A differential equation model for the stage theory of color perception

Abstract

We propose a novel differential equation model assuming the stage theory (trichromatic theory and opponent-process theory) for color inputs based on our previous model for light-dark inputs, which is an extension of the lateral inhibition model proposed by Peskin (Partial Differential Equations in Biology: Courant Institute of Mathematical Sciences Lecture Notes, New York, 1976). First, we consider the stationary problem in our novel model for color inputs and show that the output of our model can be described by a convolution integral. Moreover, we derive the necessary and sufficient conditions for the appearance of Mexican hat-type integral kernels in the outputs of our model for color inputs. Second, we demonstrate numerical results for simple color inputs and provide a theoretical prediction that the self-control mechanism exerted at horizontal cells, in conjunction with the opponent-colors (red-green, yellow-blue, and light-dark (or white-black)) plays an important role for the occurrence and non-occurrence of typical color contrasts.

Appendices

Appendices

1.1 Appendix A: Color contrasts

In Sect. 4, we focus on the input corresponding to Fig. 4g and show numerical results on the occurrence and non-occurrence of color contrasts, which are contingent on certain parameters. Furthermore, we also demonstrate numerical results that correspond to the other examples described in Fig. 4. The parameters of our model are specified in Table 1, $\gamma ^j = 1.0$, and $d_h^j = 100.0$. Figure 8 shows the inputs with parameter $\ell = 10$.

Fig. 8

Two-dimensional images of the input (49) with $\ell = 10$. a $({\hat{I}}^{L}, {\hat{I}}^{M}, {\hat{I}}^{S}) = (1,0,0)$. b $({\hat{I}}^{L}, {\hat{I}}^{M}, {\hat{I}}^{S}) = (0,1,1)$. c $({\hat{I}}^{L}, {\hat{I}}^{M}, {\hat{I}}^{S}) = (0,1,0)$. d $({\hat{I}}^{L}, {\hat{I}}^{M}, {\hat{I}}^{S}) = (1,0,1)$. e $({\hat{I}}^{L}, {\hat{I}}^{M}, {\hat{I}}^{S}) = (1,1,0)$

Figure 9 shows numerical profiles of $U^{rg}(x_1,L/2)$, $U^{yb}(x_1,L/2)$, $U^{ld}(x_1,L/2)$ for the input shown in Fig. 8. Similar to the above numerical results of Fig. 8a, we have $U^{rg}(x_1,L/2) \not = 0$ or $U^{yb}(x_1,L/2) \not =0$ in the region ${\hat{\varOmega }}$. That is, purely gray colors in ${\hat{\varOmega }}$ are depicted as the other colors.

Fig. 9

a–e Show profiles of $U_p^{rg}(x_1,L/2)$, $U_p^{yb}(x_1,L/2)$, and $U^{ld}(x_1,L/2)$ for the input shown in Figs. 8a–e, respectively. Solid, dashed, and dotted lines indicate $U_p^{yb}(x_1, L/2)$, $U_p^{rg}(x_1, L/2)$, and $U_p^{ld}(x_1, L/2)$, respectively

As shown in Fig. 9e, $U^{rg}_p(L/2, L/2) = 0$ and $U^{yb}_p(L/2, L/2) > 0$. This implies that the purely gray square that is surrounded by a blue color is perceived to be slightly yellow. Conversely, from the numerical result shown in Fig. 8e, $U^{rg}_p(L/2, L/2) = 0$ and $U^{yb}_p(L/2, L/2) < 0$. That is, the purely gray squares within a yellow region is perceived to be slightly blue.

On the other hand, in Fig. 9a–d, the signs of $U^{rg}_p(L/2, L/2)$ and $U^{yb}_p(L/2, L/2)$ are either the same or opposite. From the results shown in Fig. 9a, b, both signs of $U^{rg}_p(L/2, L/2)$ and $U^{yb}_p(L/2, L/2)$ are negative and positive, respectively. This implies that the purely gray is slightly depicted to be a mixed color of green-blue in the former case and a mixed color of red-yellow in the latter case. In particular, because we have $|U^{rg}_p(L/2, L/2)| > |U^{yb}_p(L/2, L/2)|$ from Fig. 9a, b, it is conjectured that the influence of green and red is stronger than that of blue and yellow.

