Coupled Model Neurons Research Articles

Cross-Channel Color Image Encryption Scheme Based on Discrete Memristive Coupled Neurons and DWT Compression

To address the consumption and security of color images for transmission and storage, a cross-channel color image encryption scheme based on a discrete memristive coupled neuron model and DWT compression is designed in this article. Firstly, the dynamics of the discrete memristive coupled neuron system are analyzed and found to possess the hyperchaotic phenomenon, which provides sufficient security for the encryption scheme. Secondly, the color image processed by discrete wavelet transform (DWT) has a quarter of the previous capacity. Then, the color image is combined with a Hash function, and the resulting Hash sequence is given the initial value of the hyperchaotic system. Next, a particle swarm foraging algorithm (PSFA) is designed to better disrupt the correlation in the RGB channel. Finally, a complementary DNA coding rule is implemented for the further encryption of color images. Simulation results show that even with DWT lossy compression, the recovered image can be clearly seen. The performance analysis illustrates that under the hyperchaotic system, the proposed encryption algorithm brings higher security for color images.

Dynamics and synchronization control of fractional conformable neuron system.

Dynamic analysis, electrical coupling and synchronization control of the conformable FitzHugh-Nagumo neuronal models have been presented in this work. Firstly, equations of the Adomian-Decomposition-Method and conformable neuron model have been introduced. The Adomian-Decomposition-Method has been employed for the numerical simulation analysis, since it converges fast and provides serial solutions. Fractional order and external current stimulus have been considered as bifurcation parameters and their effects on neuron model dynamics have been examined in detail. Then, the electrical coupling of the two conformable neuronal models without any controller has been revealed and the significance of the coupling strength parameter has been evaluated. To eliminate impact of the coupling strength parameter on synchronization status of neurons, Lyapunov control method has been employed for synchronization control. In the last step, the numerical simulation studies have been experimentally verified using the Texas Instrument Delfino digital signal processor board. Numerical simulation results together with experimental validation have showed that the types of dynamics of the related neuron model are not affected from the change of the fractional order of conformable derivative, but the frequency of the dynamic response of the neuronal model is changed from the alteration of the fractional order. The frequency of response of the neuron model increases with decreasing values of the fractional order. On the other hand, if there is no synchronization control method, the coupled neuron models exhibit response ranging from synchronous to asynchronous depending on the sign and value of the coupling parameter. Additionally, decreasing values of the fractional order may allow the coupled neurons to enter the synchronous state more quickly due to increasing frequency of response of the neuronal model. Finally, the coupled neuron models could exhibit synchronous behavior, that is determined by calculating the standard deviation results, regardless of the value of the coupling parameter by using the Lyapunov control method.

Effects of electric field on vibrational resonances in Hindmarsh–Rose neuronal systems for signal detection

Changes in the concentration of charged ions in neurons can generate induced electric fields, which can further modulate cell membrane potential. In this paper, Fourier coefficients are used to investigate the effect of electric field on vibrational resonance for signal detection in a single neuron model and a bidirectionally coupled neuron model, respectively. The study found that the internal electric field weakens vibrational resonance by changing two factors, membrane potential and phase-locked mode, while the periodic external electric field of an appropriate frequency significantly enhances the vibrational resonance, suggesting that the external electric field may play a constructive role in the detection of weak signals in the brain and neuronal systems. Furthermore, when the coupling of two neurons is considered, the effect of the electric field on the vibrational resonance is similar to that of a single neuron. The paper also illustrates the effect of electric field coupling on vibrational resonance. This study may provide a new theoretical basis for understanding information encoding and transmission in neurons.

Coexistence of infinitely many patterns and their control in heterogeneous coupled neurons through a multistable memristive synapse.

The phenomenon of hidden heterogeneous extreme multistability is rarely reported in coupled neurons. This phenomenon is investigated in this contribution using a model of a 2D FitzHugh-Nagumo neuron coupled with a 3D Hindmarsh-Rose neuron through a multistable memristive synapse. The investigation of the equilibria revealed that the coupled neuron model is equilibrium free and, thus, displays a hidden dynamics. Some traditional nonlinear analysis tools are used to demonstrate that the heterogeneous neuron system is able to exhibit the coexistence of an infinite number of electrical activities involving both periodic and chaotic patterns. Of particular interest, a noninvasive control method is applied to suppress all the periodic coexisting activities, while preserving only the desired chaotic one. Finally, an electronic circuit of the coupled neurons is designed in the PSpice environment and used to further support some results of the theoretical investigations.

