A multi-scene HNN construction method and adaptive synchronization method

Abstract

The application discloses a multi-scene HNN construction method and an adaptive synchronization method, wherein the non-polynomial memristor constructed by the multi-scene HNN construction method not only guarantees the smoothness of the right side of a dynamic equation, but also maintains the constant level of method complexity; based on the non-polynomial memristor, three scenes of memristor HNN are constructed, and rich dynamic behaviors are found from the HNN, including controllable one-way expansion multi-vortex attractor, controllable grid multi-vortex attractor, chaos coexistence caused by initial offset and periodic coexistence caused by initial offset. The adaptive synchronization method constructs an adaptive state observer and a synchronization controller, selects any two scenes of the memristor HNN as a master system and a slave system in the three scenes of the memristor HNN, adopts the synchronization controller to synchronize the master system and the slave system based on the adaptive state observer, and can well simulate the synchronization between different regions of the brain.

Description

Technical Field

[0001] This invention relates to the field of neurodynamics technology, and in particular to a multi-scenario HNN construction method and an adaptive synchronization method. Background Technology

[0002] The Hopfield Neural Network (HNN) was proposed based on the Ising model, which explains associative memory. It possesses continuous dynamics and can help understand human memory. In recent years, researchers have introduced memristors into HNNs in different ways. By adjusting the memristor synaptic weights, they have discovered complex dynamic behaviors in the neural network, such as multiple vortex attractors and extreme multistabilities.

[0003] In existing technologies, researchers have successfully obtained controllable multi-scroll memristor HNNs by introducing a class of memristors containing signed polynomials. While the signed function greatly simplifies circuit design, its discontinuous mathematical characteristics also bring a series of problems. A memristor containing a signed polynomial undoubtedly contains a series of "bad points." At these "bad points," determining the existence and uniqueness of solutions to the differential equations is directly unattainable. Secondly, many adaptive synchronization control methods require the system to satisfy the Lipschitz condition. Furthermore, some researchers have used continuous basis functions instead of the signed function, which, while overcoming the "bad point" problem, undoubtedly increases the complexity of circuit implementation. Summary of the Invention

[0004] This invention aims to address at least one of the technical problems existing in the prior art. To this end, this invention proposes a multi-scenario HNN construction method and an adaptive synchronization method that can maintain constant-level method complexity while generating rich dynamic behaviors.

[0005] In a first aspect, embodiments of the present invention provide a multi-scenario HNN construction method, the multi-scenario HNN construction method comprising:

[0006] Constructing a non-polynomial memristor is as follows:

[0007]

[0008] Where a, b, c, d represent the parameters of the non-polynomial memristor, i m Let W(x) represent the output current of the non-polynomial memristor, W(x) represent the memconductivity, and v m Let represent the external input voltage, x represent the internal state variables of the memristor, dt represent the derivative with respect to time t, and f(·) represent the internal state function of the memristor. The expression for f(x) is:

[0009]

[0010] Among them, f odd This indicates that it is used to induce an odd number of vortices, f even This indicates that it is used to induce an even number of scrolls, odd-scrolls indicates an odd number of scrolls, and even-scrolls indicates an even number of scrolls;

[0011] Based on the non-polynomial memristor, three scenarios of memristor HNN are constructed; wherein, the three scenarios of memristor HNN include memristor synaptic weight HNN, HNN under electromagnetic radiation, and memristor synaptic weight HNN under electromagnetic radiation.

[0012] Compared with the prior art, the first aspect of the present invention has the following beneficial effects:

[0013] The non-polynomial memristor constructed by this method not only ensures the smoothness of the right side of the dynamic equation but also maintains constant-level method complexity. Based on the non-polynomial memristor, three scenarios of memristor HNNs are constructed: memristor synaptic weighted HNN, HNN under electromagnetic radiation, and memristor synaptic weighted HNN under electromagnetic radiation. The memristor synaptic weighted HNN and the HNN under electromagnetic radiation can successfully generate controllable multi-vortex attractors with similar shapes, while the memristor synaptic weighted HNN under electromagnetic radiation can generate controllable grid multi-vortex attractors. Rich dynamic behaviors were discovered from these HNNs, including controllable unidirectional expanding multi-vortex attractors, controllable grid multi-vortex attractors, chaotic coexistence caused by initial offset, and periodic coexistence caused by initial offset.

[0014] According to some embodiments of the present invention, the memristor synaptic weighted HNN is constructed as follows:

[0015]

[0016] Where a, b, c, d represent the parameters of the memristor synaptic weighted HNN, x1, x2, x3, and x4 represent the membrane potentials of neurons 1, 2, 3, and 4, respectively, I1, I2, and I3 represent the external stimulation currents, k represents the coefficient, and tanh(·) represents the neuron activation function. and Let x1, x2, x3, and x4 represent the derivatives with respect to x4, respectively.

[0017] According to some embodiments of the present invention, the HNN under electromagnetic radiation is constructed as follows:

[0018]

[0019] Where a, b, c, d represent the parameters of the HNN under electromagnetic radiation, x1, x2, x3, and x4 represent the membrane potentials of neurons 1, 2, 3, and 4, respectively, I1, I2, and I3 represent the external stimulation currents, k represents the coefficient, and tanh(·) represents the neuron activation function. and Let x1, x2, x3, and x4 represent the derivatives with respect to x4, respectively.

[0020] According to some embodiments of the present invention, the memristor synaptic weighted HNN under the influence of electromagnetic radiation is constructed as follows:

[0021]

[0022] Where a1, b1, c1, d1, a2, b2, c2, d2 represent the parameters of the memristor synaptic weighted HNN under the influence of electromagnetic radiation, x1, x2, x3, x4, and x5 represent the membrane potentials of neurons 1, 2, 3, 4, and 5, respectively, I1, I2, and I3 represent the external stimulation currents, k1 and k2 represent coefficients, and tanh(·) represents the neuron activation function. and Let x1, x2, x3, x4, and x5 represent the derivatives with respect to x5, respectively.

