Undamped Oscillations Generated by Hopf Bifurcations in Fractional-Order Recurrent Neural Networks With Caputo Derivative

Tipping prediction of a class of large-scale radial-ring neural networks

Understanding the emergence and evolution of collective dynamics in large-scale neural networks remains a complex challenge. This paper seeks to address this gap by applying dynamical systems theory, with a particular focus on tipping mechanisms. First, we introduce a novel (n+mn)-scale radial-ring neural network and employ Coates' flow graph topological approach to derive the characteristic equation of the linearized network. Second, through deriving stability conditions and predicting the tipping point using an algebraic approach based on the integral element concept, we identify critical factors such as the synaptic transmission delay, the self-feedback coefficient, and the network topology. Finally, we validate the methodology's effectiveness in predicting the tipping point. The findings reveal that increased synaptic transmission delay can induce and amplify periodic oscillations. Additionally, the self-feedback coefficient and the network topology influence the onset of tipping points. Moreover, the selection of activation function impacts both the number of equilibrium solutions and the convergence speed of the neural network. Lastly, we demonstrate that the proposed large-scale radial-ring neural network exhibits stronger robustness compared to lower-scale networks with a single topology. The results provide a comprehensive depiction of the dynamics observed in large-scale neural networks under the influence of various factor combinations.

Stability and Bifurcation Exploration of Delayed Neural Networks With Radial-Ring Configuration and Bidirectional Coupling

For decades, studying the dynamic performances of artificial neural networks (ANNs) is widely considered to be a good way to gain a deeper insight into actual neural networks. However, most models of ANNs are focused on a finite number of neurons and a single topology. These studies are inconsistent with actual neural networks composed of thousands of neurons and sophisticated topologies. There is still a discrepancy between theory and practice. In this article, not only a novel construction of a class of delayed neural networks with radial-ring configuration and bidirectional coupling is proposed, but also an effective analytical approach to dynamic performances of large-scale neural networks with a cluster of topologies is developed. First, Coates' flow diagram is applied to acquire the characteristic equation of the system, which contains multiple exponential terms. Second, by means of the idea of the holistic element, the sum of the neuron synapse transmission delays is regarded as the bifurcation argument to investigate the stability of the zero equilibrium point and the beingness of Hopf bifurcation. Finally, multiple sets of computerized simulations are utilized to confirm the conclusions. The simulation results expound that the increase in transmission delay may cause a leading impact on the generation of Hopf bifurcation. Meanwhile, the number and the self-feedback coefficient of neurons are also playing significant roles in the appearance of periodic oscillations.

Dynamical Modeling and Qualitative Analysis of a Delayed Model for CD8 T Cells in Response to Viral Antigens

Although the immune effector CD8 T cells play a crucial role in clearance of viruses, the mechanisms underlying the dynamics of how CD8 T cells respond to viral infection remain largely unexplored. Here, we develop a delayed model that incorporates CD8 T cells and infected cells to investigate the functional role of CD8 T cells in persistent virus infection. Bifurcation analysis reveals that the model has four steady states that can finely divide the progressions of viral infection into four states, and endows the model with bistability that has ability to achieve the switch from one state to another. Furthermore, analytical and numerical methods find that the time delay resulting from incubation period of virus can induce a stable low-infection steady state to be oscillatory, coexisting with a stable high-infection steady state in phase space. In particular, a novel mechanism to achieve the switch between two stable steady states, time-delay-based switch, is proposed, where the initial conditions and other parameters of the model remain unchanged. Moreover, our model predicts that, for a certain range of initial antigen load: 1) under a longer incubation period, the lower the initial antigen load, the easier the virus infection will evolve into severe state; while the higher the initial antigen load, the easier it is for the virus infection to be effectively controlled and 2) only when the incubation period is small, the lower the initial antigen load, the easier it is to effectively control the infection progression. Our results are consistent with multiple experimental observations, which may facilitate the understanding of the dynamical and physiological mechanisms of CD8 T cells in response to viral infections.

