Wave propagation and spiral wave formation in a Hindmarsh–Rose neuron model with fractional-order threshold memristor synaps

2020 ◽  
Vol 34 (17) ◽  
pp. 2050157 ◽  
Author(s):  
Karthikeyan Rajagopal ◽  
Anitha Karthikeyan ◽  
Sajad Jafari ◽  
Fatemeh Parastesh ◽  
Christos Volos ◽  
...  
Keyword(s):  
Wave Propagation ◽  
Fractional Order ◽  
Electromagnetic Induction ◽  
Neuron Model ◽  
Spiral Wave ◽  
Spiral Waves ◽  
Bifurcation Diagrams ◽  
External Current ◽  
Proposed Model ◽  
External Stimuli

In this paper, a modified Hindmarsh–Rose neuron model is presented, which has a fractional-order threshold magnetic flux. The dynamics of the model is investigated by bifurcation diagrams and Lyapunov exponents in two cases of presence and absence of the external electromagnetic induction. Then the emergence of the spiral waves in the network of the proposed model is studied. To find the effects of different factors on the formation and destruction of spiral waves, the external current, the coupling strength and the external stimuli amplitude are varied. It is observed that all of these parameters have significant impacts on the spiral waves. Furthermore, the external electromagnetic induction influences the existence of spiral waves in specific external current values.

scholarly journals A Discrete Huber-braun Neuron Model: From Nodal Properties to Network Performance

2021 ◽  
Author(s):  
Shaoba He ◽  
Karthikeyan Rajagopal ◽  
Anitha Karthikeyan ◽  
Ashokkumar Sriniva
Keyword(s):  
Wave Propagation ◽  
Network Performance ◽  
Neuron Model ◽  
Coupling Effect ◽  
Single Layer ◽  
Simple Wave ◽  
Spiral Waves ◽  
Neuron Models ◽  
Positive Role ◽  
Multiple Wave

Abstract Many of the well-known neuron models are continuous time systems with complex mathematical definitions. Literatures have shown that a discrete mathematical model can effectively replicate the complete dynamical behaviour of a neuron with much reduced complexity. Hence, we propose a new discrete neuron model derived from the Huber-Braun neuron with two additional slow and subthreshold currents alongside the ion channel currents. We have also introduced temperature dependent ion channels to study its effects on the firing pattern of the neuron. With bifurcation and Lyapunov exponents we showed the chaotic and periodic regions of the discrete model. Further to study the complexity of the neuron model, we have used the sample entropy algorithm. Though the individual neuron analysis gives us an idea about the dynamical properties, it’s the collective behaviour which decides the overall behavioural pattern of the neuron. Hence, we investigate the spatiotemporal behaviour of the discrete neuron model in single- and two-layer network. We have considered noise and obstacles as the two important factor which changes the excitability of the neurons in the network. When there is no noise or obstacle, the network display simple wave propagation with highly excitable neurons. Literatures have shown that spiral waves can play a positive role in breaking through quiescent areas of the brain as a pacemaker by creating a coherence resonance behaviour. Hence, we are interested in studying the induced spiral waves in the network. In this condition when an obstacle is introduced the wave propagation is disturbed and we could see multiple wave re-entry and spiral waves. Now when we consider only noise with no obstacle, for selected noise variances the network supports wave re-entry. By introducing an obstacle in this noisy network, the re-entry soon disappears, and the network soon enters idle state with no resetting. In a two-layer network when the obstacle is considered only in one layer and stimulus applied to the layer having the obstacle, the wave re-entry is seen in both the layer though the other layer is not exposed to obstacle. But when both the layers are inserted with an obstacle and stimuli also applied to the layers, they behave like independent layers with no coupling effect. This in a two-layer network stimulus play an important role in spatiotemporal dynamics of the network. Similar noise effects like the single layer network are also seen in the two-layer network.