Also, Fig. 9c, d demonstrate $U^{rg}_p(L/2, L/2)>0$, $U^{yb}_p(L/2, L/2)<0$ and $U^{rg}_p(L/2, L/2)<0$, $U^{yb}_p(L/2, L/2)>0$, respectively. This implies that the purely gray squares is slightly perceived as a mixed color of red-blue in the former and mixed color of green-yellow in the latter case. Similar to what was described in Fig. 9a and b, c and d demonstrate $|U^{rg}_p(L/2, L/2)| > |U^{yb}_p(L/2, L/2)|$, suggesting that the influence of red and green are stronger than that of blue and yellow. Thus, the above results are equivalent to the visual impressions observed in Fig. 4, supporting the ability of our model to explain typical color contrasts.

Moreover, we focus on the two-dimensional influence in our model output. Here, we provide an input (49) with $\ell = 50$ and $({\hat{I}}^{L}, {\hat{I}}^{M}, {\hat{I}}^{S}) = (0,0,1)$, as shown in Fig. 10a. Figure 10b shows a two-dimensional contour profile of the output $U^{yb}_p$, which demonstrates that values of $U_p^{yb}$ at the four corners marked by $\bigstar$ are higher than ones on the boundary of a given purely gray rectangular region. This result is obtained by using the shape of two-dimensional corners and implies that our model can be used to analyze other contrast phenomena such as the Vasarely effect [72].

Fig. 10

a A two-dimensional image of the input (49) with $\ell = 50$ and $({\hat{I}}^{L}, {\hat{I}}^{M}, {\hat{I}}^{S}) = (0,0,1)$. b A two-dimensional profile with contours of the output $U_p^{yb}$. The marks $\bigcirc$, $\blacksquare$, $\blacktriangle$, $\bullet$, $\blacklozenge$, $\blacktriangledown$, and $\bigstar$ indicate contour values $-0.5$, $-0.57$, $-0.64$, 0, 0.07, 0.14, and 0.21, respectively

1.2 Appendix B: Opponent-colors induced by afterimages

We consider opponent-color induced by afterimages using two-dimensional color inputs in the region $\varOmega$ shown in Fig. 11a. In the beginning, observers watch the red rectangular region closely amidst a on the green background, as shown in the left panel of Fig. 11a. After the red region and the green background are removed as shown in the right panel of Fig. 11a, the observers perceive opponent-colors for red and green. That is, an afterimage on the red region can be perceived as green, while an afterimage on the green region can be perceived as red. Here we give the following inputs $I^i(t,x_1, x_2)$ ($i=L, M, S$) in order to discuss the occurrence and non-occurrence of afterimages observed in Fig. 11a:

$$\begin{aligned} I^L(t,x_1, x_2)&= {\left\{ \begin{array}{ll} {\hat{I}}(x_1, x_2), &{}\quad 0\le t \le T_0, \\ 1, &{}\quad T_0< t \le T_1, \end{array}\right. } \nonumber \\ I^M(t,x_1, x_2)&= {\left\{ \begin{array}{ll} 1 - {\hat{I}}(x_1, x_2), &{}\quad 0\le t \le T_0, \\ 1, &{}\quad T_0< t \le T_1, \end{array}\right. } \nonumber \\ I^S(t,x_1, x_2)&= {\left\{ \begin{array}{ll} 0, &{}\quad 0\le t \le T_0, \\ 1, &{}\quad T_0 < t \le T_1, \end{array}\right. } \end{aligned}$$

(50)

where $T_0$ and $T_1$ are the switching time of patterns and the finishing time for numerical computation, respectively. In our numerical simulations, we specify $T_0 = 5.0$ and $T_1 = 10.0$. ${\hat{I}}(x_1, x_2)$ is defined by

$$\begin{aligned} {\hat{I}}(x_1, x_2) = {\left\{ \begin{array}{ll} 1, &{}\quad (x_1, x_2) \in {\hat{\varOmega }}, \\ 0, &{}\quad (x_1, x_2) \in \varOmega \backslash {\hat{\varOmega }}, \end{array}\right. } \end{aligned}$$

where we specify the parameter $\ell$ as $\ell =50.0$ in Fig. 11a. We also use the parameters shown in Table 1, the control parameter $\gamma ^j = 1.0$ ($j=rg,yb,ld$), and the diffusion coefficient $d_h^j = 100.0$.

