Finite-time complete periodic synchronization of memristive neural networks with mixed delays

In this paper we study the oscillatory behavior of a new class of memristor based neural networks with mixed delays and we prove the existence and uniqueness of the periodic solution of the system based on the concept of Filippov solutions of the differential equation with discontinuous right-hand side. In addition, some assumptions are determined to guarantee the globally exponentially stability of the solution. Then, we study the adaptive finite-time complete periodic synchronization problem and by applying Lyapunov–Krasovskii functional approach, a new adaptive controller and adaptive update rule have been developed. A useful finite-time complete synchronization condition is established in terms of linear matrix inequalities. Finally, an illustrative simulation is given to substantiate the main results.

In this paper we study the oscillatory behavior of a new class of memristor based neural networks with mixed delays and we prove the existence and uniqueness of the periodic solution of the system based on the concept of Filippov solutions of the differential equation with discontinuous right-hand side. In addition, some assumptions are determined to guarantee the globally exponentially stability of the solution. Then, we study the adaptive finite-time complete periodic synchronization problem and by applying Lyapunov-Krasovskii functional approach, a new adaptive controller and adaptive update rule have been developed. A useful finite-time complete synchronization condition is established in terms of linear matrix inequalities. Finally, an illustrative simulation is given to substantiate the main results.
Recurrent neural network (RNN) is a deep learning model characterized by a topology that allows all connections between neurons, and by the fact each neuron usually has a complicated structure because of a large number of parallel connections with a diversity of axon lengths 1,2 . In addition, RNNs are well known for their capacity of classification, detecting regularities and their ability to learn. They can play the role of memory through feedback and are perfectly able to receive sensory data from our future agent 3,4 . In particular, continuous time RNNs (CTRNNs) are RNNs modeled by dynamical systems in the form of differential equation; they combine machine learning and physical modeling [5][6][7] . In fact, CTRNNs are mathematically easier to analyze, and continuous formulation offers more flexibility in adapting the system to the problem and adding constraints. Actually, researchers are attracted to the mathematical properties of RNNs, namely, the nature of solutions, stability and the oscillation properties 8,9 . Indeed, the dynamic properties of RNNs have been deeply discussed and several important works have been provided [10][11][12][13] . In particular, RNNs which exhibit periodic oscillation have been used to encode information in the vibration phase and to model many systems in many domains such as celestial mechanics, nonlinear vibration, electromagnetic theory, engineering, robotics [14][15][16][17][18][19] . In addition, the synchronization problem consists of analyzing the behavior between two systems: driver (or master) and responder (or slave) and could be seen in different real systems such as secure communication, information science, chaos generators and signal generators design, image processing, biological systems 20,21 . In fact, neuronal synchronization becomes one of the most attractive subjects in neuroscience, it indicates that the specific states of all the neurons in the neural networks converge to a common value [22][23][24][25] .
To make these oscillating neurons, researchers are interested in the memristor component that is a combination of memory and resistor 26,27 . Chua pointed out that the behaviour of memristor is somewhat similar to the synapses in the human brain 28 , and it can potentially offer both high connectivity and high density needed for efficient computing compared to other storage. A memristive neural networks (MNN) is described in Fig. 1. In addition, memristor studies show that MNN exhibits the feature of pinched hysteresis which means that a lag occurs between the application and the removal of a field and its subsequent effect, just as the neurons in the human brain have [29][30][31] . Some studies have been discussed to analyze the dynamic behaviour of MNN and a lot of researches were released [32][33][34][35] .
Hence, one can ask what is the impact of the delays (time-varying and distributed delay) for the stability and the synchronization of the periodic solution of MNNs. In Ref. 10 , authors investigate whether periodic solutions exist, are unique and stable for a large class of memristor-based neural networks with time-varying delays. Moreover, a novel and useful finite-time complete synchronization condition is obtained in terms of linear matrix inequalities to ensure the synchronization goal in Ref. 36 .
In this work, we extend these studies and the mathematical model of MNN with mixed delays. In fact, we analyse the stability of equilibrium points with executing significant results of the period behavior of the system. After that, we study the phenomena of synchronization from the point of view of the theory and control. In the considered system, the weights are discontinuous; the classical definition of the solution for differential equations cannot apply here. Therefore we shall propose the Filippov solution to handle this problem. Filippov developed a solution to the differential equation with a discontinuous right-hand side 37 . Based on this definition, a differential equation with a discontinuous right-hand side has the same solution set as a differential inclusion. Our contribution consists to investigate the existence and exponential stability of the periodic solutions for memristorbased neural networks with time-varying delays in the leakage term. The stability properties of this model are then analyzed and we show that the solutions of this new linear system converge to a periodic and stable limit cycle. The main novelty of our contribution lies in solving the problems of stability and synchronization and we demonstrate the impact of the mixed delays. Also results enhance and extend earlier studies on neural network dynamical systems with a continuous or discontinuous right-hand side that are memristor-based or conventional.
The rest of this paper is organized as follows. A delayed memristor-based neural networks is presented and some necessary definitions are given in "Model description and preliminaries" section. In "Uniqueness and global exponential stability" section, we introduce the Filippov's solution for our system and the existence of periodic solutions of the system. Our approaches are based on the differential inclusions and topological degree theories in set-valued analysis. In "Finite-time periodic synchronization" section, we shall study the uniqueness and global exponential stability of the w-periodic solution for the system. Especially, when the system is autonomous, we will give the sufficient conditions, uniqueness and global exponential stability of equilibrium point of the proposed system. Moreover, we designed novel control algorithms for the finite-time periodic synchronization to select neurons to pin the designed controller. In "Conclusions" section, a numerical example is obtained to show the effectiveness of the theoretical results given in the previous sections. It should be mentioned that the main results of this paper are Theorems 1-5.