Dynamics of pulses and spiral waves in excitable media with an anomalous diffusion

2016 ◽  
Vol 30 (20) ◽  
pp. 1650127 ◽  
Author(s):  
Guoyong Yuan ◽  
Xueping Bao ◽  
Shiping Yang ◽  
Guangrui Wang ◽  
Shaoying Chen
Keyword(s):  
Fractional Order ◽  
Anomalous Diffusion ◽  
Spiral Wave ◽  
Fractional Diffusion ◽  
Spiral Waves ◽  
Excitable Media ◽  
Diffusion Parameter ◽  
Positive Formula ◽  
Normal Diffusion ◽  
The One

Spiral waves and pulses in the excitable medium with an anomalous diffusion are studied. In the medium with an one-sided fractional diffusion in the [Formula: see text]-direction and a normal diffusion in the [Formula: see text]-direction, a pulse, traveling along the positive [Formula: see text]-direction, has a smaller velocity, which is different from the diffusion of a source in the other media. Its propagating velocity is a linear and increasing function of the square root of diffusion parameter, whose increasing rate depends on the fractional order. A minimal value of the diffusion parameter is needed for successfully propagating pulses, and the threshold becomes large with a decrease of the fractional order. For pulse trains, the frequency-locked bands are shifted along the increasing direction of the perturbation period when the fractional order is decreased. In the propagating process of a spiral wave, the tip drift is induced by the one-sided fractional diffusion, which may be explained by analyzing the SV area in front of the tip.

scholarly journals Stability Analysis of a Fractional-Order High-Speed Supercavitating Vehicle Model with Delay

Machines ◽  
2021 ◽  
Vol 9 (7) ◽  
pp. 129
Author(s):  
Phuc Thinh Doan ◽  
Phuc Duc Hong Bui ◽  
Mai The Vu ◽  
Ha Le Nhu Ngoc Thanh ◽  
Shakhawat Hossain
Keyword(s):  
Fractional Order ◽  
High Speed ◽  
Vehicle Speed ◽  
Bifurcation Diagrams ◽  
Order Model ◽  
New Model ◽  
Delay Term ◽  
Supercavitating Vehicle ◽  
Proposed Model ◽  
Mathematical Equations

A novel fractional-order model (FOM) of a high-speed super-cavitating vehicle (HSSV) with the nature of memory is proposed and investigated in this paper. This FOM can describe the behavior of the HSSV superior to the integer-order model by the memory effects of fractional-order derivatives. The fractional order plays the role of the advection delay, which is ignored in most of the prior studies. This new model takes into account the effect of advection delay while preserving the nonlinearity of the mathematical equations. It allows the analysis of nonlinear equations describing the vehicle with ease when eliminating the delay term in its equations. By using the fractional order to avoid the approximation of the delay term, the proposed FOM can also preserve the nature of the time delay. The numerical simulations have been carried out to study the behavior of the proposed model through the transient responses and bifurcation diagrams concerning the fractional-order and vehicle speed. The bifurcation diagrams provide useful information for a better control and design of new supper super-cavitating vehicles. The similar behaviors between the proposed model and prior ones validate the FOM while some discrepancies suggest that more appropriate controllers should be designed based on this new model.

Formation of multi-armed spiral waves in neuronal network induced by adjusting ion channel conductance

2015 ◽  
Vol 29 (07) ◽  
pp. 1550043 ◽  
Author(s):  
Chunni Wang ◽  
Jun Ma ◽  
Bolin Hu ◽  
Wuyin Jin
Keyword(s):  
Ion Channels ◽  
Neuron Model ◽  
Nearest Neighbor ◽  
Spiral Wave ◽  
Spiral Waves ◽  
Channel Conductance ◽  
Transient Period ◽  
Regular Network ◽  
Realistic Neuron ◽  
Armed Spiral

The Hodgkin–Huxley neuron model is used to describe the local dynamics of nodes in a two-dimensional regular network with nearest-neighbor connections. Multi-armed spiral waves emerge when a group of spiral waves rotate the same core synchronously. Here we have numerically investigated how multi-armed spiral waves are formed in such a system. Under the appropriate conditions, multi-armed spiral waves were able to develop as a result of adjusting the conductance of ion channels of particular neurons in the network. In a realistic neuron model, it can be practiced by blocking potassium of ion channels embedded in the membrane of neurons. For example, decreasing the potassium channel conductance in some neurons with a certain transient period can lead to the development of a group of double spirals in a localized area of the network. Furthermore, decreasing the excitability and the external forcing current to zero led to the growth of these double spirals and the formation of a stable multi-armed spiral wave that occupied the network under inhomogeneity.

scholarly journals SUPPRESSING SPIRAL WAVE TURBULENCE IN A SIMPLE FRACTIONAL-ORDER DISCRETE NEURON MAP USING IMPULSE TRIGGERING

