## 1 Introduction  
Measurements of electroencephalogram (EEG) signals have been applied to estimate human mental states such as the depressed state [1] and motivation [2]. In these studies, event-related potentials (ERPs) were used for estimating the human mental state. ERPs are EEG responses that are time-locked to the onsets of the events such as visual and/or auditory stimuli. As the amplitudes of ERPs are commonly small compared to the noise and spontaneous activity, many EEG epochs that are time-locked to the onset of events must be averaged to detect the ERPs. Therefore, it is necessary to know the timing of the events to use the ERPs for estimating the mental state. However, it is difficult to determine exactly when people watch or listen in the real environment. Thus, the use of ERPs for mental state estimation in real environments has been limited. However, [PERSON] et al. succeeded in measuring the eye-fixation-related potential (EFRP), which is a type of ERP, during walking in a real environment [3]. The EFRP is observed synchronously with saccade offset (SO) timing [4]. As the SO timing coincides with the onset time of perceiving a visual object, the timing can be used as the onset of the timing when the objects are perceived, even in real environments. In particular, the lambda response, which is part of the EFRP, is a positive component that is observed in the occipital region approximately 80 ms after the SO timing [4]. The amplitude of the lambda response is known to change according to the workload [5] and attention levels that are dedicated to visual objects [6]. Owing to these characteristics, the lambda response can be used as an indicator to predict the occurrence of visual inspection errors that are caused by changes in visual attention to the inspected object as a result of prolonged inspection [7]. Furthermore, it has been applied to workload estimation while driving a car [8]. Electrooculograms (EOGs) or eye-tracking data have commonly been used to detect SO timing [3; 4; 5]. It is advantageous to use EOG data to determine the SO timing for measuring the EFRP because the same recording systems can measure both EEG and EOG data. Moreover, the EOG can be measured by simply placing electrodes around the eyes. In previous research using EOG data for this purpose, the SO timing was determined visually as the time point of change from saccade to fixation [9]. However, this visual determination (VD) method is time-consuming and restricted to offline analysis. Therefore, the SO timing must be determined automatically to use the EFRP responses to estimate mental states in real time. Moreover, the SO timing may be inconsistent among detectors partially due to fatigue owing to long periods of visual inspection, especially when analyzing EFRPs with large datasets. The inconsistency of the SO timing determination causes decreases in the EFRP amplitude in the averaged EEG data over EEG epochs that are time-locked to the determined SO timing [10]. Therefore, the improvement in consistency by automating the determination of the SO timing will expand the applicability of state estimation in real-world settings using EFRP. A previous study proposed an automatic method that can detect saccade timing to obtain EFRP responses using the EOG signal [3]. In this method, independent components (ICs) reflecting vertical and horizontal EOGs are first identified using independent component analysis (ICA). Next, the SO timings are detected based on the peak velocity of these ICs (hereafter, this previous method is referred to as the peak velocity (PV) method). Using this method, the authors successfully observed EFRPs from EEG data that were measured in the real environment, i.e., while walking in the city. Although the PV method enables the saccade timing to be detected automatically in real environments, applying neural network (NN) models to detect SO timing may improve performance. In recent years, NN models have been used to analyze EEG and EOG data for various purposes.