Stability and Hopf bifurcation analysis of a fractional-order ring-hub structure neural network with delays under parameters delay feedback control

In this paper, a fractional-order two delays neural network with ring-hub structure is investigated. Firstly, the stability and the existence of Hopf bifurcation of proposed system are obtained by taking the sum of two delays as the bifurcation parameter. Furthermore, a parameters delay feedback controller is introduced to control successfully Hopf bifurcation. The novelty of this paper is that the characteristic equation corresponding to system has two time delays and the parameters depend on one of them. Selecting two time delays as the bifurcation parameters simultaneously, stability switching curves in $ (\tau_{1}, \tau_{2}) $ plane and crossing direction are obtained. Sufficient criteria for the stability and the existence of Hopf bifurcation of controlled system are given. Ultimately, numerical simulation shows that parameters delay feedback controller can effectively control Hopf bifurcation of system.

Boundary delineation in transrectal ultrasound images for region of interest of prostate

Accurate and robust prostate segmentation in transrectal ultrasound (TRUS) images is of great interest for ultrasound-guided brachytherapy for prostate cancer. However, the current practice of manual segmentation is difficult, time-consuming, and prone to errors. To overcome these challenges, we developed an accurate prostate segmentation framework (A-ProSeg) for TRUS images. The proposed segmentation method includes three innovation steps: (1) acquiring the sequence of vertices by using an improved polygonal segment-based method with a small number of radiologist-defined seed points as prior points; (2) establishing an optimal machine learning-based method by using the improved evolutionary neural network; and (3) obtaining smooth contours of the prostate region of interest using the optimized machine learning-based method. The proposed method was evaluated on 266 patients who underwent prostate cancer brachytherapy. The proposed method achieved a high performance against the ground truth with a Dice similarity coefficient of 96.2% ± 2.4%, a Jaccard similarity coefficient of 94.4% ± 3.3%, and an accuracy of 95.7% ± 2.7%; these values are all higher than those obtained using state-of-the-art methods. A sensitivity evaluation on different noise levels demonstrated that our method achieved high robustness against changes in image quality. Meanwhile, an ablation study was performed, and the significance of all the key components of the proposed method was demonstrated.

Intelligent contour extraction approach for accurate segmentation of medical ultrasound images

Introduction: Accurate contour extraction in ultrasound images is of great interest for image-guided organ interventions and disease diagnosis. Nevertheless, it remains a problematic issue owing to the missing or ambiguous outline between organs (i.e., prostate and kidney) and surrounding tissues, the appearance of shadow artifacts, and the large variability in the shape of organs. Methods: To address these issues, we devised a method that includes four stages. In the first stage, the data sequence is acquired using an improved adaptive selection principal curve method, in which a limited number of radiologist defined data points are adopted as the prior. The second stage then uses an enhanced quantum evolution network to help acquire the optimal neural network. The third stage involves increasing the precision of the experimental outcomes after training the neural network, while using the data sequence as the input. In the final stage, the contour is smoothed using an explicable mathematical formula explained by the model parameters of the neural network. Results: Our experiments showed that our approach outperformed other current methods, including hybrid and Transformer-based deep-learning methods, achieving an average Dice similarity coefficient, Jaccard similarity coefficient, and accuracy of 95.7 ± 2.4%, 94.6 ± 2.6%, and 95.3 ± 2.6%, respectively. Discussion: This work develops an intelligent contour extraction approach on ultrasound images. Our approach obtained more satisfactory outcome compared with recent state-of-the-art approaches . The knowledge of precise boundaries of the organ is significant for the conservation of risk structures. Our developed approach has the potential to enhance disease diagnosis and therapeutic outcomes.

A Robust and Explainable Structure-Based Algorithm for Detecting the Organ Boundary From Ultrasound Multi-Datasets

Detecting the organ boundary in an ultrasound image is challenging because of the poor contrast of ultrasound images and the existence of imaging artifacts. In this study, we developed a coarse-to-refinement architecture for multi-organ ultrasound segmentation. First, we integrated the principal curve-based projection stage into an improved neutrosophic mean shift-based algorithm to acquire the data sequence, for which we utilized a limited amount of prior seed point information as the approximate initialization. Second, a distribution-based evolution technique was designed to aid in the identification of a suitable learning network. Then, utilizing the data sequence as the input of the learning network, we achieved the optimal learning network after learning network training. Finally, a scaled exponential linear unit-based interpretable mathematical model of the organ boundary was expressed via the parameters of a fraction-based learning network. The experimental outcomes indicated that our algorithm 1) achieved more satisfactory segmentation outcomes than state-of-the-art algorithms, with a Dice score coefficient value of 96.68 ± 2.2%, a Jaccard index value of 95.65 ± 2.16%, and an accuracy of 96.54 ± 1.82% and 2) discovered missing or blurry areas.

