# Complex Dynamical Behavior of Locally Active Discrete Memristor-Coupled Neural Networks with Synaptic Crosstalk: Attractor Coexistence and Reentrant Feigenbaum Trees

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## Abstract

**:**

## 1. Introduction

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_{6}diode effect for neuromorphic computation by inducing synaptic plasticity through light signal activation. Obviously, their work has great significance. However, it was found that the synaptic plasticity of the memristor was much stronger. It could even simulate the information-processing patterns of the human brain [37,38]. This is very helpful in studying the various rich behaviors of neural networks. It is significant to construct a model of a memristor-coupled neural network. Therefore, many researchers have studied the complex dynamical behavior of memristor-coupled neural networks [39,40]. The majority of the research in recent years has been focused on the continuous type of memristor [41,42,43,44,45,46]. There are relatively few studies on discrete memristors [47]. Ren et al. [48] proposed a new discrete memristor model, built a new 3D hyperchaotic map by coupling, and investigated its dynamical behavior. Lu et al. [49] built a discrete memristor with sinusoidal conductance to apply it to a coupled Rulkov neuron map. The effects of coupling strength and control parameters on synchronization were investigated. Ma et al. [50] proposed a local active discrete memristor model and performed a series of dynamical analyses. It provided a good basis for further analyzing discrete memristors. Therefore, the research of discrete memristors and their application to discrete neural networks is very attractive and valuable [51,52].

## 2. Mathematical Model

#### 2.1. Model of the Memristor

_{n}) is the memductance, which is the reciprocal of the memristance; φ

_{n}is the state flux of the memristor, which is the magnetic flux, and its relation to charge is included in the definition of the memristor; v

_{n}denotes the input voltage; and i

_{n}denotes the output current. The internal state function is denoted by F(φ,v), which is a Lipschitz function. a, b, and c are the three constant parameters of the memristor, which are used to regulate the performance of the memristor in this research. The values of the three constant parameters were set after a lot of experiments to achieve a better performance of the memristor. The parameters are set as a = 0.1, b = 1, and c = 9.

_{n}= Asin(ω

_{n}). A = 0.11, ω = 0.06, and φ

_{0}= 0; the characteristic curve of v

_{n}and i

_{n}with the number of iterations is presented in Figure 1a. The v

_{n}-i

_{n}curves of the local active discrete memristor model are illustrated in Figure 1b–d. A = 0.11, f = 0 in Figure 1b. A = 0.11 and ω = 0.06 in Figure 1c. ω = 0.06 and f = 0 in Figure 1d. It can be seen that the curve shows a contraction of the pinch loop through the origin. When the frequency ω grows from 0.06 to 0.12, the hysteresis loop monotonically decreases in area. It gradually shrinks to a single-valued function with increasing frequency. As the amplitude A grows from 0.09 to 0.11, the hysteresis loop area monotonically enlarges. Obviously, the presented model satisfies the conditions of the generalized memristor.

#### 2.2. DMCAN Map

_{1}and k

_{2}are damping factors for neuronal refractoriness. The parameters a and b are positive constants, and f(u) = (1 + exp(−u/c)) − 1 is a logarithmic function for the steepness parameter c > 0.

_{1}= 0.65, k

_{2}= 0.11, a = 1, b = 0.8, and c = 0.05 and the initial values x

_{0}= 0.2, y

_{0}= −0.12 were set. The phase diagram shown in Figure 3a was plotted. k

_{1}ranges from 0.2 to 1.8 and other values remain constant, the BD of which is shown in Figure 3b and the LEs of which is shown in Figure 3d. k

_{1}= 0.62, and the chaotic sequence of the model is shown in Figure 3c.

_{1}and b

_{2}are the crosstalk between the two memristors.

