Rulkov Neuron Research Articles

Necessary Conditions for Complete Synchronization of a Coupled Chaotic Aihara Neuron Network with Electrical Synapses

In this paper, the synchronization problem of a chaotic Aihara neuron network is considered. Based on the master stability function (MSF) analysis, some necessary conditions are proposed to investigate the complete synchronization of the electrical coupled chaotic Aihara neuron networks. Especially, the two-dimensional parameter-space plots are obtained numerically to visualize the possible synchronized region. For the chaotic Rulkov neuron network, the numerical simulations of the MSF show that complete synchronization of the electrical coupled Aihara neuron network can be achieved when the coupling strength lies in a suitable interval, while it is highly improbable to attain complete synchronization when all neurons are in the chaotic bursting. Moreover, the complete synchronization cannot be reached with further increase or decrease of coupling strength. These results are probably common features as far as discrete-time neuron networks are concerned. Finally, based on the drive-response active control method, an LMI-based criterion is obtained to ensure complete synchronization of the available chaotic Aihara neuron network. The spatiotemporal patterns measured by the state distribution of neurons are employed to validate the effectiveness of our control scheme.

Fourier analysis of a delayed Rulkov neuron network

We have analyzed the synchronization of some different networks of chaotic Rulkov neurons with an electrical coupling that contains a delay. We have developed an algorithm to compute a certain delay whose result is to improve the synchronization of the network when it was slightly synchronized, or to get synchronized when it was desynchronized. Our general approach has been to use tools from signal analysis, such as Fourier and wavelet transforms. With these tools, we have characterized the behavior of the neurons for different parameters in frequency and time-frequency domains. The algorithm has been applied for two well-known network models: the small-world and Erdös-Rényi. We have also tested the algorithm by using non-homogeneous neurons affected with a parametric noise.

Bounded noise-induced coherence resonance in a single Rulkov neuron

Based on the chaotic Rulkov model, we investigate the impacts of bounded noise on firing activity of the neural system. First, the chaotic neural model and bounded noise are introduced and analyzed. Then, different firing patterns of the neural model driven by bounded noise are present. Finally, taking the coherence parameter R to measure the regularity of firing behavior, it is demonstrated that the regularity of neural membrane potential could achieve optimal by adjusting either the intensity of bounded noise or its frequency to an appropriate value, which means that the phenomenon of coherence resonance occurs. Moreover, it is found that coherence resonance can also be induced by another kind of bounded noise, called sine-Wiener noise. The results of this paper are beneficial to explore the functional role of noise on neural system.

Synchronization of Rulkov neuron networks coupled by excitatory and inhibitory chemical synapses.

This paper takes into account a neuron network model in which the excitatory and the inhibitory Rulkov neurons interact each other through excitatory and inhibitory chemical coupling, respectively. Firstly, for two or more identical or non-identical Rulkov neurons, the existence conditions of the synchronization manifold of the fixed points are investigated, which have received less attention over the past decades. Secondly, the master stability equation of the arbitrarily connected neuron network under the existence conditions of the synchronization manifold is discussed. Thirdly, taking three identical Rulkov neurons as an example, some new results are presented: (1) topological structures that can make the synchronization manifold exist are given, (2) the stability of synchronization when different parameters change is discussed, and (3) the roles of the control parameters, the ratio, as well as the size of the coupling strength and sigmoid function are analyzed. Finally, for the chemical coupling between two non-identical neurons, the transversal system is given and the effect of two coupling strengths on synchronization is analyzed.

Noise-induced bursting and chaos in the two-dimensional Rulkov model

We study an effect of random disturbances on the discrete two-dimensional Rulkov neuron model. We show that close to the Neimark–Sacker bifurcation, the increasing noise can cause the transition from the noisy quiescence with small-amplitude oscillations near the stable equilibria to the stochastic bursting with large-amplitude spikes. Mean values and variations of the interspike intervals are studied in dependence of the noise intensity. To study the noise-induced bursting, the analytical approach based on the stochastic sensitivity functions technique and confidence ellipses method is applied. On the basis of the largest Lyapunov exponents, we show how the noise-induced transition from the quiescence to stochastic bursting regime is accompanied by the transformation of dynamics from regular to chaotic.

