Bifurcation analysis and diverse firing activities of a modified excitable neuron model

Memristor-coupled dual-neuron mapping model: initials-induced coexisting firing patterns and synchronization activities

Synaptic plasticity makes memristors particularly suitable for simulating the connection synapses between neurons that describe magnetic induction coupling. By applying a memristor to the synaptic coupling between two map-based neuron models, a memristor-coupled dual-neuron mapping (MCDN) model is proposed in this article. The MCDN model has a line fixed point set associated with the memristor initial state, which is always unstable for the model parameters and memristor initial state of interest. Complex spiking/bursting firing patterns and their transitions are disclosed using some dynamical analysis means. The numerical results show that these spiking/bursting firings are significantly relied on the memristor initial state, demonstrating the coexistence of firing patterns. Moreover, the initial effects of complete synchronization are explored for the homogeneous MCDN model, and it is clarified that in addition to being related to the coupling strength, the synchronization activities are extremely dependent on the initial states of the memristor and neurons. Finally, these numerical results are confirmed by the FPGA-based hardware experiments.

Dynamical effects of memristive electromagnetic induction on a 2D Wilson neuron model

Electromagnetic induction plays a crucial impact on the firing activity of biological neurons, since it exists along with the mutual effect between membrane potential and ions transport. Flux-controlled memristor is an available candidate in characterizing the electromagnetic induction effect. Different from the previously reported literature, a non-ideal flux-controlled memristor with cosine mem-conductance function is employed to determine the periodic magnetization and leakage flux processes in neurons. Thereafter, a three-dimensional (3D) memristive Wilson (m-Wilson) neuron model is constructed under the consideration of this kind of electromagnetic induction. Numerical simulations are performed by multiple numerical tools, which demonstrate that the 3D m-Wilson neuron model can generate abundant firing activities. Interestingly, coexisting firing activities, antimonotonicity, and firing frequency regulation are discovered under special parameter settings. Furthermore, a PCB-based analog circuit is designed and hardware measurements are executed to verify the numerical simulations. These explorations in numerical and hardware surveys might provide insights to regulate the firing activities by appropriate electromagnetic induction.

Energy controls wave propagation in a neural network with spatial stimuli

Nervous system has distinct anisotropy and some intrinsic biophysical properties enable neurons present various firing modes in neural activities. In presence of realistic electromagnetic fields, non-uniform radiation activates these neurons with energy diversity. By using a feasible model, energy function is obtained to predict the growth of synaptic connections of these neurons. Distribution of average value of the Hamilton energy function vs. intensity of noisy disturbance can predict the occurrence of coherence resonance, which the neural activities show high regularity by applying noisy disturbance with moderate intensity. From physical viewpoint, the average energy value has similar role average power for the neuron. Non-uniform spatial disturbance is applied and energy is injected into the neural network, statistical synchronization factor is calculated to predict the network synchronization stability and wave propagation. The intensity for field coupling is adaptively controlled by energy diversity between adjacent neurons. Local energy balance will terminate further growth of the coupling intensity; otherwise, heterogeneity is formed in the network due to energy diversity. Furthermore, memristive channel current is introduced into the neuron model for perceiving the effect of electromagnetic induction and radiation, and a memristive neuron is obtained. The circuit implement of memristive circuit depends on the connection to a magnetic flux-controlled memristor into the mentioned neural circuit in an additive branch circuit. The connection and activation of this memristive neural network are controlled under external spatial electromagnetic radiation by capturing enough field energy. Continuous energy collection and exchange generate energy diversity and synaptic connection is created to regulate the synchronous firing patterns and energy balance.

Three-dimensional memristive Morris-Lecar model with magnetic induction effects and its FPGA implementation

To characterize the magnetic induction flow induced by neuron membrane potential, a three-dimensional (3D) memristive Morris-Lecar (ML) neuron model is proposed in this paper. It is achieved using a memristor induction current to replace the slow modulation current in the existing 3D ML neuron model with fast-slow structure. The magnetic induction effects on firing activities are explained by the spiking/bursting firings with period-adding bifurcation and periodic/chaotic spiking-bursting patterns, and the bifurcation mechanisms of the bursting patterns are elaborated using the fast-slow analysis method to create two bifurcation sets. In particular, the 3D memristive ML model can also exhibit the homogeneous coexisting bursting patterns when switching the memristor initial states, which are effectively illustrated by the theoretical analysis and numerical simulations. Finally, a digitally FPGA-based hardware platform is developed for the 3D memristive ML model and the experimentally measured results well verify the numerical ones.