[0023] Secondly, embodiments of the present invention also provide an adaptive synchronization method, applying the multi-scenario HNN construction method described in any of the above claims, including:

[0024] Construct an adaptive state observer and a synchronization controller;

[0025] In the three scenarios, any two of the memristor HNNs are selected as the master system and the slave system, respectively.

[0026] Based on the adaptive state observer, the synchronization controller is used to synchronize the master system and the slave system.

[0027] Compared with the prior art, the second aspect of the present invention has the following beneficial effects:

[0028] Since synchronization plays a crucial role in memory processing, synchronization between brain regions supports working and long-term memory by facilitating communication between neurons and promoting neuronal plasticity. Therefore, to simulate synchronization between different brain regions, this method constructs an adaptive state observer and a synchronization controller. Two memristor HNNs from three different scenarios are selected as the master and slave systems. Based on the adaptive state observer, the synchronization controller synchronizes the master and slave systems. The constructed adaptive state observer and synchronization controller enable fast synchronization response and good synchronization effect between the master and slave systems, effectively simulating synchronization between different brain regions.

[0029] According to some embodiments of the present invention, an adaptive state observer is constructed as follows:

[0030]

[0031] in, Represents system state variables. Let A represent the estimation of the unknown nonlinear part, U represent the known linear part, L represent the known external control input, and L and P represent the gain matrices. Y represents the observed quantity, C represents the known linear component, and γ represents the coefficient.

[0032] According to some embodiments of the present invention, an RBF neural network is used to fit the estimate of the unknown nonlinear part, and the fitting expression of the unknown nonlinear part is:

[0033]

[0034] in, Represents the weight matrix. This represents the Gaussian kernel function.

[0035] According to some embodiments of the present invention, the main system is represented as follows:

[0036]

[0037] Where X = (x1, x2, ..., x n ) T f i (X) represents a nonlinear function. and It represents the differential.

[0038] According to some embodiments of the present invention, the slave system is represented in the following manner:

[0039]

[0040] Where Y = (y1, y2, ..., y n ) T g i (Y) represents a nonlinear function, d i (t) represents the external disturbance, i = 1, 2, ..., n, let d i (t)| <D i u i Indicates external control input. and It represents the differential.

[0041] According to some embodiments of the present invention, the step of using the synchronization controller to synchronize the master system and the slave system includes:

[0042] The error function between the master system and the slave system is calculated as follows:

[0043]

[0044] Where Ε=(ε1,ε2,...,ε n ) T ;

[0045] If the slave system and the master system are synchronized, then And the sliding surface ε of the error function i Represented as:

[0046]

[0047] Where c i c represents the coefficient. i >0, τ represents the independent variable;

[0048] If s i =0, then and

[0049] according to The target of the synchronization controller is set as follows: To synchronize the slave system and the master system. Attached Figure Description

[0050] The above and / or additional aspects and advantages of the present invention will become apparent and readily understood from the description of the embodiments taken in conjunction with the following drawings, in which:

[0051] Figure 1 This is a flowchart of a multi-scenario HNN construction method according to an embodiment of the present invention;

[0052] Figure 2 This is a topology diagram of a memristor HNN in three scenarios according to an embodiment of the present invention;

[0053] Figure 3 This is a phase diagram showing different vortex numbers in scenario one of an embodiment of the present invention;

[0054] Figure 4 This is a phase diagram showing different vortex numbers in scenario two of an embodiment of the present invention;

[0055] Figure 5 This is a phase diagram showing different vortex numbers in scenario three of an embodiment of the present invention;

[0056] Figure 6This is a flowchart of an adaptive synchronization method according to an embodiment of the present invention;

[0057] Figure 7 This is a schematic diagram of an adaptive synchronization method according to an embodiment of the present invention;

[0058] Figure 8 This is an experimental result diagram of master-slave system synchronization according to an embodiment of the present invention;

[0059] Figure 9 This is a histogram analysis result diagram of an embodiment of the present invention;

[0060] Figure 10 This is a schematic diagram of a robustness test for data loss according to an embodiment of the present invention;

[0061] Figure 11 This is an experimental result diagram of scenario one implemented using the technical solution of this embodiment of the present invention;

[0062] Figure 12 This is an experimental result diagram of scenario two implemented using the technical solution of this embodiment of the present invention;

[0063] Figure 13 This is an experimental result diagram of image encryption (decryption) using the technical solution of this embodiment of the present invention. Detailed Implementation

[0064] Embodiments of the present invention are described in detail below. Examples of these embodiments are shown in the accompanying drawings, wherein the same or similar reference numerals denote the same or similar elements or elements having the same or similar functions throughout. The embodiments described below with reference to the accompanying drawings are exemplary and are only used to explain the present invention, and should not be construed as limiting the present invention.

[0065] In the description of this invention, the use of terms such as "first," "second," etc., is for the purpose of distinguishing technical features only and should not be construed as indicating or implying relative importance, or implicitly indicating the number of technical features indicated, or implicitly indicating the order of the technical features indicated.

[0066] In the description of this invention, it should be understood that the orientation descriptions, such as up, down, etc., are based on the orientation or positional relationship shown in the drawings and are only for the convenience of describing this invention and simplifying the description, and are not intended to indicate or imply that the device or element referred to must have a specific orientation, or be constructed and operated in a specific orientation, and therefore should not be construed as a limitation of this invention.

[0067] In the description of this invention, it should be noted that, unless otherwise explicitly defined, terms such as "setting," "installation," and "connection" should be interpreted broadly, and those skilled in the art can reasonably determine the specific meaning of the above terms in this invention in conjunction with the specific content of the technical solution.

[0068] In existing technologies, researchers have successfully obtained controllable multi-scroll memristor HNNs by introducing a class of memristors containing signed polynomials. While the signed function greatly simplifies circuit design, its discontinuous mathematical characteristics also bring a series of problems. A memristor containing signed polynomials undoubtedly contains a series of "bad points." At these "bad points," determining the existence and uniqueness of solutions to the differential equations is directly unattainable. Secondly, many adaptive synchronization control methods require the system to satisfy the Lipschitz condition; although this overcomes the "bad point" problem, it undoubtedly increases the complexity of circuit implementation.