Ultrasound Prostate Segmentation Using Adaptive Selection Principal Curve and Smooth Mathematical Model

Accurate prostate segmentation in ultrasound images is crucial for the clinical diagnosis of prostate cancer and for performing image-guided prostate surgery. However, it is challenging to accurately segment the prostate in ultrasound images due to their low signal-to-noise ratio, the low contrast between the prostate and neighboring tissues, and the diffuse or invisible boundaries of the prostate. In this paper, we develop a novel hybrid method for segmentation of the prostate in ultrasound images that generates accurate contours of the prostate from a range of datasets. Our method involves three key steps: (1) application of a principal curve-based method to obtain a data sequence comprising data coordinates and their corresponding projection index; (2) use of the projection index as training input for a fractional-order-based neural network that increases the accuracy of results; and (3) generation of a smooth mathematical map (expressed via the parameters of the neural network) that affords a smooth prostate boundary, which represents the output of the neural network (i.e., optimized vertices) and matches the ground truth contour. Experimental evaluation of our method and several other state-of-the-art segmentation methods on datasets of prostate ultrasound images generated at multiple institutions demonstrated that our method exhibited the best capability. Furthermore, our method is robust as it can be applied to segment prostate ultrasound images obtained at multiple institutions based on various evaluation metrics.

Dynamical Bifurcation for a Class of Large-Scale Fractional Delayed Neural Networks With Complex Ring-Hub Structure and Hybrid Coupling

Real neural networks are characterized by large-scale and complex topology. However, the current dynamical analysis is limited to low-dimensional models with simplified topology. Therefore, there is still a huge gap between neural network theory and its application. This article proposes a class of large-scale neural networks with a ring-hub structure, where a hub node is connected to n peripheral nodes and these peripheral nodes are linked by a ring. In particular, there exists a hybrid coupling mode in the network topology. The mathematical model of such systems is described by fractional-order delayed differential equations. The aim of this article is to investigate the local stability and Hopf bifurcation of this high-dimensional neural network. First, the Coates flow graph is employed to obtain the characteristic equation of the linearized high-dimensional neural network model, which is a transcendental equation including multiple exponential items. Then, the sufficient conditions ensuring the stability of equilibrium and the existence of Hopf bifurcation are achieved by taking time delay as a bifurcation parameter. Finally, some numerical examples are given to support the theoretical results. It is revealed that the increasing time delay can effectively induce the occurrence of periodic oscillation. Moreover, the fractional order, the self-feedback coefficient, and the number of neurons also have effects on the onset of Hopf bifurcation.

Pattern Formation in a Reaction-Diffusion BAM Neural Network With Time Delay: (k1, k2) Mode Hopf-Zero Bifurcation Case

This article investigates the joint effects of connection weight and time delay on pattern formation for a delayed reaction-diffusion BAM neural network (RDBAMNN) with Neumann boundary conditions by using the (k₁,k₂) mode Hopf-zero bifurcation. First, the conditions for k₁ mode zero bifurcation are obtained by choosing connection weight as the bifurcation parameter. It is found that the connection weight has a great impact on the properties of steady state. With connection weight increasing, the homogeneous steady state becomes inhomogeneous, which means that the connection weight can affect the spatial stability of steady state. Then, the specified conditions for the k₂ mode Hopf bifurcation and the (k₁,k₂) mode Hopf-zero bifurcation are established. By using the center manifold, the third-order normal form of the Hopf-zero bifurcation is obtained. Through the analysis of the normal form, the bifurcation diagrams on two parameters' planes (connection weight and time delay) are obtained, which contains six areas. Some interesting spatial patterns are found in these areas: a homogeneous periodic solution, a homogeneous steady state, two inhomogeneous steady state, and two inhomogeneous periodic solutions.

Bifurcations of a Fractional-Order Four-Neuron Recurrent Neural Network with Multiple Delays

This paper investigates the bifurcation issue of fractional-order four-neuron recurrent neural network with multiple delays. First, the stability and Hopf bifurcation of the system are studied by analyzing the associated characteristic equations. It is shown that the dynamics of delayed fractional-order neural networks not only depend heavily on the communication delay but also significantly affects the applications with different delays. Second, we numerically demonstrate the effect of the order on the Hopf bifurcation. Two numerical examples illustrate the validity of the theoretical results at the end.

