Stability analysis of the Hindmarsh–Rose neuron under electromagnetic induction

Nonlinear Dynamics - We consider the Hindmarsh–Rose neuron model modified by taking into account the effect of electromagnetic induction on membrane potential. We study the impact of the...     

Adaptive synchronization of time delay Hindmarsh–Rose neuron system via self-feedback

Synchronization of two mismatched time delay Hindmarsh?CRose neuron systems with self-feedback is investigated. Based on the Lyapunov stability theory and the adaptive control theory, a linear adaptive feedback controller and parameter estimation update law are proposed, and the sufficient conditions for synchronization of the two mismatched systems with chaotic bursting behavior are obtained. The correctness of the proposed methods is rigorously demonstrated. Finally, numerical simulations are employed to verify the effectiveness of the proposed scheme.     

Dynamical behavior and network analysis of an extended Hindmarsh–Rose neuron model

Nonlinear Dynamics - In this paper, the extended Hindmarsh–Rose neuron model, which considers the slow intracellular exchange of calcium ions between its store and the cytoplasm, is studied....     

Synthesizing attractors of Hindmarsh–Rose neuronal systems

In this paper a periodic parameter-switching scheme is applied to the Hindmarsh–Rose neuronal system to synthesize certain attractors. Results show numerically, via computer graphic simulations, that the obtained synthesized attractor belongs to the class of all admissible attractors for the Hindmarsh–Rose neuronal system and matches the averaged attractor obtained with the control parameter replaced with the averaged switched parameter values. This feature allows us to imagine that living beings are able to maintain vital behavior while the control parameter switches so that their dynamical behavior is suitable for the given environment.     

Lag synchronization of multiple identical Hindmarsh–Rose neuron models coupled in a ring structure

Lag synchronization of multiple identical Hindmarsh–Rose neuron systems coupled in a ring structure is investigated. In the coupled systems, each neuron receives signals only via synaptic strength from the nearest neighbors. Based on the Lyapunov stability theory, the sufficient conditions for synchronization of the multiple systems with chaotic bursting behavior can be obtained. The synchronization condition about the control parameter g is also obtained by numerical method. Finally, numerical simulations are provided to show the effectiveness of the developed methods.     

Spatial evolution of Hindmarsh–Rose neural network with time delays

Nonlinear Dynamics - Spatial relations between neurons in the network with time delays play a crucial role in determining dynamics of the system. During the development of the nervous system,...     

Codimension-two bifurcation analysis in two-dimensional Hindmarsh–Rose model

In this paper, we analyze the codimension-2 bifurcations of equilibria of a two-dimensional Hindmarsh–Rose model. By using the bifurcation methods and techniques, we give a rigorous mathematical analysis of Bautin bifurcation. The main result is that no more than two limit cycles can be bifurcated from the equilibrium via Hopf bifurcation; sufficient conditions for the existence of one or two limit cycles are obtained. This paper also shows that the model undergoes a Bogdanov–Takens bifurcation which includes a saddle-node bifurcation, an Andronov–Hopf bifurcation, and a homoclinic bifurcation. In some case, the globally asymptotical stability is discussed.     

Hidden dynamics in a fractional-order memristive Hindmarsh–Rose model

Nonlinear Dynamics - By showing that there is no any memristor in the integer-order Hindmarsh–Rose (H–R) model, the slow ion channel is remodeled by a fractional-order memristor, where...     

Bifurcations and chaos in a two-dimensional discrete Hindmarsh–Rose model

In this paper, the dynamics of a two-dimensional discrete Hindmarsh–Rose model is discussed. It is shown that the system undergoes flip bifurcation, Neimark–Sacker bifurcation, and 1:1 resonance by using a center manifold theorem and bifurcation theory. Furthermore, we present the numerical simulations not only to illustrate our results with the theoretical analysis, but also to exhibit the complex dynamical behaviors, including orbits of period 3, 6, 15, cascades of period-doubling bifurcation in orbits of period 2, 4, 8, 16, quasiperiodic orbits, and chaotic sets. These results obtained in this paper show far richer dynamics of the discrete Hindmarsh–Rose model compared with the corresponding continuous model.     

Memristor-based oscillatory behavior in the FitzHugh–Nagumo and Hindmarsh–Rose models

Nonlinear Dynamics - The neural firing activities related to information coding maintaining the information transmission vary qualitatively considering the electromagnetic induction. The firing of...     

Chaos bursting synchronization of mismatched Hindmarsh–Rose systems via a single adaptive feedback controller

Chaotic bursting synchronization of mismatched Hindmarsh–Rose neuron systems is investigated. Based on the Lyapunov stability theory, an adaptive feedback control scheme for the synchronization of the neuron systems is proposed when partially parameters of the response system are unknown and different with those of the drive system. Furthermore, in the proposed scheme, only a single adaptive feedback controller is needed, which is efficient and easy to implement. Finally, numerical simulations are provided to show the effectiveness of the developed methods.     

Multiple scales and normal forms in a ring of delay coupled oscillators with application to chaotic Hindmarsh–Rose neurons

We study the appearance and stability of spatiotemporal periodic patterns like phase-locked oscillations, mirror-reflecting waves, standing waves, in-phase or antiphase oscillations, and coexistence of multiple patterns, in a ring of bidirectionally delay coupled oscillators. Hopf bifurcation, Hopf–Hopf bifurcation, and the equivariant Hopf bifurcation are studied in the viewpoint of normal forms obtained by using the method of multiple scales which is a kind of perturbation technique, thus a clear bifurcation scenario is depicted. We find time delay significantly affects the dynamics and induces rich spatiotemporal patterns. With the help of the unfolding system near Hopf–Hopf bifurcation, it is confirmed in some regions two kinds of stable oscillations may coexist. These phenomena are shown for the delay coupled limit cycle oscillators as well as for the delay coupled chaotic Hindmarsh–Rose neurons.     

