A method for improving the synchronization capability of a neuron circuit by coupling a non-volatile memristor

Abstract

The application discloses a method for improving the synchronization ability of a neuron circuit by using a non-volatile memristor coupling. On the basis of an existing HR and tabu-learning neuron model, a non-volatile switchable memristor model is introduced to simulate long-term and short-term plasticity switchable synapses, and a memristor-coupled double-neuron model is established based on the same. The non-volatile parameter of the memristor is used to realize neuron synchronization control in a neuron coupling system for the first time. When the parameter of the memristor is volatile, a coupling neuron system needs a large coupling strength to realize synchronization. When the parameter of the memristor is non-volatile, the coupling neuron system can realize synchronization under a small coupling strength.

Description

Technical Field

[0001] This invention belongs to the field of synchronization control technology in nonlinear dynamics. It utilizes the biomimetic properties of memristors as artificial synapses. By introducing the coupling effect of memristor synapses into two heterogeneous neuron models, when the memristor switches to non-volatile mode, it can be used to simulate the long-term plasticity of neuronal synapses and improve the synchronization ability of neurons. Background Technology

[0002] Neurons in the brain interact through synapses, facilitating brain function, with synaptic plasticity playing a crucial role in information processing. Synaptic plasticity enables adaptation to the environment, learning, and memory, and is generally categorized into long-term and short-term plasticity. Short-term plasticity involves transient changes in synaptic efficacy, while long-term plasticity is characterized by persistent changes in synaptic strength and is considered the foundation of memory storage in the human brain. In recent years, applications of models such as Hindmarsh-Rose (HR), tabu-learning, and FitzHugh-Nagumo (FHN) neurons and their synapses have revealed the important role of synaptic plasticity in the synchronous activity of neurons in different brain regions and in interneuronal information transmission, demonstrating various synchronous and firing behaviors of neurons.

[0003] Memristors possess powerful nonlinearity, plasticity, memory, and storage properties, describing a rich functional relationship between charge and magnetic flux while also exhibiting inherent biomimetic characteristics. Therefore, using memristors to simulate biological synapses in neurons helps to reproduce neuronal synchronization and firing activity. Non-volatility is a crucial characteristic of memristors, representing their unique memory storage capability, which can be used to simulate the long-term plasticity of neurons. Currently, memristors have important applications in simulating neuronal synapses, synaptic plasticity, brain memory storage and synaptic weight update mechanisms, as well as steady-state plasticity.

[0004] Memristor synapses between neurons play a decisive role in synchronization, a key mechanism for the brain to achieve higher functions such as perception, motor control, and memory integration. Currently, memristor synapses significantly influence the synchronization of memristor-coupled neural networks by considering coupling strength, initial values ​​of the memristor, the form of the memristor equation, and the control of local active parameters. However, the impact of memristor synaptic plasticity on neuronal synchronization ability has been neglected. Furthermore, the integrity of brain function is related to neuronal heterogeneity; therefore, the heterogeneity caused by the differences in real-world neurons should also be taken into account.

[0005] Therefore, how to simulate the synchronization and dynamic behavior of synaptic plasticity (short-term plasticity and long-term plasticity) in heterogeneous neural networks, and to analyze in detail the relationship between the synchronization and firing activities of neurons and the memristor characteristics of non-volatile switchable memristors, is a problem that urgently needs to be solved by those skilled in the art. Summary of the Invention

[0006] 1. Purpose of the invention

[0007] To enhance the synchronization capability of neuronal circuits using non-volatile memristors, the main steps include:

[0008] Step 1: Consider a class of models that implement non-volatile switchable memristors through parameter tuning.

[0009] Step 2: Analyze the switchable mechanism by which memristors achieve non-volatility

[0010] Step 3: Consider a general model of coupled heterogeneous neurons based on a non-volatile switchable magnetically controlled memristor.

[0011] Step 4: Constructing synchronization judgment criteria based on Lyapunov function stability analysis and Lipschitz's law

[0012] Step 5: Consider two heterogeneous neuron coupling models based on non-volatile switchable magnetically controlled memristors, and verify them from the perspective of dynamic simulation.

[0013] Step 6: Consider an evaluation index for the synchronization of memristor-coupled heterogeneous neurons under different memristor characteristics from the perspective of numerical simulation.

[0014] Step 7: Consider a heterogeneous neuron coupling circuit based on a non-volatile switchable magnetically controlled memristor for simulation verification.

