CN115936085A - Multi-scene HNN construction method and self-adaptive synchronization method - Google Patents
Multi-scene HNN construction method and self-adaptive synchronization method Download PDFInfo
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Abstract
The invention discloses a multi-scene HNN construction method and a self-adaptive synchronization method, wherein a non-polynomial memristor constructed by the multi-scene HNN construction method not only ensures the smoothness of the right side of a dynamic equation, but also keeps the method complexity of a constant level; based on a non-polynomial memristor, memristive HNNs of three scenes are constructed, and rich dynamic behaviors are found from the HNNs, wherein the rich dynamic behaviors comprise controllable unidirectional expansion multi-scroll attractors, controllable grid multi-scroll attractors, chaotic coexistence caused by initial offset and periodic coexistence caused by initial offset. According to the self-adaptive synchronization method, the self-adaptive state observer and the synchronous controller are constructed, the memristors HNNs of any two scenes are selected from the memristors HNNs of the three scenes to serve as the master system and the slave system, the master system and the slave system are synchronized by the synchronous controller based on the self-adaptive state observer, and synchronization between different areas of a brain can be well simulated.
Description
Technical Field
The invention relates to the technical field of neurodynamics, in particular to a multi-scene HNN construction method and a self-adaptive synchronization method.
Background
HNNs (Hopfield neural networks) were proposed based on an interpretation of the Ising model of associative memory. It has continuous dynamics and can help understand the memory of the human brain. In recent years, researchers introduce memristors into HNN in different ways, and by adjusting synaptic weights of the memristors, they find that complex dynamic behaviors such as multi-scroll attractors, extreme multistability and the like exist in a neural network.
In the prior art, researchers successfully obtained controllable multi-scroll memristor HNN by introducing a type of memristor containing sign polynomials. Although the sign function greatly simplifies the circuit design, its discrete mathematical nature also presents a series of problems. A memristor containing a sign polynomial implicitly contains a series of "bad" dots. At these "bad" points, the judgment of the existence and uniqueness of the differential equation solution is directly unavailable. Secondly, many adaptive synchronous control methods require the system to meet the Lipschitz condition. Furthermore, there are researchers using continuous basis functions instead of sign functions, which, while overcoming the problem of "bad" spots, undoubtedly increase the complexity of the circuit implementation.
Disclosure of Invention
The present invention is directed to solving at least one of the problems of the prior art. Therefore, the invention provides a multi-scene HNN construction method and a self-adaptive synchronization method, which can keep the method complexity of constant level and generate abundant dynamic behaviors.
In a first aspect, an embodiment of the present invention provides a multi-scenario HNN construction method, where the multi-scenario HNN construction method includes:
constructing a non-polynomial memristor as:
wherein a, b, c, d represent parameters of the non-polynomial memristor, i m Representing the output current of the non-polynomial memristor, W (x) representing the memristor, v m Representing the external input voltage, x representing the memristor internal state variable, dtThe derivation of the time t is shown, f (-) represents a memristor internal state function, and the expression of f (x) is as follows:
wherein f is odd The representation being used to induce an odd number of scrolls, f even Indicating that it is used to induce an even number of scrolls, odd-scrolls indicating an odd number of scrolls, even-scrolls indicating an even number of scrolls;
constructing memristors HNNs of three scenes based on the non-polynomial memristor; the memristive HNNs of the three scenes comprise memristive synaptic weights HNNs, HNNs under electromagnetic radiation and memristive synaptic weights HNNs under the action of the electromagnetic radiation.
Compared with the prior art, the first aspect of the invention has the following beneficial effects:
the non-polynomial memristor constructed by the method not only ensures the smoothness of the right side of the dynamic equation, but also keeps the method complexity of a constant level; constructing memristors HNNs of three scenes based on a non-polynomial memristor; the memristive HNNs in the three scenes comprise memristive synaptic weight HNNs, HNNs under electromagnetic radiation and memristive synaptic weight HNNs under the action of the electromagnetic radiation, the memristive synaptic weight HNNs and the HNNs under the electromagnetic radiation can successfully generate controllable multi-scroll attractors with similar shapes, the memristive synaptic weight HNNs under the action of the electromagnetic radiation can generate controllable grid multi-scroll attractors, and abundant dynamic behaviors including controllable one-way expansion multi-scroll attractors, controllable grid multi-scroll attractors, chaotic coexistence caused by initial offset and periodic coexistence caused by initial offset are found from the HNNs.
According to some embodiments of the invention, the memristive synaptic weight HNN is constructed as:
wherein a, b, c, d represent parameters of the memristive synaptic weight HNN, x 1 、x 2 、x 3 And x 4 Representing the membrane potentials of neurons 1,2, 3 and 4, respectively, I 1 、I 2 And I 3 Representing the external stimulus current, k representing the coefficient, tanh (-) representing the neuron activation function,
and &>
Respectively represent a pair x 1 、x 2 、x 3 And x 4 A differential of (c).
According to some embodiments of the invention, the HNNs under electromagnetic radiation are constructed as:
wherein a, b, c, d represent the parameters of HNN under the electromagnetic radiation, x 1 、x 2 、x 3 And x 4 Representing the membrane potentials of neurons 1,2, 3 and 4, respectively, I 1 、I 2 And I 3 Representing the external stimulus current, k representing the coefficient, tanh (-) representing the neuron activation function,
and &>
Respectively represent a pair x 1 、x 2 、x 3 And x 4 Differentiation of (2).
According to some embodiments of the invention, the memristive synaptic weights HNN under the effect of the electromagnetic radiation are constructed as:
wherein, a 1 ,b 1 ,c 1 ,d 1 ,a 2 ,b 2 ,c 2 ,d 2 A parameter, x, representing the memristive synaptic weight HNN under the effect of said electromagnetic radiation 1 、x 2 、x 3 、x 4 And x 5 Representing the membrane potentials of neurons 1,2, 3, 4 and 5, respectively, I 1 、I 2 And I 3 Represents the external stimulus current, k 1 And k 2 Representing the coefficients, tanh (-) representing the neuron activation function,
and &>
Respectively represent a pair x 1 、x 2 、x 3 、x 4 And x 5 Differentiation of (2).
In a second aspect, an embodiment of the present invention further provides a self-adaptive synchronization method, which applies any one of the above multi-scenario HNN construction methods, including:
constructing an adaptive state observer and a synchronous controller;
selecting memristors HNNs of any two scenes from the memristors HNNs of the three scenes as a master system and a slave system respectively;
synchronizing the master system and the slave system with the synchronization controller based on the adaptive state observer.
