Birth and death of spiral waves in a network of Hindmarsh–Rose neurons with exponential magnetic flux and excitable media

doi:10.1016/j.amc.2019.02.041

Applied Mathematics and Computation

Volume 354, 1 August 2019, Pages 377-384

https://doi.org/10.1016/j.amc.2019.02.041 Get rights and content

Highlights

•
An improved Hindmarsh–Rose model with external electromagnetic radiation and discontinuous exponential flux coupling is introduced.
•
We propose a two-dimensional network of such Hindmarsh–Rose neurons.
•
The spatiotemporal patterns of the network in different cases are presented.
•
The effective factors on the formation of spiral waves are investigated.

Abstract

Spiral waves are spatiotemporal patterns, observed in complex networks of biological oscillators such as heart and brain. In this paper, at first a modified Hindmarsh–Rose model with external electromagnetic radiation and discontinuous exponential flux coupling is introduced. Then a 2-dimensional network of the proposed model is constructed. The spatiotemporal patterns of the network are investigated in two different cases of periodic and quasi-periodic external magnetic excitations, by varying the external force parameters. As a consequence, the effective factors on the formation and elimination of spiral waves are reported.

Introduction

Recently complex dynamical networks which are large sets of connected fundamental units, have attracted increasing attention in a variety of fields such as social, biological, mathematical, and engineering sciences [1]. One of the fascinating and remarkable phenomena in complex dynamical networks is displaying collective behaviors. Collective behavior is a significant property of complex systems and refers to the fact that the whole is more than the sum of components. Since complex networks are naturally spatiotemporal systems, pattern formation is observed in these systems [2]. Spatial patterns such as target patterns, spiral waves and scroll waves are examples of patterns emerging in complex systems such as Reaction–diffusion systems and neuronal networks [3], [4], [5].

Neuronal networks are complex dynamical networks that can exhibit many spatially structured activity states such as spatiotemporal chaos [6], stochastic resonance [7], synchronization [8], [9], [10], chimera states [11], [12], [13] and spiral waves [14], [15]. The network structure, the neural model and even considering the perturbation and noise, which is an intrinsic property of neuronal networks [16], [17], can effect on the patterns formation. Investigating these phenomena can bring new insights on neurons functioning [18]. Among the presented tools for studying the neuronal network, the mechanism of wave propagation is one of the most efficacious one [19]. Spiral wave is a kind of collective behaviors and can be found prevalently in nature [20], [21]. In fact, Spiral waves are a special propagation of non-linear waves that rotate around a center (known as a seed) which determines the waves dynamics [22].

It is of great importance to investigate the dynamics of spiral waves since they have been observed in the mammalian neocortex [23] and also cardiac arrhythmia [24]. It has been demonstrated that both atrial fibrillation and ventricular fibrillation occur as a result of spiral waves [25], [26]. A spiral seed in the heart fiber spins with a frequency higher than the hearts natural frequency and makes the heartbeat irregular [24]. Therefore, it can cause fibrillation. Thus, modeling and recognizing the structure of spiral waves formation can help designing methods to eliminate the spiral waves and control the fibrillations [27].

Many works have been done on the formation and analysis of spiral waves in neuronal networks in the recent years. Wang et. al. [28] studied the influence of the network structure on the stability of spiral waves. They observed that by adding random links to the regular network, the spiral waves start to disappear and instead other spatial patterns are formed. Rostami et. al. [29] considered the interaction between the obstacle and the wavefronts and found that this interaction can lead to spiral waves and reentry. Gabi and Ogawa [30] investigated spiral wave and periodic traveling wave instability in partial differential equations model of cardiac excitation and showed the pattern of the front and the back interaction in one and two dimensions. There are also some schemes for suppressing and controlling the spiral waves [31], [32]. As an example, Yuan et.al. [32] applied a one-channel feedback to control the spiral waves in the complex Ginzburg Landau equation.

In this paper we study the effect of an external electromagnetic excitation in a Hindmarsh–Rose neuronal network on spiral waves formation. We use exponential magnetic induction in our model which refers to ion currents flow across the membrane.

The paper is organized as follows. In the following section, the mathematical model of the modified HR model and neuronal network is described. Section 3 describes the numerical method and presents the resulting dynamics. Finally, conclusions are given in Section 4.