[0015] The specific process of step 1 above includes: deriving a general model of a magnetically controlled memristor based on Ohm's law, i.e., the principle of Chua's memristor:

[0016] (1)

[0017] Where V and I represent the voltage across the memristor and the current flowing through the memristor, respectively; It is a memorized derivative; It is the magnetic flux of the memristor. The first derivative of magnetic flux with respect to time; function It is related to the materials of electrical components and the corresponding physical operating mechanisms. To maintain the form of the memory derivative function... Always positive, we introduce a simple quadratic term memorization form. Where p and q are set to 3 and 5 respectively; consider the function To achieve non-volatile switchability, an absolute value function is used. To achieve this memristor characteristic, the final mathematical model of the non-volatile switchable memristor is as follows:

[0018] (2)

[0019] Where V and I represent the voltage and current flowing through the memristor, respectively; p and q are the gains controlling the magnitude of the memristor current; and g and m are the memristor characteristic parameters, i.e., the non-volatile modulation parameters. In reality, the memristor parameters can be adjusted according to the actual situation. For example, when the parameters are set to m=1 and g=1, the memristor model is a non-volatile memristor; when m=1 and g=3, the memristor model is a volatile memristor. Considering the convergence of the memristor, the parameter g must always satisfy g>0.

[0020] Then, step 2 is performed to analyze the switchable mechanism by which the memristor achieves non-volatility. Non-volatility can be classified using the point-of-op (POP) diagram. If the number of negative slope intersections on the POP diagram is greater than or equal to 2, it is non-volatile; otherwise, it is volatile. Setting equation (1) V=0, MATLAB can be used to clearly demonstrate... and The relationship between them is the power outage diagram. (See attached diagram.) Figure 1 As shown in (a), when m=1 and g=1, the positive slope line and the two negative slope lines are respectively... The zero axis intersects, forming three zero points Q in POP. −1 Q0 and Q1 (i.e. When V=0 The relationship between g and g is such that two of the points are intersections of negative slopes, therefore the memristor exhibits non-volatility. When the value of g increases beyond 2, such as g=3, it exhibits volatile behavior. (Appendix) Figure 1 The analytical method is the same when g is fixed and m is changed in (b).

[0021] Furthermore, to simplify the above analysis process, we take the distance L between the negative slope curve and the origin as the key feature of the non-volatile switching of this model, and obtain the relationship between the distance L between the zero point of the negative slope curve and the origin, which can intuitively reflect the non-volatility of the memristor, as shown below:

[0022] (3)

[0023] Where L is the intersection point Q of the negative slope curve in the power outage diagram and the x-axis. i (i≠0), the distance between the memristor and the origin Q0 of the coordinate axis. When L=0, the memristor can be identified as a volatile memristor, which can be used to simulate the short-term plasticity of neuronal synapses; when L>0, the memristor can be identified as a non-volatile memristor, which can be used to simulate the long-term plasticity of neuronal synapses.

[0024] Furthermore, in step 3, a general model of coupled heterogeneous neurons based on a non-volatile switchable magnetically controlled memristor is proposed. X and Y represent the state variables of the two neurons, respectively, and F(X) and G(Y) are the function models of the two heterogeneous neurons. The general memristor model is introduced as a synapse to connect the two neurons. The connection topology of the heterogeneous neurons is shown in the attached figure. Figure 2 As shown, the general mathematical model for memristor synaptic coupling of heterogeneous neurons is as follows:

[0025] (4)

[0026] Where X and Y are the state variables of the two neurons, respectively, and φ is the state variable of the memristor; k is the coupling strength connecting the neurons, β=1; F(X) and G(Y) are two neuron function models considering boundedness and differentiability; H(φ) is a function determined by the non-volatile switchable memristor. This model will be applied to the analysis process in step 4.

[0027] Furthermore, the detailed process of step 4 includes: transforming the general coupling model proposed in step 3 into an error function model; deriving the critical coupling strength required for synchronization of heterogeneous neurons using Lipschitz's theorem; verifying the derivation results based on Lyapunov function stability analysis; and roughly deriving the final critical coupling strength based on numerical simulation results, thus deriving a method for improving the synchronization capability of neuronal circuits using non-volatile memristor coupling. The specific process is as follows:

[0028] (1) Using the general memristor-coupled heterogeneous neuron model proposed in step 3, the error function of the model is expressed as follows:

[0029] (5)

[0030] Where X and Y are the state variables of the two neurons, and e = XY represents the synchronization error. This represents the first derivative of the synchronization error function with respect to time t.

[0031] (2) Assume that the error system in equation (5) satisfies the Lipschitz condition, that is:

[0032] (6)

[0033] Where L2 is the Lipschitz constant, and F(X)−G(X) represents the deviation of functions F and G near X. Since simulation results show that heterogeneous neurons are asymptotically synchronized, the error can be ignored when the coupling strength is sufficiently large. Therefore, we conclude that by integrating the above expressions, we can always determine a suitable Lipschitz constant L0 such that it satisfies the condition ||F(X)−G(X)||≤L0||XY||.

[0034] (7)

[0035] This further shows that equation (5) will satisfy the exponential rate asymptotic stability, that is, neurons can achieve synchronization.

[0036] (3) Using the error function V(t,e)=e T e, as the Lyapunov function of the error system, proves the above conclusion. The derivative of this function along the direction of the error system can be expressed as:

[0037] (8)

[0038] Therefore, we can obtain:

[0039] (9)

[0040] Where L0 is the Lioschitz constant, which can usually be obtained by calculating the infinite norm of the Jacobian matrix (i.e., L0 = ||J||). ∞ When L0 < 2kW(φ) and t→∞, the Lyapunov function satisfies V(t,e)→0.