[0069] To address the aforementioned issues, this invention constructs a non-polynomial memristor, which not only ensures the smoothness of the right-hand side of the dynamic equation but also maintains constant-level method complexity. Based on the non-polynomial memristor, three scenarios of memristor HNNs are constructed. These three scenarios include a memristor synaptic weight HNN, an HNN under electromagnetic radiation, and a memristor synaptic weight HNN under electromagnetic radiation. The memristor synaptic weight HNN and the HNN under electromagnetic radiation can successfully generate controllable multi-vortex attractors with similar shapes, while the memristor synaptic weight HNN under electromagnetic radiation can generate controllable grid multi-vortex attractors. Rich dynamic behaviors were discovered from these HNNs, including controllable unidirectional expanding multi-vortex attractors, controllable grid multi-vortex attractors, chaotic coexistence induced by initial offset, and periodic coexistence induced by initial offset.

[0070] Reference Figure 1 This invention provides a multi-scenario HNN construction method, characterized in that the multi-scenario HNN construction method includes, but is not limited to, steps S110 to S120:

[0071] Step S110: Based on memristor theory, construct a non-polynomial memristor as follows:

[0072]

[0073] Where a, b, c, d represent the parameters of the non-polynomial memristor, i m Let W(x) represent the output current of the non-polynomial memristor, and let W(x) represent the memconductance. m Let represent the external input voltage, x represent the internal state variables of the memristor, dt represent the derivative with respect to time t, and f(·) represent the internal state function of the memristor. The expression for f(x) is:

[0074]

[0075] Among them, f odd This indicates that it is used to induce an odd number of vortices, f even This indicates that an even number of scrolls were used to induce the scrolls; odd-scrolls indicate an odd number of scrolls, and even-scrolls indicate an even number of scrolls. odd and f even It is a piecewise function and has the following form:

[0076]

[0077]

[0078] Where A, F, and N are parameters, especially N, which can be used to control the number of scrolls, and f odd It is used to induce an odd number of vortices, in contrast to f. even It was used to induce an even number of vortices. In the following sections, A... m and F m They will be set to 1 and 0.5 respectively, and their values ​​will remain unchanged.

[0079] Step S120: Based on the non-polynomial memristor, construct memristor HNNs for three scenarios; wherein, the memristor HNNs for the three scenarios include memristor synaptic weight HNN, HNN under electromagnetic radiation, and memristor synaptic weight HNN under electromagnetic radiation.

[0080] Specifically, refer to Figure 2 This embodiment is based on 3D HNN, employing three different methods to model the memristor HNN and study its dynamic characteristics. The original 3D HNN can be represented as:

[0081]

[0082] Where x1, x2, and x3 represent the membrane potentials of neurons 1, 2, and 3, respectively. I1, I2, and I3 represent the external stimulation currents. In this embodiment, the external stimulation currents are uniformly set to 0.

[0083] Scene 1: Figure 2 Figure a in the diagram is a schematic of a memristor synaptic weighted HNN. As is well known, the conductance of a memristor changes with external input voltage or current, which is very similar to the synapses of biological neurons. Therefore, using memristors to replace the connection weights between neurons is more physiologically sound. Moreover, experiments have shown that this can lead to richer dynamic behaviors. By replacing the connection weights between neuron 1 and neuron 2 with their conductance, the memristor synaptic weighted HNN (i.e., the new 4D memristor HNN) is constructed as follows:

[0084]

[0085] Where a, b, c, d represent the parameters of the memristor synaptic weighted HNN, x1, x2, x3, and x4 represent the membrane potentials of neurons 1, 2, 3, and 4, respectively, I1, I2, and I3 represent the external stimulation currents, k represents the coefficient, and tanh(·) represents the neuron activation function. and Let x1, x2, x3, and x4 represent the derivatives with respect to x4, respectively.

[0086] Scene 2: Figure 2 Figure b in the diagram is a schematic of an HNN under electromagnetic radiation. When a neuron is exposed to a magnetic field, its membrane magnetic flux changes, generating an induced current. To simulate the effect of neuron 2 being exposed to a magnetic field, a memristor current is introduced. This yields another type of 4D memristor HNN under electromagnetic radiation. The construction of the HNN under electromagnetic radiation is as follows:

[0087]

[0088] Where a, b, c, d represent the parameters of the HNN under electromagnetic radiation, x1, x2, x3, and x4 represent the membrane potentials of neurons 1, 2, 3, and 4, respectively, I1, I2, and I3 represent the external stimulation currents, k represents the coefficient, and tanh(·) represents the neuron activation function. and Let x1, x2, x3, and x4 represent the derivatives with respect to x4, respectively.

[0089] Scene 3: Figure 2 Figure c in the diagram is a schematic of a memristor synaptic weighted HNN under electromagnetic radiation. Based on the first two scenarios, when memristors are used to replace neuron connection weights and neurons are exposed to electromagnetic radiation, the memristor synaptic weighted HNN under electromagnetic radiation is constructed as follows:

[0090]

[0091] Where a1, b1, c1, d1, a2, b2, c2, d2 represent the parameters of the memristor synaptic weighted HNN under electromagnetic radiation, x1, x2, x3, x4, and x5 represent the membrane potentials of neurons 1, 2, 3, 4, and 5, respectively, I1, I2, and I3 represent the external stimulation currents, k1 and k2 represent coefficients, and tanh(·) represents the neuron activation function. and Let x1, x2, x3, x4, and x5 represent the derivatives with respect to x5, respectively.

[0092] In this embodiment, by constructing a non-polynomial memristor, not only is the smoothness of the right side of the dynamic equation guaranteed, but the method complexity is also maintained at a constant level. Based on the non-polynomial memristor, three scenarios of memristor HNNs are constructed. Among them, the three scenarios of memristor HNNs include memristor synaptic weight HNN, HNN under electromagnetic radiation, and memristor synaptic weight HNN under electromagnetic radiation. The memristor synaptic weight HNN and the HNN under electromagnetic radiation can successfully generate controllable multi-vortex attractors with similar shapes. The memristor synaptic weight HNN under electromagnetic radiation can generate controllable grid multi-vortex attractors. Rich dynamic behaviors were found in these HNNs, including controllable unidirectional expanding multi-vortex attractors, controllable grid multi-vortex attractors, chaotic coexistence caused by initial offset, and periodic coexistence caused by initial offset.

