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The pattern formation in heterogeneous excitable media is a common phenomenon for spatiotemporal systems. In this paper, a bi-layer neuronal network is studied in which the two layers are connected using electromagnetic field coupling with two types of coupling gradients (i.e., step-like and cone-like). It is observed that when the central intensity of a gradient fieid is small, cone-like has less destructive effect on the target wave of the first layer compared to step-like. To further study the influence of environmental factors on pattern formation, the central intensity and external stimulation in the gradient field are continuously increased. The results show that the larger central intensity of the gradient field destroys the the target wave of the first layer and may form a spiral wave, and a larger external stimulation is more effective for inducing spiral waves in the second layer. Finally, synchronization factors are used to predict the pattern and formation mechanism of the spiral wave, and it is found that spiral waves are more likely to occur at the second layer with smaller values and the opposite at the first layer. [ABSTRACT FROM AUTHOR]

Both time delay and coupling form are the most important factors in neural networks. The properties of firing rate oscillation and cluster synchronization induced by time delay are studied in random network of different coupling neurons. In previous work, the firing rate oscillation of cortical network was observed at the presence of three factors (time delay, weak sinusoidal signal, and noise). Here, we found that the firing rate oscillation can be induced only by the time delay, and the spike train can be propagated at a certain interval time, which is consistent with the value of delay time. Furthermore, the phenomenon of cluster synchronization occurs in random network, which may originates from network structure, and this connection between the neurons trigger spikes within a time-restricted window, resulting in cluster synchronization between corresponding neurons. These numerical results provide a potential theoretical basis for certain pathological brain rhythms associated with epileptic seizures. [ABSTRACT FROM AUTHOR]

The feed-forward neural networks are the basis and have been widely applied on modern deep learning models, wherein connection strength between neurons plays a critical role in weak signal propagation and neural synchronization. In this paper, a four-variable Hindmarsh–Rose (HR) neural model is presented by introducing an additive variable as magnetic flow which changes the membrane potential via a memristor. The improved HR neurons in the feed-forward multilayer (four and eight layers) networks are investigated. The effects of electromagnetic radiation, synaptic weight and noise intensity on the propagation of the subthreshold excitatory postsynaptic current (EPSC) signal and the neural synchronization are discussed. It is found that when the system is in a weak magnetic field, the subthreshold EPSC signal can be successfully transmitted to the post-layers. Moreover, the neural synchronization of each layer is affected by electromagnetic radiation in the network, and with the help of noise the constant input current will transmit to the post-layers in a stable periodic synchronous form. Our findings provide a possible mechanism for enhancing the subthreshold signal propagation and triggering the neural synchronization in the nervous system. [ABSTRACT FROM AUTHOR]

Abstract In this paper, based on a chain Hindmarsh–Rose (HR) neural network under the action of electromagnetic field, the effects of connection strength between adjacent neurons on the wave propagation are investigated by utilizing numerical simulations. When the connection strength is increased via the decreasing of distance from central neuron, it is found that the firing rates of neurons in chain HR neural network are increased, and the velocity of wave propagation also becomes fast with the increasing of connection strength maximum. The chemical autapse imposed on the central neuron has a great influence on the firing rates of neurons and the wave propagation with different autaptic intensities. The firing rates of neurons are high, and many neurons can stand the excited state by increasing the field coupling strength. However, when the connection strength is decreased via the decreasing of distance from central neuron, the influences of connection strength maximum on the wave propagation are very small. The synchronization factor of the chain HR neural network is investigated by changing the maximum of connection strength, the autaptic intensity, and the field coupling intensity, respectively. It is found that the larger the field coupling strength is, the better the synchronization of neurons in the chain neural network will be, and the firing rates of neurons are high. [ABSTRACT FROM AUTHOR]

Abstract The feed-forward neural network is an artificial neural network, which is used extensively in deep learning models, wherein synaptic weight and characteristic time play very important role in information moves. In this paper, based on a feed-forward multilayer (ten layers) Hindmarsh–Rose (HR) neural network, the effects of synaptic weight and characteristic time on the signal propagation are investigated under the cases of continuous and transient external stimulated current, respectively. In the presence of continuous external stimulated current triggering the discharge of neurons, it is found that a random input signal driven by Gaussian white noise can be transmitted from input layer to next layers, and the propagation of weak spike train is gradually disappeared in the following layers when the synaptic weight is small. However, by choosing the appropriate values of synaptic weight and characteristic time, the mean firing rate of neurons in output layer is increased and the synchronization of neural firing in the following layers can be triggered. In the presence of transient (a short period) stimulated current triggering the discharge of neurons on the input layer, the firing rate of neurons cannot be transmitted from the input layer to the following layers with a small synaptic weight. Moreover, with the increasing of the synaptic weight, the mean firing rates of neurons in the following layers are higher than that in input layer, and the neurons in the following layers can be excited. [ABSTRACT FROM AUTHOR]

