuron model utilized as a square neuronal network, showing the remarkable multistructure of dynamical patterns ranging from characteristic spiral wave domains of spatiotemporal phase coherence to regions of hyperchaos. The proposed model comprises a hyperbolic memductance function as the monotone differentiable magnetic flux. Hindmarsh-Rose neurons with an external electromagnetic excitation are considered in three different cases: no excitation, periodic excitation, and quasiperiodic excitation. We performed an extensive study of the neuronal dynamics including calculation of equilibrium points, bifurcation analysis, and Lyapunov spectrum. We have found the property of antimonotonicity in bifurcation scenarios with no excitation or periodic excitation and identified wide regions of hyperchaos in the case of quasiperiodic excitation. Furthermore, the formation and elimination of the spiral waves in each case of external excitation with respect to stimuli parameters are investigated. We have identified novel forms of Hindmarsh-Rose bursting dynamics. Our findings reveal multipartite spiral wave formations and symmetry breaking spatiotemporal dynamics of the neuronal model that may find broad practical applications.
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Affiliation(s)
| | - Fatemeh Parastesh
- Department of Biomedical Engineering, Amirkabir University of Technology (Tehran Polytechnic), 424 Hafez Ave., Tehran 15875-4413, Iran
| | - Hamed Azarnoush
- Department of Biomedical Engineering, Amirkabir University of Technology (Tehran Polytechnic), 424 Hafez Ave., Tehran 15875-4413, Iran
| | - Boshra Hatef
- Neuroscience Research Center, Baqiyatallah University of Medical Sciences, Tehran 14359-16471, Iran
| | - Sajad Jafari
- Nonlinear Systems and Applications, Faculty of Electrical and Electronics Engineering, Ton Duc Thang University, Ho Chi Minh City, Vietnam
| | - Vesna Berec
- Institute of Nuclear Sciences, Vinca, PO Box 522, 11001 Belgrade, Serbia
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Collective responses in electrical activities of neurons under field coupling. Sci Rep 2018; 8:1349. [PMID: 29358677 PMCID: PMC5778049 DOI: 10.1038/s41598-018-19858-1] [Citation(s) in RCA: 83] [Impact Index Per Article: 13.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/14/2017] [Accepted: 01/09/2018] [Indexed: 11/08/2022] Open
Abstract
Synapse coupling can benefit signal exchange between neurons and information encoding for neurons, and the collective behaviors such as synchronization and pattern selection in neuronal network are often discussed under chemical or electric synapse coupling. Electromagnetic induction is considered at molecular level when ion currents flow across the membrane and the ion concentration is fluctuated. Magnetic flux describes the effect of time-varying electromagnetic field, and memristor bridges the membrane potential and magnetic flux according to the dimensionalization requirement. Indeed, field coupling can contribute to the signal exchange between neurons by triggering superposition of electric field when synapse coupling is not available. A chain network is designed to investigate the modulation of field coupling on the collective behaviors in neuronal network connected by electric synapse between adjacent neurons. In the chain network, the contribution of field coupling from each neuron is described by introducing appropriate weight dependent on the position distance between two neurons. Statistical factor of synchronization is calculated by changing the external stimulus and weight of field coupling. It is found that the synchronization degree is dependent on the coupling intensity and weight, the synchronization, pattern selection of network connected with gap junction can be modulated by field coupling.
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Rostami Z, Jafari S. Defects formation and spiral waves in a network of neurons in presence of electromagnetic induction. Cogn Neurodyn 2018; 12:235-254. [PMID: 29564031 DOI: 10.1007/s11571-017-9472-y] [Citation(s) in RCA: 38] [Impact Index Per Article: 6.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/04/2017] [Revised: 12/04/2017] [Accepted: 12/29/2017] [Indexed: 11/30/2022] Open
Abstract
Complex anatomical and physiological structure of an excitable tissue (e.g., cardiac tissue) in the body can represent different electrical activities through normal or abnormal behavior. Abnormalities of the excitable tissue coming from different biological reasons can lead to formation of some defects. Such defects can cause some successive waves that may end up to some additional reorganizing beating behaviors like spiral waves or target waves. In this study, formation of defects and the resulting emitted waves in an excitable tissue are investigated. We have considered a square array network of neurons with nearest-neighbor connections to describe the excitable tissue. Fundamentally, electrophysiological properties of ion currents in the body are responsible for exhibition of electrical spatiotemporal patterns. More precisely, fluctuation of accumulated ions inside and outside of cell causes variable electrical and magnetic field. Considering undeniable mutual effects of electrical field and magnetic field, we have proposed the new Hindmarsh-Rose (HR) neuronal model for the local dynamics of each individual neuron in the network. In this new neuronal model, the influence of magnetic flow on membrane potential is defined. This improved model holds more bifurcation parameters. Moreover, the dynamical behavior of the tissue is investigated in different states of quiescent, spiking, bursting and even chaotic state. The resulting spatiotemporal patterns are represented and the time series of some sampled neurons are displayed, as well.