[0093] To better illustrate this, the following analysis and experiments were conducted in this embodiment:

[0094] 1. Equilibrium Point Analysis

[0095] Equilibrium point analysis is an effective method for studying dynamic behavior. Setting the left-hand side of the equations for the memristor synaptic weighted HNN, the HNN under electromagnetic radiation, and the memristor synaptic weighted HNN under electromagnetic radiation to 0 and performing Gaussian elimination, we obtain the trajectory equations for variables x2 and x3. Their specific form can be described by the following equation:

[0096]

[0097]

[0098]

[0099] Numerical solutions for x2 and x3 under memristor synaptic weighted HNN, HNN under electromagnetic radiation, and memristor synaptic weighted HNN under electromagnetic radiation were obtained using graphical and numerical methods, and then all roots were found. After linearization at the equilibrium point, the Jacobian matrix of the system under memristor synaptic weighted HNN, HNN under electromagnetic radiation, and memristor synaptic weighted HNN under electromagnetic radiation was obtained, and the results are shown below.

[0100]

[0101] J 11 =1.3-2.3tanh 2 (x1),J 12 =1.2tanh 2 (x2)-1.2,J 13 =0.2-0.2tanh 2 (x3),J 14 =0,J 21 =2-2tanh2 (x1),J 22 =-1 + k(a + bf(x4))(1 - tanh 2 (x2)),J 23 =1.1 - 1.1tanh 2 (x3),J 24 =kbtanh(x2)f'(x4),J 31 =5.5tanh 2 (x1) - 5.5,J 32 =0,J 33 =1.1tanh 2 (x3) - 2.1,J 34 =0,J 41 =0,J 42 =c - ctanh 2 (x2),J 43 =0,J 44 =-df'(x4).

[0102]

[0103] where J 11 =1.3 - 2.3tanh 2 (x1),J 12 =1.2tanh 2 (x2) - 1.2,J 13 =0.2 - 0.2tanh 2 (x3),J 14 =0,J 21 =2 - 2tanh 2 (x1),J 22 =-tanh 2 (x2)+k(a + bf(x4)),J 23 =1.1 - 1.1tanh 2 (x3),J 24 =kbx2f'(x4),J 31 =5.5tanh 2 (x1) - 5.5,J 32 =0,J 33 =1.1tanh 2 (x3) - 2.1,J 34 =0,J 41 =0,J 42 =c,J 43 =0,J 44 =-df'(x4).

[0104]

[0105] J 11 =1.3-2.3tanh 2 (x1),J 12 =1.2tanh 2 (x2)-1.2,J 13 =0.2-0.2tanh 2 (x3),J 14 =0,J 15 =0,J 21 =2-2tanh 2 (x1),J 22 = -1 + k1(a1 + b1f(x4))(1 - tanh 2 (x2))+k2(a2+b2f(x5)),J 23 =1.1-1.1tanh 2 (x3),J 24 =k1b1 tanh(x2)f'(x4),J 25 =k2b2x2f'(x5),J 31 =5.5tanh 2 (x1)-5.5,J 32 =0,J 33 =1.1tanh 2 (x3)-2.1,J 34 =0,J 35 =0,J 41 =0,J 42 =c1-c1 tanh 2 (x2),J 43 =0,J 44 =-d1f'(x4),J 45 =0,J 51 =0,J 52 =c2,J 53 =0,J 54 =0,J 55 =-d2f'(x5).

[0106] Let J be the Jacobian matrix of each system. The eigenvalues ​​corresponding to the equilibrium points can be obtained by solving the characteristic equation det(λE-J)=0.

[0107] 2. Experiment.

[0108] To intuitively understand the dynamic behavior of the system, specific parameters are substituted into this section. The internal state variable function f(x) of the memristor is f... oddGiven that parameter N is 1, the equilibrium points and corresponding eigenvalues ​​for the three scenarios can be calculated using the above equilibrium point analysis. According to Shil'nikov's theorem in the literature "C. Silva, "Shil'nikov's theorem-a tutorial," IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, vol. 40, no. 10, pp. 675–682, 1993," if there exists a theorem satisfying... If the system has a pure real eigenvalue δ and two complex conjugate eigenvalues ​​α+βi with δα<0, then chaos will occur at the equilibrium point. Based on the value of the equilibrium point x1, this paper roughly classifies all equilibrium points into three types: upper equilibrium point, middle equilibrium point, and lower equilibrium point. For scenario one, the parameters are set as a = 0.58, b = 0.02, c = 2.7, d = 1.1, and k = 2.15. Through calculation, the middle equilibrium point is (0,0,0,x). 41 The lower equilibrium point is (-0.5233, -0.3430, 1.6236, x). 42 And the upper equilibrium point is (0.5012, 0.3292, -1.5430, x). 43 ). x 41 x 42 and x 43 It is the root of f(x) = 0, f(x) = -0.81, and f(x) = 0.78. Since N = 1, there are 3 × 5 equilibrium points. For scenario two, let the parameters a = 0.58, b = 0.02, c = 2.3, d = 1.22, and k = 0.4. The equilibrium point is (0, 0, 0, x). 41 The lower equilibrium point is (-0.5157, -0.3383, 1.5959, x). 42 And the upper equilibrium point is (0.5123, 0.3362, -1.5835, x). 43 ). x 41 x 42 and x 43 It is the root of f(x) = 0, f(x) = -0.6145, and f(x) = 0.8893. Similar to the previous scenario, there are now 3 × 5 equilibrium points. Scenario 3 has one more dimension than the previous scenarios. Let the parameters a1 = 0.6, b1 = 0.01, c1 = 2.725, d1 = 1.8, k1 = 2.1, a2 = 0.5, b2 = 0.02, c2 = 1.6, d2 = 0.59, k2 = 0.1. The equilibrium point is (0, 0, 0, x). 41 ,x 51 The lower equilibrium point is (-0.4950, -0.3252, 1.5205, x).42 ,x 52 And the upper equilibrium point is (0.4876, 0.3205, -1.4935, x). 43 ,x 53 ). x 41 x 42 and x 43 It is a root of f(x) = 0, f(x) = -0.4757, and f(x) = 0.4692. 51 x 52 and x 53 It is a root of f(x) = 0, f(x) = -0.8820, and f(x) = 0.8692. By calculating the eigenvalues ​​and applying Shil'nikov's theorem, it can be determined that in these three cases, all middle equilibrium points are unstable Index-1 saddle points, and all upper and lower equilibrium points are unstable Index-2 saddle points.