[0041] Furthermore, the critical coupling strength k at which neurons achieve synchronization can be obtained. c The relationship between the constant and the Lipschitz constant is as follows:

[0042] (10)

[0043] Where L0 is the Lipschitz constant. Since the Lipschitz constant L0 is determined by the neuron's internal dynamics function, this constant is fixed for a specific neuron.

[0044] Furthermore, since the synchronicity of the memristor-coupled neurons is related to the function W(φ), which is controlled by the variable φ, when the coupled neurons are in a synchronized state (i.e., e=X−Y=0), the voltage V across the neurons will be approximately 0. At this point, it is necessary to determine the evolution of φ, which simplifies to the analysis of the interrupted electrogram in step 2. Based on the kinetic function in equation (1), the evolution of φ can be written as the following mathematical function:

[0045] (11)

[0046] Furthermore, by solving equation (11), we can obtain the functional conditions satisfied by the state variable φ:

[0047] (12)

[0048] Here, coefficients C1 and C2 depend on the initial value of φ. Since g is greater than zero, φ may converge to different constant values ​​as t approaches ∞, specifically:

[0049] (13)

[0050] The value of C2 (−1≤C2≤1) is directly proportional to the initial value of φ.

[0051] Furthermore, by substituting equation (13) into equation (10), the critical synchronization value k can be derived. c And obtain k c The distance L mentioned in step 2 satisfies the following equation.

[0052] (14)

[0053] The value of L is given in equation (2) in step 2. Since the Lipschitz constant L0 is determined only by the neuron's internal functions, it is a fixed value for a specific neuron. Given the critical coupling strength k... c When the memristor is in a volatile state, L0 remains almost constant and can be calculated using numerical simulation. For fixed memristor parameters (p=3, q=5), it can be concluded that for a non-volatile memristor (L≥1), the required critical coupling strength will always be less than the critical coupling strength under the action of a volatile memristor synapse.

[0054] Furthermore, considering the universality of non-volatile switchable memristors in heterogeneous neuron coupling models in step 5, existing heterogeneous neuron models are introduced into equation (4) to analyze and verify the results of theoretical derivation from a dynamic perspective. First, heterogeneous tabu-tabu neuron models with mismatched memristor coupling parameters and different types of HR-tabu neuron models are introduced into step 3 to obtain the two mathematical models of heterogeneous neurons based on non-volatile switchable magnetically controlled memristors considered in step 4. Specifically, the first model uses parameters w in completely identical tabu neurons. i As a heterogeneous parameter forming the heterogeneity of neurons, the second model considers replacing the second neuron in the former model with an HR neuron model to form the heterogeneity of neurons.

[0055] Tabu learning neurons, developed from Hopfield neural networks, can accurately simulate various synchronization and firing behaviors of neurons. Different parameters w... i The internal functions of the Tabu neurons in the given case are respectively taken as X and Y in equation (4), i.e., F(X) and G(Y), to obtain the following Tabu-Tabu heterogeneous neuron coupling model based on memristors:

[0056] (15)

[0057] Where x 1,2 y 1,2 These represent the dimensionless state variables of the neuron, and also its membrane potential and internal variables; k is the coupling strength connecting the neurons, expressed using the coupling term k(e+fφ). 2 This is used to reflect the electromagnetic induction effect with quadratic characteristics. Other parameters are fixed as a = 0.2, b = 0.3, c = 0.5, d = 1, w² = 5.2, p = 3, q ​​= 5, β = 1. Through w... 1,2 The differences reflect heterogeneity.

[0058] The second-order Hindmarsh-Rose (HR) neuron model has low complexity but can accurately simulate and predict frequency-current relationships. It is often used to simulate various synchronization and firing behaviors of neurons. Substituting the HR neuron model into X, F(X) corresponding to the general model in equation (4) yields the HR-tabu heterogeneous neuron coupling model based on memristors as shown below:

[0059] (16)

[0060] Where x1 and y1 represent the membrane potential variable and the recovery variable related to intracellular current of the HR neuron, respectively. Other parameters can be fixed as a1 = 1, b1 = 3, c1 = 1, d1 = 5, a2 = 0.2, b2 = 0.3, c2 = 0.5, d2 = 1, w = 5.2, p = 3, q ​​= 5 and β = 1.

[0061] Furthermore, in step 6, an evaluation index is considered to analyze the correlation of memristor-coupled heterogeneous neurons under different characteristics from the perspective of numerical simulation, so as to clearly demonstrate the synchronization ability of neurons under different memristor characteristics. The results of neuronal synchronization ability under different memristor characteristics are visualized and analyzed to obtain the attached... Figure 3 4, 5, and appendix Figure 6The corresponding variables and their error time series plots. The synchronization factor R, from the perspective of mean-field theory, characterizes the correlation of time series, and is defined as:

[0062] (17)

[0063] Where N represents the number of sequences in the system, for the neuron coupling model proposed above, N = 2. , t i and t e Indicates the start and end times of the calculation. Clearly, the larger the value of R, the higher the correlation of the sequences. When R=1, it indicates that the coupled neuron model has achieved complete synchronization.