Exponentially Stable Periodic Oscillation and Mittag-Leffler Stabilization for Fractional-Order Impulsive Control Neural Networks With Piecewise Caputo Derivatives

It is well known that the conventional fractional-order neural networks (FONNs) cannot generate nonconstant periodic oscillation. For this point, this article discusses a class of impulsive FONNs with piecewise Caputo derivatives (IPFONNs). By using the differential inclusion theory, the existence of the Filippov solutions for a discontinuous IPFONNs is investigated. Furthermore, some decision theorems are established for the existence and uniqueness of the (periodic) solution, global exponential stability, and impulsive control global stabilization to IPFONNs. This article achieves four key issues that were not solved in the previously existing literature: 1) the existence of at least one Filippov solution in a discontinuous IPFONN; 2) the existence and uniqueness of periodic oscillation in a nonautonomous IPFONN; 3) global exponential stability of IPFONNs; and 4) impulsive control global Mittag-Leffler stabilization for FONNs.

Large-Scale Neural Networks With Asymmetrical Three-Ring Structure: Stability, Nonlinear Oscillations, and Hopf Bifurcation

A large number of experiments have proved that the ring structure is a common phenomenon in neural networks. Nevertheless, a few works have been devoted to studying the neurodynamics of networks with only one ring. Little is known about the dynamics of neural networks with multiple rings. Consequently, the study of neural networks with multiring structure is of more practical significance. In this article, a class of high-dimensional neural networks with three rings and multiple delays is proposed. Such network has an asymmetric structure, which entails that each ring has a different number of neurons. Simultaneously, three rings share a common node. Selecting the time delay as the bifurcation parameter, the stability switches are ascertained and the sufficient condition of Hopf bifurcation is derived. It is further revealed that both the number of neurons in the ring and the total number of neurons have obvious influences on the stability and bifurcation of the neural network. Ultimately, some numerical simulations are given to illustrate our qualitative results and to underpin the discussion.

Artificial neural networks: a practical review of applications involving fractional calculus

In this work, a bibliographic analysis on artificial neural networks (ANNs) using fractional calculus (FC) theory has been developed to summarize the main features and applications of the ANNs. ANN is a mathematical modeling tool used in several sciences and engineering fields. FC has been mainly applied on ANNs with three different objectives, such as systems stabilization, systems synchronization, and parameters training, using optimization algorithms. FC and some control strategies have been satisfactorily employed to attain the synchronization and stabilization of ANNs. To show this fact, in this manuscript are summarized, the architecture of the systems, the control strategies, and the fractional derivatives used in each research work, also, the achieved goals are presented. Regarding the parameters training using optimization algorithms issue, in this manuscript, the systems types, the fractional derivatives involved, and the optimization algorithm employed to train the ANN parameters are also presented. In most of the works found in the literature where ANNs and FC are involved, the authors focused on controlling the systems using synchronization and stabilization. Furthermore, recent applications of ANNs with FC in several fields such as medicine, cryptographic, image processing, robotic are reviewed in detail in this manuscript. Works with applications, such as chaos analysis, functions approximation, heat transfer process, periodicity, and dissipativity, also were included. Almost to the end of the paper, several future research topics arising on ANNs involved with FC are recommended to the researchers community. From the bibliographic review, we concluded that the Caputo derivative is the most utilized derivative for solving problems with ANNs because its initial values take the same form as the differential equations of integer-order.

An efficient numerical scheme for the simulation of time-fractional nonhomogeneous Benjamin-Bona-Mahony-Burger model

The Benjamin-Bona-Mahony-Burger (BBM-Burger) equation is important for explaining the unidirectional propagation of long waves in nonlinear dispersion systems. This manuscript proposes an algorithm based on cubic B-spline basis functions to study the nonhomogeneous time fractional model of BBM-Burger via Caputo derivative. The discretization of fractional derivative is achieved by L1 formula, while the temporal and spatial derivatives are interpolated by means of Crank-Nicolson and forward finite difference scheme together with B-spline basis functions. The performance of the Cubic B-spline scheme (CBS) is examined by three test problems with homogeneous initial and boundary conditions. The obtained results are found to be in good agreement with the exact solutions. The behaviour of travelling wave is studied and presented in the form of tables and graphics for various values of α and t. A linear stability analysis, based on the von Neumann scheme, shows that the CBS is unconditionally stable. Moreover, the accuracy of the scheme is quantified by computing error norms.