Multistability of fractional-order memristor-coupled heterogeneous neurons and its hardware realization

There is heterogeneity among different neurons, and the activities of neurons are greatly different, so the coupling between heterogeneous neurons can show richer dynamic phenomena, which is of great significance in understanding the neural function of the human brain. Unfortunately, in many studies of memristive coupled neurons, researchers have considered two adjacent identical neurons, but ignored the heterogeneous neurons. In this paper, two models are chosen, i.e. a Hindmarsh-Rose neuron model and a Hopfield neuron model, which are very different from each other. The proposed fractional-order linear memristor and fractional-order hyperbolic memristor simulated neural synapses are introduced into the two heterogeneous neuron models, considering not only the coupling between the two neurons, but also the coupling between single neurons. The self-coupling of neurons, a five-dimensional fractional memristive coupled heterogeneous neuron model, is established. In the numerical simulation of the new neuron model, the phase diagrams, bifurcation diagrams, Lyapunov exponent diagrams, and attraction basins are used to demonstrate the changes in coupling strength and other parameters in the memristive coupled heterogeneous neuron model, the new neuron model performance coexistence of different attractors. On the other hand, by changing the initial state of the system while keeping the relevant parameters of the system unchanged, the multistable phenomenon of the coupled heterogeneous neuron model can be observed. Using the phase diagram, the coexistence of different periods, and the phenomenon of period and chaos can be clearly observed. The coexistence of different attractor states can also be observed in the attractor basin. This has many potential implications for studying dynamic memory and information processing in neurons. Uncovering different types of multistable states from a dynamical perspective can provide an insight into the role of multistable states in brain information processing and cognitive function. Finally, the neuron model is implemented based on the micro control unit of the advanced RISC machine, and the phase diagram is observed under some parameters of the coupled neuron model on an oscilloscope. The experimental results show the validity of the theoretical analysis.

Complex dynamics from heterogeneous coupling and electromagnetic effect on two neurons: Application in images encryption

This paper studies the effect of the electromagnetic flux on the dynamics of an introduced model of heterogeneous coupled neurons. Analytical investigation of the coupled neurons revealed that the obtained model is equilibrium free thus displays hidden firing activities. Based on the Helmholtz theorem, it is demonstrated that the coupled neurons possess a Hamilton energy, which enables to keep the electrical activity of the coupled neurons. Numerical simulations based on the fourth-order Runge-Kutta formula have enabled us to find a range of the electromagnetic induction strength where the model exhibits hysteretic dynamics. That hysteresis justifies the coexistence of two different firing activities for the same parameters captured. This latter behavior is further supported using bifurcation diagrams, the graph of the maximum Lyapunov exponent, phase portraits, time series, and attraction basins as arguments. Beside, the STM32F407ZE microcontroller development board is exploited for the digital implementation of the proposed model. The results of microcontroller implementation perfectly supported the results of the numerical simulation of bistability. Finally, a compressive sensing approach is used to compress and encrypt digital images based on the sequences of the above coupled-neurons model. The plain color image is decomposed into R, G, and B components. The DWT is applied to each component to obtain the corresponding sparse components. Confusion keys are obtained from the proposed coupled neurons to scramble each sparse component. The measurement matrixes obtained from the coupled neurons sequence are used to compress the confused sparse matrices corresponding to R, G, and B components. Each component is quantified, and a diffusion step is then applied to improve the randomness and consequently the information entropy.

Global and local reduced models for interacting, heterogeneous agents.