[0064] Finally, in step 7, based on Kirchhoff's circuit laws, the circuit characteristics of the operational amplifier, and the voltage-current relationship of the circuit elements, an equivalent circuit model of a non-volatile switchable memristor-coupled HR-tabu neuron is proposed and simulated for verification. Its equivalent circuit equation can be derived as follows:

[0065] (18)

[0066] Where R0 = 10kΩ, C i =100μF. Other circuit parameters can be set as follows: R a1 =10kΩ, R b1 =3.33kΩ, R c1 =10kΩ, R d1 =2kΩ, R a2 =50kΩ, R b2 =33.3kΩ, R c2 =20kΩ, R d2 =10kΩ, R β =20kΩ, R m =10kΩ, E=1V.

[0067] Finally, draw the circuit schematic in PSIM software as follows: Figure 10 As shown, a physical realizability simulation experiment was conducted using PSIM software, and the results were compared and verified with the numerical solution method using MATLAB.

[0068] As can be seen from the above technical solution, compared with the prior art, the present invention discloses a method for improving the synchronization capability of neuronal circuits by using non-volatile memristor coupling, which has the following beneficial effects:

[0069] 3. Beneficial effects:

[0070] This invention proposes a method to switch memristors between volatile and non-volatile modes. Specifically, it achieves non-volatile switchability through changes in two internal parameters of the memristor, which is related to the plasticity switching of neuronal synapses. These parameters are then introduced as coupling synapses into two different heterogeneous neuron models, establishing two models of non-volatile switchable memristors coupled to heterogeneous neurons. Next, the correlation of neuronal sequences under the coupling effects of memristors with different memristor characteristics (volatile and non-volatile) is analyzed from the perspective of dynamic numerical simulation. It is observed that the correlation of neuronal sequences differs when the memristor switches to different memristor characteristics. Specifically, when the coupling strength is fixed, heterogeneous neurons under the action of non-volatile memristor synapses exhibit better synchronization ability compared to volatile neurons, which is another way to improve neuronal synchronization. Finally, the mechanism of this process is analyzed and verified using Lipschitz's theorem, Lyapunov function stability theory, and circuit simulation experiments. Since synaptic plasticity of neurons is closely related to long-term learning, memory, and synchronization of neurons in the human hippocampus, these results and analytical methods help to understand and study the neural signal transmission and processing mechanisms of heterogeneous neurons under synaptic coupling. Attached Figure Description

[0071] Figure 1 The diagram shows the de-energization diagram of the memristor and the relationship between the minimum distance L between the intersection point of the negative slope and the origin in the gm dual-parameter spatial domain.

[0072] Figure 2 The diagram shown is a memristor-coupled heterogeneous neuron structure.

[0073] Figure 3 The figure shows the coupling strength-correlation coefficient curves of tabu-tabu neurons under different memristor parameters (characteristics).

[0074] Figure 4 The diagram shows the critical coupling strength of tabu-tabu neurons under different memristor parameters (characteristics).

[0075] Figure 5 The figure shows the sequence correlation coefficients of HR-tabu neurons under different memristor parameters (characteristics).

[0076] Figure 6 Neuronal membrane potential variables and their error time series diagrams under different memristor parameters (characteristics)

[0077] Figure 7 The figure shows the correlation of HR-tabu neurons under different memristor parameters (characteristics) in the two-parameter spatial domain.

[0078] Figure 8The figure shown represents the critical coupling strength of HR-tabu neurons under different memristor parameters (characteristics).

[0079] Figure 9 The critical coupling strength value for synchronization derived using Lipschitz's theorem and numerical simulation.

[0080] Figure 10 Non-volatile switchable memristor coupled HR-tabu neuron circuit model diagram

[0081] Figure 11 Simulation timing diagram of non-volatile switchable memristor coupled HR-tabu neuron circuit Detailed Implementation

[0082] To provide a clearer understanding of the objectives, features, and advantages of this invention, the following detailed description is provided in conjunction with the accompanying drawings and specific embodiments:

[0083] This invention discloses a method for improving the synchronization capability of neuronal circuits using non-volatile memristor coupling. The method includes enabling the memristor to switch between volatile and non-volatile states using only two parameters stored within it; formulating the key characteristics of this non-volatile switching process using a power-off diagram; and finally, presenting the results in… Figure 1 In detail, with m=1 fixed, the memristor power-off diagrams generated under different parameters g are shown below. Figure 1 As shown in (a), when m=1 and g=1, the positive slope line and the two negative slope lines are respectively... The zero axis intersects, forming three zero points Q in POP. −1 Q0 and Q1 (i.e. (The relationship with φ when V=0). Two of these are intersections of negative slopes, therefore the memristor exhibits non-volatility. When the value of g increases beyond 2, such as g=3, it exhibits volatile behavior. A fixed g=2 corresponds to... Figure 1 (b) Analytical methods and Figure 1 (a) Consistent. Generally speaking, when the parameters satisfy 2m<g(g> When the value is 0, the memristor is volatile, simulating the short-term plasticity of a neuron's synapse; otherwise, it is non-volatile, simulating the long-term plasticity of a neuron. Next, the key change characteristic between the intersection of negative and positive slopes, namely the distance L, is formalized to obtain a theoretically valid mg two-parameter planar graph of distance L. It can be seen that the light-colored diagonal line in the middle is divided into volatile (left) and non-volatile (right) parameter spaces on either side.