[0109] The above analysis shows that the number of equilibrium points is directly proportional to N. According to Shil'nikov theory, as the number of equilibrium points increases, the size of the attractor also increases. From the system's phase diagram, it is easy to see that when the equilibrium points satisfy f′(x⁴) > 0, vortices will appear. In summary, when f(x) = f odd In scenario one and scenario two, the system will generate 2N+1 double spirals. In scenario three, the system will generate (2N+1)×(2N+1) double spirals. Conversely, when f(x) = f even In scenario one and scenario two, the system will generate 2N+2 double vortices. In scenario three, it will generate (2N+2)×(2N+2) double vortices. Figure 3 and Figure 4 The phase diagrams are for scenarios one and two, respectively, when 1, 2, 3, 4, 5, and 6 double spirals are generated. Figure 5 Phase diagrams were generated for scenario three with 1×1, 2×2, 3×3, 4×4, 5×5, and 6×6 double spirals, respectively. It can be observed that the number of spirals can be well controlled by adjusting N. Specifically:

[0110] Figure 3 To generate phase diagrams with different numbers of vortices for scenario one, Figure 3 Figures (a), (b), and (c) in the figure correspond to f. odd ; Figure 3 Figures (d), (e), and (f) in the figure correspond to f. even Among them, Figure (a) shows 1 double spiral, Figure (b) shows 3 double spirals, Figure (c) shows 5 double spirals, Figure (d) shows 2 double spirals, Figure (e) shows 4 double spirals, and Figure (f) shows 6 double spirals.

[0111] Figure 4To generate phase diagrams with different numbers of vortices for scenario two. Figure 4 Figures (a), (b), and (c) in the figure correspond to f. odd ; Figure 4 Figures (d), (e), and (f) in the figure correspond to f. even Among them, Figure (a) shows 1 double spiral, Figure (b) shows 3 double spirals, Figure (c) shows 5 double spirals, Figure (d) shows 2 double spirals, Figure (e) shows 4 double spirals, and Figure (f) shows 6 double spirals.

[0112] Figure 5 To generate phase diagrams with different numbers of vortices for scenario three, Figure 5 Figures (a), (b), and (c) in the figure correspond to f. odd ; Figure 5 Figures (d), (e), and (f) in the figure correspond to f. even Among them, Figure (a) shows a 1×1 double spiral, Figure (b) shows a 3×3 double spiral, Figure (c) shows a 5×5 double spiral, Figure (d) shows a 2×2 double spiral, Figure (e) shows a 4×4 double spiral, and Figure (f) shows a 6×6 double spiral.

[0113] Reference Figure 6 The present invention also provides an adaptive synchronization method, wherein the memristor HNN for the three scenarios in the above-mentioned multi-scenario HNN construction method includes, but is not limited to, steps S210 to S230:

[0114] Step S210: Construct an adaptive state observer and a synchronization controller;

[0115] Step S220: Select any two memristor HNNs from the three scenarios as the master system and slave system, respectively.

[0116] Step S230: Based on the adaptive state observer, a synchronization controller is used to synchronize the master system and the slave system.

[0117] 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 neuronal plasticity. Therefore, in order to simulate synchronization between different brain regions, this method constructs an adaptive state observer and a synchronization controller. Any two memristor HNNs from three scenarios are selected as the master system and slave system. Based on the adaptive state observer, the synchronization controller synchronizes the master system and slave system. The constructed adaptive state observer and synchronization controller enable the master system and slave system to respond quickly and achieve good synchronization results, effectively simulating synchronization between different brain regions.

[0118] In some embodiments, the adaptive state observer is constructed as follows:

[0119]

[0120] in, Represents system state variables. Let A represent the estimation of the unknown nonlinear part, U represent the known linear part, L represent the known external control input, and L and P represent the gain matrices. Y represents the observed quantity, C represents the known linear component, and γ represents the coefficient.

[0121] In some embodiments, an RBF neural network is used to fit the estimate of the unknown nonlinear component, and the fitting expression for the unknown nonlinear component is:

[0122]

[0123] in, Represents the weight matrix. This represents the Gaussian kernel function.

[0124] In some embodiments, the main system is represented as follows:

[0125]

[0126] Where X = (x1, x2, ..., x n ) T f i (X) represents a nonlinear function. and It represents the differential.

[0127] In some embodiments, the system is represented as follows:

[0128]

[0129] Where Y = (y1, y2, ..., y n ) T g i (Y) represents a nonlinear function, d i (t) represents the external disturbance, i = 1, 2, ..., n, let |d i (t)| <D i u i Indicates external control input. and It represents the differential.

[0130] In some embodiments, a synchronization controller is used to synchronize the master system and the slave system, including:

[0131] The error function between the master system and the slave system is calculated as follows:

[0132] Ε=YX

[0133] Where Ε=(ε1,ε2,...,ε n ) T ;

[0134] If synchronizing from the primary system and the secondary system, then And the sliding surface ε of the error function i Represented as:

[0135]

[0136] Where c i c represents the coefficient. i >0, τ represents the independent variable;

[0137] If s i =0, then and

[0138] according to Set the target of the synchronization controller as follows: This is to synchronize the slave system and the master system.