#### 2.3. Fixed-Point Characterization

_{1}, b

_{2}) = (1.045, 0.8, 0.05, 0.62, 0.28, 0.11, 0.1, 0.1, 0.1). The equilibrium point calculated from the above equation and the given parameters is (8, 8, 8, 8, 8, 0, 0). The equilibrium equations are as follows:

## 3. Hidden Dynamical Analysis of DMCAN with Multi-Stability

#### 3.1. Variation of the Crosstalk Strength b_{2}

_{0}, y

_{0}, z

_{0}, m

_{0}, f

_{0}, g

_{0}) = (0.1, 0.1, 0, 0.1, 0, 0) and (x

_{0}, y

_{0}, z

_{0}, m

_{0}, f

_{0}, g

_{0}) = (0.1, −0.1, 0, −0.1, 0, 0). The parameters are specified as (a, b, c, d, e, h, k, b

_{1}) = (1.18, 0.8, 0.05, 0.62, 0.4, 0.11, 0.1, 0.1). b

_{2}takes values in the range (0, 0.8). The corresponding BD and LEs are shown in Figure 5.

_{2}are 0.01, 0.456, and 0.779, the phase diagrams and the chaotic sequences of the coexisting attractors of different topologies can be obtained. There is still good correspondence. Figure 6a,b shows the coexistence of a chaotic attractor and a chaotic attractor. Figure 6c,d shows the coexistence of a chaotic attractor and a multi-periodic attractor. Figure 6e,f shows the coexistence of a multi-periodic attractor and a multi-periodic attractor. The three groups of coexisting phenomena nicely demonstrate the system’s complex dynamical behavior. The multiple stability is verified for the system.

#### 3.2. Variation of the Coupling Coefficient k between Neurons

_{0}, y

_{0}, z

_{0}, m

_{0}, f

_{0}, g

_{0}) = (0.1, 0.1, 0, 0.1, 0, 0) and (x

_{0}, y

_{0}, z

_{0}, m

_{0}, f

_{0}, g

_{0}) = (0.1, −0.1, 0, −0.1, 0, 0). The parameters are specified as (a, b, c, d, e, h, b

_{1}, b

_{2}) = (1.18, 0.8, 0.05, 0.62, 0.75, 0.11, 0.1, 0.1). The range of k is taken to be (0, 0.25). Then, the corresponding BD and LEs are shown in Figure 7.

#### 3.3. Variation of Neural Network System Parameter e

_{0}, y

_{0}, z

_{0}, m

_{0}, f

_{0}, g

_{0}) = (0.2, 0.2, 0.3, 0.3, 0.1, 0.1) and (x

_{0}, y

_{0}, z

_{0}, m

_{0}, f

_{0}, g

_{0}) = (0.2, 0.2, −0.3, 0.3, 0.1, 0.1) separately. The parameters are set to (a, b, c, d, h, k, b

_{1}, b

_{2}) = (1.045, 0.8, 0.05, 0.62, 0.11, 0.1, 0.1, 0.1). The value of e ranges from (0, 0.8). The corresponding BD and LEs are depicted in Figure 9.

## 4. State Transfer and Complexity

#### 4.1. State Transfer

_{0}, y

_{0}, z

_{0}, m

_{0}, f

_{0}, g

_{0}) = (0.1, 0.1, 0, 0.1, 0, 0) and the parameters are set to (a, b, c, d, e, h, k, b

_{1}, b

_{2}) = (1.18, 0.8, 0.05, 0.62, 0.4, 0.11, 0.1, 0.1, 0.584). The chaotic sequence is shown in Figure 12a, where it can be seen that the system realizes the transition between chaotic and periodic states in a small range. Figure 12b,c is the phase diagram of the corresponding state.

#### 4.2. SE Complexity Analysis

_{1}and b

_{2}. The largest value of SE is 0.9 when parameters b

_{1}and b

_{2}are varied, and an improved chaotic performance of the system is achieved. The above initial values as well as the parameters remain unchanged. With b

_{1}= 0.1 and b

_{2}= 0.1 set, the complexity SE with initial values x

_{0}and z

_{0}is shown in Figure 13b. The complexity varying with initial values x

_{0}and f

_{0}is shown in Figure 13c. It can be seen that the maximum value of SE is still around 0.9, and the complexity is high in most regions. The red region with high parameter interval complexity should be chosen as much as possible in the figure. The system may generate chaotic attractors in this region when taking the relevant parameters, while the blue region has lower parameter interval complexity. The analysis shows that the system is more complex.