Section snippets

The model

Up to now, many neuron models have been proposed to analyze the dynamic properties of neurons. The pioneer study of electrical activities of a neuron relates to Hodgkin and Huxley that was applied on nerve conduction in the squid large axon [33]. After that some simplification of this physiologically realistic model were proposed such as FitzHugh model [34], Hindmarsh–Rose model [35], Izhikevich model [36] and etc. The Hindmarsh–Rose model has been studied increasingly, since it contains

Numerical results

To numerically analyze the formation of spiral wave in the MHRN, we construct a network of 100 × 100 neurons. The coupling strength is considered as $D = 0.4$ and two stimulations are set with start time and end time of 30 and 31 for first stimulation and 60 and 61 for second stimulation. We investigate the patterns of the waves that are formed by applying different I_ext and external periodical force on the main diameter of the plane with different values of A and ω. I_ext is the level of the

Conclusion

In this paper firstly a new modified magnetic Hindmarsh–Rose neuron model was proposed with the external electromagnetic radiation and discontinuous flux coupling. The magnetic flux coupling was introduced exponentially to be applicable in rapid changes of excitation potentials. Then we constructed a square array network of this model with nearest-neighbor connection. We applied two different external magnetic excitations, periodic and quasi-periodic, and investigated the network patterns by

Acknowledgments

The authors acknowledge Prof. GuanRong Chen, Department of Electronic Engineering, City University of Hong Kong for suggesting helpful references.

References (42)

S. Boccaletti et al.
Complex networks: Structure and dynamics
Phys. Rep.
(2006)
J.-X. Chen et al.
Dynamics of scroll waves with time-delay propagation in excitable media
Commun. Nonlinear Sci. Numer. Simul.
(2018)
Y.H. Zheng et al.
Spatiotemporal patterns and chaotic burst synchronization in a small-world neuronal network
Physica A
(2008)
M. Jalili
Spike phase synchronization in multiplex cortical neural networks
Physica A
(2017)
F. Parastesh et al.
Synchronizability of two neurons with switching in the coupling
Appl. Math. Comput.
(2019)
X. Huang et al.
Spiral wave dynamics in neocortex
Neuron
(2010)
Z. Rostami et al.
Taking control of initiated propagating wave in a neuronal network using magnetic radiation
Appl. Math. Comput.
(2018)
G. Yuan et al.
Feedback-controlled dynamics of spiral waves in the complex Ginzburg–-Landau equation
Nonlinear Dyn.
(2017)
A.L. Hodgkin et al.
Propagation of electrical signals along giant nerve fibres
Proc. R. Soc. Lond. Ser. B
(1952)
E.M. Izhikevich
Neural excitability, spiking and bursting
Int. J. Bifurcat. Chaos
(2000)

X. Liao et al.

Pattern formation in oscillatory complex networks consisting of excitable nodes

Phys. Rev. E

(2011)

K. Khaothong et al.

Effect of a modified sinusoidal forcing on spiral wave in a simulated reaction-diffusion system

J. Phys. Conf. Ser.

(2017)

D. Mahanta et al.

Spirals in a reaction–diffusion system: dependence of wave dynamics on excitability

Phys. Rev. E

(2018)

D. Guo et al.

Frequency-difference-dependent stochastic resonance in neural systems

Phys. Rev. E

(2017)

Q. Wang et al.

Synchronous bursts on scale-free neuronal networks with attractive and repulsive coupling

PLoS One

(2011)

T. Chouzouris et al.

Chimera states in brain networks: empirical neural vs. modular fractal connectivity

Chaos

(2018)

S. Majhi et al.

Chimera states in a multilayer network of coupled and uncoupled neurons

Chaos

(2017)

Z. Wei et al.

Nonstationary chimeras in a neuronal network

Europhys. Lett.

(2018)

Y. Deng et al.

Parametric wave induces straight drift of spiral waves in excitable medium

Europhys. Lett.

(2017)

Y. Yao et al.

Impact of bounded noise on the formation and instability of spiral wave in a 2d lattice of neurons

Sci. Rep.

(2017)

M. Perc

Thoughts out of noise

Eur. J. Phys.