Compared with the prior art, the second aspect of the invention has the following beneficial effects:
since synchronization plays an important role in memory processing, synchronization between brain regions supports working memory and long-term memory by facilitating communication between neurons and promoting plasticity of neurons. Therefore, in order to simulate synchronization between different brain regions, the method constructs an adaptive state observer and a synchronous controller, selects memristors HNNs of any two scenes from the memristors HNNs of the three scenes as a master system and a slave system, adopts the synchronous controller to synchronize the master system and the slave system based on the adaptive state observer, can enable the master system and the slave system to have quick synchronous response and good synchronous effect through the constructed adaptive state observer and synchronous controller, and can well simulate synchronization between different brain regions.
According to some embodiments of the invention, the adaptive state observer is constructed as:
wherein the content of the first and second substances,
represents a system state variable, <' > based on>
Represents an estimate of the unknown non-linear portion, A represents the known linear portion, U represents the known external control input, L and P represent gain matrices, and->
Y represents the observed quantity, C represents the known linear portion, and γ represents a coefficient.
According to some embodiments of the invention, the estimate of the unknown non-linear part is fitted using an RBF neural network, the fitted expression for the unknown non-linear part being:
wherein the content of the first and second substances,
represents a weight matrix, <' > based on>
Representing a gaussian kernel function.
According to some embodiments of the invention, the host system is represented by:
wherein, X = (X) 1 ,x 2 ,...,x n ) T ，f i (X) represents a non-linear function,
and & ->
The differential is indicated.
According to some embodiments of the invention, the slave system is represented by:
wherein, Y = (Y) 1 ,y 2 ,...,y n ) T ，g i (Y) represents a non-linear function, d i (t) represents an external disturbance, i =1,2 i (t)|<D i ，u i Which represents an external control input, is,
and & ->
The differential is indicated.
According to some embodiments of the invention, said synchronizing said master system and said slave system with said synchronization controller comprises:
calculating an error function between the master system and the slave system as:
wherein, Ε = (epsilon) 1 ,ε 2 ,...,ε n ) T ；
If the slave system and the master system are synchronized, then
And the sliding mode surface epsilon of the error function i Expressed as:
wherein c is i Represents a coefficient, c i >0, τ represents an independent variable;
if s i If not =0, then
And->
According to
Setting a target of the synchronization controller to ^ based>
To synchronize the slave system with the master system.
Drawings
The above and/or additional aspects and advantages of the present invention will become apparent and readily appreciated from the following description of the embodiments, taken in conjunction with the accompanying drawings of which:
FIG. 1 is a flow chart of a multi-scenario HNN construction method according to an embodiment of the present invention;
FIG. 2 is a topological diagram of memristors HNNs of three scenarios in accordance with an embodiment of the present invention;
FIG. 3 is a phase diagram illustrating the generation of different scroll numbers in scenario one of the present invention;
FIG. 4 is a phase diagram illustrating the generation of different scroll numbers in scenario two according to an embodiment of the present invention;
FIG. 5 is a phase diagram illustrating the generation of different scroll numbers according to a third embodiment of the present invention;
FIG. 6 is a flow chart of an adaptive synchronization method according to an embodiment of the present invention;
FIG. 7 is a schematic diagram of an adaptive synchronization method according to an embodiment of the invention;
FIG. 8 is a diagram of experimental results of master-slave system synchronization according to an embodiment of the present invention;
FIG. 9 is a graph of the results of a histogram analysis according to an embodiment of the present invention;
FIG. 10 is a diagram illustrating a robustness test against data loss according to an embodiment of the present invention;
FIG. 11 is a diagram illustrating experimental results in scenario one of implementation of the solution of the present embodiment according to an embodiment of the present invention;
fig. 12 is a diagram of experimental results in a scenario two implemented by using the technical solution of the present embodiment according to an embodiment of the present invention;
fig. 13 is a diagram showing the result of an experiment for encrypting (decrypting) an image using the technical solution of the present embodiment according to an embodiment of the present invention.
Detailed Description
Reference will now be made in detail to embodiments of the present invention, examples of which are illustrated in the accompanying drawings, wherein like or similar reference numerals refer to the same or similar elements or elements having the same or similar function throughout. The embodiments described below with reference to the accompanying drawings are illustrative only for the purpose of explaining the present invention, and are not to be construed as limiting the present invention.
In the description of the present invention, if there are first, second, etc. described, it is only for the purpose of distinguishing technical features, and it is not understood that relative importance is indicated or implied or that the number of indicated technical features is implicitly indicated or that the precedence of the indicated technical features is implicitly indicated.
In the description of the present invention, it should be understood that the orientation or positional relationship referred to, for example, the upper, lower, etc., is indicated based on the orientation or positional relationship shown in the drawings, and is only for convenience of description and simplification of description, but does not indicate or imply that the device or element referred to must have a specific orientation, be constructed in a specific orientation, and be operated, and thus should not be construed as limiting the present invention.
In the description of the present invention, it should be noted that unless otherwise explicitly defined, terms such as setup, installation, connection, etc. should be understood in a broad sense, and those skilled in the art can reasonably determine the specific meanings of the above terms in the present invention by combining the detailed contents of the technical solutions.
In the prior art, researchers have successfully obtained controllable multi-scroll memristors HNN by introducing a class of memristors comprising sign polynomials. Although the sign function greatly simplifies the circuit design, its discrete mathematical nature also presents a series of problems. A memristor containing a sign polynomial undoubtedly contains a series of "bad" dots. At these "bad" points, the judgment of the existence and uniqueness of the differential equation solution is directly unavailable. Secondly, many adaptive synchronous control methods require the system to meet the Lipschitz condition, which overcomes the problem of "bad" spots, but undoubtedly increases the complexity of circuit implementation.
In order to solve the problems, the invention not only ensures the smoothness of the right side of the dynamic equation, but also keeps the method complexity of constant level by constructing the non-polynomial memristor; constructing memristors HNNs of three scenes based on a non-polynomial memristor; the memristive HNNs in the three scenes comprise memristive synaptic weight HNNs, HNNs under electromagnetic radiation and memristive synaptic weight HNNs under the action of the electromagnetic radiation, the memristive synaptic weight HNNs and the HNNs under the electromagnetic radiation can successfully generate controllable multi-scroll attractors with similar shapes, the memristive synaptic weight HNNs under the action of the electromagnetic radiation can generate controllable grid multi-scroll attractors, and abundant dynamic behaviors including controllable one-way expansion multi-scroll attractors, controllable grid multi-scroll attractors, chaotic coexistence caused by initial offset and periodic coexistence caused by initial offset are found from the HNNs.