[0084] Then, the memristor is introduced into the heterogeneous neuron model, resulting in a structural diagram of a non-volatile switchable memristor coupled to a heterogeneous neuron. (See [link]). Figure 2The two neurons with different gray levels represent two heterogeneous neurons that are connected to each other via synapses. The synapse model is shown in the enlarged ellipse between the two models. Consider introducing memristors as artificial synapses to achieve the connection between neurons, with direct coupling as the coupling method.

[0085] The corresponding general mathematical model for coupling is as follows:

[0086] (19)

[0087] Where X and Y are the state variables of two neurons, respectively; F(X) and G(Y) are the function models of two heterogeneous neurons; kW(φ)(YX) shows the direct coupling form with W(φ) as the memristor function and coupling strength k. Subsequently, the two heterogeneous neuron mathematical models are substituted to obtain two different memristor-coupled heterogeneous neuron mathematical models, as shown in equations (4) and (5).

[0088] To assess the synchronization between two memristor coupled neurons, a synchronization factor R is defined based on the time-series correlation of the two coupled neurons, according to mean-field theory:

[0089] (20)

[0090] Where N represents the number of sequences in the system, and for the neuron coupling model proposed above, N=2. , t i and t e Indicates the start and end times of the calculation. .

[0091] Dynamic numerical simulation and theoretical derivation:

[0092] The mathematical model of equation (4) was solved in Matlab using the Runge-Kutta method, and the synchronization factor R under different coupling strengths k was calculated, as shown below. Figure 3 The coupling strength-correlation curves are shown. Figures (a), (b), (c), and (d) show control implementations with different parameters, corresponding to w1=0.9 and 1.5 respectively. Non-volatile memristors (m=1, g=0.2, 0.5, 2, see...) were used in the experiment. Figure 3 (a); g=2, m=1, 3, 6, see Figure 3 (b) and volatile memristors (m=1, g=3, 4, see Figure 3 (a)(c); g=2, m =0.1, 0.5, see Figure 3(b)(d)). The results show that for non-volatile memristors, the R value reaches 1 faster, while for volatile memristors it is slower. Therefore, non-volatile memristors facilitate synchronization between non-identical neurons. When the parameter mismatch between two coupled neurons decreases, Figure 3 (c)(d) When the parameter mismatch between the two coupled neurons decreases (w1=1.5, w2=5.2), the R value still approaches 1 faster, although in this case, the R value is more likely to exceed that in cases of larger parameter mismatch. Based on the above analysis, it can also be concluded that the synchronization process of heterogeneous neurons always satisfies progressive synchronization.

[0093] Furthermore, in order to more accurately study the synchronization ability of heterogeneous tabu learning neurons coupled with memristors under different memristor characteristics, Figure 4 The key coupling strength k was recorded. c The synchronization factor R = 0.98 ( Figure 3 (The dashed line in the middle). Figure 4 (a) shows that, given parameters m=0.5, 1.0, 1.5 and w1=0.9, w2=5.2, k c The relationship with g. As the value of g increases, k... c It also increases accordingly; when g is greater than 2m, k c It stabilizes at approximately 0.93. Since the memristor transitions from non-volatile to volatile when g is greater than 2m, the synchronization capability decreases abruptly as the memristor becomes volatile. These results are obtained through… Figure 4 (b) Given parameters g=1.0, 2.0, 3.0 and w1=0.9, w2=5.2, k c The relationship with m was verified, where k is greater than g / 2 (i.e., non-volatile). c The value decreases from a constant of approximately 0.93 to a small value. Meanwhile, for cases with small neuron parameter mismatches (w1=1.5, w2=5.2), besides the critical coupling strength k for synchronization... c Aside from changing to 0.26 instead of 0.93, the other results are similar. Figure 4 (c)(f) show the critical coupling strength k when the memristor is in a non-volatile state, for w1 = 0.9 and 1.5. c The relationship with distance L (L>1). Clearly, as distance L increases, the synchronization threshold k... c The synchronization decreases as the distance L increases when the memristor is non-volatile.

[0094] Furthermore, the synchronization effect of non-volatile memristors on another heterogeneous neuron type (HR-tabu neurons) is considered. Figure 5As shown in (a) and (b), the relationship between the synchronization factor R and the coupling strength k under different parameters g and m indicates that when g ≤ 2m (for non-volatile memristors, i.e., m = 1, g = 0.1, 0.5, 2), R increases towards 1 more rapidly; while when g > 2m (for volatile memristors, i.e., m = 1, g = 3, 4), R increases towards 1 more slowly. This suggests that two different neurons can more easily achieve synchronization through coupling via a non-volatile memristor.