Effect of Higher Order Terms on the Bifurcation Structure of Coupled Rulkov Neurons

Three kinds of bifurcations of two coupled Rulkov neurons with electrical synapses are investigated in this paper. The critical normal forms are derived based on the center manifold theorem and the normal form theory. For the flip and the Neimark–Sacker bifurcation, the quartic terms and above in the normal forms are defined as higher order terms, which originate from the Taylor expansion of the original system. Then the effects of the quartic and quintic terms on the flip and the Neimark–Sacker bifurcation structure are discussed, which verifies that the normal form is locally topologically equivalent to the original system for the infinitesimal 4-sphere of initial conditions and tiny perturbation on the bifurcation curve. By the flip-Neimark–Sacker bifurcation analysis, a novel firing pattern can be found which is that the orbit oscillates between two invariant cycles. Two disconnected cardioid cycles also appear, which makes one, two, three, four, etc. turns happen before closure. Finally, we present a global bifurcation structure in the parameter space and exhibit the distribution of the periodic, quasi-periodic and chaotic firing patterns of the coupled neuron model.

Synchronization Dynamics of Two Heterogeneous Chaotic Rulkov Neurons with Electrical Synapses

Two heterogeneous chaotic Rulkov neurons with electrical synapses are investigated in this paper. First, we study the ability of the second neuron to modify the dynamics of the first neuron. It is shown that when the parameters of the first neuron are located at the vicinity of the Neimark–Sacker bifurcation curves the first firing neuron can be controlled into the quiescent state when coupled with the second neuron. While the parameters of the first neuron are near the flip bifurcation curves the first firing neuron cannot be suppressed. Second, we discuss burst synchronization for two bursting neurons and two tonic spiking neurons. It is shown that two heterogeneous chaotic Rulkov neurons with tonic spiking firing cannot reach anti-phase synchronization under the inhibitory coupling, which is different from the property of nonchaotic Rulkov neurons. Finally, we show that for two bursting neurons if the coupling is strong enough then burst synchronization can be converted into spike synchronization. However, complete synchronization cannot be achieved for any strong coupling.

Synchronization of two identical and non-identical Rulkov models

In this paper, the synchronization of two chaotic Rulkov map-based neurons is taken into account. Firstly, based on the master stability function (MSF) analysis, the complete synchronization of two electrical coupled chaotic Rulkov neurons is investigated in detail. The two-dimensional parameter-space plot that displays directly the values of the MSF in different colors is numerically obtained. The numerical values of the MSF show that the two electrical coupled Rulkov neurons are likely to achieve the complete synchronization when each single neuron is in a silent state or a period-1 bursting state, while are unable to reach the complete synchronous state when each single neuron is in a chaotic bursting state or a spiking state. Secondly, Pearson’s correlation coefficient is employed to measure the synchronization degree, which demonstrates the nonexistence of the complete synchronization for non-identical electrical coupled Rulkov neurons. Importantly, the complete synchronization can not be reached with the increase of the electrical coupling strength, which is different from the continuous-time neuronal models. Finally, based on the active control method, a synchronization scheme is presented to study the complete synchronization for two Rulkov neurons no matter whether they are identical or not. The scheme is also applied to investigate the anticipated synchronization and the lag synchronization for any two Rulkov neurons. Numerical simulations verify the correctness of our analytical results and the effectiveness of our methods.

Complete synchronization of coupled Rulkov neuron networks

Based on the master stability function (MSF) analysis, the complete synchronization of coupled chaotic Rulkov neuron networks is investigated in detail. The two-dimensional parameter-space plots that display directly the values of the MSF in different colors are numerically obtained. For the electrical coupled Rulkov neuron network, the values of MSF are all positive when the single Rulkov neuron is in chaotic bursting state or spiking state, which means that complete synchronization of the electrical coupled Rulkov neuron network can not attain. Importantly, a specific inner linking function is found to make the values of the MSF negative, which means that the necessary condition of complete synchronization is satisfied. Through numerical simulations, the existence of complete synchronization is verified for Rulkov neuron network with the very specific inner linking function that has been employed. In addition, the number of nodes of networks and the explicit thresholds for the coupling strength that guarantees the necessary condition of complete synchronization are derived in three typical regular networks. More interestingly, the same route of spatiotemporal patterns transition is found for Rulkov neurons in three typical regular networks.