Referring to fig. 1, an embodiment of the present invention provides a multi-scene HNN construction method, which is characterized in that the multi-scene HNN construction method includes, but is not limited to, steps S110 to S120:
step S110, constructing a non-polynomial memristor according to a memristor theory as follows:
wherein a, b, c, d represent parameters of a non-polynomial memristor, i m Represents the output current of a non-polynomial memristor, W (x) represents a memristor, v m Representing the external input voltage, x representing the memristor internal state variable, dt representing derivation of time t, f (-) representing the memristor internal state function, and f (x) having the expression:
wherein f is odd The representation being used to induce an odd number of scrolls, f even Indicating that it is used to induce an even number of scrolls, odd-scrolls indicating an odd number of scrolls, even-scrolls indicating an even number of scrolls. f. of odd And f even Is a piecewise function and has the following form:
wherein A, F and N are parameters, in particular N can be used to control the number of scrolls, F odd Is used to induce an odd number of scrolls, as opposed to f even Is used to induce an even number of scrolls. In the following paragraphs, A m And F m Will be set to 1 and 0.5 respectively and its value will not change.
Step S120, constructing memristors HNNs in three scenes based on the non-polynomial memristors; the memristive HNNs of the three scenes comprise memristive synaptic weights HNNs, HNNs under electromagnetic radiation and memristive synaptic weights HNNs under the action of the electromagnetic radiation.
Specifically, referring to fig. 2, in this embodiment, based on the 3D HNN, three different methods are used to perform memristive HNN modeling and study the dynamic characteristics thereof. The original 3D HNN may be represented as:
wherein x is 1 、x 2 And x 3 Representing the membrane potential of neurons 1,2 and 3, respectively. I is 1 、I 2 And I 3 Respectively, represent the external stimulus current. In the present embodiment, the external stimulus current is set to 0 uniformly.
Scene one: the graph a in fig. 2 is a schematic diagram of memristive synaptic weights HNN. It is well known that the memristor's memristor changes due to an external input voltage or current, which is very similar to the synapse of a biological neuron. Therefore, using memristors instead of connection weights between neurons would be more physiological. Moreover, experiments show that the method can bring more abundant dynamic behaviors. The memristive synaptic weight HNN (i.e., the new 4D memristive HNN) is constructed by memristing instead of the connection weight between neuron 1 and neuron 2:
wherein a, b, c and d represent parameters of memristive synaptic weights HNN, and x 1 、x 2 、x 3 And x 4 Respectively, membrane potentials of neurons 1,2, 3 and 4, I 1 、I 2 And I 3 Representing the external stimulus current, k representing the coefficient, tanh (-) representing the neuron activation function,
and &>
Respectively represent a pair x 1 、x 2 、x 3 And x 4 Differentiation of (2). />
Scene two, a diagram b in fig. 2 is a schematic diagram of HNN under electromagnetic radiation. When a neuron is exposed to a magnetic field, its membrane flux changes to generate an induced current. To simulate the effects of exposure of neuron 2 to a magnetic field, a memristive current is introduced. Thus obtaining the 4D memristor HNN under another electromagnetic radiation, and constructing the HNN under the electromagnetic radiation as follows:
wherein a, b, c and d represent the parameters of HNN under electromagnetic radiation, and x 1 、x 2 、x 3 And x 4 Representing the membrane potentials of neurons 1,2, 3 and 4, respectively, I 1 、I 2 And I 3 Representing the external stimulus current, k representing the coefficient, tanh (-) representing the neuron activation function,
and &>
Respectively represent a pair x 1 、x 2 、x 3 And x 4 Differentiation of (2).
In a third scenario, a diagram c in fig. 2 is a schematic diagram of memristive synaptic weights HNN under the action of electromagnetic radiation. Based on the former two scenarios, when memristors are used for replacing neuron connection weights and neurons are exposed to electromagnetic radiation, the memristor synapse weights HNN under the action of the electromagnetic radiation are constructed as follows:
wherein, a 1 ,b 1 ,c 1 ,d 1 ,a 2 ,b 2 ,c 2 ,d 2 A parameter representing the memristive synaptic weight HNN under the influence of electromagnetic radiation,x 1 、x 2 、x 3 、x 4 and x 5 Respectively, membrane potentials of neurons 1,2, 3, 4 and 5, I 1 、I 2 And I 3 Represents the external stimulus current, k 1 And k 2 Representing the coefficients, tanh (-) representing the neuron activation function,
and &>
Respectively represent a pair x 1 、x 2 、x 3 、x 4 And x 5 A differential of (c).
In the embodiment, by constructing the non-polynomial memristor, not only is the smoothness of the right side of the dynamic equation ensured, but also the method complexity of the constant level is maintained; constructing memristors HNNs of three scenes based on a non-polynomial memristor; the memristive HNNs in the three scenes comprise memristive synaptic weight HNNs, HNNs under electromagnetic radiation and memristive synaptic weight HNNs under the action of the electromagnetic radiation, the memristive synaptic weight HNNs and the HNNs under the electromagnetic radiation can successfully generate controllable multi-scroll attractors with similar shapes, the memristive synaptic weight HNNs under the action of the electromagnetic radiation can generate controllable grid multi-scroll attractors, and abundant dynamic behaviors including controllable one-way expansion multi-scroll attractors, controllable grid multi-scroll attractors, chaotic coexistence caused by initial offset and periodic coexistence caused by initial offset are found from the HNNs.
For better illustration, the following analyses and experiments were performed in this example:
1. analysis of equilibrium points
Balance point analysis is an effective means to study dynamic behavior. Making the left side of the above equation for the memristive synaptic weight HNN, HNN under electromagnetic radiation, and HNN under electromagnetic radiation 0 and performing Gaussian elimination to obtain the variable x 2 And x 3 The trajectory equation of (1). Specific forms thereof can be described by the following formula:
solving the memristive synapse weight HNN, HNN under electromagnetic radiation and x under the memristive synapse weight HNN under the action of electromagnetic radiation by using a graphical method and a numerical method 2 And x 3 And solving the numerical value of (2) to further obtain all roots. After linearization at the equilibrium point, a memristive synapse weight HNN, HNN under electromagnetic radiation and a jacobian matrix of a system under the memristive synapse weight HNN under the action of the electromagnetic radiation are obtained, and the results are shown below.