Nonlinear mechanism for the enhanced bursting activities induced by fast inhibitory autapse and reduced activities by fast excitatory autapse

The paradoxical phenomena that excitatory modulation does not enhance but reduces or inhibitory modulation not suppresses but promotes neural firing activities have attracted increasing attention. In the present study, paradoxical phenomena induced by both fast excitatory and inhibitory autapses in a "Fold/Big Homoclinic" bursting are simulated, and the corresponding nonlinear and biophysical mechanisms are presented. Firstly, the enhanced conductance of excitatory autapse induces the number of spikes per burst and firing rate reduced, while the enhanced inhibitory autapse cause both indicators increased. Secondly, with fast-slow variable dissection, the burst of bursting is identified to locate between a fold bifurcation and a big saddle-homoclinic orbit bifurcation of the fast subsystem. Enhanced excitatory or inhibitory autapses cannot induce changes of both bifurcation points, i.e., burst width. However, width of slow variable between two successive spikes within a burst becomes wider for the excitatory autapse and narrower for the inhibitory autapse, resulting in the less and more spikes per burst, respectively. Last, the autaptic current of fast autapse mainly plays a role during the peak of action potential, differing from the slow autaptic current with exponential decay, which can play roles following the peak of action potential. The fast excitatory autaptic current enhances the amplitude of the action potential and reduces the repolarization of the action potential to lengthen the interspike interval (ISI) of the spiking of the fast subsystem, resulting in the wide width of slow variable between successive spikes. The fast inhibitory autaptic current reduces the amplitude of action potential and ISI of spiking, resulting in narrow width of slow variable. The novel example of the paradoxical responses for both fast modulations and nonlinear mechanism extend the contents of neurodynamics, which presents potential functions of the fast autapse.

Adaptive synapse-based neuron model with heterogeneous multistability and riddled basins

Biological neurons can exhibit complex coexisting multiple firing patterns dependent on initial conditions. To this end, this paper presents a novel adaptive synapse-based neuron (ASN) model with sine activation function. The ASN model has time-varying equilibria with the variation of externally applied current and its equilibrium stability involves transitions between stable and unstable points through fold and Hopf bifurcations, resulting in complex distributions of attractive regions with heterogeneous multi-stability. Globally coexisting heterogeneous behaviors are studied by bifurcation diagram, phase portrait, dynamical distribution, and basin of attraction. The results show that the number of coexisting heterogeneous attractors can be up to 12, but for a simple neuron model, such a large number of coexisting heterogeneous attractors has not been reported in the relevant literature. Most interestingly, the ASN model also has riddled-like complex basins of attraction and four illustrative examples are depicted by the phase portraits with small changes of the initial conditions. Besides, the ASN model is implemented using a simple microcontroller platform, and various heterogeneous coexisting attractors are acquired experimentally to validate the numerical results.

Electromagnetic induction effects on electrical activity within a memristive Wilson neuron model

Neurons can exhibit abundant electrical activities due to physical effects of various electrophysiology environments. The electromagnetic induction flows can be triggered by changes in neuron membrane potential, which can be equivalent to a memristor applying on membrane potential. To imitate the electromagnetic induction effects, we propose a three-variable memristor-based Wilson neuron model. Using several kinetic analysis methods, the memristor parameter- and initial condition-related electrical activities are explored intensively. It is revealed that the memristive Wilson neuron model can display rich electrical activities, including the asymmetric coexisting electrical activities and antimonotonicity phenomenon. Finally, using off-the-shelf discrete components, an analog circuit on a hardware level is implemented to verify the numerically simulated coexisting electrical activities. Studying these rich electrical activities in neurons can build the groundwork to widen the neuron-based engineering applications.

