# Episodic Memory and Information Recognition Using Solitonic Neural Networks Based on Photorefractive Plasticity

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Biological vs. Solitonic Neural Networks

- Intensity of input signals;
- Persistence of signals along a specific path.

## 3. Architecture of a Learning SNN

## 4. Mathematical Model of the SNN

_{Bias}on the input values x

_{i}. W is not constant and can change according to the learning of the unit. As can be observed, the matrix is nonlinear, which means it depends on X values as well. Therefore, the resolution of Equation (1) occurs recursively, as the learning process is “slow”—that is, it requires reinforcing over time the information channel to be stored.

## 5. A 4-Bit SNN Working as an Episodic Memory

_{j=1,3}is

_{j=2}is instead

## 6. Comparison with the SNN Dynamics

_{e}representing the electric field present along the extraordinary direction of the crystal, which is the sum of the bias and the locally induced screening.

#### 6.1. Learning and Memorization Processes

- ▪
- Single input beam, which corresponds to a single illuminated pixel in a 4 × 4 pixel matrix (1-digit, 4-bit numbers);
- ▪
- Two input beams, which correspond to two illuminated pixels (2-digit, 4-bit numbers);
- ▪
- Three input beams, which correspond to three illuminated pixels (3-digit, 4-bit numbers).

#### 6.1.1. 1-Digit, 4-Bit Numbers

#### 6.1.2. 2-Digit, 4-Bit Numbers

#### 6.1.3. 3-Digit, 4-Bit Numbers

#### 6.2. Materials and Memorization

_{3}). Often, these materials show an intrinsic memory, in the sense that the photoexcited charges can be localized in states that can hardly relax. For example, bulk crystals of lithium niobate showed extremely slow dielectric relaxations, considered almost permanent: soliton waveguides in these crystals might be still active months after their writing [22].

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Diagram of the functional parallelism between BNNs and SNNs. Both systems are able to modify their structure, to process the incoming information and to store them. The structural variations are due, in the biological case, to variations in the density of neurotransmitters, while in the solitonic case in the density of electrical charges.

**Figure 2.**Schematic architecture of an SNN used for bit-to-bit storage of information. (

**A**) A single neural unit structure is constituted by a single X-junction; (

**B**) one processing level in the network consists of several units in parallel; (

**C**) successive levels of the network are connected in series with each other; (

**D**) even-order levels have two fewer units. The suppression of these units is replaced with the insertion of the total reflections at the lateral edges of the substrate.

**Figure 3.**Structural scheme of the SNN used as an episodic memory. W represents the computational weight of the node. W

^{(1)}is the computational weight relative to the node of the first neuron in Layer 1; W

^{(2)}is the computational weight relative to the node of the second neuron in Layer 2; W

^{(3)}is the computational weight relative to the node of the neuron in Layer 2; W

^{(4)}is the computational weight relative to the node of the first neuron in Layer 3; W

^{(5)}is the computational weight relative to the node of the first neuron in Layer 3. The inputs at the entrance to the network are x

_{i}while the signals at the end of the whole processing are y

_{i}.

**Figure 4.**Represents the evolution of the intensities in the three layers with respect to the two input configurations, 1−0−0−0 and 0−0−0−1 (1-digit, 4-bit numbers), for the three cases of SJR: (

**a**) SJR = 0.7−0.3; (

**b**) SJR = 0.8−0.2; (

**c**) SJR = 0.9−0.1.

**Figure 5.**Represents the energy flow in a three-layer network learning two input numbers, 1−0−0−1 and 0−1−1−0, for the three SJRs: (

**a**) SJR = 0.7−0.3; (

**b**) SJR = 0.8−0.2; (

**c**) SJR = 0.9−0.1.

**Figure 6.**Represents the evolution of the intensities in the three layers with respect to the three input configurations, 1−1−1−0 and 0−1−1−1, for the three cases of imbalance: (