Wherein J 11 ＝1.3-2.3tanh 2 (x 1 ),J 12 ＝1.2tanh 2 (x 2 )-1.2,J 13 ＝0.2-0.2tanh 2 (x 3 ),J 14 ＝0,J 21 ＝2-2tanh 2 (x 1 ),J 22 ＝-1+k(a+bf(x 4 ))(1-tanh 2 (x 2 )),J 23 ＝1.1-1.1tanh 2 (x 3 ),J 24 ＝kbtanh(x 2 )f'(x 4 ),J 31 ＝5.5tanh 2 (x 1 )-5.5,J 32 ＝0,J 33 ＝1.1tanh 2 (x 3 )-2.1,J 34 ＝0,J 41 ＝0,J 42 ＝c-ctanh 2 (x 2 ),J 43 ＝0,J 44 ＝-df'(x 4 )。
Wherein J 11 ＝1.3-2.3tanh 2 (x 1 ),J 12 ＝1.2tanh 2 (x 2 )-1.2,J 13 ＝0.2-0.2tanh 2 (x 3 ),J 14 ＝0,J 21 ＝2-2tanh 2 (x 1 ),J 22 ＝-tanh 2 (x 2 )+k(a+bf(x 4 )),J 23 ＝1.1-1.1tanh 2 (x 3 ),J 24 ＝kbx 2 f'(x 4 ),J 31 ＝5.5tanh 2 (x 1 )-5.5,J 32 ＝0,J 33 ＝1.1tanh 2 (x 3 )-2.1,J 34 ＝0,J 41 ＝0,J 42 ＝c,J 43 ＝0,J 44 ＝-df'(x 4 )。
Wherein J 11 ＝1.3-2.3tanh 2 (x 1 ),J 12 ＝1.2tanh 2 (x 2 )-1.2,J 13 ＝0.2-0.2tanh 2 (x 3 ),J 14 ＝0,J 15 ＝0,J 21 ＝2-2tanh 2 (x 1 ),J 22 ＝-1+k 1 (a 1 +b 1 f(x 4 ))(1-tanh 2 (x 2 ))+k 2 (a 2 +b 2 f(x 5 )),J 23 ＝1.1-1.1tanh 2 (x 3 ),J 24 ＝k 1 b 1 tanh(x 2 )f'(x 4 ),J 25 ＝k 2 b 2 x 2 f'(x 5 ),J 31 ＝5.5tanh 2 (x 1 )-5.5,J 32 ＝0,J 33 ＝1.1tanh 2 (x 3 )-2.1,J 34 ＝0,J 35 ＝0,J 41 ＝0,J 42 ＝c 1 -c 1 tanh 2 (x 2 ),J 43 ＝0,J 44 ＝-d 1 f'(x 4 ),J 45 ＝0,J 51 ＝0,J 52 ＝c 2 ,J 53 ＝0,J 54 ＝0,J 55 ＝-d 2 f'(x 5 )。
And J is a Jacobian matrix of each system, and the eigenvalue corresponding to the balance point can be obtained by solving an eigen equation det (lambda E-J) =0.
2. And (5) carrying out experiments.
In this section, specific parameters are substituted in order to intuitively understand the dynamic behavior of the system. The function f (x) of the state variable inside the memristor is f odd On the premise that the parameter N is 1, the balance point and the corresponding characteristic values in the three scenes can be calculated by using the balance point analysis. According to the Shil 'nikov theorem in the documents "C.Silva," Shil' nikov's the item-a tubular, "IEEE Transactions on Circuits and Systems I: fundamental therapeutics and Applications, vol.40, no.10, pp.675-682,1993", if any, satisfies the Shil' nikov theorem
And δ α<A pure real eigenvalue δ of 0 and two complex conjugate eigenvalues α + β i, the system will exhibit chaos at the equilibrium point. According to the equilibrium point x 1 All equilibrium points are broadly classified herein into three types: an upper balance point, a middle balance point, and a lower balance point. For scenario one, the setting parameters a =0.58, b =0.02, c =2.7, d =1.1, and k =2.15. By calculation, the middle equilibrium point is (0, x) 41 ). The lower equilibrium point is (-0.5233, -0.3430,1.6236 42 ) And the upper equilibrium point is (0.5012, 0.3292, -1.5430 43 )。x 41 、x 42 And x 43 Is the root of f (x) =0, f (x) = -0.81, and f (x) = 0.78. Since N is 1, there are 3 × 5 equilibrium points. For scenario two, let parameters a =0.58, b =0.02, c =2.3, d =1.22, k =0.4. The middle equilibrium point is (0, x) 41 ). The lower equilibrium point is (-0.5157, -0.3383,1.5959 42 ) And the upper balance point is (0.5123, 0.3362, -1.5835, x 43 )。x 41 、x 42 And x 43 Is the root of f (x) =0, f (x) = -0.6145, and f (x) = 0.8893. Similar to the scene, there are 3 × 5 balance points at this time. Scene three has one more dimension than the previous scene. Let parameter a 1 ＝0.6、b 1 ＝0.01、c 1 ＝2.725、d 1 ＝1.8、k 1 ＝2.1、a 2 ＝0.5、b 2 ＝0.02、c 2 ＝1.6、d 2 ＝0.59、k 2 =0.1. The middle equilibrium point is (0, x) 41 ,x 51 ). The lower equilibrium point is (-0.4950, -0.3252,1.5205, x 42 ,x 52 ) And the upper balance point is (0.4876, 0.3205, -1.4935, x) 43 ,x 53 )。x 41 、x 42 And x 43 Is the root of f (x) =0, f (x) = -0.4757 and f (x) = 0.4692. x is the number of 51 、x 52 And x 53 Is the root of f (x) =0, f (x) = -0.8820, and f (x) = 0.8692. By calculating the eigenvalues and the Shil' nikov theorem, it was determined that in all three cases, all the middle equilibrium points were unstable Index-1 saddle points, and all the upper and lower equilibrium points were unstable Index-2 saddle points.