[0139] To facilitate understanding by those skilled in the art, a set of preferred embodiments is provided below:

[0140] Due to synaptic plasticity, the connection weights between neurons are not constant. Furthermore, not all internal variables of a neural network can be directly observed by other neural networks; for example, memristor synaptic weights in scenario one and electromagnetic radiation in scenario two are considered difficult to observe directly. To simulate synchronization between different brain regions, this embodiment designs an adaptive synchronization method.

[0141] I. Observer Design.

[0142] The general form of the system to be observed can be expressed as:

[0143]

[0144] Where X represents the internal state, A represents the known linear component, F(X) represents the unknown nonlinear component, and U represents the known external control input. It is important to note that these are all matrices. This embodiment improves the method for calculating the gain matrix L based on existing technology, and its corresponding observer can be represented by the following formula.

[0145]

[0146] in, and It is an estimate of the system's state variables and the unknown nonlinear components. Furthermore... Since neural networks can fit any nonlinear function, this embodiment uses an RBF neural network to fit F(X), that is... in It is a weight matrix and It is a Gaussian kernel function. Let the optimal weights be W. * Therefore, the final fitting result of the nonlinear term is... Observation error is denoted as And its derivative has the following form:

[0147]

[0148] in To prove and It is uniform and ultimately bounded, and the corresponding Lyapunov function is constructed as follows:

[0149]

[0150] Then, its derivative is shown below.

[0151]

[0152] Based on the above formula, the online learning rules of the RBF neural network are set as follows: Where η > 0. Substituting this into the above equation, we get:

[0153]

[0154] Let R = M(C) T C+M) -1 [M(C T C+M) -1 ] T Substituting this into the above equation yields the following equation:

[0155]

[0156] in, Therefore, in order to make and For a uniform system to eventually become bounded, it must satisfy the following condition:

[0157]

[0158] Since making Q negative is less feasible, we cannot directly solve for Q by setting Q < 0. Assume the equations of the observed system are smooth, and due to the boundedness of chaos, the Jacobian matrix of F(X) satisfies the following condition:

[0159]

[0160] The operator '≤' is used to determine the element-wise less than. Based on the conclusion in the literature "Y. Wang, R. Rajamani, and DMBevly, 'Observer design for parameter varying differentiable nonlinear systems, with application to slip angle estimation,' IEEE Transactions on Automatic Control, vol. 62, no. 4, pp. 1940–1945, 2017," the following formula can be obtained:

[0161]

[0162] When the following conditions are met, which make the solution easier,

[0163]

[0164] The following equation will hold true:

[0165]

[0166] Finally, it was proven that the observer is uniform and eventually bounded.

[0167] II. Synchronous Controller Design.

[0168] The general form of the main system can be described by the following formula:

[0169]

[0170] Where X = (x1, x2, ..., x n ) T And f i (X) is a nonlinear function.

[0171] The system can be described in its general form by the following formula:

[0172]

[0173] Where Y = (y1, y2, ..., y n ) T And g i (Y) is a nonlinear function. d i (t) represents the external disturbance and let |d i (t)| <D i u i It is an external control input. Its error function is defined by the following formula:

[0174] Ε = Y - X

[0175] where Ε = (ε1, ε2,..., ε n ) T . To synchronize the slave system with the master system, the error between the master and slave systems must be zero. In other words the sliding surface ε of the error function i can be expressed as:

[0176]

[0177] where c i > 0. When s i = 0 then it means that so now the goal of controller design becomes to make Take the Lyapunov function as:

[0178]

[0179] Then its derivative with respect to the independent variable t is:

[0180]

[0181] Because g i (Y) is unknown, so in this embodiment, an RBF neural network is also used to fit this function. After weight learning, the fitting result of the RBF neural network can be described by the following formula:

[0182]

[0183] where W is the weight matrix of the RBF neural network and let W * be the optimal weight. x is the external input of the neural network and its form is δ the error remaining after neural network fitting and let |δ| < B. G is the Gaussian kernel function. So equation 52 can be transformed into:

[0184]

[0185] The zero controller u i has the following form:

[0186]

[0187] where is the estimated weight matrix during neural network learning and let η > 0. Substituting it into equation 54, the following formula can be deduced:

[0188]

[0189] when And η>D+B, then Established. At this time. In order to For this to be valid, a corresponding Lyapunov function needs to be constructed to derive the learning pattern of the weights.

[0190]

[0191] Where λ>0. Then its derivative with respect to the independent variable t is:

[0192]

[0193] When satisfied And η>D+B, thereby and Therefore, this is established. Finally, the learning law of the weights was derived, and the stability of the controller was proven.

[0194] To better illustrate this, the following simulation experiment was conducted in this embodiment:

[0195] Taking the synchronization between scenario one and scenario two as an example, the schematic diagram is as follows: Figure 7 As shown, in this diagram, 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 the memristor HNN in scenario one be the master system, and the memristor HNN in scenario two be the slave system. Their parameters are the same as before. Figure 3 and Figure 4 The experimental parameters are kept consistent. Gains L and P can be solved using the LMI toolbox. For the main system, the solution is:

[0196]

[0197] The solution obtained from the system is as follows:

[0198]

[0199] Controller parameters η and c i The values ​​were set to 5 and 1 respectively. The simulation results are as follows: Figure 8 As shown, the solid line represents the output of the main system, and the dashed line represents the output of the slave system. The experimental results show that the synchronization response of this embodiment is fast and the effect is good.

[0200] This invention also provides three scenarios for HNN image encryption applications, including:

[0201] Due to the large size and high correlation between pixels in images, traditional encryption algorithms such as DES and AES are inefficient and lack strong resistance to attacks. Chaotic image encryption, however, offers advantages such as high efficiency, simple implementation, and strong resistance to attacks due to the pseudo-randomness of chaos. The HNNs for the three scenarios designed in this application exhibit complex dynamic behaviors, which can significantly improve the security of chaotic encryption. The image encryption algorithm mainly includes four steps. First, random sequences S1, S2, and S3 are generated using any one of the three scenario HNNs, with lengths consistent with the number of pixels in the image to be encrypted. Second, a forward addition modulo-divergence algorithm is performed using S1, followed by a reverse addition modulo-divergence algorithm using S2. Third, S3 is used as a mapping rule to scramble the image. Finally, a forward multiplication modulo-divergence algorithm is performed using S1, followed by a reverse multiplication modulo-divergence algorithm using S2. The decryption process is the inverse operation of encryption.