Effects of astrocyte on weak signal detection performance of Hodgkin–Huxley neuron

Nonlinear Dynamics - By virtue of recent developments in brain measurement technology, it is now recognized that information processing in brain includes not only neurons but also astrocytes. For...     

Measure Valued Solutions of the 2D Keller–Segel System

We develop a theory of global measure-valued solutions for the classical Keller–Segel model. These solutions are obtained considering the limit of solutions of a regularized problem. We also prove that different regularizations yield different limit measures in the case in which classical solutions of the Keller–Segel system are not globally defined in time.     

Simulating the electric activity of FitzHugh–Nagumo neuron by using Josephson junction model

The resistive-capacitive-inductance Josephson junction (RCLSJ) model can simulate the electric activities of neurons. In this paper, the RCLSJ system is controlled to reproduce the dynamical properties of the FitzHugh?CNagumo system neuron by using the improved adaptive synchronization scheme. Improved Lyapunov functions with two controllable gain coefficients (??,??) is constructed, and the controller is approached analytically to realize linear generalized synchronization defined as $x=k\hat{x}+C$ . The summation of error function during the process of synchronization and the power consumption of controller are calculated in the dimensionless model to measure the effect of the two gain coefficients (??,??) by selecting different constants (k,C) to represent different kinds of generalized synchronization. The results are approached as follows: (1) the power consumption of the controller is independent of the selection of the two gain coefficients (??,??); (2) the synchronization region is marked in the phase space of the two gain coefficients; (3) the power consumption of controller is dependent on the selection of constants (k,C), smaller power consumption of the controller is required with larger k at fixed C; larger power consumption costs with larger C at fixed k. The specific case for C=0,k=1 is also discussed to understand the case for complete synchronization.     

A construction of new exact periodic wave and solitary wave solutions for the 2D Ginzburg–Landau equation

Using a uniform algebraic method, new exact solitary wave solutions and periodic wave solutions for 2D Ginzburg–Landau equation are obtained. Moreover, three-dimensional and two-dimensional graphics of some solutions have been plotted.     

Circuit implementation and finite-time synchronization of the 4D Rabinovich hyperchaotic system

This paper studies the problem of the circuit implementation and the finite-time synchronization for the 4D (four-dimensional) Rabinovich hyperchaotic system. The electronic circuit of 4D hyperchaotic system is designed. It is rigorously proven that global finite-time synchronization can be achieved for hyperchaotic systems which have uncertain parameters.     

Algorithm and implementation of soil–tire contact analysis code based on dynamic FE–DE method

One of the fundamental problems in terramechanics is soil–tire system. Past achievements on this topic can be observed in various literatures. Fast development on CPU power of PC system enables us to apply numerical methods to this basic subject. Among others, finite element method (FEM) has been applied to simple problems of soil–tire system not only in 2D but also in 3D approach. However, it is noted that the current FEM technology cannot handle “singular” boundary conditions with sufficient accuracy of analysis. Typical example of this limitation can be seen in an application to traction tire–soil contact problems, where the contact point of tire lug tip behaves as the singular point of stress field. On the other hand, distinct or discrete element method (DEM) has in essence the capability of analyze microscopic deformation (or flow) of soil as many researchers have already been demonstrated. It is noted that DEM suffers large calculation time that is consumed not only at contact check between particulate elements but also at incremental time step. In our present study, we try to combine both merit of FEM and DEM together in order to analyze the soil–tire system interaction, where, for example, a tire and deep soil layer are modeled as FEM and soil surface layer as DEM. We propose simple algorithm of this FE–DE coupled method and sample program is developed that can solve some basic terramechanics problems in order to verify our idea. The obtained result shows qualitatively sufficient accuracy.     

2D FE–DEM analysis of tractive performance of an elastic wheel for planetary rovers

We have been developing a simulation program for use with soil–wheel interaction problems by coupling Finite Element Method (FEM) and Discrete Element Method (DEM) for which a wheel is modeled by FEM and soil is expressed by DEM. Previous two-dimensional FE–DEM was updated to analyze the tractive performance of a flexible elastic wheel by introducing a new algorithm learned from the PID-controller model. In an elastic wheel model, four structural parts were defined using FEM: the wheel rim, intermediate part, surface layer, and wheel lugs. The wheel rigidity was controlled by varying the Young’s Modulus of the intermediate part. The tractive performance of two elastic wheels with lugs for planetary rovers of the European Space Agency was analyzed. Numerical results were compared with experimentally obtained results collected at DLR Bremen, Germany. The FE–DEM result was confirmed to depict similar behaviors of tractive performance such as gross tractive effort, net traction, running resistance, and wheel sinkage, as in the results of experiments. Moreover, the tractive performance of elastic wheels on Mars was predicted using FE–DEM. Results clarified that no significant difference of net traction exists between the two wheels.     

A 2D DEM–LBM study on soil behaviour due to locally injected fluid

Leakage from underground pipes could result in foundations being undermined and cause damage to adjacent infrastructure. Soil particles surrounding the leaking area could be mobilised, displaced, and even washed out of the soil matrix by the leaking fluid, generating a void or cavity. A two-dimensional simulation using a coupled discrete element method and lattice Boltzmann method (DEM–LBM) has been used to investigate the behaviour of a soil bed subject to a locally injected fluid, which represents a leak in a pipe. Various values of inter-particle surface energy were also adopted to model the mechanical effects of cohesive particles. The results suggest that the inter-particle surface energy greatly influences the bed response with respect to the leaking fluid, including the excess pressure initiating the cavity, the cavity shape and its evolution rate.