**a**) SJR = 0.7−0.3; (

**b**) SJR = 0.8−0.2; (

**c**) SJR = 0.9−0.1.

**Figure 7.**The solver code is divided into three sections: the first one writes a balanced network based on X-junction units by solving the nonlinear wave equations of four light beams within a nonlinear medium. The output of this section corresponds to the refractive index mapping of the junction, which is now used as the initial structure for the propagation of N signal beams, which are injected into 4 input arms (i.e., an N-digit, 4-bit number). The signal beams would slightly modify specific channels of the network arms. By recursive feedbacks, the network slowly learns the inserted information. The last phase is then the testing one: using a single iteration, further “unknown” signals enter the trained refractive index mapping and generate specific output patterns, depending on whether they are recognized or not.

**Figure 8.**Perfectly balanced SNN network. At this stage, the network is ready to accept input information and change according to them.

**Figure 9.**Training and testing phases are reported for 1-digit, 4-bit numbers. In the first raw, the training phases are reported. In the following raws, the testing experiments are reported. Those for which training and testing numbers coincide are framed in red.

**Figure 10.**Amplitudes of the output signals for the different training numbers. As shown, for each training number, there is only one channel whose amplitude exceeds a threshold value identified by the dashed horizontal line.

**Figure 11.**Training and testing phases are reported for 2−digit, 4−bit numbers. In the first raw, the training phases are reported. In the following raws, the testing experiments are reported. Those for which training and testing numbers coincide are framed in red.

**Figure 12.**Amplitudes of the output signals for the different training 2-digit, 4-bit numbers. As shown, for each training number, there is only one case in which both outputs exceed the threshold (dashed horizontal line): (

**a**–

**f**) report the cases of validation after having trained the possible combinations of channels respectively. However, the network is also able to recognize single-trained digits.

**Figure 13.**Training and testing phases are reported for 3-digit, 4-bit numbers. In the first raw, the training phases are reported. In the following raws, the testing experiments are reported. Those for which training and testing numbers coincide are framed in red.

**Figure 14.**Amplitudes of the output signals for the different training 2-digit, 4-bit numbers. As shown, for each training number, there is only one case in which both outputs exceed the threshold (dashed horizontal line): (

**a**–

**d**) report the cases of validation after having trained the possible combinations of channels respectively. However, the network is able to recognize also single trained digits.

**Table 1.**Unbalance obtained between the two channels of the soliton neuron after 20 iterations of the mathematical model as a function of the SAT parameter.

SAT | SJR: Single Junction Ratio |
---|---|

1 | 0.7−0.3 |

2 | 0.8−0.2 |

3 | 0.9−0.1 |

**Table 2.**Results of the simulation outputs of the mathematical model for 1-digit, 4-bit numbers and for different SJR values: (a) SJR = 0.7−0.3, (b) SJR = 0.8−0.2, and (c) SJR = 0.9−0.1.

(a) | Single Junction Ratio: 0.7−0.3 | (b) | Single Junction Ratio: 0.8−0.2 | (c) | Single Junction Ratio: 0.9−0.1 | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

channel | 1 | 2 | 3 | 4 | channel | 1 | 2 | 3 | 4 | channel | 1 | 2 | 3 | 4 | ||

INPUT | 1 | 0 | 0 | 0 | INPUT | 1 | 0 | 0 | 0 | INPUT | 1 | 0 | 0 | 0 | ||

OUTPUT | 0.224 | 0.274 | 0.134 | 0.368 | OUTPUT | 0.132 | 0.198 | 0.089 | 0.581 | OUTPUT | 0.054 | 0.109 | 0.040 | 0.797 | ||

INPUT | 0 | 1 | 0 | 0 | INPUT | 0 | 1 | 0 | 0 | INPUT | 0 | 1 | 0 | 0 | ||

OUTPUT | 0.251 | 0.524 | 0.056 | 0.169 | OUTPUT | 0.133 | 0.670 | 0.003 | 0.193 | OUTPUT | 0.063 | 0.837 | 0.000 | 0.100 | ||