Large collections of coupled, heterogeneous agents can manifest complex dynamical behavior presenting difficulties for simulation and analysis. However, if the collective dynamics lie on a low-dimensional manifold, then the original agent-based model may be approximated with a simplified surrogate model on and near the low-dimensional space where the dynamics live. Analytically identifying such simplified models can be challenging or impossible, but here we present a data-driven coarse-graining methodology for discovering such reduced models. We consider two types of reduced models: globally based models that use global information and predict dynamics using information from the whole ensemble and locally based models that use local information, that is, information from just a subset of agents close (close in heterogeneity space, not physical space) to an agent, to predict the dynamics of an agent. For both approaches, we are able to learn laws governing the behavior of the reduced system on the low-dimensional manifold directly from time series of states from the agent-based system. These laws take the form of either a system of ordinary differential equations (ODEs), for the globally based approach, or a partial differential equation (PDE) in the locally based case. For each technique, we employ a specialized artificial neural network integrator that has been templated on an Euler time stepper (i.e., a ResNet) to learn the laws of the reduced model. As part of our methodology, we utilize the proper orthogonal decomposition (POD) to identify the low-dimensional space of the dynamics. Our globally based technique uses the resulting POD basis to define a set of coordinates for the agent states in this space and then seeks to learn the time evolution of these coordinates as a system of ODEs. For the locally based technique, we propose a methodology for learning a partial differential equation representation of the agents; the PDE law depends on the state variables and partial derivatives of the state variables with respect to model heterogeneities. We require that the state variables are smooth with respect to model heterogeneities, which permit us to cast the discrete agent-based problem as a continuous one in heterogeneity space. The agents in such a representation bear similarity to the discretization points used in typical finite element/volume methods. As an illustration of the efficacy of our techniques, we consider a simplified coupled neuron model for rhythmic oscillations in the pre-Bötzinger complex and demonstrate how our data-driven surrogate models are able to produce dynamics comparable to the dynamics of the full system. A nontrivial conclusion is that the dynamics can be equally well reproduced by an all-to-all coupled and by a locally coupled model of the same agents.

A new locally active memristive synapse-coupled neuron model

In this paper, a new type of non-volatile locally active memristor with bi-stability is proposed by injecting appropriate voltage pulses to realize a switching mechanism between two stable states. It is found that the memristive parameters of the new memristor can affect the local activity, which has been rarely reported, and this phenomenon is explained based on mathematical analyses and numerical simulations. Then, a locally active memristive coupled neuron model is constructed using the proposed memristor as a connecting synapse. The parameter-associated dynamical behaviors are revealed by bifurcation plots, phase plane portraits and dynamical evolution maps. Moreover, the bi-stability phenomenon of the new coupled neuron model is disclosed by local attraction basins, and the periodic burster and multi-scroll chaotic burster are found if a multi-level pulse current is used to imitate a periodical external stimulus on the neurons. The Hamiltonian energy function is calculated and analyzed with or without external excitation. Finally, the neuronal circuit is designed and implemented, which can mimic electrical activity of the neurons and is useful for physical applications. The experimental results captured from the analog circuit are consistent well with the numerical simulation results.

Transient dynamics and multistability in two electrically interacting FitzHugh-Nagumo neurons.

We analyze the existence of chaotic and regular dynamics, transient chaos phenomenon, and multistability in the parameter space of two electrically interacting FitzHugh-Nagumo (FHN) neurons. By using extensive numerical experiments to investigate the particular organization between periodic and chaotic domains in the parameter space, we obtained three important findings: (i) there are self-organized generic stable periodic structures along specific directions immersed in a chaotic portion of the parameter space; (ii) the existence of transient chaos phenomenon is responsible for long chaotic temporal evolution preceding the asymptotic (periodic) dynamics for particular parametric combinations in the parameter space; and (iii) the existence of various multistable domains in the parameter space with an arbitrary number of attractors. Additionally, we also prove through numerical simulations that chaos, transient chaos, and multistability prevail even for different coupling strengths between identical FHN neurons. It is possible to find multistable attractors in the phase and parameter spaces and to steer them apart by increasing the asymmetry in the coupling force between neurons. Such a strategy can be essential to experimental matters, as setting the right parameter ranges. As the FHN model shares the crucial properties presented by the more realistic Hodgkin-Huxley-like neurons, our results can be extended to high-dimensional coupled neuron models.

Suppressing spiral waves in a lattice array of coupled neurons using delayed asymmetric synapse coupling

Abstract The coupling between neuronal oscillator plays a significant role in their network performance. When the coupling is asymmetric in an electrical synapse connection, the entire dynamical behaviour of the neuron model changes. Such asymmetric synapse coupling on neuron models exposed to magnetic flux induction will display more complex behaviours. Hence, we propose a coupled neuron model considering flux coupling with asymmetric electrical synapse. The various dynamical properties of the coupled neuron model are explored by considering the coupling coefficients and flux coupling constant as the control parameters. The coexistence of at least chaotic with chaotic or chaotic with periodic or periodic with periodic firing patterns are identified in the new model for different values of coupling strengths. Such coexisting firing patters are also seen for discrete values of the flux coupling coefficient. The investigation is expanded further to the network performance of such coupled neurons considering the coupling coefficient as the parameter of discussion. A simple applied plane wave is disturbed and rotating spiral waves are formed in the network for different values of the coupling strength. We have shown that by introducing magnetic flux coupling into the neuron model, we could suppress the spiral waves in the network. As an alternative by considering a coupled neuron model without flux coupling and with delayed asymmetric electrical synapse coupling, we showed that the spiral waves are completely suppressed effectively for selected values of delay. Using delayed coupling comparatively more effective than flux coupling for supressing spiral waves in the network.