From the above analysis, it can be seen that the number of equilibrium points is proportional to N. According to the Shil' nikov theory, the size of the attractor increases as the equilibrium point increases. It can be easily found from the phase diagram of the system that when the balance point satisfies f' (x) 4 )>At 0, wrap will occur. In summary, when f (x) = f odd Then, for scene one and scene two, the system will generate 2N +1 double vortex volumes. And for scene three, the system will generate (2N + 1) × (2N + 1) double vortex volumes. In contrast, when f (x) = f even Then the system in scene one and scene two will generate 2n +2 double scrolls. And scene three will generate (2N + 2) × (2N + 2) double vortex volumes. Fig. 3 and 4 are phase diagrams when scene one and scene two generate 1,2, 3, 4, 5, and 6 double scrolls, respectively. Fig. 5 is a phase diagram when 1 × 1,2 × 2, 3 × 3, 4 × 4, 5 × 5, and 6 × 6 double scrolls are generated in a scene three, respectively. It can be seen that the number of volumes can be well controlled as long as N is adjusted. Wherein:
FIG. 3 is a phase diagram of the generation of different scroll numbers in scene one, where graphs (a), (b) and (c) in FIG. 3 correspond to f odd (ii) a In FIG. 3, the graphs (d), (e) and (f) correspond to f even Wherein, diagram (a) shows 1 double scroll, diagram (b) shows 3 double scrolls, diagram (c) shows 5 double scrolls, diagram (d) shows 2 double scrolls, diagram (e) shows 4 double scrolls, and diagram (f) shows 6 double scrolls;
FIG. 4 is a phase diagram of the generation of different scroll numbers in scenario two, FIG. 4Corresponding to (a), (b) and (c) in (f) odd (ii) a In FIG. 4, the graphs (d), (e) and (f) correspond to f even Wherein, diagram (a) shows 1 double scroll, diagram (b) shows 3 double scrolls, diagram (c) shows 5 double scrolls, diagram (d) shows 2 double scrolls, diagram (e) shows 4 double scrolls, and diagram (f) shows 6 double scrolls;
FIG. 5 is a phase diagram of the three scenarios with different scroll numbers, wherein the diagram (a), the diagram (b) and the diagram (c) in FIG. 5 correspond to f odd (ii) a In FIG. 5, the graphs (d), (e) and (f) correspond to f even In the drawings, fig. 1 × 1 double scrolls, fig. 3 × 3 double scrolls, fig. 5 × 5 double scrolls, fig. 2 × 2 double scrolls, fig. 4 × 4 double scrolls, and fig. 6 × 6 double scrolls are shown in fig. (a), fig. (b), and fig. 4 × 4 double scrolls.
Referring to fig. 6, an embodiment of the present invention further provides a self-adaptive synchronization method, where memristors HNN of three scenarios in the HNN construction method based on any one of the multiple scenarios include, but are not limited to, steps S210 to S230:
step S210, constructing a self-adaptive state observer and a synchronous controller;
step S220, selecting memristors HNNs of any two scenes from the memristors HNNs of the three scenes as a master system and a slave system respectively;
and step S230, synchronizing the master system and the slave system by adopting a synchronous controller based on the adaptive state observer.
In steps S210 to S230 of some embodiments, since synchronization plays an important role in memory processing, synchronization between brain regions supports working memory and long-term memory by promoting communication between neurons and promoting plasticity of neurons. Therefore, in order to simulate synchronization between different brain regions, the method constructs the adaptive state observer and the synchronous controller, selects memristors HNNs of any two scenes from memristors HNNs of three scenes as a master system and a slave system, adopts the synchronous controller to synchronize the master system and the slave system based on the adaptive state observer, enables the master system and the slave system to have fast synchronous response and good synchronous effect through the constructed adaptive state observer and synchronous controller, and can well simulate synchronization between different brain regions.
In some embodiments, the adaptive state observer is constructed as:
wherein the content of the first and second substances,
represents a system state variable, <' > or>
Represents an estimate of the unknown nonlinear portion, A represents the known linear portion, U represents the known external control input, L and P represent gain matrices, and->
Y represents an observed quantity, C represents a known linear portion, and γ represents a coefficient.
In some embodiments, the estimate of the unknown non-linear part is fitted using an RBF neural network, the fitted expression for the unknown non-linear part being:
wherein the content of the first and second substances,
represents a weight matrix, based on the weight of the reference signal>
Representing a gaussian kernel function. />
In some embodiments, the host system is represented by:
wherein, X = (X) 1 ,x 2 ,...,x n ) T ，f i (X) represents a non-linear function,
and & ->
The differential is indicated.
In some embodiments, the slave system is represented by:
wherein, Y = (Y) 1 ,y 2 ,...,y n ) T ，g i (Y) represents a non-linear function, d i (t) represents an external disturbance, i =1,2 i (t)|<D i ，u i Which represents an external control input, is,
and & ->
The differential is indicated.
In some embodiments, synchronizing the master system and the slave system with a synchronization controller includes:
calculating an error function between the master system and the slave system as:
Ε＝Y-X
wherein, Ε = (epsilon) 1 ,ε 2 ,...,ε n ) T ；
If the slave system and the master system are synchronized, then
And the sliding mode surface epsilon of the error function i Expressed as:
wherein c is i Represents a coefficient, c i >0, τ represents an independent variable;
if s i If not =0, then
And->
According to
Setting a target of a synchronization controller to ^ er>
To synchronize the slave system with the master system.
To facilitate understanding by those skilled in the art, the following sets of preferred embodiments are provided:
due to synaptic plasticity, the connection weights between neurons are not invariant. On the other hand, not all internal variables of the neural network can be directly observed by other neural networks, e.g., memristive synaptic weights in scenario one and electromagnetic radiation in scenario two are considered difficult to directly observe. In order to simulate the synchronization between different areas of the brain, the present embodiment designs an adaptive synchronization method.
1. And (4) designing an observer.
The general form of the system to be observed can be expressed as:
where X is the internal state, A is the known linear part, F (X) is the unknown nonlinear part, and U is the known external control input. It is to be noted that they are matrices. The present embodiment is based on the prior art, and the solving method of the gain matrix L is improved, and the corresponding observer can be represented by the following formula.