In this paper, the condition of completely nearest-neighbor couplings is introduced into the coupled Hindmarsh-Rose neurons in two-dimensional arrays. It is found that the stable rotating spiral wave can be developed and the transition of spiral wave in the coupled Hindmarsh-Rose neurons are investigated. The factor of synchronization is defined to investigate the development and instability of the spiral wave. Furthermore, the external injected current, coupling coefficients and other decisive bifurcation parameter r and χ, are endowed with different values to study the transition of spiral wave by analyzing the factor of synchronization and the snapshots of the activator. It is found that the critical sudden change points in the curve for factor of synchronization often indicates sudden transition of spiral wave, the instability or death of the spiral wave. The snapshots are also plotted to confirm the results from the curve of the factor of synchronization. Finally, the noise-induced instability and chaotic logistic map-induced instability of spiral wave are investigated and discussed. [ABSTRACT FROM AUTHOR]

Ca2+ spiral wave is observed in a large number of cell types. Based on a Ca2+ model presented by Atri et al., the propagation and synchronization of the intracellular Ca2+ spiral waves are numerically studied and some interesting results are obtained. (i) The largest Lyapunov exponents versus bifurcation parameter is calculated to investigate the oscillation of Ca2+ and it is found that almost all of the largest Lyapunov exponents are negative except for some others that are very close to zero (with an order of magnitude about 10-4 to 10-5). (ii) The two controllable parameters - the concentration of inositol 1,4,5-trisphosphate (IP3) and the diffusion coefficient - play an important role in inducing and supporting the rotating spiral wave, and the critical thresholds for supporting Ca2+ spiral wave are identified by calculating the statistical factor of synchronization in two-dimensional space. (iii) The driving layer (layer-1) can activate the Ca2+ oscillations in the driven layer (layer-2), and Ca2+ spiral wave in the driven layer can also be developed to synchronize the Ca2+ spiral wave in the drive with appropriate coupling intensity no matter whether mono-directional or mutual coupling of Ca2+ is being used. (iv) An optimal synchronization is observed for an intermediate amount of noise in the presence of Gaussian white noise, and for this noise amount, the Ca2+ waves in the two layers reach complete synchronization most quickly, it is a resonance-like behavior due to noise. [ABSTRACT FROM AUTHOR]

Breakup of spiral wave in the Hindmarsh-Rose neurons with nearest-neighbour couplings is reported. Appropriate initial values and parameter regions are selected to develop a stable spiral wave and then the Gaussian coloured noise with different intensities and correlation times is imposed on all neurons to study the breakup of spiral wave, respectively. Based on the mean field theory, the statistical factor of synchronization is defined to analyse the evolution of spiral wave. It is found that the stable rotating spiral wave encounters breakup with increasing intensity of Gaussian coloured noise or decreasing correlation time to certain threshold. [ABSTRACT FROM AUTHOR]

Abstract: In a multicellular system of rat hepatocytes and even in an intact liver, cytoplasmic calcium oscillations are synchronized and highly coordinated. In this paper, the mean-field coupling term has been introduced to describe the coupling flux, which is more efficient than gap junctional coupling terms. An optimal coupling strength and an optimal stimulation level for the synchronization of the coupled system have been observed in this paper. Moreover, it has been proved that these results are independent of the cells number. Interestingly, it has been observed that the intracellular noise and the extracellular noise have different effects on the synchronization of the coupled system. [Copyright &y& Elsevier]

Abstract: The frequency of free cytosolic calcium concentration ([Ca2+]) oscillations elicited by a given agonist concentration differs between individual hepatocytes. However, in multicellular systems of rat hepatocytes and even in the intact liver, [Ca2+] oscillations are synchronized and highly coordinated. In this paper, we have investigated theoretically the effects of gap junction permeable to calcium and of the total Ca2+ channel number located on endoplasmic reticulum on intercellular synchronization. Figures of ratio between mean oscillating frequency of coupled cells describe visually the process of phase-locking. By virtue of a set of phase analysis, we can observe a gradual transition from synchronous behavior to nonsynchronous behavior. Furthermore, a signal-to-noise ratio in two dimensional parameter space (coupling strength-total Ca2+ channel number) has suggested that, coherence resonance will occur for appropriate noise and coupling. [Copyright &y& Elsevier]