The effect of oxygen concentration on the coupled neurons: Rich spiking patterns and synchronization

The dynamical microenvironments play a crucial role in neuronal spiking patterns. In this paper, we investigated the effect of oxygen concentration on different synchronous spiking patterns of two coupled neuron models by including dynamical ion concentration. Two coupling modes of electrical diffusive coupling and potassium diffusive coupling were considered. In these two cases, oxygen concentration exhibited an important role in the synchronous spiking patterns between two coupled neurons, and extremely rich electrical activities were observed. For the potassium diffusive coupling, differential synchronous patterns of oscillation state (OS), synchronous epileptic seizure state (SSZ) and synchronous spreading depression state (SSD) as well as SZ and SD bursting states were generated. For the electrical diffusive coupling, differential synchronous patterns of resting state (RS), SSZ and SSD were observed.

Spatiotemporal synchronization in asymmetrically bidirectionally coupled neurons over a complex network

The spatiotemporal synchronous activity of delay coupled chaotic systems in a network is very important to understand the communications through neurons of the human body. In this paper, we present a homogeneous one-dimensional delay coupled neuron model in a complex network. We design a network model in which the links are stochastically updated at every time step. In this model, the nodes of the network are bidirectionally asymmetrically coupled instead of usual symmetric coupling. We further analyze the impacts of dynamic random updating of links and delay term on the synchronization phenomenon of the network. Synchronized fixed points are found in the networks of neurons numerically. Analytically, the stability range of synchronized fixed point and period-2 orbits are determined which are in very good agreement with the numerical simulation results. Moreover, numerical simulation results have been presented and are found to be in very good concurrence with our analytical result. Interestingly, we have found that in our model synchronized period-2 and period-4 dynamics appear for weak coupling which were not observed earlier. In addition, we have plotted the time series data and maximum Lyapunov exponent to confirm the existence of synchronized chaos.

Synchronous firing frequency dependence in unidirectional coupled neuronal networks with chemical synapses

Rhythmic brain waves caused by the synchronous firing activity of neurons are believed to be relevant for higher brain functions such as the attention and sleep-wake state switching. In the brain, cortical neurons, including interneurons, are anatomically and functionally diverse and can fire at different frequencies. However, it still remains unclear how such cortical networks that comprise a variety of neurons create synchronous activity. The present study examined entrainments of firing activity in cortical networks. To explore how cortical networks causes synchronous firing and to elucidate the mechanism for synchronous activity, we performed numerical simulations of synchronous firing behaviors observed in a unidirectional coupled neuron model, mimicking one of the minimum motifs in cortical networks. Furthermore, we observed bifurcations of periodic oscillations, which cause various phase locking states, in the coupled neuron model. We analyzed these bifurcations to investigate the effects of differences in membrane excitability, synaptic properties, and model structural complexity on the synchronous firing frequency. We found that the post-synaptic neuron could more readily attain a phase locking state at higher or lower frequencies than the intrinsic firing frequency of the pre-synaptic neuron for excitatory and inhibitory synaptic inputs, respectively. This finding suggests that synaptic coupling properties might determine the entrainable frequency range (phase-locked range) of the synchronous firing behavior. Thus, the specific rhythmic waves evoked by brain activity may be attributable to the synaptic coupling properties constituting local circuits.

Mixed mode oscillations and phase locking in coupled FitzHugh-Nagumo model neurons.

We study the dynamics of a low-dimensional system of coupled model neurons as a step towards understanding the vastly complex network of neurons in the brain. We analyze the bifurcation structure of a system of two model neurons with unidirectional coupling as a function of two physiologically relevant parameters: the external current input only to the first neuron and the strength of the coupling from the first to the second neuron. Leveraging a timescale separation, we prove necessary conditions for multiple timescale phenomena observed in the coupled system, including canard solutions and mixed mode oscillations. For a larger network of model neurons, we present a sufficient condition for phase locking when external inputs are heterogeneous. Finally, we generalize our results to directed trees of model neurons with heterogeneous inputs.