Wherein the content of the first and second substances,
and &>
Is an estimate of the system state variables and the unknown nonlinear part. Furthermore +>
Since the neural network can fit any non-linear function, the present embodiment uses the RBF neural network to fit F (X), that is
Wherein->
Is the weight matrix and->
Is a gaussian kernel function. Recording the optimal weight as W * . The final fit of the non-linear term is therefore ≥ based on the result of the final fit>
Observation error is recorded as->
And its derivative has the following form:
wherein
To prove->
And &>
Is uniform and finally bounded, the corresponding lyapunov function is constructed as follows:
then, the derivatives thereof are as follows.
According to the above formula, the RBF neural network on-line learning rule is set as
Wherein eta>0. Bringing it into the formula: />
Let R = M (C) T C+M) -1 [M(C T C+M) -1 ] T And bringing it into the above formula can be deduced from the following formula:
wherein the content of the first and second substances,
so in order to make->
And &>
Uniform final bounding, the following equation needs to be satisfiedConditions are as follows:
since the feasibility of letting Q negative is weak, it is not straightforward to have Q <0 solve. Assuming that the equation of the observed system is smooth, and due to the chaotic bounding nature, the Jacobian matrix of F (X) satisfies the following condition:
where the operator '≦' is less by element. The following can be concluded from the literature "Y.Wang, R.Rajamani, and D.M.Bevly," Observer design for parameter varying differential nonlinear systems, "IEEE Transactions on Automatic Control, vol.62, no.4, pp.1940-1945, 2017":
when the following more resolvable condition is satisfied,
the following equation can be established:
finally, the observer is proved to be uniform and finally bounded.
2. And (4) designing a synchronous controller.
The general form of the host system can be described by the following formula:
wherein, X = (X) 1 ,x 2 ,...,x n ) T And f i (X) is a non-linear function.
The general form of the slave system can be described by the following equation:
wherein Y = (Y) 1 ,y 2 ,...,y n ) T And g is i (Y) is a non-linear function. d i (t) is an external disturbance and let | d i (t)|<D i 。u i Is an external control input. The error function is defined by:
Ε＝Y-X
wherein, Ε = (epsilon) 1 ,ε 2 ,...,ε n ) T . In order to synchronize the slave system and the master system, the error between the master and slave systems must be 0. In other words
Slip form surface epsilon of error function i Can be expressed as:
wherein, c i >0. When s is i Is not less than 0
That means that>
The goal of the controller design now becomes to have £ greater>
Taking the Lyapunov function as:
its derivative with respect to the argument t is then:
because of g i (Y) is unknown, so the present embodiment also uses the RBF neural network to fit this function. After weight learning, the fitting result of the RBF neural network can be described by the following formula:
where W is the weight matrix of the RBF neural network and let W be * Is the most suitable weight. x is an external input to the neural network and is in the form of
Error still existing after fitting of delta neural network and leading | delta tint<B. G is a gaussian kernel function. Equation 52 can be transformed into:
zero controller u i Having the form:
wherein
Estimate weight matrix when learning for neural network and let η>0. Substituting this into equation 54 may derive the following:
when in use
And η>D + B, then->
This is true. At this time->
To make->
If the weight is established, a corresponding Lyapunov function needs to be constructed so as to derive the learning rule of the weight.
Wherein λ >0. Its derivative with respect to the argument t is then:
when it is satisfied with
And η>D+B，/>
Thereby->
And &>
And then this is true. And finally, deducing the learning rule of the weight and completing the proving of the stability of the controller.
For better illustration, the following simulation experiments were performed in this example:
taking the synchronization between the first scene and the second scene as an example, a schematic diagram is shown in fig. 7, where master represents the master system, slave represents the slave system, observer _ m represents the master system observer, observer _ m represents the slave system observer, and controller _ a represents the synchronization controller. Let memristor HNN in scene one be the master system, and memristor HNN in scene two be the slave system. Their parameters were consistent with the experimental parameters previously described in fig. 3 and 4. The gains L and P may be solved by means of an LMI toolbox. For the main system, the result of the solution is:
for the slave system, the result of the solution is:
controller parameters η and c i Set to 5 and 1, respectively. The simulation result is shown in fig. 8, in which the solid line represents the output result of the main system, and the dotted line represents the output result of the slave system.
The embodiment of the invention also provides an image encryption application of HNN in three scenes, which comprises the following steps:
because the image has the characteristics of large size, high correlation among pixels and the like, the traditional encryption algorithms such as DES, AES and the like have weak efficiency and anti-attack capability. The chaotic image encryption has the advantages of high efficiency, simple realization, strong attack resistance and the like due to the chaotic pseudo-randomness. The HNNs in the three scenes designed by the application have complex dynamic behaviors, and the safety of chaotic encryption can be obviously improved. The image encryption algorithm mainly comprises four steps. Firstly, the HNNs of any one of the three scenes are used for generating random sequences S1, S2 and S3, and the length of the random sequences is consistent with the number of pixels of an image to be encrypted. Second, a forward addition modulo diffusion algorithm is performed using S1, and then a reverse addition modulo diffusion algorithm is performed using S2. Third, the picture is scrambled using S3 as a mapping rule. And finally, performing forward multiplication modular diffusion algorithm by using S1, and performing reverse multiplication modular diffusion algorithm by using S2. The decryption process is the inverse of the encryption.
And (3) carrying out security analysis on the image encryption algorithm:
(1) And (3) testing randomness: taking the random sequence generated by the 5D multi-scroll memristor HNN in scenario three as an example, NIST P800-22 random test was performed. The test results are shown in Table 1. The test result shows that 15 tests of NIST P800-22 successfully pass, and the random sequence is proved to have good randomness and can be used for image encryption.
(2) Histogram analysis: an ideal encryption algorithm should make the gray scale distribution uniform and leave no valuable statistical information to the attacker. As can be seen from fig. 9, after the encryption, the gray distribution of the picture is changed from very uneven to even. Therefore, the image encryption is intuitively proved to have good anti-attack capability through histogram analysis. Among them, a diagram (a) in fig. 9 is an original image, a diagram (b) in fig. 9 is an encrypted image, a diagram (c) in fig. 9 is an original image histogram, and a diagram (d) in fig. 9 is an encrypted image histogram.
