## 1 Introduction
For example, the use of recurrent neural network (RNN) models for EEG and EOG data improved the accuracy of detecting drowsiness while driving [11]. Furthermore, a combination of convolutional neural networks (CNNs) and RNN models was used for EOG data to improve the accuracy of sleep stage classification [12]. Thus, NN models have been applied to EEG and EOG data and have improved the classification accuracies compared to traditional methods. Among the NN models, RNN models have also demonstrated potential for change point detection. For example, an RNN model was proposed to predict the change points from data of sensors attached to industrial equipment parts [13]. Moreover, an RNN model outperformed the previous method in detecting the change points in multiple time-series load data of an electric power company for power outage analysis [14]. As the SO timing can be considered as a change point from saccade to fixation detection, these previous studies have suggested that RNN models can also be applied to SO timing detection. In general, a large amount of training data are required to improve the detection accuracy of NN models. As a large amount of EOG and EEG data are rarely available for SO timing detection, to the best of our knowledge, no study has applied an NN model to detect the SO timing automatically by relying only on EOG signals. In our previous study, we obtained a relatively large EEG and EOG dataset while participants periodically traversed their eyes from left to right in a task that mimicked visual inspection (2500 trials with 50 participants in total) [7]. These relatively large datasets provide the possibility of using NN models to detect SO timing, which may yield better performance than the VD and PV methods. The purpose of this study was to develop a novel NN model to predict the SO timing using the dataset that was obtained in the previous research [7]. The labels indicating the SO timing were created by the VD method and the labeled data were used for supervised learning. We employed a gated recurrent unit (GRU)--an RNN derivative model that is suitable for handling time series data [15]. In recent years, other models that are suitable for handling time-series data, such as the Transformer [16] and Seq2 Seq [17], have been proposed. However, these new methods are more complex than RNN models [16; 17]. Seq2 Seq uses RNNs as encoders and decoders, and, therefore, it is computationally heavier than RNN models. The heavy computational cost is also an issue in the Transformer because it includes tens of millions of parameters [18]. As this is the first study to detect saccade offsets automatically using NN models, we used an RNN model, which is relatively computationally light and easy to apply. The trained RNN model was evaluated in terms of the (1) error between the SO timing determined by the proposed and VD methods and (2) the consistency of the detected SO timing across trials. For the first evaluation procedure, we verified whether the proposed RNN model correctly predicted the SO timing by calculating the difference between the SO timing that was predicted by the proposed model and that obtained by the VD method. The second evaluation procedure was intended to account for the possible inconsistency in the SO timing across trials, which may occur owing to fatigue or spreading the work over several days because of the lengthy determination process. As ERPs are observed by averaging multiple trials relative to the event onset timing, the consistent determination of the SO timing increases the mean amplitudes. Conversely, inconsistent determination across trials decreases the mean amplitudes, which may affect the reliability of the use of EFRPs for mental state estimation. Thus, the evaluation of the inconsistency is important in terms of practical use. We adopted the mean amplitudes of the lambda response as the consistency index and compared the values across the proposed, VD, and PV methods.