## 3 Methodology
### Analysis and Structure of the Bio-Inspired Retinal Model
#### 3.1.3 Outer Plexiform Layer  
The synaptic behavior between neurons can be modeled as an electrical circuit, as explained in Equation (2). Therefore, in accordance with the model presented in [29], the membrane potential of a cone \(c(k,t)\) at location \(k\) corresponds to the flux of incident photons in an electrical circuit with resistance \(r_{c}\) and capacitance \(C_{c}\), excited by a voltage source with an internal resistance \(r\). The gap junctions (dendrite and axon connections) between cones are modeled by \(R_{c}\), between position \(k\) and the neighboring positions \(k-1\) and \(k+1\). \[C_{c}\frac{d}{dt}\ c(k,t)=\frac{s(k,t)-c(k,t)}{r_{c}}+\frac{c(k-1,t)-c(k,t)}{R_ {c}}+\frac{c(k+1,t)-c(k,t)}{R_{c}} \tag{2}\]  
The solution to Equation (2) is obtained by successively taking the Fourier transforms with respect to the discrete variable \(k\) and the continuous variable \(t\), resulting in the transfer function of the conical circuit presented below:  
\[\mathrm{C}(f_{s},f_{t})=\frac{1}{1+2\alpha_{c}\cdot(1-\cos(2\pi af_{s}))+j2 \pi\tau_{c}f_{t}}S(f_{s},f_{t}) \tag{3}\]  
where \(f_{s}\) and \(f_{t}\) are the spatial and temporal frequencies, respectively, while \(\alpha_{c}=r_{c}/R_{c}\) is the space containing the gap junctions and \(\tau\) is the time constant of the cell membrane. From Equation (3), it follows that the photoreceptor layer functions as a spatiotemporal low-pass filter of non-separable variables. In the process of chromatic adaptation through negative feedback to the photoreceptors, the horizontal cells play a fundamental role in modulating the information processed by the cones, which exhibit behavior analogous to what is described by Equation (3), whose coefficients are \(\alpha_{h}=r_{h}/R_{h}\) and \(\tau_{h}=r_{h}C_{h}\), as presented in the following equation:  
\[F_{h}(f_{s},f_{t})=\frac{1}{1+2\alpha_{h}\cdot(1-\cos(2\pi f_{s}))+j2\pi\tau_{ h}f_{t}}S(f_{s},f_{t}) \tag{4}\]  
In summary, given that horizontal cells regulate abrupt changes in luminance, it can be concluded that they function as a low-pass filter in the frequency domain. In this context, the study conducted by [30] introduces an additional gain \(\beta\) in Equations (3) and (4), as presented in Equations (5) and (6), respectively. In these equations, \(\beta_{ph}\) represents the gain of \(F_{ph}\), which is typically maintained at 0 but can be increased up to 1 to enhance the dynamic signal. Conversely, \(\beta_{h}\) denotes the gain of the filter \(F_{h}\), which is generally set at 1 but can be decreased down to 0 to extract only the contours  
Figure 4: Photoreceptor model response (different \(V_{0}\) values). (**a**) Original image. (**b**) Processed image with \(V_{0}=0.3\). (**c**) Processed image with \(V_{0}=0.4\). (**d**) Processed image with \(V_{0}=0.5\). (**e**) Processed image with \(V_{0}=0.6\). (**f**) Processed image with \(V_{0}=0.7\). (**g**) Processed image with \(V_{0}=0.8\). (**h**) Processed image with \(V_{0}=0.9\). (**i**) Processed image with \(V_{0}=1\). of the system. Consequently, low-pass filters can be considered within the OPL, which comprises horizontal cells and photoreceptors [26], enabling their behavior to be modeled with relative simplicity. The model for photoreceptors \(F_{ph}(f_{s},f_{t})\) is expressed by Equation (5), whereas the model for horizontal cells \(F_{h}(f_{s},f_{t})\) is determined by Equation (6). Both functions depend on the spatial frequency \(f_{s}\) and temporal frequency \(f_{t}\). \[F_{ph}(f_{s},f_{t})=\frac{1}{1+\beta_{ph}+2\alpha_{ph}(1-\cos(2\pi f_{s}))+j2 \pi\tau_{ph}f_{t}} \tag{5}\]  
\[F_{h}(f_{s},f_{t})=\frac{1}{1+\beta_{h}+2\alpha_{h}(1-\cos(2\pi f_{s}))+j2\pi \tau_{h}f_{t}} \tag{6}\]  
Despite having the same structure, Equations (5) and (6) differ in certain terms. In these equations, \(\beta_{ph}\) represents the gain of the filter \(F_{ph}\), which is generally set at 0, while \(F_{h}\) denotes the gain of the filter S, typically set at 0 to extract only the contours of the system.