[0095] Then, the specific differences are shown through time-series visualization of the variables. Figure 6 (a)-(h) illustrate the variations in membrane potentials x1, x2, and synchronization error ∆x(t) in the HR-tabu neural network at given coupling strengths (k = 0.5 and k = 10). The former corresponds to non-volatile memristor coupling (m = 1, g = 1), and the latter to volatile memristor coupling (m = 1, g = 3). Clearly, neurons in non-volatile memristor coupling exhibit smaller synchronization errors.

[0096] Next, in order to more intuitively and comprehensively demonstrate the relationship between memristor characteristics and the synchronization of the HR-tabu neural network, Figure 7 Figures (a) and (b) illustrate the relationship between the synchronization factor R of two coupled neurons and the parameters g and k when m=1 (or g=2). The purple areas in the figures mark parameters with larger synchronization factors R (R≥0.98), which determine the critical coupling value k for synchronization given parameters g or m. c Since the parameters m and g determine the plasticity of a synapse, the synchronization of a neural network is affected by synaptic plasticity. Under the influence of non-volatile synapses, the HR-Tabu neural network can achieve a higher synchronization factor R with a smaller coupling strength. Conversely, when the synapse is in a volatile state, a higher coupling strength is required to achieve synchronization.

[0097] Consistent with the above, it demonstrates that according to Figure 8 (a)( Figure 8 (b) gives the synchronization critical value k when m=1 (parameter m corresponds to parameter g=2). c The relationship between the memristor's non-volatility and the parameter g, and the effect of this non-volatility on synchronization, is the same as in the case of coupled heterogeneous tabu-tabu neurons. The critical coupling strength k for synchronization... c First, it increases with increasing g, and then reaches its maximum value (k) when the memristor transitions from a non-volatile state (g≤2m) to a volatile state (g>2m). c =1), such as Figure 8 As shown in (a), the value of m is 0.5, 1.0, or 1.5. Simultaneously, when the memristor transitions from a volatile to a non-volatile state, the synchronization threshold k... cAs m increases, it exhibits a quadratic function trend.

[0098] Subsequently, through the derivation of the synchronization judgment criteria based on the stability analysis of Lyapunov functions and Lipschitz's law in step 4, and combined with numerical simulation methods, the results were obtained. Figure 9 The results are shown, where the graph represents simulation data and the solid line represents theoretical data. Specifically, given m=0.5, 1.0, 1.5 (or g=1, 2, 3) respectively, the synchronous critical coupling strength k obtained from formula (13) is... c The relationship with g and Figure 9 The numerical results in (a) and (b) are very consistent. Meanwhile, on the double logarithmic coordinate axis based on formula (15), the critical value k... c Relationship with distance L and Figure 9 The numerical results in (c) also show a very good agreement, with the parameter w1 in f1(x) set to 0.9 and 1.5 respectively. Clearly, the parameter mismatch between the two coupled neurons affects the absolute value of the synchronization critical coupling strength (related to L0), but does not affect the critical value k. c The relationship between L and L.

[0099] Finally, based on Kirchhoff's circuit laws, the circuit characteristics of the operational amplifier, and the voltage-current relationship of the circuit elements, an equivalent circuit model of a non-volatile switchable memristor-coupled HR-tabu neuron can be derived as follows:

[0100] (twenty one)

[0101] Where R0 = 10kΩ, Ci = 100μF; other circuit parameters can be set as follows: Ra1 = 10kΩ, Rb1 = 3.33kΩ, Rc1 = 10kΩ, Rd1 = 2kΩ, Ra2 = 50kΩ, Rb2 = 33.3kΩ, Rc2 = 20kΩ, Rd2 = 10kΩ, Rβ = 20kΩ, Rm = 10kΩ, E = 1V.

[0102] Based on the above mathematical model, a schematic diagram of a non-volatile switchable memristor-coupled HR-tabu neuron circuit is designed, as follows: Figure 10As shown, it mainly includes: (a) HR neuron circuit model, (b) tabu learning neuron circuit model, (c) tabu learning neuron activation function circuit model, and (d) non-volatile switchable memristor circuit model. Specifically, the outputs x2 and -W(φ) from circuits (b) and (d) are used as partial coupling terms input into circuit (a); the outputs x1 and W(φ) from circuits (a) and (d) are used as partial coupling terms input into circuit (b); the tabu-learning neuron activation function -f(x) from (c) is input into circuit (b); and the memristor equivalent circuit module from (d) serves as a coupling module, outputting coupling terms -W(φ) and W(φ) to circuits (a) and (b) respectively.