Astrocytic modulation on neuronal electric mode selection induced by magnetic field effect

Astrocytes as well as electromagnetic induction have been primarily considered as main factors in regulating neuronal firing patterns in the recent decade. In this work, an improved neuron-astrocyte model in consideration of the modulation of astrocytes and the electromagnetic induction is employed to explore the extend to which both of the factors affect the firing modes of the neurons. The "alternation mode", defined as the alternative of neural normal spiking mode with the high-frequency bursting-like mode, clearly shows the functions of astrocytes on neurons. Moreover, the firing pattern of the neuron becomes more abnormal when astrocytes are hyper-excitable, the reason why the abnormal coupling of the astrocyte leads to the "alternation mode" of the neuron have been studied. In addition, the effect of electromagnetic induction manifests nonlinear characteristic towards neurons, complex firing modes of neurons are observed in the weaker field and a switching mode consists with quiescent and spiking mode appears when there is a higher stronger field. This approved model can reveal the normal or abnormal electric activities of neuron considered electromagnetic induction induced by the degree of excitability of the astrocyte. These results can provide potential understanding about the effects of astrocyte on neuronal activity when the coupling of electromagnetic field is considered.

Anti-control of periodic firing in HR model in the aspects of position, amplitude and frequency

This paper proposes a novel controller to control position, amplitude and frequency of periodic firing activity in Hindmarsh-Rose model based on Hopf bifurcation theory which is composed of linear control gain and nonlinear control gain. First, we select the activation of the fast ion channel as control parameter. Based on explicit criterion of Hopf bifurcation, a series of conditions are obtained to derive the linear gains of controller responsible for control of the location where the periodic firing activity occurs. Then, based on the control parameter, a series of conditions are obtained to derive the nonlinear gains of controller responsible for controlling the amplitude and frequency of periodic firing activity by using center manifold and normal form. Finally, the numerical experiments show that our controller can make the periodic firing activity occur at designed value and control the amplitude and frequency of periodic firing activity by adjusting nonlinear control gain of controller.

Dynamics of a Large-Scale Spiking Neural Network with Quadratic Integrate-and-Fire Neurons

Since the high dimension and complexity of the large-scale spiking neural network, it is difficult to research the network dynamics. In recent decades, the mean-field approximation has been a useful method to reduce the dimension of the network. In this study, we construct a large-scale spiking neural network with quadratic integrate-and-fire neurons and reduce it to a mean-field model to research the network dynamics. We find that the activity of the mean-field model is consistent with the network activity. Based on this agreement, a two-parameter bifurcation analysis is performed on the mean-field model to understand the network dynamics. The bifurcation scenario indicates that the network model has the quiescence state, the steady state with a relatively high firing rate, and the synchronization state which correspond to the stable node, stable focus, and stable limit cycle of the system, respectively. There exist several stable limit cycles with different periods, so we can observe the synchronization states with different periods. Additionally, the model shows bistability in some regions of the bifurcation diagram which suggests that two different activities coexist in the network. The mechanisms that how these states switch are also indicated by the bifurcation curves.

Multiple Firing Patterns in Coupled Hindmarsh-Rose Neurons with a Nonsmooth Memristor

A model is introduced by coupling two three-dimensional Hindmarsh-Rose models with the help of a nonsmooth memristor. The firing patterns dependent on the external forcing current are explored, which undergo a process from adding-period to chaos. The stability of equilibrium points of the considered model is investigated via qualitative analysis, from which it can be gained that the model has diversity in the number and stability of equilibrium points for different coupling coefficients. The coexistence of multiple firing patterns relative to initial values is revealed, which means that the referred model can appear various firing patterns with the change of the initial value. Multiple firing patterns of the addressed neuron model induced by different scales are uncovered, which suggests that the discussed model has a multiscale effect for the nonzero initial value.