[0202] Security analysis of the above image encryption algorithm:

[0203] (1) Randomness test: Taking the random sequence generated by the 5D multi-vortex memristor HNN in scenario 3 as an example, the NIST P800-22 randomness test was performed. The test results are shown in Table 1. The test results show that all 15 tests of NIST P800-22 were successfully passed, proving that the random sequence has good randomness and can be used for image encryption.

[0204] (2) Histogram Analysis: An ideal encryption algorithm should ensure a uniform grayscale distribution, leaving no valuable statistical information for attackers. From Figure 9 As can be seen, after encryption, the grayscale distribution of the image changes from highly uneven to uniform. Therefore, histogram analysis intuitively demonstrates that this image encryption has good resistance to attacks. Among other things, Figure 9 Figure (a) in the figure is the original image. Figure 9 Figure (b) in the image is an encrypted image. Figure 9 Figure (c) in the figure is the histogram of the original image. Figure 9 Figure (d) in the figure is the histogram of the encrypted image.

[0205] (3) Information entropy analysis: Information entropy can measure the randomness of information. The higher the information entropy, the stronger the randomness. The information entropy calculated by encrypting a 512×512 Lena image using the algorithm in this embodiment is 7.9994. The references “S. Zhang, J. Zheng, X. Wang, Z. Zeng, and S. He, “Initial offset boosting coexisting attractors in memristive multi-double-scroll hopfield neural network,” NonlinearDynamics, vol. 102, no. 4, pp. 2821–2841, 2020” and “Q. Lai, Z. Wan, H. Zhang, and G. Chen, “Design and analysis of multi-scroll memristive hopfield neural network with adjustable memductance and application to image encryption,” IEEE Transactions on Neural Networks and Learning Systems, pp. 1–14, 2022” have citations of 7.9977 and 7.9978 respectively, indicating that this embodiment has better encryption security.

[0206] (4) Correlation Analysis: Due to the high correlation between adjacent pixels in an image, attackers can use this characteristic to infer the grayscale values ​​of adjacent pixels, thereby recovering the entire plaintext image. Therefore, a good encryption algorithm must break the correlation between adjacent pixels. The correlation coefficients of the original 512×512 Lena image were calculated to be 0.9848, 0.9695, and 0.9587, respectively. The correlation coefficients of the encrypted image were -0.0039, -0.0100, and -0.0021, respectively. The calculated data shows that the algorithm has good decorrelation capabilities.

[0207] (5) Differential Attack Analysis: Typically, the number of variable pixel rates (NPCR) and uniform average variation intensity (UACI) can measure the encryption algorithm's ability to resist differential attacks. The ideal values ​​for NPCR and UACI are 99.6094% and 33.4635%, respectively. In this embodiment, the average values ​​of NPCR and UACI for the encrypted 512×512 Lena image are 99.6103% and 33.4621%, respectively. Since these values ​​are very close to the ideal values, it can be demonstrated that this algorithm has excellent resistance to differential attacks.

[0208] (6) Robustness to data loss: Since transmitted images may suffer data loss, it is necessary to have the ability to decrypt lost encrypted images into clear images. In this embodiment, some pixels of the encrypted image are artificially set to zero to simulate data loss. Figure 10 This demonstrates that the image encryption algorithm exhibits excellent resistance to data loss. Specifically, Figure 10 Figure (a) in the figure is the original image. Figure 10 Figure (b) in the image shows an encrypted image with lost data. Figure 10 Figure (c) shows the decrypted image with data loss. The minimum 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 of the original image and the decrypted image after data loss are 1744.8782 and 15.7710, respectively. Both the experimental images and statistical results show that the algorithm has good robustness to data loss.

[0209] Table 1

[0210] 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

[0211] The FPGA circuit implementation in this embodiment includes:

[0212] This embodiment implements the circuit design of the memristor HNN for the three scenarios described above using an FPGA. The integrated development environment (IDE) is Vivado 2018.3. This embodiment uses floating-point IP cores provided by Vivado, including addition, subtraction, multiplication, and division IP cores, comparison IP cores, floating-point to fixed-point IP cores, and fixed-point to floating-point IP cores. The floating-point standard is the IEEE 754 standard, which more precisely includes one sign bit, eight exponent bits, and 23 fraction bits. The FPGA development board used is the ALINX AX7Z100, driven by the Xilinx ZYNQ7000 series chip XC7Z100-2FFG900. Considering the number of output pins on the development board and the maximum value of the system output, the fixed-point precision is 14 bits, including one sign bit, four integer bits, and nine fraction bits. In addition, this embodiment also uses the Cordic IP core to implement the sine function. Since the operating range of this IP core is [-π, π], this embodiment designs an adaptive phase shift circuit to implement the required sine function. It is worth mentioning that, to improve the efficiency of the algorithm, this embodiment uses a piecewise function as shown in the following equation to approximate the hyperbolic tangent operation, instead of the cumbersome exponential operation. Therefore, the parameters of these memristor HNN systems need to be fine-tuned to obtain the ideal multi-vortex. For scenario one, except for parameter k=2, the other parameters are the same as before. Figure 3The experimental parameters remain consistent. For scenario two, except for parameter k = 0.5, the other system parameters are the same as before. Figure 4 The experimental parameters remained consistent.

[0213]

[0214] For the FPGA implementation of memristor HNN in three scenarios, this embodiment uses the RK4 algorithm for iterative solution. Since the RK4 iteration requires calculating four predicted slopes, this embodiment naturally designs four states, which can be named K1, K2, K3, and K4. This design uses a counter to ensure state transitions after a fixed number of clock cycles. The solution iteration starts in state K1, and jumps to the next state K2 after the counter is full. Similarly, K2 jumps to K3, eventually reaching K4. When in state K4 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 point, the FPGA sends a write request signal to the digital-to-analog converter, thereby outputting the final result to the oscilloscope. Due to the robustness of digital circuits, it can be observed that... Figure 11 and Figure 12 The circuit implementation results and Figure 3 and Figure 4 The MATLAB simulation results are exactly the same.