Model description and preliminaries
In this paper, we shall investigate the following memristive neural networks with time-varying delay: where n ≥ 1 represents the number of neurons in the network, x i (t) correponds to the potential membrane of the neuron i at time t, the a i is a positive constant rate with which the i th neuron will reset its potential to the resting state in isolation when it is disconnected. In addition, f j , g j , h j and φ j denote the activation functions of (1)  www.nature.com/scientificreports/ jth neuron at time t, b ij (t), c ij (t), p ij (t) are the synaptic connection weight of the unit j on the unit i at time t, J i is the input unit i and τ ij (t) corresponds to the transmission delay of the ith unit along the axon of the jth unit at time t and is continuously differentiable function satisfying where τ = max 1≤i,j≤n max t∈[0,ω] , τ ij (t) , τ is a nonnegative constant, b ij (t) , c ij (t − τ ij (t)) and p ij (t) are memristive synaptic weights. Basing on memristor feature and the current-voltage characteristic, we write: . . , n are all constants.
Throughout this paper, we always assume the following hypothesis: , p ij (.), τ ij (.) and J i (.) are continuous and w-periodic functions.

Definition 2
We say that a square matrix is an M-matrix if it has all nonpositive elements outside the diagonal and all positive successive principal minors 38 . (1) There exist n positive constants α 1 , α 2 , . . . α n such that α i m ii + n j=1,j� =i α j m ji > 0, i = 1, . . . , n.
(2) There exist n positive constants β 1 , β 2 , . . . β n such that β i m ii + n j=1,j� =i β j m ij > 0, i = 1, . . . , n. where ⊂ R n is the set of points where V is not differentiable and N ⊂ R n is an arbitrary set with measure zero 41 . In the following, for a continuous ω-periodic function f(t) defined on R , we define
The initial states proposed for system (1) are as follow Consider x t ∈ C([−τ , 0], R n ) described by x t (s) = x(t + s), −τ ≤ s ≤ 0 , and the initial states (10) can be rewritten as Suppose that A ⊂ R n , then x → φ(x) is said a set-valued map from A to R n , if for every point x ∈ A , there exists a nonempty set φ(x) ⊂ R n . We call a set-valued map φ with nonempty values, an upper semicontinuous at