Numerical Bifurcation Analysis of Pacemaker Dynamics in a Model of Smooth Muscle Cells

Evidence from experimental studies shows that oscillations due to electro-mechanical coupling can be generated spontaneously in smooth muscle cells. Such cellular dynamics are known as pacemaker dynamics. In this article, we address pacemaker dynamics associated with the interaction of [Formula: see text] and [Formula: see text] fluxes in the cell membrane of a smooth muscle cell. First we reduce a pacemaker model to a two-dimensional system equivalent to the reduced Morris-Lecar model and then perform a detailed numerical bifurcation analysis of the reduced model. Existing bifurcation analyses of the Morris-Lecar model concentrate on external applied current, whereas we focus on parameters that model the response of the cell to changes in transmural pressure. We reveal a transition between Type I and Type II excitabilities with no external current required. We also compute a two-parameter bifurcation diagram and show how the transition is explained by the bifurcation structure.

Complex bifurcation analysis and synchronization optimal control for Hindmarsh-Rose neuron model under magnetic flow effect

In this contribution, the complex behaviour of the Hindmarsh-Rose neuron model under magnetic flow effect (mHR) is investigated in terms of bifurcation diagrams, Lyapunov exponent plots and time series when varying only the electromagnetic induction strength. Some exciting phenomena are found including, for instance, various firings patterns by applying appropriate magnetic strength and Hopf-fold bursting through fast-slow bifurcation. In addition to this, the interesting phenomenon of Hopf bifurcation is examined in the model. Thus, we prove that Hopf bifurcation occurs in this memristor-based HR neuron model when an appropriately chosen magnetic flux varies and reaches its critical value. Furthermore, one of the main results of this work was the optimal control approach to realize the synchronization of two mHR. The main advantage of the proposed optimal master-slave synchronization from a control point of view is that, in the practical application, the electrical activities (quiescent, bursting, spiking, period and chaos states) of a neuron can be regulated by a pacemaker (master) associated with biological neuron (slave) to treat some diseases such as epilepsy. A suitable electronic circuit is designed and used for the investigations. PSpice based simulation results confirm that the electrical activities and synchronization between coupled neurons can be modulated by electromagnetic flux.

Various firing activities and finite-time synchronization of an improved Hindmarsh-Rose neuron model under electric field effect

Nowadays, it is important to realize systems that can model the electrical activity of neurons taking into account almost all the properties of the intracellular and extracellular environment in which they are located. It is in this sense that we propose in this paper, the improved model of Hindmarsh-Rose (HR) which takes into account the fluctuation of the membrane potential created by the variation of the ion concentration in the cell. Considering the effect of the electric field that is produced on the dynamic behavior of neurons, the essential properties of the model such as equilibrium point and its stability, bifurcation diagrams, Lyapunov spectrum, frequency spectra, time series of the membrane potential and phase portraits are thoroughly investigated. We thus prove that Hopf bifurcation occurs in this system when the parameters are chosen appropriately. We also observe that by varying specific parameters of the electric field, the model presents a very rich and striking event, namely hysteresis phenomenon, which justifies the coexistence of multiple attractors. Besides, by applying a suitable sinusoidal excitation current, we prove that the neuron under electric field effect can present several important electrical activities including quiescent, spiking, bursting and even chaos. We propose the improved HR model under electric field effect (mHR) to study the finite-time synchronization between two neurons when performing synapse coupling across the membrane potential and the electric field coupling. As a result, we find that the synchronization between the two neurons is weakly influenced by the variation of the intensity of the electric field coupling while it is strongly impacted when the intensity of the synapse coupling is modified. From these results, it is obvious that the electric field can be another effective bridge connection to encourage the exchange and coding of the signal. Using the finite-time synchronization algorithm, we theoretically quantify the synchronization time between these neurons. Finally, Pspice simulations are presented to show the feasibility of the proposed model as well as that of the developed synchronization strategy.

Energy dependence on discharge mode of Izhikevich neuron driven by external stimulus under electromagnetic induction

Energy supply plays a key role in metabolism and signal transmission of biological individuals, neurons in a complex electromagnetic environment must be accompanied by the absorption and release of energy. In this paper, the discharge mode and the Hamiltonian energy are investigated within the Izhikevich neuronal model driven by external signals in the presence of electromagnetic induction. It is found that multiple electrical activity modes can be observed by changing external stimulus, and the Hamiltonian energy is more dependent on the discharge mode. In particular, there is a distinct shift and transition in the Hamiltonian energy when the discharge mode is switched quickly. Furthermore, the amplitude of periodic stimulus signal has a greater effect on the neuronal energy compared to the angular frequency, and the average Hamiltonian energy decreases when the discharge rhythm becomes higher. Based on the principle of energy minimization, the system should choose the minimum Hamiltonian energy when maintaining various trigger states to reduce the metabolic energy of signal processing in biological systems. Therefore, our results give the possible clues for predicting and selecting appropriate parameters, and help to understand the sudden and paroxysmal mechanisms of epilepsy symptoms.