INPUT | 0 | 0 | 1 | 0 | INPUT | 0 | 0 | 1 | 0 | INPUT | 0 | 0 | 1 | 0 | ||

OUTPUT | 0.169 | 0.056 | 0.505 | 0.271 | OUTPUT | 0.193 | 0.003 | 0.670 | 0.133 | OUTPUT | 0.100 | 0.000 | 0.837 | 0.063 | ||

INPUT | 0 | 0 | 0 | 1 | INPUT | 0 | 0 | 0 | 1 | INPUT | 0 | 0 | 0 | 1 | ||

OUTPUT | 0.368 | 0.134 | 0.271 | 0.227 | OUTPUT | 0.581 | 0.089 | 0.195 | 0.135 | OUTPUT | 0.796 | 0.041 | 0.108 | 0.055 |

**Table 3.**Results of the simulation outputs of the mathematical model for 2-digit, 4-bit numbers and for different SJR values: (a) SJR = 0.7−0.3, (b) SJR = 0.8−0.2, and (c) SJR = 0.9−0.1.

(a) | Single Junction Ratio: 0.7−0.3 | (b) | Single Junction Ratio: 0.8−0.2 | (c) | Single Junction Ratio: 0.9−0.1 | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

channel | 1 | 2 | 3 | 4 | channel | 1 | 2 | 3 | 4 | channel | 1 | 2 | 3 | 4 | ||

INPUT | 1 | 0 | 0 | 1 | INPUT | 1 | 0 | 0 | 1 | INPUT | 1 | 0 | 0 | 1 | ||

OUTPUT | 0.5747 | 0.4122 | 0.4285 | 0.5846 | OUTPUT | 0.6641 | 0.3066 | 0.3345 | 0.6948 | OUTPUT | 0.7849 | 0.1639 | 0.1977 | 0.8536 | ||

INPUT | 0 | 1 | 1 | 0 | INPUT | 0 | 1 | 1 | 0 | INPUT | 0 | 1 | 1 | 0 | ||

OUTPUT | 0.4212 | 0.586 | 0.5717 | 0.4211 | OUTPUT | 0.3236 | 0.6887 | 0.666 | 0.3218 | OUTPUT | 0.1851 | 0.827 | 0.8063 | 0.1816 | ||

INPUT | 1 | 0 | 1 | 0 | INPUT | 1 | 0 | 1 | 0 | INPUT | 1 | 0 | 1 | 0 | ||

OUTPUT | 0.3902 | 0.336 | 0.6588 | 0.615 | OUTPUT | 0.3314 | 0.2209 | 0.7648 | 0.7029 | OUTPUT | 0.1933 | 0.1017 | 0.8785 | 0.8266 | ||

INPUT | 0 | 1 | 0 | 1 | INPUT | 0 | 1 | 0 | 1 | INPUT | 0 | 1 | 0 | 1 | ||

OUTPUT | 0.6121 | 0.6637 | 0.3377 | 0.3865 | OUTPUT | 0.6999 | 0.7731 | 0.2227 | 0.3043 | OUTPUT | 0.8264 | 0.8869 | 0.1029 | 0.1839 | ||

INPUT | 1 | 1 | 0 | 0 | INPUT | 1 | 1 | 0 | 0 | INPUT | 1 | 1 | 0 | 0 | ||

OUTPUT | 0.508 | 0.792 | 0.2054 | 0.4946 | OUTPUT | 0.3635 | 0.8366 | 0.1482 | 0.6518 | OUTPUT | 0.1944 | 0.9057 | 0.0787 | 0.8212 | ||

INPUT | 0 | 0 | 1 | 1 | INPUT | 0 | 0 | 1 | 1 | INPUT | 0 | 0 | 1 | 1 | ||

OUTPUT | 0.4946 | 0.2054 | 0.7704 | 0.5296 | OUTPUT | 0.6518 | 0.1482 | 0.8086 | 0.3915 | OUTPUT | 0.8213 | 0.0787 | 0.8804 | 0.2196 |

**Table 4.**Results of the simulation outputs of the mathematical model for 3-digit, 4-bit numbers and for different SJR values: (a) SJR = 0.7−0.3, (b) SJR = 0.8−0.2, and (c) SJR = 0.9−0.1.