Fractals ◽  
2021 ◽  
pp. 2140030
Author(s):  
KARTHIKEYAN RAJAGOPAL ◽  
SHIRIN PANAHI ◽  
MO CHEN ◽  
SAJAD JAFARI ◽  
BOCHENG BAO
Keyword(s):  
Fractional Order ◽  
Neuron Model ◽  
Spiral Wave ◽  
Wave Turbulence ◽  
Effective Parameters ◽  
Neuron Models ◽  
Collective Behaviors ◽  
One Dimensional ◽  
Dynamical Characteristics ◽  
Significant Interest

One-dimensional (1D) map-based neuron models are of significant interest according to their simplicity of simulation and ability to mimic real neurons’ complex behaviors. A fractional-order 1D neuron map is proposed in this paper. Dynamical characteristics of the model are analyzed by obtaining bifurcation diagrams and the Lyapunov exponents’ diagram. Furthermore, emerging the spiral wave as one of the most important collective behaviors is studied in a 2D lattice consisting of this new FO neuron model. The outcome of changing stimuli, coupling strength, and fractional-order parameter as the effective parameters is examined in this network. Moreover, an efficient way of suppressing the spiral wave has been investigated using impulse triggering.

Turbulent electrical activity at sharp-edged inexcitable obstacles in a model for human cardiac tissue

2014 ◽  
Vol 307 (7) ◽  
pp. H1024-H1035 ◽  
Author(s):  
Rupamanjari Majumder ◽  
Rahul Pandit ◽  
A. V. Panfilov
Keyword(s):  
Wave Propagation ◽  
Cardiac Tissue ◽  
Spiral Wave ◽  
Spiral Waves ◽  
Delayed Rectifier ◽  
High Frequency Stimulation ◽  
Secondary Importance ◽  
Ionic Model ◽  
Delayed Rectifier Current ◽  
Human Cardiac Tissue

Wave propagation around various geometric expansions, structures, and obstacles in cardiac tissue may result in the formation of unidirectional block of wave propagation and the onset of reentrant arrhythmias in the heart. Therefore, we investigated the conditions under which reentrant spiral waves can be generated by high-frequency stimulation at sharp-edged obstacles in the ten Tusscher-Noble-Noble-Panfilov (TNNP) ionic model for human cardiac tissue. We show that, in a large range of parameters that account for the conductance of major inward and outward ionic currents of the model [fast inward Na+ current ( INa), L—type slow inward Ca2+ current ( ICaL), slow delayed-rectifier current ( IKs), rapid delayed-rectifier current ( IKr), inward rectifier K+ current ( IK1)], the critical period necessary for spiral formation is close to the period of a spiral wave rotating in the same tissue. We also show that there is a minimal size of the obstacle for which formation of spirals is possible; this size is ∼2.5 cm and decreases with a decrease in the excitability of cardiac tissue. We show that other factors, such as the obstacle thickness and direction of wave propagation in relation to the obstacle, are of secondary importance and affect the conditions for spiral wave initiation only slightly. We also perform studies for obstacle shapes derived from experimental measurements of infarction scars and show that the formation of spiral waves there is facilitated by tissue remodeling around it. Overall, we demonstrate that the formation of reentrant sources around inexcitable obstacles is a potential mechanism for the onset of cardiac arrhythmias in the presence of a fast heart rate.

scholarly journals Multistability and Formation of Spiral Waves in a Fractional-Order Memristor-Based Hyperchaotic Lü System with No Equilibrium Points

2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Bo Yan ◽  
Shaobo He ◽  
Shaojie Wang
Keyword(s):  
Fractional Order ◽  
Initial Conditions ◽  
Equilibrium Points ◽  
Spiral Wave ◽  
Spiral Waves ◽  
High Complexity ◽  
Coexisting Attractors ◽  
Lü System ◽  
Hyperchaotic Lu System ◽  
Lu System

Multistablity analysis and formation of spiral wave in the fractional-order nonlinear systems is a recent hot topic. In this paper, dynamics, coexisting attractors, complexity, and synchronization of the fractional-order memristor-based hyperchaotic Lü system are investigated numerically by means of bifurcation diagram, Lyapunov exponents (LEs), chaos diagram, and sample entropy (SampEn) algorithm. The results show that the system has rich dynamics and high complexity. Meanwhile, coexisting attractors in the system are observed and hidden dynamics are illustrated by changing the initial conditions. Finally, the network based on the system is built, and the emergence of spiral waves is investigated and chimera states are observed.