Ionic channel blockage in stochastic Hodgkin-Huxley neuronal model driven by multiple oscillatory signals

Ionic channel blockage and multiple oscillatory signals play an important role in the dynamical response of pulse sequences. The effects of ionic channel blockage and ionic channel noise on the discharge behaviors are studied in Hodgkin-Huxley neuronal model with multiple oscillatory signals. It is found that bifurcation points of spontaneous discharge are altered through tuning the amplitude of multiple oscillatory signals, and the discharge cycle is changed by increasing the frequency of multiple oscillatory signals. The effects of ionic channel blockage on neural discharge behaviors indicate that the neural excitability can be suppressed by the sodium channel blockage, however, the neural excitability can be reversed by the potassium channel blockage. There is an optimal blockage ratio of potassium channel at which the electrical activity is the most regular, while the order of neural spike is disrupted by the sodium channel blockage. In addition, the frequency of spike discharge is accelerated by increasing the ionic channel noise, the firing of neuron becomes more stable if the ionic channel noise is appropriately reduced. Our results might provide new insights into the effects of ionic channel blockages, multiple oscillatory signals, and ionic channel noises on neural discharge behaviors.

Multiple bifurcations and coexistence in an inertial two-neuron system with multiple delays

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.

Coexistence of firing patterns and its control in two neurons coupled through an asymmetric electrical synapse

In this paper, the effects of asymmetry in an electrical synaptic connection between two neuronal oscillators with a small discrepancy are studied in a 2D Hindmarsh-Rose model. We have found that the introduced model possesses a unique unstable equilibrium point. We equally demonstrate that the asymmetric electrical couplings as well as external stimulus induce the coexistence of bifurcations and multiple firing patterns in the coupled neural oscillators. The coexistence of at least two firing patterns including chaotic and periodic ones for some discrete values of coupling strengths and external stimulus is demonstrated using time series, phase portraits, bifurcation diagrams, maximum Lyapunov exponent graphs, and basins of attraction. The PSpice results with an analog electronic circuit are in good agreement with the results of theoretical analyses. Of most/particular interest, multistability observed in the coupled neuronal model is further controlled based on the linear augmentation scheme. Numerical results show the effectiveness of the control strategy through annihilation of the periodic coexisting firing pattern. For higher values of the coupling strength, only a chaotic firing pattern survives. To the best of the authors' knowledge, the results of this work represent the first report on the phenomenon of coexistence of multiple firing patterns and its control ever present in a 2D Hindmarsh-Rose model connected to another one through an asymmetric electrical coupling and, thus, deserves dissemination.

Temperature effect on memristive ion channels

Neuron shows distinct dependence of electrical activities on membrane patch temperature, and the mode transition of electrical activity is induced by the patch temperature through modulating the opening and closing rates of ion channels. In this paper, inspired by the physical effect of memristor, the potassium and sodium ion channels embedded in the membrane patch are updated by using memristor-based voltage gate variables, and an external stimulus is applied to detect the variety of mode selection in electrical activities under different patch temperatures. It is found that each ion channel can be regarded as a physical memristor, and the shape of pinched hysteresis loop of memristor is dependent on both input voltage and patch temperature. The pinched hysteresis loops of two ion-channel memristors are dramatically enlarged by increasing patch temperature, and the hysteresis lobe areas are monotonously reduced with the increasing of excitation frequency if the frequency of external stimulus exceeds certain threshold. However, for the memristive potassium channel, the AREA1 corresponding to the threshold frequency is increased with the increasing of patch temperature. The amplitude of conductance for two ion-channel memristors depends on the variation of patch temperature. The results of this paper might provide insights to modulate the neural activities in appropriate temperature condition completely, and involvement of external stimulus enhance the effect of patch temperature.