Existence of periodic solution
In the rest of this section we will investigate the existence of periodic solutions of the generalized memristor system.
By the differential equation system (1), we consider the set-valued maps as follow: for t ∈ R and i, j = 1, 2, . . . , n, ] are all closed, convex and compact for every t ∈ R and i, j = 1, . . . , n.
A function x(t) is said to be a solution of system (1) on [0, T) with initial condition (7) or (8), if x(t) is absolutely continuous on any compact interval of [0, T] and satisfies differential inclusions: In the following, we discuss dynamical behavior of system (1) using the following set-valued version of the Mawhin coincidence theorem. Lemma 2 (Mawhin coincidence theorem 42 ) Suppose that φ : R × R n → K ν (R n ) is upper semicontinuous and ω -periodic in t. If the following conditions hold: www.nature.com/scientificreports/ (1) There exists a bounded open set � ⊂ C ω , the set of all continuous, ω-periodic functions: R → R n , such that for any ∈ (0, 1) , each ω-periodic function x(t) of the inclusion If a matrix O ≥ 0 then all elements of O are greater than or equal to 0, and similarly we can describe O > 0 . It follows that for given vectors or matrices O and P, O ≥ P (or O > P ) and similarly that O − P ≥ 0 (or O − P > 0 ). After that, we give sufficient conditions to guarantee the existence of periodic solutions for the memristive neural network.
Then there exists at least one ω-periodic solution of system (1).
Step 2 we shall use contradiction to demonstrate the condition (2) in Lemma 2. Let us consider that there exists a solution u = (u 1 , u, . . . , u n ) T ∈ ∂� ∩ R n of the inclusion Then u is a constant vector on R n such that |u i | = ζ i for i ∈ (1, 2, .., n). Therefore, we have for Then, there exists some t * ∈ [0, ω) such that It follows Therefore (I − S)ζ * ≤ θ , which contradicts the fact (I − S)ζ * > θ and the condition 2 of Lemma 2 holds.
It follows that
The proof is finished.

Uniqueness and global exponential stability
Now, we will prove the uniqueness and global exponential stability of the ω-periodic solution for the system (1). Mainly, when the system (1) is considered autonomous, we will find the sufficient conditions on the existence, uniqueness and global exponential stability of fixed point of the system.
The function |y i (t)| is locally Lipschitz continuous in y i on R . Hence, the Clarke's generalized gradient of function |y i (t)| at y i (t) is  V (t + s), then we get (43)    www.nature.com/scientificreports/ Therefore for all i = 1, . . . , n . Thus, for any t > 0, Hence, the ω-periodic solution x * (t) of system (1) is globally exponentially stable. Then, the periodic solution x * (t) of system (1) is unique. The proof is complete.
Next, we demonstrate the existence and global exponential stability of the equilibrium point for autonomous neural network model (1). (H4) and (H5) for system (1).
Firstly, for autonomous system (1), using Theorems 1 and 3 we can get the following result.
Thus x * = x * (0) is an equilibrium point of system (1) and x * is unique and globally exponentially stable.
Using Lemma 3, for any two vectors u = (u 1 , . . . , u n ) T ∈ R n and v = (v 1 , . . . , v n ) T ∈ R n , we have where and 0 < σ < 1 . Thus, the map Ŵ : R n → R n is a contraction mapping on R n . It follows that, there is a unique fixed point u * ∈ R n such that u * ∈ Ŵ(u * ) , i.e., where i = 1, . . . , n , and u is unique, we obtain that system (1) has a unique equilibrium x * .
Thus, following the proof of Theorem 1, we prove easily that equilibrium x * of system (1) is globally exponentially stable.

