# Differential Neural Network-Based Nonparametric Identification of Eye Response to Enforced Head Motion

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## Abstract

**:**

## 1. Introduction

- a novel modeling strategy is proposed for the ocular response on head movements based on a spiking DNN with no parameters;
- a new aggregated system is used to confirm the validity of the proposed model. It consists of an experimental system with a motion platform, inertial sensors, an eye-tracking device for acquiring data, and a neural network for processing it.

## 2. Description of Vestibular–Ocular Connection

## 3. Modeling Ocular Response to Enforced Acceleration

## 4. Formulation of Spiking-Differential-Neural-Network-Based Model

**Theorem**

**1.**

**Proof**

**of**

**Theorem 1.**

## 5. Modeling Process and Experimental Validation

## 6. Numerical Simulation

## 7. Conclusions

## 8. Patents

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

VOR | Vestibular–Ocular Reflex |

DNN | Differential Neural Network |

SDNN | Spiking Differential Neural Network |

MSE | Mean Square Error |

MAE | Mean Absolute Error |

sMAE | Standardized Mean Absolute Error |

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**Figure 2.**Experimental setup for collecting the ocular response to the controlled accelerated movements.

**Figure 3.**Identification with Izhikevich activation function for high-frequency rotations: (

**a**)—recorded head rotation; (

**b**)—identification error; (

**c**)—recorded data and identification results comparison; (

**d**)—evolution of weights.

**Figure 4.**Identification with sigmoidal activation function for high-frequency rotations: (

**a**)—recorded head rotation; (

**b**)—identification error; (

**c**)—recorded data and identification results comparison; (

**d**)—evolution of weights.

**Figure 5.**Identification with Izhikevich activation function for low-frequency rotations: (

**a**)—recorded head rotation; (

**b**)—identification error; (

**c**)—recorded data and identification results comparison; (

**d**)—evolution of weights.

**Figure 6.**Identification with sigmoidal activation function for low-frequency rotations: (

**a**)—recorded head rotation; (

**b**)—identification error; (

**c**)—recorded data and identification results comparison; (

**d**)—evolution of weights.

Parameter | Izhikevich | Sigmoidal |
---|---|---|

Matrix A | $20\times diag(-1,-2)$ | $20\times diag(-2,-2)$ |

Matrix P | $1575.9\times diag(60,40)$ | $1575.9\times diag(60,40)$ |

Matrix ${K}_{1}$ | $0.15\times diag(10,1)$ | $0.0001\times diag(20,10)$ |

Matrix ${K}_{2}$ | $0.15\times diag(1,1)$ | $0.0001\times diag(20,10)$ |

Matrix ${W}_{1}\left(0\right)$ | $20\times \left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right]$ | $0.1\times \left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right]$ |

Matrix ${W}_{2}\left(0\right)$ | $20\times \left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right]$ | $20\times \left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right]$ |

Identifier Type | High-Frequency Data | Low-Frequency Data | ||||
---|---|---|---|---|---|---|

MSE | MAE | sMAE | MSE | MAE | sMAE | |

Izhikevich | 0.000186 | 0.008948 | 0.119975 | 0.000187 | 0.009647 | 0.140333 |

Sigmoidal | 0.000710 | 0.021099 | 0.282897 | 0.000588 | 0.021496 | 0.312143 |

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**MDPI and ACS Style**

Chairez, I.; Mukhamedov, A.; Prud, V.; Andrianova, O.; Chertopolokhov, V.
Differential Neural Network-Based Nonparametric Identification of Eye Response to Enforced Head Motion. *Mathematics* **2022**, *10*, 855.
https://doi.org/10.3390/math10060855

**AMA Style**

Chairez I, Mukhamedov A, Prud V, Andrianova O, Chertopolokhov V.
Differential Neural Network-Based Nonparametric Identification of Eye Response to Enforced Head Motion. *Mathematics*. 2022; 10(6):855.
https://doi.org/10.3390/math10060855

**Chicago/Turabian Style**

Chairez, Isaac, Arthur Mukhamedov, Vladislav Prud, Olga Andrianova, and Viktor Chertopolokhov.
2022. "Differential Neural Network-Based Nonparametric Identification of Eye Response to Enforced Head Motion" *Mathematics* 10, no. 6: 855.
https://doi.org/10.3390/math10060855