Pulse-coupled neural network and its optimization for segmentation of electrical faults with infrared thermography

This paper describes a method for segmenting electrical equipment fault with infrared thermography by using pulse-coupled neural networks. The pulse coupled neuron model used in PCNN is an optimization of the original neural model, in order to easily control and alter the behavior of neuron activity. In the receptive field, the image and its gradient information are regarded as the inputs of PCNN. The method for adjusting the value of linking coefficient is then derived from the principle of maximum likelihood estimate in the region of temporal pulse outputs, thus building the inner relationship between the parameters and image statistics. Besides, the edge constraint method, which is integrated into the pulse generator, is designed to alter the behavior of the neighboring neurons to be captured. It is shown that the addition of the constraint to the model increases the possibility of desired fault region segmentation. Finally, several experimental results, especially performed on the electrical equipment fault images, show that the proposed model has better performance than some existing PCNN-based models in the performance of image segmentation.

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.

A critical firing rate associated with tonic-to-bursting transitions in synchronized gap-junction coupled neurons

A transition between tonic and bursting neuronal behaviors is studied using a linear chain of three electrically coupled model neurons. Numerical simulations show that, depending on their individual dynamical states, the neurons first synchronize either in a tonic or in a bursting regime. Additionally, a characteristic firing rate, mediating tonic-to-bursting transitions in networked neurons, is found to be associated with a firing rate encountered in the single neuron’s equivalent transition. A few cases describing this peculiar phenomenon are presented.

Effect of Phase Response Curve Shape and Synaptic Driving Force on Synchronization of Coupled Neuronal Oscillators.

The role of the phase response curve (PRC) shape on the synchrony of synaptically coupled oscillating neurons is examined. If the PRC is independent of the phase, because of the synaptic form of the coupling, synchrony is found to be stable for both excitatory and inhibitory coupling at all rates, whereas the antisynchrony becomes stable at low rates. A faster synaptic rise helps extend the stability of antisynchrony to higher rates. If the PRC is not constant but has a profile like that of a leaky integrate-and-fire model, then, in contrast to the earlier reports that did not include the voltage effects, mutual excitation could lead to stable synchrony provided the synaptic reversal potential is below the voltage level the neuron would have reached in the absence of the interaction and threshold reset. This level is controlled by the applied current and the leakage parameters. Such synchrony is contingent on significant phase response (that would result, for example, by a sharp PRC jump) occurring during the synaptic rising phase. The rising phase, however, does not contribute significantly if it occurs before the voltage spike reaches its peak. Then a stable near-synchronous state can still exist between type 1 PRC neurons if the PRC shows a left skewness in its shape. These results are examined comprehensively using perfect integrate-and-fire, leaky integrate-and-fire, and skewed PRC shapes under the assumption of the weakly coupled oscillator theory applied to synaptically coupled neuron models.

Dynamic transition on the seizure-like neuronal activity by astrocytic calcium channel block

The involvement of astrocytes in neuronal firing dynamics is becoming increasingly evident. In this study, we used a classical hippocampal tripartite synapse model consisting of soma-dendrite coupled neuron models and a Hodgkin–Huxley-like astrocyte model, to investigate the seizure-like firing in the somatic neuron induced by the over-expressed neuronal N-methyl-d-aspartate (NMDA) receptors. Based on this model, we further investigated the effect of the astrocytic channel block on the neuronal firing through a bifurcation analysis. Results show that blocking inositol-1,4,5-triphosphate(IP3)-dependent calcium channel in astrocytes efficiently suppresses the astrocytic calcium oscillation, which in turn suppresses the seizure-like firing in the neuron.

Dynamic random links enhance diversity-induced coherence in strongly coupled neuronal systems

We investigate the influence of diversity on the temporal regularity of spiking in a ring of coupled model neurons. We find diversity-induced coherence in the spike events, with an optimal amount of parametric heterogeneity at the nodal level yielding the greatest regularity in the spike train. Further, we investigate the system under random spatial connections, where the links are both dynamic and quenched, and in all the cases we observe marked diversity-induced coherence. We quantitatively find the effect of coupling strength and random rewiring probability, on the optimal coherence that can be achieved under diversity. Our results indicate that the largest coherence in the spike events emerge when the coupling strength is high, and when the underlying connections are mostly random and dynamically changing.