## 5. DSP Implementation

_{2}) = (0.1, 0.01). The remaining parameters are the same as in Section 3.3. The phase diagram in Figure 15a can be obtained. Figure 15b then shows its corresponding final experimental result realized by the DSP oscilloscope. The DSP hardware realization is described in Figure 16.

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**(

**a**) DC V-I plot over the region −2 < φ < 2; (

**b**) the memristor POP plot with five balancing points.

**Figure 3.**Dynamical analysis of Aihara neuron simulation plots: (

**a**) phase diagram; (

**b**) BD; (

**c**) chaotic sequence; and (

**d**) LEs.

**Figure 5.**The BD and LEs for different initial values when the crosstalk coefficient b

_{2}changes. (

**a**,

**b**) values are (0.1, 0.1, 0, 0.1, 0, 0); (

**c**,

**d**) values are (0.1, −0.1, 0, −0.1, 0, 0).

**Figure 6.**Phase diagram and chaotic sequences of attractor coexistence. (

**a**,

**b**) b

_{2}= 0.01; (

**c**,

**d**) b

_{2}= 0.456; and (

**e**,

**f**) b

_{2}= 0.779.

**Figure 7.**The BD and LEs for different initial values for variations in the coupling parameter k. (

**a**,

**b**) values are (0.1, 0.1, 0, 0.1, 0, 0); (

**c**,

**d**) values are (0.1, −0.1, 0, −0.1, 0, 0).

**Figure 8.**Phase diagram and chaotic sequences of attractor coexistence. (

**a**,

**b**) k = 0.025; (

**c**,

**d**) k = 0.16; (

**e**,

**f**) k = 0.1149.

**Figure 9.**BD and LEs for different initial values when the coefficient e varies. (

**a**,

**b**) values are (0.2, 0.2, 0.3, 0.3, 0.1, 0.1); (

**c**,

**d**) values are (0.2, 0.2, −0.3, 0.3, 0.1, 0.1).

**Figure 11.**The change of the value of the system parameter a; the BD of e shows the Feigenbaum tree phenomenon. (

**a**) a = 1.145; (

**b**) a = 1.16; (

**c**) a = 1.175; (

**d**) a = 1.18; (

**e**) a = 1.185; and (

**f**) a = 1.19.

**Figure 12.**The state transfer in the DMCAN map. (

**a**) the chaotic sequence map; (

**b**,

**c**) the corresponding attractor phase maps.

**Figure 13.**Three-dimensional SE complexity images of the system, (

**a**) varying with crosstalk coefficients b

_{1}, b

_{2}; (

**b**) varying with initial values x

_{0}and z

_{0}; and (

**c**) varying with initial values x

_{0}and f

_{0}.

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## Share and Cite

**MDPI and ACS Style**

Liu, D.; Wang, K.; Cao, Y.; Lu, J.
Complex Dynamical Behavior of Locally Active Discrete Memristor-Coupled Neural Networks with Synaptic Crosstalk: Attractor Coexistence and Reentrant Feigenbaum Trees. *Electronics* **2024**, *13*, 2776.
https://doi.org/10.3390/electronics13142776

**AMA Style**

Liu D, Wang K, Cao Y, Lu J.
Complex Dynamical Behavior of Locally Active Discrete Memristor-Coupled Neural Networks with Synaptic Crosstalk: Attractor Coexistence and Reentrant Feigenbaum Trees. *Electronics*. 2024; 13(14):2776.
https://doi.org/10.3390/electronics13142776

**Chicago/Turabian Style**

Liu, Deheng, Kaihua Wang, Yinghong Cao, and Jinshi Lu.
2024. "Complex Dynamical Behavior of Locally Active Discrete Memristor-Coupled Neural Networks with Synaptic Crosstalk: Attractor Coexistence and Reentrant Feigenbaum Trees" *Electronics* 13, no. 14: 2776.
https://doi.org/10.3390/electronics13142776