(2006)

Cited by (48)

Chimera-like state in the bistable excitatory-inhibitory cortical neuronal network
2024, Chaos, Solitons and Fractals
In recent years, the coexistence of different states in the neural system has attracted widespread interest. Researchers have found a coexisting state of spiking and resting in homogeneous networks, which is known as the chimera-like state. The real cortical network is a much more complex and heterogeneous network. Therefore, the excitatory-inhibitory cortical neuronal network is constructed based on Hodgkin-Huxley neuronal model in this paper, and the chimera-like state is further investigated in the heterogeneous network. It is found that the chimera-like state is related to the balance between excitatory and inhibitory synaptic currents. The excitatory coupling current can counteract the initial condition effect and promote synchronized firing of neurons in the network. The inhibitory coupling current desynchronizes the network and thus induces synaptic noise, resulting in an inverse bell-shaped dependence of the change in the number of spiking neurons. We analyzed the underlying mechanisms of synaptic noise in the phase plane diagram and found it has asymmetry for the neuronal state transition. In addition, neurons with low degrees have a higher probability of undergoing state transitions. Finally, we verified that the chimera-like state is robust to network topology and initial conditions. The results provide a new insight into neuronal interactions in heterogeneous networks and might help to reveal the mechanisms of coexistence of different states in the cortical network.
Spiral waves in fractal dimensions and their elimination in λ − ω systems with less damaging intervention
2024, Chaos, Solitons and Fractals
The theory of spiral waves in excitable media is considered an important topic in several dynamical systems. They have been observed in various physicochemical and biological systems such as the Belousov-Zhabotinsky chemical reactions, lethal arrhythmia in the heart, amoebas Dictyostelium discoideium, disinhibited mammalian neocortex retinal spreading depression, and seizures in the brain. In this study, spiral waves in an excitable system are analytically studied in fractal dimensions based on the theory of integration and differentiation for a non-integer dimensional space. We consider the generic oscillatory model $λ - ω$ system characterized by a rotational symmetry where spiral waves are expected to occur. It was observed that for lower fractal dimensions with respect to unity, spiral waves may be suppressed in an excitable media without destroying the propagation of normal waves. This may be considered as an alternative approach to experimental and numerical approaches found in the literature. Their eliminations are important since they forbid the transition to a chaotic state in excitable media. Our approach could contribute to an improved therapy of clinical conditions such as atrial fibrillation and suppressing arrhythmias in cardiac models.
Dynamics, synchronization and traveling wave patterns of flux coupled network of Chay neurons
2024, BioSystems
Studies in the literature have demonstrated the significance of the synchronization of neuronal electrical activity for signal transmission and information encoding. In light of this importance, we investigate the synchronization of the Chay neuron model using both theoretical analysis and numerical simulations. The Chay model is chosen for its comprehensive understanding of neuronal behavior and computational efficiency. Additionally, we explore the impact of electromagnetic induction, leading to the magnetic flux Chay neuron model. The single neuron model exhibits rich and complex dynamics for various parameter choices. We explore the bifurcation structure of the model through bifurcation diagrams and Lyapunov exponents. Subsequently, we extend our study to two coupled magnetic flux Chay neurons, identifying mode locking and structures reminiscent of Arnold’s tongue. We evaluate the stability of the synchronized manifold using Lyapunov theory and confirm our findings through simulations. Expanding our study to networks of diffusively coupled flux Chay neurons, we observe coherent, incoherent, and imperfect chimera patterns. Our investigation of three network types highlights the impact of network topology on the emergent dynamics of the Chay neuron network. Regular networks exhibit diverse patterns, small-world networks demonstrate a critical transition to coherence, and random networks showcase synchronization at specific coupling strengths. These findings significantly contribute to our understanding of the synchronization patterns exhibited by the magnetic flux Chay neuron. To assess the synchronization stability of the Chay neuron network, we employ master stability function analysis.
Polarized electric field-induced drift of spiral waves in discontinuous cardiac media
2023, Chaos, Solitons and Fractals
The clinical application of high-voltage defibrillation to eliminate spiral waves in the cardiac possesses profound adverse effects. There is a demand for alternative approaches with lower intensity, such as utilizing low-intensity electric fields to induce the drift of spiral waves. Phase singularity (PS) denotes the tip of spiral wave. Under the influence of the electric fields, PS can drift towards non-excited boundaries for the purpose of eliminating spiral waves. Unlike previous investigations, here we consider discontinuities in myocardial tissue caused by cellular structures, where the myocardial tissue consists of interlaced rod-shaped cells. We apply different electric fields to the spiral waves in this discontinuous myocardial tissue and found following conclusions: (1) The constant electric field possesses the capacity to induce the drift of the PS in the direction that is anti-parallell to the electric field. (2) The polarized electric field of specific intensity and angular frequency can induce the linear drift of PS. (3) Interestingly, the drift of the spiral wave in a predictable direction can be controlled by varying the phase difference of the polarized electric field as well as the initial phase of the spiral wave. We discuss the effect of electric fields on spiral waves in discontinuous myocardial media through numerical simulations. Our study may provide some insight into the practical application of low-energy elimination of spiral waves in the cardiac.
Complex dynamics in a fractional order nephron pressure and flow regulation model
2023, BioSystems
Cardiovascular diseases can be attributed to irregular blood pressure, which may be caused by malfunctioning kidneys that regulate blood pressure. Research has identified complex oscillations in the mechanisms used by the kidney to regulate blood pressure. This study uses established physiological knowledge and earlier autoregulation models to derive a fractional order nephron autoregulation model. The dynamical behaviour of the model is analyzed using bifurcation plots, revealing periodic oscillations, chaotic regions, and multistability. A lattice array of the model is used to study collective behaviour and demonstrates the presence of chimeras in the network. A ring network of the fractional order model is also considered, and a diffusion coupling strength is adopted. A basin of synchronization is derived, considering coupling strength, fractional order or number of neighbours as parameters, and measuring the strength of incoherence. Overall, the study provides valuable insights into the complex dynamics of the nephron autoregulation model and its potential implications for cardiovascular diseases.
Analysis of dynamical robustness of multilayer neuronal networks with inter-layer ephaptic coupling at different scales
2022, Applied Mathematical Modelling
Citation Excerpt :
Network theory has become an effective and popular tool to describe structural interdependence among interacting units in neuroscience, physics, chemistry, biology and engineering. One of the interests in complex network is concerned about collective dynamics of networked systems consisting of dynamical units [1–5]. On the basis of dynamical units, coupling schemes and complex topology structure, various collective dynamics arise in networked system [6–8].
In nervous system, a non-synaptic communication between neurons, known as the ephaptic coupling, emerges depending on electromagnetic induction which is induced by extracellular electric fields. In this paper, a multilayer neuronal network is constructed by adopting ephaptic coupling between layers, and dynamical robustness of multilayer neuronal networks with inter-layer ephaptic coupling is analyzed at different scales. The macroscopic oscillation of the whole network occupies the middle ground between mesoscopic oscillation of intermediate layer and that of the top (or bottom) layer. Strong inter-layer ephaptic coupling is capable of suppressing the resonance like phenomena of oscillation at mesoscale and macroscale. At weak electrical coupling, dynamical robustness of intermediate layer is stronger than that of top and bottom layers, and is even stronger than that of the whole network. While at strong electrical coupling, dynamical robustness of each layer is comparable to that of the whole network. The inter-layer ephaptic coupling could enhance dynamical robustness of each layer and the entire multilayer neuronal network, which is in stark contrast to electrical coupling with the tendency to spoil the dynamical robustness. Dynamical robustness of the whole network is always weaker than that of intermediate layer, but is stronger than that of top and bottom layers with increasing inter-layer ephaptic coupling. The firing modes of neurons before the critical ratio is analyzed at microscale. The ratio of inactive neurons switches the firing patterns of the active neuron among different firing patterns in multilayer neuronal network. The dynamics of multilayer neuronal network with inter-layer ephaptic coupling is verified in the analog circuit built on Multisim. This study provides new clues to understand mechanism of collective phenomenon in realistic neuronal systems.