Finite-time periodic synchronization
In this section, we will examine the finite-time synchronization problem of delayed memristive neural networks. For this purpose, we consider the delayed memristive neural network model (1) as the drive system, and a controlled response system is modeled by the following functional differential equation: where y i (t) is the controller to be designed.

Definition 7
The memristive neural network (1) is said to be completely synchronized onto (53) in finite time if by designing a suitable controller v i (t) to system (53), there exists a constant t 1 >0 ( t 1 depends on the initial value), satisfying We take e i (t) = x i (t) − y i (t) the error term. Then, one can obtain the following result.

Theorem 5
We consider that then system (1) exists at least one w-periodic solution. If there exists a positive definite matrix S satisfying  The master model (1) and the slave model (53) are state-dependent switching systems, hence, we can divide the error system into the following four cases at time t.
Case 1 If |x i (t)| > T i , |y i (t)| ≥ T i, at time t, then the master system (1) and the slave system (53) decrease respectively, to the following models: and Correspondingly, the error system can be written as L e t u s d e n ot e f j (e j (t)) = f j (x j (t)) − f j (y j (t)) ; g j (e j (t − τ )) = g j (x j (t − τ )) − g j (y j (t − τ )) a n d h j (e j (t)) = h j (x j (t)) − h j (y j (t)) . Under assumption (H2), evaluating the derivation of V(t) along the trajectory of error system gives (56) (58) Case 2 Let |x i (t)| > T i , |y i (t)| > T i at time t, then the master system (1) and the slave system (53) decrease to the following systems: and Hence, we obtain the following error system Similarly, we write According to Lemmas 1, it follows (59) Ph(e(t)) + u(t) (60) (62) www.nature.com/scientificreports/ . Case 3 If |x i (t)| > Ti , |y i (t)| ≤ T i at time t, then the master system (1) and the slave system (53) reduce to (60) and (61). Correspondingly, the error system can be written as evaluating the derivation of V(t) along the trajectory of (68), we have In consideration of the definition l i and Z 2 , one has V i (t) ≤ − √ 2kV 1 2 (t). Case 4. Let |x i (t)| ≤ T i , |y i (t)| > T i at time t, then the master system (1) and the slave system (53) reduce to (60) and (62). Then, we obtain the following error system: www.nature.com/scientificreports/ an M-matrix. Then theorem 4 holds and the system has a unique equilibrium point x * , which is globally exponentially stable. After simulation of these two systems using matlab Toobox,we obtain the graphical illustration Figs. 2 and 3 shows the periodic dynamic behaviors of the output of the two neurons which are in accordance with theoretical results.
To prove the effectiveness of our result on finite-time synchronization we consider the master system the above simulated example and the following system is the slave.
Let consider the following response RNN: y i (t) = −a i x i (t) + 2 j=1 b ij f j (y j (t)) + 2 j=1 c ij g j (y j (t − τ j )) +

Conclusions
In this paper, we study a memristive recurrent neural networs by giving assumptions for the existence and uniqueness of periodic solution. In addition, we detemine sufficient conditions that ensure the global exponential stability of this solution. Further more, we garantee the finite-synchronization problem of delayed memristive by determining several assymptions.
Meanwhile, the theoretical proposed model can be tested in practical issues like brain computing interface, image processing, pattern recognition and intelligent control. In our ongoing future works, the proposed neural  www.nature.com/scientificreports/ network model will be adjusted to analyze the electroencephalography (EEG) data for implementing continuous vigilance estimation using EEG signals acquired by wearable dry electrodes in both simulated and real driving environments. Also, MNN synchronization and EEG signals can be combined to study the brain dynamics at rest following a perturbation.  www.nature.com/scientificreports/

Data availability
The data that support the findings of this study are available from author Hajer Brahmi but restrictions apply to the availability of these data, which were used under license for the current study, and so are not publicly available. Data are however available from the authors upon reasonable request and with permission of the author Hajer Brahmi.