[0215] Furthermore, this embodiment also includes an image encryption circuit that allows the FPGA to directly encrypt (decrypt) images. The FPGA-based image encryption implementation mainly comprises four macroscopic states: initial state, encryption state, waiting state, and display state. Each state transition is controlled by a counter N. The initial state is set to skip the transitional states of the chaotic system. When counter N reaches N1, the initial state transitions to the encryption state. Based on the previously implemented chaotic signal generator, the system sends an enable signal and the solution result to the outside world each time it completes a solution. The enable signal activates the random number generation module to generate an 8-bit random number using the incoming chaotic signal to complete the encryption. When counter N reaches N2, encryption is complete, and the encryption state transitions to the waiting state. In this case, if the display enable signal arrives, the state continues to transition to the display state. In this state, the FPGA outputs the original image (encrypted image) and the encrypted image (decrypted image) to the I / O device. Figure 13 Experimental results of image encryption (decryption) based on FPGA image encryption circuit are presented. Figure 13 Figure (b) shows a scenario where decryption fails due to an incorrect key. Wherein:

[0216] Figure 11 To achieve the experimental results in Scenario 1 using the technical solution of this embodiment, wherein, Figure 11 The middle figure (a) is a global view of the experiment. Figure 11 The middle figure (b) shows f(x) = f odd And the 3-vortex attractor when N=1, Figure 11 The middle figure (c) shows f(x) = f even And the 4-vortex attractor when N=1.

[0217] Figure 12 To achieve the experimental results in Scenario 2 using the technical solution of this embodiment, wherein, Figure 12 The middle figure (a) is a global view of the experiment. Figure 12 The middle figure (b) shows f(x) = f odd And the 3-vortex attractor when N=1, Figure 12 The middle figure (c) shows f(x) = f even And the 4-vortex attractor when N=1.

[0218] Figure 13 The experimental results of image encryption (decryption) using the technical solution of this embodiment are shown below. Figure 13 The middle image (a) shows the original image and the encrypted image. Figure 13 Image (b) shows the encrypted image and the incorrectly decrypted image. Figure 13 The middle image (c) shows the encrypted and decrypted images.

[0219] The embodiments of the present invention have been described in detail above with reference to the accompanying drawings. However, the present invention is not limited to the above embodiments. Within the scope of knowledge possessed by those skilled in the art, various changes can be made without departing from the spirit of the present invention.

Claims

1. A method for constructing HNNs for multiple scenarios, characterized in that, The multi-scenario HNN construction method is applied to image encryption, including: Constructing a non-polynomial memristor is as follows: in, This represents the parameters of the non-polynomial memristor. This represents the output current of the non-polynomial memristor. Indicates memory guidance, Indicates the external input voltage. This represents the internal state variables of the memristor. Indicates time Perform differentiation. This represents the internal state function of the memristor. The expression is: in, This indicates that it was used to induce an odd number of vortices. This indicates that it is used to induce an even number of scrolls, odd-scrolls indicates an odd number of scrolls, and even-scrolls indicates an even number of scrolls; Based on the aforementioned non-polynomial memristor, three scenarios of memristor HNNs are constructed; wherein, the three scenarios of memristor HNNs include memristor synaptic weighted HNNs, HNNs under electromagnetic radiation, and memristor synaptic weighted HNNs under electromagnetic radiation, specifically: The memristor synaptic weighted HNN is constructed as follows: ； The HNN under the electromagnetic radiation is constructed as follows: ； The memristor synaptic weighted HNN under the influence of electromagnetic radiation is constructed as follows: ； in, The parameters representing the memristor synaptic weighted HNN or the parameters of the HNN under electromagnetic radiation are... , , , and These represent the membrane potentials of neurons 1, 2, 3, 4, and 5, respectively. , and Indicates external stimulus current. , and Represents the coefficient. Represents the neuron activation function. , , , and They represent respectively to , , , and The differential, The parameters represent the memristor synaptic weights HNN under the influence of the electromagnetic radiation. Using any of the three scenarios of the memristor HNN, a random sequence for image encryption is generated.

2. An adaptive synchronization method, characterized in that, The multi-scenario HNN construction method described in claim 1 includes: Construct an adaptive state observer and a synchronization controller; In the three scenarios, any two of the memristor HNNs are selected as the master system and the slave system, respectively. Based on the adaptive state observer, the synchronization controller is used to synchronize the master system and the slave system.

3. The adaptive synchronization method according to claim 2, characterized in that, The adaptive state observer is constructed as follows: in, Represents system state variables. This represents the estimate of the unknown nonlinear component. Represents the known linear part. This represents known external control inputs. and Represents the gain matrix. , Representing the observed quantity, Represents the known linear part. Represents the coefficient.

4. The adaptive synchronization method according to claim 3, characterized in that, The unknown nonlinear component is estimated using an RBF neural network, and the fitting expression for the unknown nonlinear component is as follows: in, Represents the weight matrix. This represents the Gaussian kernel function.

5. The adaptive synchronization method according to claim 2, characterized in that, The main system is represented as follows: in, , Represents a nonlinear function. , and It represents the differential.

6. The adaptive synchronization method according to claim 5, characterized in that, The slave system is represented as follows: in, , Represents a nonlinear function. Indicates external disturbance. ,make , Indicates external control input. , and It represents the differential.

7. The adaptive synchronization method according to claim 6, characterized in that, The method of using the synchronization controller to synchronize the master system and the slave system includes: The error function between the master system and the slave system is calculated as follows: in, ; If the slave system and the master system are synchronized, then And the sliding surface of the error function Represented as: in Represents the coefficient. , Indicates the independent variable; like ,but and ; according to The target of the synchronization controller is set as follows: This is to synchronize the slave system and the master system.

Images