Period-adding bifurcation of neural firings induced by inhibitory autapses with time-delay

Neural firing rhythm plays an important role in achieving the function of a nervous system. Neurons with autapse, which starts and ends in the same cell, are widespread in the nervous system. Previous results of both experimental and theoretical studies have shown that autaptic connection plays a role in influencing dynamics of neural firing patterns and has a significant physiological function. In the present study, the dynamics of a neuronal model, i.e., Rulkov model with inhibitory autapse and time delay, is investigated, and compared with the dynamics of neurons without autapse. The bifurcations with respect to time-delay and the coupling strength are extensively studied, and the time series of membrane potentials is also calculated to confirm the bifurcation analysis. It can be found that with the increase of time-delay and/or the coupling strength, the period-adding bifurcation of neural firing patterns can be induced in the Rulkov neuron model. With the increase of the period number of the firing rhythm, the average firing frequency increases. When time-delay and/or coupling strength are/is greater than their/its corresponding certain thresholds/threshold, the average firing frequency is higher than that of the neuron without autapse. Furthermore, new bursting patterns, which appear at suitable time delays and coupling strengths, can be well interpreted with the dynamic responses of an isolated single neuron to a negative square current whose action time, duration, and strength are similar to those of the inhibitory coupling current modulated by the coupling strength and time delay. The bursts of neurons with autapse show the same pattern as the square negative current-induced burst of the isolated single neuron when the time delay corresponds to the phase. The bifurcation structure of the neural firing rhythm of the neuron without autapse can be obtained with the fast-slow dissection method. The dynamic responses of the isolated bursting neuron to the negative square current are acquired by using the fast-slow variable dissection method, which can help to recognize the new rhythms induced by the external negative pulse current applied at different phases. The new rhythm patterns are consistent with those lying in the period-adding bifurcations. The results not only reveal that the inhibitory autapse can induce typical nonlinear phenomena such as the period-adding bifurcations, but also provide the new phenomenon that the inhibitory autapse can enhance the firing frequency, which is different from previous viewpoint that inhibitory effect often reduces the firing frequency. These findings further enrich the understanding of the nonlinear phenomena induced by inhibitory autapse.

Eliminating synchronization in bistable networks

The problem of controlling synchrony in bistable networks, which possess coherent and incoherent attractors in a certain range of parameters, is considered. Along with the known Kuramoto-type models, we introduce the bistable networks consisting of all-to-all coupled noisy FitzHugh-Nagumo neurons as well as chaotic Rulkov neurons. We suggest two different algorithms to switch the bistable networks from the stable coherent state to the stable incoherent state. One of them is an act-and-wait control method, which utilizes the mean field measurements and homogeneous time delayed feedback perturbations with the periodically switched on and off feedback gain. We show that this algorithm is efficient for the globally coupled populations. Another algorithm is based on the multisite coordinated reset stimulation. The algorithm is nonfeedback, but it uses inhomogeneous perturbations and is efficient even for the networks with a complex scale-free topology. In addition to the numerical analysis of finite-size networks, the analytical results for the Kuramoto-type models in the thermodynamic limit are presented.

Electronic circuit implementation of the chaotic Rulkov neuron model

Numerical integration is the most common and straightforward approach in computational neuroscience for the study of biological neuron models based on ordinary differential equations. For some purposes, numerical simulations are not enough due to the multiple bottlenecks in computer architectures. However, when electronic circuits are used to simulate in real time large arrays of coupled neurons, the simulations are much faster than the computer simulations. We present here an electronic implementation of a map-based neuron model, a chaotic Rulkov neuron model, that can be easily transferred on a large scale integration circuit and thus provide a framework for the simulation of large networks of neurons. The Rulkov model is a map-based neuron model that has a surprising abundance of features, such as periodic and chaotic spiking and bursting. The discrete time dynamics allows to tune the time scale of the circuit to the needs of the specific application. Since the circuit described here only uses 18 MOS transistors, it offers new perspectives for building large networks of neurons in a single device. This is very relevant for the analysis of large networks of coupled neurons in order to investigate its dynamics over the network and its synchronization properties.