Abstract: The frequency of free cytosolic calcium concentration ([Ca2+]) oscillations elicited by a given agonist concentration differs between individual hepatocytes. However, in multicellular systems of rat hepatocytes and even in the intact liver, [Ca2+] oscillations are synchronized and highly coordinated. In this paper, we have investigated theoretically the gap junction permeable to calcium and to IP3 on intercellular synchronization by means of a mathematical model, respectively. It is shown that gap junction permeable to calcium and to IP3 are effective on synchronizing calcium oscillations in coupled hepatocytes. Our theoretical results are similar either for the case of Ca2+ acting as coordinating messenger or for the case of IP3 as coordinating messenger. There exists an optimal coupling strength for a pair of connected hepatocytes. Appropriate coupling strength and IP3 level can induce various harmonic locking of intercellular [Ca2+] oscillations. Furthermore, a phase diagram in two-dimensional parameter space of the coupling strength and IP3 level (or the velocity of IP3 synthesis) has been predicted, in which the synchronization region is similar to Arnol''d tongue. [Copyright &y& Elsevier]

Most external stimuli, including sound, temperature, and illumination, exhibit spatially heterogeneous, and different amplitudes of the same signal are received by neurons at different positions in the neural network. To address this issue, we constructed a grid-like neural network using memristive FitzHugh-Nagumo neurons. The neuronal responses depend on the spatially distributed stimuli, with the stimulus amplitudes being determined by the distance from the central area. Consequently, complete synchronization occurs in the network comprising periodic neurons, chaotic neurons, and their hybrid forms. Periodic patterns maintain the highest Hamilton energy whereas the lowest Hamilton energy appears in chaotic neurons. In a network consisting of chaotic neurons, the synchronization threshold is larger compared to the other types. In particular, the periodic neurons with the highest energy oscillations can regulate the low-energy chaotic neurons into periodic patterns. Similar conclusions are drawn in a chain-like network. The results advance the understanding of the synchronization mechanisms in the presence of spatial heterogeneity. • The effects of spatially heterogeneous stimuli on collective behaviors are investigated within the regular networks. • A network comprising both chaotic firing and chaotic firing neurons is found to synchronize easily. • Periodic firing neurons with higher energy can regulate the chaotic neurons with lower energy into periodic patterns. [ABSTRACT FROM AUTHOR]

• A scale-free neuronal network with small-world property is proposed to study the effects of spike-timing-dependent plasticity (STDP) and time delay. • Appropriate time delay and STDP maximum weights can induce network synchronization. • Characteristics of small-world, scale-free and small-world property scale-free networks are compared. Spike-timing-dependent plasticity (STDP) is one of the important rules for the change of synaptic weights between neurons in biological nervous systems. In this paper, we study the effect of STDP on the synchronization phenomenon induced by time delay in the neuronal network which is the scale-free network with small-world property, and nodes of the network are constructed by Izhikevich neuron and connected by chemical synapses. For appropriate time delay values, there exists an optimal range of STDP maximum weight value in which the synchronization of the network is better, and in addition the synchronization is decreased with the increasing of STDP maximum weight value. The network with high synchronization has a centralized distribution of synaptic weights within it, while conversely, an unsynchronized network has a more discrete distribution of synaptic weights. When the STDP maximum weight value is too small, the collective firing pattern of network is not affected by synaptic current, and the synchronization of the network is also not affected. Interestingly, comparing with the small-world network and the scale-free network, it is found that a network has a smaller range of optimal STDP maximum weight values when the network is of larger average clustering coefficient, shorter average shortest path length, and higher small-world property. Our results can illuminate the potential significance of STDP for information processing and transmission in the nervous system. [ABSTRACT FROM AUTHOR]

Noise and time delay are ubiquitous in various physical and biological systems. In this paper, we studied the phase synchronization and mode transition of oscillation mode induced by time delays and noises in a coupled FitzHugh–Nagumo (FHN) neural system. In the case of coupled neurons with single time delay, it is found that the mode oscillation of neurons induced by the noise and time delay undergoes a successive transitions from the synchronization, the out-of phase, the alternating phase-drift and anti-phase state, to the anti-phase state. In the case of coupled neurons with two time delays, when the two delays are comparable, the large amplitude oscillation of potential is in alternating synchronous and asynchronous state in the absence of noises, and the small amplitude oscillation of potential is always synchronized. The oscillation behaviors of coupled neurons with two time delays are similar to the cases with a single time delay in the presence of noises. When one of two time delays is largely dominant over the other, the neurons show completely different dynamic behaviors. • Synchronization and mode transition are studied in coupled FitzHugh–Nagumo system. • A successive mode transitions can be induced by single time delay. • The dynamic behaviors of system with multiple time delays are very different. [ABSTRACT FROM AUTHOR]