[0103] Using the electronic circuit simulation software PSIM, Figure 11 (a)-(h) illustrate the dynamics and synchronization error of the memristor-coupled neuron model, and these results are consistent with... Figure 6 The phenomena shown in (a)-(h) are consistent. According to equations (5) and (18), g=1 / (R) g C i ), m = 1 / (R) m C i ), k=1 / ( R k Ci). When the resistance R of the memristor g Set to 10kΩ, R k Set to 20kΩ (or R) k When the resistance is set to 1kΩ, the memristor is in a non-volatile state (corresponding to m=1, g=1, k=0.5 or 10), and the coupled neurons are in R k Unsynchronized at 20kΩ, but at R k Synchronization is achieved at 1kΩ, as shown in Figures 13(a)-(d). However, when the memristor is in a volatile state (corresponding to m=1, g=3, k=0.5 or 10), i.e. R... g Set to 3.33kΩ, R k Set to 20kΩ (or R) k When the Ω is set to 1kΩ, the coupled neurons in R k =20kΩ and R k Synchronization still failed even at 1kΩ, as shown in Figures 13(e)-(h). Therefore, when the coupled memristor is in a non-volatile state, the coupled neurons are more likely to achieve synchronization.

[0104] This invention proposes a method for enhancing the synchronization capability of neuronal circuits using non-volatile memristor coupling. By simulating the synaptic plasticity of neurons through the non-volatile nature of memristors, dynamic enhancement of neuronal synchronization is achieved, providing another neuromorphic computing method. Compared to traditional fixed coupling methods, this method utilizes the storage characteristics of memristors to significantly improve the synchronization capability of neuronal networks, providing a new technical path for brain-like computing, intelligent sensing, and neuromorphic hardware design. Experimental simulations and theoretical derivations show that in typical heterogeneous neuronal networks, this method indirectly enhances the connection strength between neurons by simulating the long-term and short-term plasticity of synapses, until neurons achieve synchronization, demonstrating significant application value. Through specific implementation methods, this paper systematically elucidates the principles and implementation methods of this invention, aiming to help understand the core ideas of the method. For those skilled in the art, modifications or equivalent substitutions can be made to the specific implementation methods without departing from the overall concept of this invention. Furthermore, adaptive adjustments can be made to the implementation methods and application scope based on the ideas of this invention. In summary, the content of this specification should not be construed as a limitation of this invention.