Adaptive Synchronization of Fractional-Order Output-Coupling Neural Networks via Quantized Output Control

This article focuses on the adaptive synchronization for a class of fractional-order coupled neural networks (FCNNs) with output coupling. The model is new for output coupling item in the FCNNs that treat FCNNs with state coupling as its particular case. Novel adaptive output controllers with logarithm quantization are designed to cope with the stability of the fractional-order error systems for the first attempt, which is also an effective way to synchronize fractional-order complex networks. Based on fractional-order Lyapunov functionals and linear matrix inequalities (LMIs) method, sufficient conditions rather than algebraic conditions are built to realize the synchronization of FCNNs with output coupling. A numerical simulation is put forward to substantiate the applicability of our results.

Dynamics Analysis and Design for a Bidirectional Super-Ring-Shaped Neural Network With n Neurons and Multiple Delays

Recently, the dynamics of delayed neural networks has always incurred the widespread concern of scholars. However, they are mostly confined to some simplified neural networks, which are only made up of a small amount of neurons. The main cause is that it is difficult to decompose and analyze generally high-dimensional characteristic matrices. In this article, for the first time, we can solve the computing issues of high-dimensional eigenmatrix by employing the formula of Coates flow graph, and the dynamics is considered for a bidirectional neural network with super-ring structure and multiple delays. Under certain circumstances, the characteristic equation of the linearized network can be transformed into the equation with integration element. By analyzing the equation, we find that the self-feedback coefficient and the delays have significant effects on the stability and Hopf bifurcation of the network. Then, we achieve some sufficient conditions of the stability and Hopf bifurcation on the network. Furthermore, the obtained conclusions are applied to design a standardized high-dimensional network with bidirectional ring structure, and the scale of the standardized high-dimensional network can be easily extended or reduced. Afterward, we propose some designing schemes to expand and reduce the dimension of the standardized high-dimensional network. Finally, the results of theories are coincident with that of experiments.

Qualitative Analysis and Bifurcation in a Neuron System With Memristor Characteristics and Time Delay

This article focuses on the hybrid effects of memristor characteristics, time delay, and biochemical parameters on neural networks. First, we propose a novel neuron system with memristor and time delays in which the memristor is characterized by a smooth continuous cubic function. Second, the existence of equilibria of this type of neuron system is examined in the parameter space. Sufficient conditions that ensure the stability of equilibria and occurrence of pitchfork bifurcation are given for the memristor-based neuron system without delay. Third, some novel criteria of the addressed neuron system are constructed for guaranteeing the delay-dependent and delay-independent stability. The specific conditions are provided for Hopf bifurcations, and the properties of Hopf bifurcation are ascertained using the center manifold reduction and the normal form theory. Moreover, there exists a phenomenon of bistability for the delayed memristor-based neuron system having three equilibria. Finally, the effectiveness of the theoretical results is demonstrated by numerical examples.

Monostability and Multistability for Almost-Periodic Solutions of Fractional-Order Neural Networks With Unsaturating Piecewise Linear Activation Functions

Since the unsaturating activation function is unbounded, more complex dynamics may exist in neural networks with this kind of activation function. In this article, monostability and multistability results of almost-periodic solutions are developed for fractional-order neural networks with unsaturating piecewise linear activation functions. Some globally Mittag-Leffler attractive sets are given, and the existence of globally Mittag-Leffler stable almost-periodic solution is demonstrated by using Ascoli-Arzela theorem. In particular, some sufficient conditions are provided to ascertain the multistability of almost-periodic solutions based on locally positively invariant set. It shows that there exists an almost-periodic solution in each positively invariant set, and all trajectories converge to this periodic trajectory in that rectangular area. Two illustrative examples are provided to demonstrate the effectiveness of the proposed sufficient criteria.

Quasi-Synchronization and Bifurcation Results on Fractional-Order Quaternion-Valued Neural Networks

In this article, the quasi-synchronization and Hopf bifurcation issues are investigated for the fractional-order quaternion-valued neural networks (QVNNs) with time delay in the presence of parameter mismatches. On the basis of noncommutativity property of quaternion multiplication results, the quaternion network has been split as four real-valued networks. A synchronization theorem for fractional-order QVNNs is derived by employing suitable Lyapunov functional candidate; furthermore, the bifurcation behavior of the hub-structured fractional-order QVNNs with time delay has been investigated. Finally, two numerical examples are provided to demonstrate the effectiveness of the theoretical results.