(3) Information entropy analysis: the entropy of information may measure the randomness of the information. Larger information entropy means stronger randomness. The information entropy calculated by encrypting a 512 × 512.Lena image using the algorithm of the present embodiment is 7.9994. Documents "S.Zhang, J.Zheng, X.Wang, Z.Zeng, and S.He", "Initial offset boosting copolymers in a spatial multi-double-conditional host network," non-linear Dynamics, vol.102, no.4, pp.2821-2841,2020 "and documents" Q.Lai, Z.Wan, H.Zhang, and G.Chen "," Design and analysis of a multi-conditional host network with an adaptation to an encryption, "IEEE Transactions on Neural Networks and principles, 991-14, pp.2" are better security Systems than the present embodiment, 20277, 77, 7.78, respectively.
(4) And (3) correlation analysis: due to the high correlation between adjacent pixels of the image, an attacker can use this property to deduce the gray values of the adjacent pixels, thereby recovering the entire plaintext image. A good encryption algorithm must break the correlation between adjacent pixels. The correlation coefficients of the 512 x 512 original Lena images were calculated to be 0.9848, 0.9695, 0.9587, respectively. The correlation coefficients of the encrypted image are-0.0039, -0.0100 and-0.0021 respectively. As can be seen from the calculated data, the algorithm has good decorrelation capability.
(5) Differential attack analysis: in general, the number of varying pixel rates (NPCR) and the uniform average variation strength (UACI) may measure the ability of an encryption algorithm to resist differential attacks. The ideal values for NPCR and UACI are 99.6094% and 33.4635%, respectively. In the present embodiment, the average values of NPCR and UACI of the encrypted 512 × 512Lena image are 99.6103% and 33.4621%, respectively. The algorithm has excellent differential attack resistance as the value is very close to the ideal value.
(6) Robustness of data loss: since data loss may occur in the transmitted image, it is necessary to have the ability to decrypt the lost encrypted image into a clear image. The present embodiment artificially sets a part of pixels of an encrypted image to zero to simulate data loss. Fig. 10 illustrates that the image encryption algorithm has excellent data loss resistance. In fig. 10, (a) is an original image, (b) is an encrypted image with data loss, and (c) is a decrypted image with data loss. The minimum values of Mean Square Error (MSE) and peak signal-to-noise ratio (PSNR) of the encrypted image without data loss and the encrypted image with data loss are 1490.1093 and 16.3988, respectively. The MSE and PSNR for the original image and the decrypted image after data loss are 1744.8782 and 15.7710, respectively. The experimental result image and the statistical result show that the algorithm has better robustness for data loss.
TABLE 1
| STATISTICAL TEST | P-VALUE | RESULT |
| Frequency | 0.2757 | PASS |
| BlockFrequency | 0.2757 | PASS |
| CumulativeSums | 0.3520 | PASS |
| Runs | 0.1223 | PASS |
| LongestRun | 0.3505 | PASS |
| Rank | 0.2757 | PASS |
| FFT | 0.8343 | PASS |
| NonOverlappingTemplate | 0.5035 | PASS |
| OverlappingTemplate | 0.6371 | PASS |
| Universal | 0.3505 | PASS |
| ApproximateEntropy | 0.0909 | PASS |
| RandomExcursions | 0.1612 | PASS |
| RandomExcursionsVariant | 0.2261 | PASS |
| Serial | 0.4373 | PASS |
| LinearComplexity | 0.7399 | PASS |
The FPGA circuit implementation of this embodiment includes:
the circuit design of the memristor HNN of the three scenes is realized through the FPGA. The Integrated Development Environment (IDE) is vivado 2018.3. In this embodiment, the floating-point IP core provided by vivado is used, and includes an addition, subtraction, multiplication, and division operation IP core, a comparison IP core, a floating-point-to-fixed-point IP core, and a fixed-point-to-floating-point IP core. The floating point number standard is the IEEE 754 standard and, more precisely, includes one sign bit, 8 exponent bits, and 23 fraction bits. The FPGA development board used was AX7Z100 provided by ALINX corporation. It is driven by Xilinx ZYNQ7000 series chip XC7Z100-2FFG 900. The fixed point number has a precision of 14 bits including a 1-bit sign bit, four integer bits and nine fractional bits in consideration of the number of output pins of the development board and the maximum value of the system output. In addition, this embodiment also uses Cordic IP core to implement the sin function. Since the operating range of the IP core is [ - π, π ], this embodiment designs an adaptive phase shift circuit to achieve the desired sin function. It should be noted that, in order to improve the efficiency of the algorithm, the present embodiment uses a piecewise function shown in the following equation to approximate hyperbolic tangent operation, rather than a tedious exponential operation. Therefore, fine tuning of the parameters of these memristive HNN systems is required to obtain the desired multi-wrap. For scenario one, the other parameters, except for parameter k =2, were consistent with the experimental parameters of fig. 3 above. For scenario two, the other system parameters, except for parameter k =0.5, were consistent with the experimental parameters of fig. 4 above.
For the FPGA implementation of the memristive HNNs in the three scenarios, the RK4 algorithm is adopted for iterative solution in the embodiment. Since iteration of RK4 requires calculation of four estimated slopes, the present embodiment naturally designs four states, which may be named as K1, K2, K3, and K4. This design uses a counter to ensure that the state jump is performed after a fixed number of clock cycles. The iteration of the solution starts from the K1 state and jumps to the next state K2 after the counter is full. Similarly, K2 jumps to K3 and eventually to K4. When in the K4 state and the counter is full, the state jumps to K1, and the final result is calculated and converted into a fixed point number. At this time, the FPGA sends a write request signal to the digital-to-analog converter, thereby outputting a final result to the oscilloscope. Due to the robustness of the digital circuits, it can be found that the circuit implementation results in fig. 11 and 12 are exactly the same as the MATLAB simulation results in fig. 3 and 4.
In addition, the embodiment also designs an image encryption circuit, so that the FPGA can directly encrypt (decrypt) the image. The image encryption implementation based on the FPGA mainly comprises four macro states, namely an initial state, an encryption state, a waiting state and a display state. The jump of each state is also controlled by a counter N, and the purpose of setting the initial state is to jump over the transition state of the chaotic system. When the counter N is noted as N1, the initial state jumps to the encrypted state. On the basis of the chaos signal generator realized in the past, each time the system completes the solution, an enabling signal and a solution result are sent to the outside. The enabling signal activates the random number generation module to generate an 8-bit random number by using the incoming chaotic signal to complete encryption. When the counter N counts N2, the encryption is completed, and the encryption state jumps to a waiting state. In this case, if the display enable signal arrives, the state continues to jump to the display state. In this state, the FPGA outputs an original image (encrypted image) and an encrypted image (decrypted image) to the IO device. Fig. 13 shows the experimental results of image encryption (decryption) based on the FPGA image encryption circuit. Diagram (b) in fig. 13 shows a scenario in which decryption fails due to a key error. Wherein:
fig. 11 is an experimental result in scenario one implemented by using the technical solution of the present embodiment, where a diagram (a) in fig. 11 is an experimental global diagram, and a diagram (b) in fig. 11 is f (x) = f odd And 3 scroll attractors at N =1, where f (x) = f in fig. 11 (c) even And 4 scroll attractors with N =1.