View all citing articles on Scopus

View full text

Birth and death of spiral waves in a network of Hindmarsh–Rose neurons with exponential magnetic flux and excitable media

Highlights

Abstract

Introduction

Section snippets

The model

Numerical results

Conclusion

Acknowledgments

Phys. Rep.

Commun. Nonlinear Sci. Numer. Simul.

Physica A

Physica A

Appl. Math. Comput.

Neuron

Appl. Math. Comput.

Nonlinear Dyn.

Proc. R. Soc. Lond. Ser. B

Int. J. Bifurcat. Chaos

Pattern formation in oscillatory complex networks consisting of excitable nodes

Phys. Rev. E

Effect of a modified sinusoidal forcing on spiral wave in a simulated reaction-diffusion system

J. Phys. Conf. Ser.

Spirals in a reaction–diffusion system: dependence of wave dynamics on excitability

Phys. Rev. E

Frequency-difference-dependent stochastic resonance in neural systems

Phys. Rev. E

Synchronous bursts on scale-free neuronal networks with attractive and repulsive coupling

PLoS One

Chimera states in brain networks: empirical neural vs. modular fractal connectivity

Chaos

Chimera states in a multilayer network of coupled and uncoupled neurons

Chaos

Nonstationary chimeras in a neuronal network

Europhys. Lett.

Parametric wave induces straight drift of spiral waves in excitable medium

Europhys. Lett.

Impact of bounded noise on the formation and instability of spiral wave in a 2d lattice of neurons

Sci. Rep.

Thoughts out of noise

Eur. J. Phys.