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Affiliation(s)
- Zahra Rostami
- Biomedical Engineering Department, Amirkabir University of Technology, Tehran, 15875-4413 Iran
| | - Sajad Jafari
- Biomedical Engineering Department, Amirkabir University of Technology, Tehran, 15875-4413 Iran
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Paraskevov AV, Zendrikov DK. A spatially resolved network spike in model neuronal cultures reveals nucleation centers, circular traveling waves and drifting spiral waves. Phys Biol 2017; 14:026003. [PMID: 28333685 DOI: 10.1088/1478-3975/aa5fc3] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
Abstract
We show that in model neuronal cultures, where the probability of interneuronal connection formation decreases exponentially with increasing distance between the neurons, there exists a small number of spatial nucleation centers of a network spike, from where the synchronous spiking activity starts propagating in the network typically in the form of circular traveling waves. The number of nucleation centers and their spatial locations are unique and unchanged for a given realization of neuronal network but are different for different networks. In contrast, if the probability of interneuronal connection formation is independent of the distance between neurons, then the nucleation centers do not arise and the synchronization of spiking activity during a network spike occurs spatially uniform throughout the network. Therefore one can conclude that spatial proximity of connections between neurons is important for the formation of nucleation centers. It is also shown that fluctuations of the spatial density of neurons at their random homogeneous distribution typical for the experiments in vitro do not determine the locations of the nucleation centers. The simulation results are qualitatively consistent with the experimental observations.
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Affiliation(s)
- A V Paraskevov
- National Research Centre "Kurchatov Institute", 123182 Moscow, Russia. Moscow Institute of Physics and Technology (State University), 141700 Dolgoprudny, Russia
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Qian Y, Zhang Z. The Fundamental Structure and the Reproduction of Spiral Wave in a Two-Dimensional Excitable Lattice. PLoS One 2016; 11:e0149842. [PMID: 26900841 PMCID: PMC4762983 DOI: 10.1371/journal.pone.0149842] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/16/2016] [Accepted: 02/05/2016] [Indexed: 11/19/2022] Open
Abstract
In this paper we have systematically investigated the fundamental structure and the reproduction of spiral wave in a two-dimensional excitable lattice. A periodically rotating spiral wave is introduced as the model to reproduce spiral wave artificially. Interestingly, by using the dominant phase-advanced driving analysis method, the fundamental structure containing the loop structure and the wave propagation paths has been revealed, which can expose the periodically rotating orbit of spiral tip and the charity of spiral wave clearly. Furthermore, the fundamental structure is utilized as the core for artificial spiral wave. Additionally, the appropriate parameter region, in which the artificial spiral wave can be reproduced, is studied. Finally, we discuss the robustness of artificial spiral wave to defects.
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Affiliation(s)
- Yu Qian
- Nonlinear Research Institute, Baoji University of Arts and Sciences, Baoji, Shaanxi, China
- * E-mail:
| | - Zhaoyang Zhang
- Department of Physics, Faculty of Science, Ningbo University, Ningbo, Zhejiang, China
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Qin H, Ma J, Wang C, Wu Y. Autapse-induced spiral wave in network of neurons under noise. PLoS One 2014; 9:e100849. [PMID: 24967577 PMCID: PMC4072706 DOI: 10.1371/journal.pone.0100849] [Citation(s) in RCA: 42] [Impact Index Per Article: 4.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/02/2014] [Accepted: 05/31/2014] [Indexed: 11/23/2022] Open
Abstract
Autapse plays an important role in regulating the electric activity of neuron by feedbacking time-delayed current on the membrane of neuron. Autapses are considered in a local area of regular network of neurons to investigate the development of spatiotemporal pattern, and emergence of spiral wave is observed while it fails to grow up and occupy the network completely. It is found that spiral wave can be induced to occupy more area in the network under optimized noise on the network with periodical or no-flux boundary condition being used. The developed spiral wave with self-sustained property can regulate the collective behaviors of neurons as a pacemaker. To detect the collective behaviors, a statistical factor of synchronization is calculated to investigate the emergence of ordered state in the network. The network keeps ordered state when self-sustained spiral wave is formed under noise and autapse in local area of network, and it independent of the selection of periodical or no-flux boundary condition. The developed stable spiral wave could be helpful for memory due to the distinct self-sustained property.
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Affiliation(s)
- Huixin Qin
- Department of Physics, Lanzhou University of Technology, Lanzhou, China
| | - Jun Ma
- Department of Physics, Lanzhou University of Technology, Lanzhou, China
| | - Chunni Wang
- Department of Physics, Lanzhou University of Technology, Lanzhou, China
| | - Ying Wu
- School of Aerospace, Xian Jiaotong University, Xian, China
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