Fig. 12 is an experimental result in scenario two implemented by using the technical solution of the present embodiment, where fig. 12 (a) is an experimental global diagram, and fig. 12 (b) is f (x) = f odd And 3 scroll attractors at N =1, where f (x) = f in fig. 12 (c) even And 4 scroll attractors with N =1.
Fig. 13 shows the experimental result of encrypting (decrypting) an image using the solution of the present embodiment, where fig. 13 (a) shows an original image and an encrypted image, fig. 13 (b) shows an encrypted image and an error decrypted image, and fig. 13 (c) shows an encrypted image and a decrypted image.
The embodiments of the present invention have been described in detail with reference to the accompanying drawings, but the present invention is not limited to the above embodiments, and various changes can be made within the knowledge of those skilled in the art without departing from the gist of the present invention.
Claims (10)
1. A multi-scene HNN construction method is characterized by comprising the following steps:
constructing a non-polynomial memristor as:
wherein a, b, c, d represent parameters of the non-polynomial memristor, i m Representing the output current of the non-polynomial memristor, W (x) representing the memristor, v m The method is characterized by comprising the following steps of (1) representing an external input voltage, x representing a memristor internal state variable, dt representing derivation of time t, f (-) representing a memristor internal state function, and f (x) being expressed as follows:
wherein f is odd The representation being used to induce an odd number of scrolls, f even Indicating being used to induce an even number of scrolls, odd-scrolls indicating an odd number of scrolls, even-scrolls indicating an even number of scrolls;
constructing memristors HNNs of three scenes based on the non-polynomial memristor; the memristive HNNs of the three scenes comprise memristive synaptic weights HNNs, HNNs under electromagnetic radiation and memristive synaptic weights HNNs under the action of the electromagnetic radiation.
2. The multi-scenario HNN construction method according to claim 1, wherein constructing the memristive synaptic weight HNN is:
wherein a, b, c, d represent parameters of the memristive synaptic weight HNN, x 1 、x 2 、x 3 And x 4 Representing the membrane potentials of neurons 1,2, 3 and 4, respectively, I 1 、I 2 And I 3 Represents the external stimulus current, k represents the coefficient, tanh (-) represents the neuron activation function,
and &>
Respectively represent a pair x 1 、x 2 、x 3 And x 4 Differentiation of (2).
3. The method for constructing the multi-scene HNN according to claim 1, wherein the HNN under the electromagnetic radiation is constructed by:
wherein a, b, c, d represent the parameters of HNN under the electromagnetic radiation, x 1 、x 2 、x 3 And x 4 Representing the membrane potentials of neurons 1,2, 3 and 4, respectively, I 1 、I 2 And I 3 Representing the external stimulus current, k representing the coefficient, tanh (-) representing the neuron activation function,
and &>
Respectively represent a pair x 1 、x 2 、x 3 And x 4 A differential of (c).
4. The multi-scenario HNN construction method according to claim 1, wherein the memristive synaptic weight HNN under the action of the electromagnetic radiation is constructed by:
wherein, a 1 ,b 1 ,c 1 ,d 1 ,a 2 ,b 2 ,c 2 ,d 2 A parameter, x, representing the memristive synaptic weight HNN under the effect of said electromagnetic radiation 1 、x 2 、x 3 、x 4 And x 5 Respectively, membrane potentials of neurons 1,2, 3, 4 and 5, I 1 、I 2 And I 3 Denotes an external stimulus current, k 1 And k 2 Representing the coefficients, tanh (-) representing the neuron activation function,
and &>
Respectively represent a pair x 1 、x 2 、x 3 、x 4 And x 5 Differentiation of (2).
5. An adaptive synchronization method, characterized in that the HNN construction method of multi-scene according to any one of claims 1 to 4 is applied, comprising:
constructing an adaptive state observer and a synchronous controller;
selecting memristors HNNs of any two scenes from the memristors HNNs of the three scenes as a master system and a slave system respectively;
synchronizing the master system and the slave system with the synchronization controller based on the adaptive state observer.
6. The adaptive synchronization method according to claim 5, characterized in that the adaptive state observer is constructed as:
wherein, the first and the second end of the pipe are connected with each other,
represents a system state variable, <' > based on>
Represents an estimate of the unknown non-linear portion, A represents the known linear portion, U represents the known external control input, L and P represent gain matrices, and->
Y represents an observed quantity, C represents a known linear portion, and γ represents a coefficient.
7. The adaptive synchronization method of claim 6, wherein an RBF neural network is used to fit the estimate of the unknown non-linear part, and the fitting expression of the unknown non-linear part is:
wherein, the first and the second end of the pipe are connected with each other,
represents a weight matrix, <' > based on>
Representing a gaussian kernel function.
8. The adaptive synchronization method according to claim 5, wherein the primary system is represented by:
wherein, X = (X) 1 ,x 2 ,...,x n ) T ，f i (X) represents a non-linear function,
and & ->
The differential is indicated.
9. The adaptive synchronization method according to claim 8, wherein the slave system is represented by:
wherein, Y = (Y) 1 ,y 2 ,...,y n ) T ，g i (Y) represents a non-linear function, d i (t) represents an external disturbance, i =1,2 i (t)|<D i ，u i It is indicative of an external control input,
and & ->
The differential is indicated.
10. The adaptive synchronization method of claim 9, wherein the synchronizing the master system and the slave system with the synchronization controller comprises:
calculating an error function between the master system and the slave system as:
Ε＝Y-X
wherein e = (epsilon) 1 ,ε 2 ,...,ε n ) T ；
If the slave system and the master system are synchronized, then
And the sliding mode surface epsilon of the error function i Expressed as:
wherein c is i Represents a coefficient, c i >0, τ represents an independent variable;
if s i =0, then
And->
According to
Setting a target of the synchronization controller to ^ based>
To synchronize the slave system with the master system. />
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