# Dynamic Analysis and FPGA Implementation of Fractional-Order Hopfield Networks with Memristive Synapse

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## Abstract

**:**

## 1. Introduction

## 2. Design of Fractional-Order HNN Model with Memristive Synapse

#### 2.1. Fractional-Order Operators

**Definition 1**

**Definition 2**

#### 2.2. Fractional-Order Memristor Model

#### 2.3. Fractional-Order Memristive HNN

#### 2.4. Stability Analysis

## 3. Complexity Through Spectral Entropy Analysis

## 4. Implementation

_{i}computes the convolution between the i-th state and $\psi \left[k\right]$. The outputs of each block are then combined to compute ${x}_{i}[k+1]$ as the difference between them. Finally, a register is used to update the state $x\left[k\right]$.

#### Implementation Results

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Pinched hysteresis loop. (

**a**) For different $\alpha $-orders when $F=1$. (

**b**) For different frequencies when $\alpha =0.97$.

**Figure 2.**Intersections of the curves ${f}_{1}({x}_{2},{x}_{4})$ (red curve) and ${f}_{2}({x}_{2},{x}_{4})$ (blue curve) under the fixed parameters $\kappa =2.0$, $p=1$, and $q=3.3$. The external input parameter I is varied across (

**a**) $I=-3$, (

**b**) $I=0$, and (

**c**) $I=4$.

**Figure 3.**Eigenvalues of the Jacobian matrix in the complex plane, evaluated at equilibrium points for different values of I. For $I=-3$, the equilibrium points are $(0.378,-0.301,0.089,-0.376,0.411)$; for $I=0$, they are $(0,0,0,0,0.5)$; and, for $I=4$, they are $(-0.504,0.426,-0.114,0.534,0.622)$.

**Figure 4.**Phase portraits of the fractional-order system (6) with $\alpha =0.97$ and initial conditions $({x}_{1}\left(0\right),{x}_{2}\left(0\right),{x}_{3}\left(0\right),{x}_{4}\left(0\right),{x}_{5}\left(0\right))=(0,0,0,1,1)$. (

**a**) 3D projection on the ${x}_{1},{x}_{4},{x}_{5}$ plane; (

**b**) 3D projection on the ${x}_{2},{x}_{3},{x}_{4}$ plane.

**Figure 5.**(

**a**) Bifurcation diagram showing the local maxima of the state ${x}_{1}$ as a function of the control parameter $\alpha $, with the blue color corresponding to initial conditions $(0,0,0,1,1)$ and the red color representing the initial conditions $(0,0,0,-1,-1)$. This represents a commensurate case, where the fractional orders for both the neuron equations and the memristor equation are identical. (

**b**) Bifurcation diagram showing an incommensurate case where the fractional orders for the neuron equations in (6) (${\alpha}_{i}$, with $i=1,2,3,4$) are fixed at ${\alpha}_{i}=0.97$, while the fractional-order derivative for the memristor equation ${\alpha}_{5}$ is varied within the range ${\alpha}_{5}\in (0.9,1)$.

**Figure 6.**Basin of attraction in the ${x}_{4}\left(0\right)-{x}_{5}\left(0\right)$ plane with ${x}_{1}\left(0\right)=0$, ${x}_{2}\left(0\right)=0$, ${x}_{3}\left(0\right)=0$, and $\alpha =0.97$, coexisting attractor above (blue) and below (red).

**Figure 7.**Phase portraits of the fractional-order system (6) with $\alpha =0.97$ and initial conditions $(0,0,0,1,2.5)$ for the blue color and $(0,0,0,-1,-2.5)$ for the red color, where (

**a**) represents ${x}_{1}$−${x}_{2}$, (

**b**) ${x}_{1}$−${x}_{3}$, (

**c**) ${x}_{1}$−${x}_{4}$, and (

**d**) ${x}_{2}$−${x}_{3}$.

**Figure 8.**Phase portraits of the fractional-order system (6) with $\alpha =0.97$ and initial conditions $(0,0,0,1,1)$ for the blue color and $(0,0,0,-1,-1)$ for the red color, where (

**a**) represents ${x}_{1}$−${x}_{2}$, (

**b**) ${x}_{1}$−${x}_{3}$, (

**c**) ${x}_{1}$−${x}_{4}$, and (

**d**) ${x}_{2}$−${x}_{3}$.

**Figure 9.**Complexity diagram for the fractional-order (6) with $\alpha =0.97$, ${x}_{1}\left(0\right)=0$, ${x}_{2}\left(0\right)=0$, ${x}_{3}\left(0\right)=0$, ${x}_{4}\left(0\right)\in [-3,3]$, and ${x}_{5}\left(0\right)\in [-3,3]$.