Application of an Element-free Galerkin Method to Water Wave Propagation Problems

2014 ◽  
Vol 60 (1-4) ◽  
pp. 87-105 ◽  
Author(s):  
Ryszard Staroszczyk
Keyword(s):  
Wave Propagation ◽  
Galerkin Method ◽  
Present Model ◽  
Free Surface Elevation ◽  
Element Free Galerkin Method ◽  
Plane Flow ◽  
Element Free Galerkin ◽  
Proposed Model ◽  
Sloping Bed ◽  
Element Free

Abstract The paper is concerned with the problem of gravitational wave propagation in water of variable depth. The problem is solved numerically by applying an element-free Galerkin method. First, the proposed model is validated by comparing its predictions with experimental data for the plane flow in water of uniform depth. Then, as illustrations, results of numerical simulations performed for plane gravity waves propagating through a region with a sloping bed are presented. These results show the evolution of the free-surface elevation, displaying progressive steepening of the wave over the sloping bed, followed by its attenuation in a region of uniform depth. In addition, some of the results of the present model are compared with those obtained earlier by using the conventional finite element method.

Analysis of a New Class of Impulsive Implicit Sequential Fractional Differential Equations

2020 ◽  
Vol 21 (6) ◽  
pp. 571-587 ◽  
Author(s):  
Akbar Zada ◽  
Sartaj Ali ◽  
Tongxing Li
Keyword(s):  
Differential Equations ◽  
Fractional Order ◽  
Fractional Differential Equations ◽  
Sufficient Conditions ◽  
Uniqueness Of Solutions ◽  
Integer Order ◽  
New Class ◽  
Proposed Model ◽  
Fractional Differential ◽  
Sequential Fractional Differential Equations

AbstractIn this paper, we study an implicit sequential fractional order differential equation with non-instantaneous impulses and multi-point boundary conditions. The article comprehensively elaborate four different types of Ulam’s stability in the lights of generalized Diaz Margolis’s fixed point theorem. Moreover, some sufficient conditions are constructed to observe the existence and uniqueness of solutions for the proposed model. The proposed model contains both the integer order and fractional order derivatives. Thus, the exponential function appearers in the solution of the proposed model which will lead researchers to study fractional differential equations with well known methods of integer order differential equations. In the last, few examples are provided to show the applicability of our main results.

scholarly journals System of Time Fractional Models for COVID-19: Modeling, Analysis and Solutions

Symmetry ◽  
2021 ◽  
Vol 13 (5) ◽  
pp. 787
Author(s):  
Olaniyi Iyiola ◽  
Bismark Oduro ◽  
Trevor Zabilowicz ◽  
Bose Iyiola ◽  
Daniel Kenes
Keyword(s):  
Fractional Order ◽  
Recovery Rate ◽  
Death Rate ◽  
Numerical Solutions ◽  
Complex Nature ◽  
Uniqueness Of Solutions ◽  
Fractional Equations ◽  
Effective Contact ◽  
Proposed Model ◽  
Fractional Models

The emergence of the COVID-19 outbreak has caused a pandemic situation in over 210 countries. Controlling the spread of this disease has proven difficult despite several resources employed. Millions of hospitalizations and deaths have been observed, with thousands of cases occurring daily with many measures in place. Due to the complex nature of COVID-19, we proposed a system of time-fractional equations to better understand the transmission of the disease. Non-locality in the model has made fractional differential equations appropriate for modeling. Solving these types of models is computationally demanding. Our proposed generalized compartmental COVID-19 model incorporates effective contact rate, transition rate, quarantine rate, disease-induced death rate, natural death rate, natural recovery rate, and recovery rate of quarantine infected for a holistic study of the coronavirus disease. A detailed analysis of the proposed model is carried out, including the existence and uniqueness of solutions, local and global stability analysis of the disease-free equilibrium (symmetry), and sensitivity analysis. Furthermore, numerical solutions of the proposed model are obtained with the generalized Adam–Bashforth–Moulton method developed for the fractional-order model. Our analysis and solutions profile show that each of these incorporated parameters is very important in controlling the spread of COVID-19. Based on the results with different fractional-order, we observe that there seems to be a third or even fourth wave of the spike in cases of COVID-19, which is currently occurring in many countries.

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