Claims

1. A method for enhancing the synchronization capability of neuronal circuits using non-volatile memristor coupling, characterized in that... Includes the following steps: Step 1: Consider a class of models that implement non-volatile switchable memristors through parameter tuning. Based on the state equation of a memristor and the related Ohm's law and Chua's theorem, a general electromagnetically controlled memristor is represented as follows: (1) Where V and I represent the voltage and current flowing through the memristor, respectively; W(φ) is the memristor derivative; and φ is the magnetic flux through the memristor. The first derivative of the magnetic flux with respect to time is represented; the function g(φ,V) is related to the material of the electrical component and the corresponding physical operation mechanism; p and q are the gains that control the magnitude of the memristor current; g and m are the memristor characteristic parameters, i.e., the non-volatile modulation parameters. Step 2: Analyze the switchable mechanism by which memristors achieve non-volatility The non-volatility of a memristor can be quickly determined using a power-off diagram. The non-volatility switchable mechanism of the memristor model proposed in the first step is formalized, yielding a relationship between the distance L from the zero point of the negative slope of the power-off diagram and the origin. This relationship visually reflects the non-volatility of the memristor, as shown below: (2) Where L is the intersection point Q of the negative slope curve in the power outage diagram and the x-axis. i (i≠0), the distance between the memristor and the origin Q0 of the coordinate axis; when L=0, the memristor can be identified as a volatile memristor, which can be used to simulate the short-term plasticity of neuronal synapses; when L>0, the memristor can be identified as a non-volatile memristor, which can be used to simulate the long-term plasticity of neuronal synapses. Step 3: Consider a general model of coupled heterogeneous neurons based on a non-volatile switchable magnetically controlled memristor. As shown below: (3) Where X and Y are the state variables of the two neurons, respectively, φ is the magnetic flux of the memristor, k is the coupling strength connecting the neurons, β=1; F(X) and G(Y) are two neuron function models that satisfy boundedness and differentiability; H(φ) is a function determined by the non-volatile switchable memristor, and this model will be applied to the analysis process in step 4. Step 4: Constructing synchronization judgment criteria based on Lyapunov function stability analysis and Lipschitz's law For the general memristor-coupled heterogeneous neuron model proposed in step 3, the synchronization conditions of the heterogeneous neuron coupling model can be approximately solved by combining Lyapunov function stability analysis and Lipschitz theorem with simulation data. The method is as follows: First, using the general memristor-coupled heterogeneous neuron model proposed in step 3, the error function of the model is expressed as follows: (4) Where e=XY represents the synchronization error. This represents the first derivative of the synchronization error function; Assume the error system satisfies the Lipschitz condition, i.e.: (5) Where L2 is the Lipschitz constant, and F(X)−G(X) represents the deviation of functions F and G near X. Based on the conditions that the two types of neurons and their associated error functions satisfy boundedness and differentiability, the inequality ||F(X)−G(X)||≤C0 can be established, where the constant C0 becomes asymptotically negligible as the error approaches zero. When X→Y, there obviously exists a Lipschitz constant L1 such that ||F(X)−G(X)||≤L1||XY|| holds. Therefore, by integrating the above expressions, we can always determine a suitable Lipschitz constant L0 such that it satisfies the condition ||F(X)−G(Y)||≤L0||XY||. (6) Equation (4) satisfies the asymptotic stability of the exponential rate; This can be proven using relevant theorems: Using the error function V(t,e)=e T As the Lyapunov function of the error system, the derivative along the direction of the error system can be expressed as: (7) Therefore, we can obtain: (8) When L0 < 2kW(φ) and t→∞, the Lyapunov function satisfies V(t,e)→0; Then, the critical coupling strength k at which neurons achieve synchronization can be obtained. c The relationship between the constant and the Lipschitz constant is as follows: (9) Since the Lipschitz constant L0 is determined by the neuron's internal dynamics function, this constant is fixed for a specific neuron. Based on the memristor dynamics function in equation (1), when neurons are synchronized, the voltage V across the memristor is approximately 0. The evolution of φ can be theoretically derived to obtain the following mathematical function: (10) Then, by solving equation (10), we can obtain the function conditions satisfied by the state variable φ: (11) The value of C2 (−1≤C2≤1) is directly proportional to the initial value of φ; Finally, by substituting equation (11) into equation (9), the critical synchronization value k can be derived. c And obtain k c The distance L mentioned in step 2 satisfies the following equation: (12) The value of parameter L is given in equation (2) in step 2. Since the Lipschitz constant L0 is determined only by the internal function of the neuron, it is a fixed value for a specific neuron. L0 can be calculated by numerical methods. For fixed memristor parameters, i.e. p = 3 and q = 5, it can be concluded that for non-volatile memristors, i.e. L ≥ 1, the required critical coupling strength will always be less than the critical coupling strength under the action of volatile memristor synapses. Step 5: Consider two heterogeneous neuron coupling models based on non-volatile switchable magnetically controlled memristors, and verify them from the perspective of dynamic simulation. Consider two cases of heterogeneous neuron coupling models: parameter mismatch in the neuron model and heterogeneity caused by different types, as shown below: (13) Equation (13) represents the parameter w coupled through the memristor. i Tabu-Tabu heterogeneous neuron coupling model under mismatch; where x 1,2 y 1,2 Let be the dimensionless state variables of the neurons, and k be the coupling strength connecting the neurons. Using the coupling term k(e+fφ) 2 This is used to reflect the electromagnetic induction effect with quadratic characteristics; other parameters are fixed at a=0.2, b=0.3, c=0.5, d=1, w²=5.2, p=3, q=5, β=1, and the result is obtained by using w... 1,2 Differentiating to reflect heterogeneity; (14) Equation (14) represents the heterogeneity caused by memristor coupling of different types of neurons, i.e., the mathematical model of HR-tabu neuron coupling based on memristor; where x1 and y1 represent the membrane potential variable of HR neurons and the recovery variable related to intracellular current of neurons, respectively. Other parameters can be fixed as a1 = 1, b1 = 3, c1 = 1, d1 = 5, a2 = 0.2, b2 = 0.3, c2 = 0.5, d2 = 1, w = 5.2, p = 3, q ​​= 5 and β = 1; Step 6: Consider an evaluation index for the synchronization of memristor-coupled heterogeneous neurons under different characteristics from the perspective of numerical simulation. The synchronization factor R, from the perspective of mean-field theory, characterizes the correlation of time series, and is defined as follows: (15) Where N represents the number of sequences in the system, for the neuron coupling model proposed above, N = 2. , t i and t e Indicates the start and end times of the calculation. Obviously, the larger the value of R, the higher the correlation of the sequences. When R=1, it means that the coupled neuron model has achieved complete synchronization. Step 7: Consider a heterogeneous neuron coupling circuit based on a non-volatile switchable magnetically controlled memristor for simulation verification. Based on Kirchhoff's circuit laws, the circuit characteristics of operational amplifiers, and the voltage-current relationship of circuit elements, an equivalent circuit model of a non-volatile switchable memristor-coupled HR-tabu neuron is given (14). The circuit equation can be derived as follows: (16) Where R0 = 10kΩ, C i =100μF; other circuit parameters can be set as follows: Ra1=10kΩ, Rb1=3.33kΩ, R c1 =10kΩ, R d1 =2kΩ, R a2 =50kΩ, R b2 =33.3kΩ, R c2 =20kΩ, R d2 =10kΩ, R β =20kΩ, R m =10kΩ, E=1V; Finally, the schematic diagram was drawn using PSIM software and a physical realizability simulation experiment was conducted to compare and verify the results with the numerical solution method in MATLAB. The above steps present a method for improving the synchronization capability of neuronal circuits by using non-volatile memristors for coupling.

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