(a) | Single Junction Ratio: 0.7−0.3 | (b) | Single Junction Ratio: 0.8−0.2 | (c) | Single Junction Ratio: 0.9−0.1 | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

channel | 1 | 2 | 3 | 4 | channel | 1 | 2 | 3 | 4 | channel | 1 | 2 | 3 | 4 | ||

INPUT | 1 | 1 | 1 | 0 | INPUT | 1 | 1 | 1 | 0 | INPUT | 1 | 1 | 1 | 0 | ||

OUTPUT | 0.6581 | 0.8561 | 0.7274 | 0.7583 | OUTPUT | 0.4993 | 0.8724 | 0.807 | 0.8213 | OUTPUT | 0.2856 | 0.9174 | 0.8994 | 0.8975 | ||

INPUT | 0 | 1 | 1 | 1 | INPUT | 0 | 1 | 1 | 1 | INPUT | 0 | 1 | 1 | 1 | ||

OUTPUT | 0.7653 | 0.7273 | 0.8416 | 0.6658 | OUTPUT | 0.8315 | 0.8083 | 0.8508 | 0.5094 | OUTPUT | 0.9078 | 0.9012 | 0.8952 | 0.2958 | ||

INPUT | 1 | 0 | 1 | 1 | INPUT | 1 | 0 | 1 | 1 | INPUT | 1 | 0 | 1 | 1 | ||

OUTPUT | 0.7305 | 0.4804 | 0.9308 | 0.8583 | OUTPUT | 0.8067 | 0.3503 | 0.959 | 0.884 | OUTPUT | 0.8973 | 0.1888 | 0.983 | 0.9309 | ||

INPUT | 1 | 1 | 0 | 1 | INPUT | 1 | 1 | 0 | 1 | INPUT | 1 | 1 | 0 | 1 | ||

OUTPUT | 0.8445 | 0.9342 | 0.4924 | 0.7288 | OUTPUT | 0.8591 | 0.9612 | 0.3675 | 0.8122 | OUTPUT | 0.8913 | 0.9827 | 0.2056 | 0.9205 |

Material | LiNbO_{3} |
---|---|

linear ordinary refractive index | n_{oo} = 2.3247 |

linear extraordinary refractive index | n_{oe} = 2.2355 |

electro-optic coefficient | r_{33} = 3.1 × 10^{−11} m/V |

electric bias | E_{bias} = 36 kV/cm |

writing beam wavelength | 532 nm |

writing beam power | 10 μW |

signal beam | 632 nm |

signal beam power | 0.5 μW |

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**MDPI and ACS Style**

Bile, A.; Tari, H.; Fazio, E.
Episodic Memory and Information Recognition Using Solitonic Neural Networks Based on Photorefractive Plasticity. *Appl. Sci.* **2022**, *12*, 5585.
https://doi.org/10.3390/app12115585

**AMA Style**

Bile A, Tari H, Fazio E.
Episodic Memory and Information Recognition Using Solitonic Neural Networks Based on Photorefractive Plasticity. *Applied Sciences*. 2022; 12(11):5585.
https://doi.org/10.3390/app12115585

**Chicago/Turabian Style**

Bile, Alessandro, Hamed Tari, and Eugenio Fazio.
2022. "Episodic Memory and Information Recognition Using Solitonic Neural Networks Based on Photorefractive Plasticity" *Applied Sciences* 12, no. 11: 5585.
https://doi.org/10.3390/app12115585