**Figure 10.**Values of SE from the time series of ${x}_{1}$ obtained via (6), when $\alpha =0.97$ and the initial conditions were set as $[{x}_{1}\left(0\right),{x}_{2}\left(0\right),{x}_{3}\left(0\right),{x}_{4}\left(0\right),{x}_{5}\left(0\right)]$=$[0,0,0,1,1]$. (

**a**) Varying the parameter $c\in [0.1,3]$. (

**b**) Varying the parameter $d\in [2,5]$.

**Figure 11.**Experimental setup for the implementation of the proposed system: (

**a**) IFM 127 VCA to 24 VDC, 30-watt power supply, model DN1020; (

**b**) National Instruments sbRIO-9626 with XC6SLX45 FPGA; and (

**c**) PROTEK 100 MHz oscilloscope, model P-2510.

**Figure 12.**Block diagram description for the implementation of the proposed system using the GL integration method. This block description represents (15), where blocks ${f}_{i}{h}^{\alpha}$ compute the dynamics of the i-th state and blocks FIR

_{i}compute the convolution between the i-th state and the binomial coefficient array $\psi \left[k\right]$.

**Figure 13.**Implementation of the convolution between the binomial coefficients and the most recent ${L}_{m}$ samples from each state. The design utilizes a single Multiply–Accumulate (MAC) structure, where a mod-${L}_{m}$ counter, referred to as “coef_counter”, is employed to iterate through all the coefficients.

**Figure 14.**Approximation to activation function by a piecewise approximation. (

**a**) Hyperbolic tangent function $tanh\left(x\right)$ (black color continuous line) and its piecewise approximation $\u03f5\left(x\right)$ (red color dashed line) described in Equation (17) for $N=4$ and $M=10$. (

**b**) Magnitude of the error in the approximation $|tanh\left(x\right)-\u03f5(x\left)\right|$.

**Figure 15.**Phase portraits registered in the P-2510 scope for the implemented fractional-order system, where (

**a**) represents ${x}_{1}$−${x}_{2}$, (

**b**) ${x}_{1}$−${x}_{3}$, (

**c**) ${x}_{1}$−${x}_{4}$, and (

**d**) ${x}_{2}$−${x}_{3}$.

**Table 1.**Equilibrium points and eigenvalues of $\mathbf{J}$ for distinct values of the external current I.

I | Equilibrium Point | ${\mathit{\lambda}}_{2,3}$ | ${\mathit{\lambda}}_{4,5}$ |
---|---|---|---|

−3 | (0.378, −0.301, 0.089, −0.376, 0.411) | 0.268 ± 5.673i | 0.485 ± 2.258i |

0 | (0, 0, 0, 0, 0.5) | 0.304 ± 6.047i | 0.696 ± 2.541i |

4 | (−0.504, 0.426, −0.114, 0.534, 0.622) | 0.237 ± 5.408i | 0.325 ± 1.976i |

**Table 2.**Resource utilization summary for the implementation of the proposed system on the XC6SLX45 FPGA.

Device Utilization | Used | Total | Percent |
---|---|---|---|

Total Slices | 6727 | 6822 | 98.6% |

Slice Registers | 16,393 | 54,576 | 30.0% |

Slice LUTs | 25,953 | 27,288 | 95.1% |

Block RAMs | 8 | 116 | 6.9% |

DSP48s | 15 | 58 | 25.9% |

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**MDPI and ACS Style**

Anzo-Hernández, A.; Zambrano-Serrano, E.; Platas-Garza, M.A.; Volos, C.
Dynamic Analysis and FPGA Implementation of Fractional-Order Hopfield Networks with Memristive Synapse. *Fractal Fract.* **2024**, *8*, 628.
https://doi.org/10.3390/fractalfract8110628

**AMA Style**

Anzo-Hernández A, Zambrano-Serrano E, Platas-Garza MA, Volos C.
Dynamic Analysis and FPGA Implementation of Fractional-Order Hopfield Networks with Memristive Synapse. *Fractal and Fractional*. 2024; 8(11):628.
https://doi.org/10.3390/fractalfract8110628

**Chicago/Turabian Style**

Anzo-Hernández, Andrés, Ernesto Zambrano-Serrano, Miguel Angel Platas-Garza, and Christos Volos.
2024. "Dynamic Analysis and FPGA Implementation of Fractional-Order Hopfield Networks with Memristive Synapse" *Fractal and Fractional* 8, no. 11: 628.
https://doi.org/10.3390/fractalfract8110628