## 2 Materials and Methods  
### AIP PES Prediction  
Artificial neural networks (ANNs) are artificial intelligence models well suited for representing nonlinear relationships in input data [22]. ANNs effectively model complex nonlinear dynamics, making them valuable in predicting variables in nonlinearly distributed parameter systems, such as temperature distributions or stock market data [23; 24]. Their ability to handle intricate nonlinearities has seen applications in diverse fields, including quality prediction in injection molding and noise reduction in synthetic aperture radar (SAR) images [25; 26]. ANNs are also widely used to predict critical variables in industrial processes through data-driven soft sensors, often combined with dual LASSO techniques to minimize model overfitting [27]. This nonlinear modeling capability makes ANNs a powerful tool for tackling conceptual challenges, including those presented by the Adaptive Information Processing (AIP) model, foundational to EMDR therapy [28]. The AIP model serves as a theoretical framework in EMDR, positing the brain as an information processing system that continuously seeks equilibrium [29]. [PERSON], the founder of EMDR, suggests that the AIP model reflects the brain's natural tendency towards survival and mental health, processing experiences adaptively under normal conditions. However, negative experiences can disrupt this system, leading to maladaptive memory storage, which may manifest as persistent psychological symptoms [27; 30; 31; 32]. These unprocessed memories are hypothesized to contribute to various mental health issues, including PTSD, depression, and chronic pain [33]. Within the AIP framework, EMDR is conceptualized as a process that combines visual stimuli with information processing time, wherein visual stimuli serve as external triggers for the brain's adaptive processing. Let \(f(v,d,s)\) represent the stimulus and \(P(t)\) the amount of processed information; the basic information processing function can be expressed as follows:  
\[ \begin{split} P(t+1)=P(t)+\Delta P\\ \Delta P=f(B(t))=f(v,d,s)\end{split}  \tag{1}\]  
where \(P(t+1)\) represents the information processed at the next time step, and \(f(v,d,s)\) is a function of speed (s\({}_{sp}\)), distance (\(d\)), and size (s\({}_{sz}\)). The Predicted Effectiveness Score (PES) relates to the information processed, such that optimal stimulus parameters maximize information processing, thereby increasing treatment efficacy. This relationship can be quantified as:  
\[E(v,d,s)=\frac{\Delta P}{I}\times 100 \tag{2}\]  
where \(E(v,d,s)\) is the PES (%) and III represents the total information. \(\Delta P\), the information processed by the stimulus, is a function of speed (\(s_{sp}\)), distance (\(d\)), and size (\(s_{sz}\)). By combining these equations, we derive a unified PES prediction formula:  
\[E(v,d,s)=\frac{f(v,d,s)}{I}\times 100 \tag{3}\]  
In this equation, \(E(v,d,s)\) represents the effectiveness of the visual stimulus, and \(f(v,d,s)\) is a function that optimizes stimulus efficacy by adjusting for deviations from ideal parameter ranges. Assuming effective information processing occurs within optimal ranges, this relationship can be modeled by a Gaussian function:  
\[f(v,d,s)=\exp\left(-\left(\frac{(v-v_{0})^{2}}{2\sigma_{v}^{2}}+\frac{(d-d_{0 })^{2}}{2\sigma_{d}^{2}}+\frac{(s-s_{0})^{2}}{2\sigma_{s}^{2}}\right)\right) \tag{4}\]  
Here, \(v_{0}\), \(d_{0}\) and \(s_{0}\) represent the optimal values for speed (\(s_{sp}\)), distance (\(d\)), and size (\(s_{sz}\)), respectively, while \(\sigma_{v}\), \(\sigma_{d}\), and \(\sigma_{s}\) are the standard deviations (tolerances) for each parameter. This Gaussian model allows for maximum effectiveness within optimal ranges and reduced scores outside them. Gaussian functions are widely used in neural networks as activation functions, aiding in uncertainty modeling and regression analysis [34; 35].