Edge-Based Fractional-Order Adaptive Strategies for Synchronization of Fractional-Order Coupled Networks With Reaction-Diffusion Terms

In this paper, spatial diffusions are introduced to fractional-order coupled networks and the problem of synchronization is investigated for fractional-order coupled neural networks with reaction-diffusion terms. First, a new fractional-order inequality is established based on the Caputo partial fractional derivative. To realize asymptotical synchronization, two types of adaptive coupling weights are considered, namely: 1) coupling weights only related to time and 2) coupling weights dependent on both time and space. For each type of coupling weights, based on local information of the node's dynamics, an edge-based fractional-order adaptive law and an edge-based fractional-order pinning adaptive scheme are proposed. Furthermore, some new analytical tools, including the method of contradiction, L'Hopital rule, and Barbalat lemma are developed to establish adaptive synchronization criteria of the addressed networks. Finally, an example with numerical simulations is provided to illustrate the validity and effectiveness of the theoretical results.

Robust Finite-Time Stabilization of Fractional-Order Neural Networks With Discontinuous and Continuous Activation Functions Under Uncertainty

This paper is concerned with robust finite-time stabilization for a class of fractional-order neural networks (FNNs) with two types of activation functions (i.e., discontinuous and continuous activation function) under uncertainty. It is worth noting that there exist few results about FNNs with discontinuous activation functions, which is mainly because classical solutions and theories of differential equations cannot be applied in this case. Especially, there is no relevant finite-time stabilization research for such system, and this paper makes up for the gap. The existence of global solution under the framework of Filippov for such system is guaranteed by limiting discontinuous activation functions. According to set-valued analysis and Kakutani's fixed point theorem, we obtain the existence of equilibrium point. In particular, based on differential inclusion theory and fractional Lyapunov stability theory, several new sufficient conditions are given to ensure finite-time stabilization via a novel discontinuous controller, and the upper bound of the settling time for stabilization is estimated. In addition, we analyze the finite-time stabilization of FNNs with Lipschitz-continuous activation functions under uncertainty. The results of this paper improve corresponding ones of integer-order neural networks with discontinuous and continuous activation functions. Finally, three numerical examples are given to show the effectiveness of the theoretical results.

Impact of leakage delay on bifurcation in high-order fractional BAM neural networks

The effects of leakage delay on the dynamics of neural networks with integer-order have lately been received considerable attention. It has been confirmed that fractional neural networks more appropriately uncover the dynamical properties of neural networks, but the results of fractional neural networks with leakage delay are relatively few. This paper primarily concentrates on the issue of bifurcation for high-order fractional bidirectional associative memory(BAM) neural networks involving leakage delay. The first attempt is made to tackle the stability and bifurcation of high-order fractional BAM neural networks with time delay in leakage terms in this paper. The conditions for the appearance of bifurcation for the proposed systems with leakage delay are firstly established by adopting time delay as a bifurcation parameter. Then, the bifurcation criteria of such system without leakage delay are successfully acquired. Comparative analysis wondrously detects that the stability performance of the proposed high-order fractional neural networks is critically weakened by leakage delay, they cannot be overlooked. Numerical examples are ultimately exhibited to attest the efficiency of the theoretical results.

Dissipativity and stability analysis of fractional-order complex-valued neural networks with time delay

As we know, the notion of dissipativity is an important dynamical property of neural networks. Thus, the analysis of dissipativity of neural networks with time delay is becoming more and more important in the research field. In this paper, the authors establish a class of fractional-order complex-valued neural networks (FCVNNs) with time delay, and intensively study the problem of dissipativity, as well as global asymptotic stability of the considered FCVNNs with time delay. Based on the fractional Halanay inequality and suitable Lyapunov functions, some new sufficient conditions are obtained that guarantee the dissipativity of FCVNNs with time delay. Moreover, some sufficient conditions are derived in order to ensure the global asymptotic stability of the addressed FCVNNs with time delay. Finally, two numerical simulations are posed to ensure that the attention of our main results are valuable.