Fig. 11

a A two-dimensional image of the input (50) with $\ell = 50.0$ This image induces an afterimage (green or red) for opponent colors of red and green, respectively. b–d One-dimensional profiles of the input $I^{i}$ ($i=L, M, S$) for $0 \le t \le T_0$ at the black line of a. A horizontal axis indicates the $x_1$-axis and a vertical axis indicates values of $I^i$. e–g Profiles $u_h^j(t,x_1,L/2)$ $(j=rg, yb, ld)$ for $t=5.0$ and $t=6.0$, which are obtained as a numerical result for inputs $I^{i}$ ($i=L, M, S$) of b–d. Black arrows indicate directions for movement of profiles

Figure 11b–d are one-dimensional profiles of the input $I^{i}$ ($i=L, M, S$) for $t=0$ at the black line of Fig. 11a. Figure 11e–g show numerical results obtained as one-dimensional profile of $u_p^j$ $(j=rb, yb, l d)$ for the input (50). From the profile of $u_p^{rg} (t,x_1,L/2)$ for $t=6.0$ in Fig. 11e, we can confirm that the opponent-colors (red-green) appear as an afterimage. On the other hand, the profiles of $u_p^{yb} (t,x_1,L/2)$ and $u_p^{l d} (t,x_1,L/2)$ for $t=6.0$ in Fig. 11f, g have values close to 0. Therefore, the occurrence of opponent-colors induced by the afterimage can be explained using our model.

Finally, we show a dependency of the size of red regions and two parameters, $\gamma ^j$ and $d_h^j$. In order to evaluate opponent-color afterimages, we define the following index:

$$\begin{aligned} A_p^{rg} = \int _{T_0}^{T_1}u_p^{rg}(t, L/2, L/2) \mathrm{d}t \end{aligned}$$

Figure 12 shows profiles of $A_p^{rg}$ for the input (50) with $\ell \in [5.0, 50.0]$ in cases with three parameter pairs $(\gamma ^j, d_h^j) = (1.0, 100.0)$, (0.1, 100.0), and (1.0, 10.0). Since an opponent-color afterimage for the input (50) indicates green, the opponent-color afterimage appears when $A_p^{rg}$ is negative. In the case of $\gamma ^j=0.1$, we have $A_p^{rg}>0$ for all $\ell \in [5.0, 50.0]$. This implies that when the control parameter $\gamma ^j$ is small, the opponent-color afterimage does not occur.

On the other hand, a profile of $A_p^{rg}$ with $d_h^j=10.0$ has negative values that are larger than the case of $d_h^j = 100.0$. This implies that when the magnitude of diffusion coefficient $d_h^j$ (the range of lateral inhibition) is decreased, the occurrence of opponent-color afterimage becomes remarkable. Therefore, our numerical results can predict that afterimages are easier to recognize in the central vision than in the peripheral vision because the fovea having an influence on the lateral inhibition that is weaker than the surrounding region of the retina.

Fig. 12

Profiles of $A_p^{rg}$. Solid, dashed, and dotted lines indicate profiles with the parameter pairs $(\gamma ^j, d_h^j) = (1.0, 100.0)$, (0.1, 100.0), and (1.0, 10.0), respectively

1.3 Appendix C: Relationship between our model equations (7), (8) and time evolution equations with convolution integrals

In the subject of neurophysiology, it has been known as experimental results that the receptive field structure of the bipolar cells and the retinal ganglion cells consist of the short-range activation and the long-range inhibition and its structure can be represented by the Mexican hat type function. After that, convolution integral models with the Mexican hat type kernel were proposed and the relation between inputs and outputs has been studied. Therefore, a major direction of mathematical modeling for retinal processing is to construct the convolution integral modeling.

In this section, we investigate the relationship between our model equations (7), (8) and time evolution equations with convolution integrals.

1.3.1 Stage 1

In the major direction of mathematical modeling for retinal processing, the convolution integral has been adopted as one of mathematical frameworks describing the blurring effect. A blurred output $U_e = U_e(x)$ for an input I(x) $(x \in {\mathbb {R}}^{n}, \ n=1,2)$ is described by the following convolution integral

$$\begin{aligned} U_e(x) = \int _{{\mathbb {R}}^{n}} G(x-y) I(y) \mathrm{d}y = (G*I)(x), \quad x \in {\mathbb {R}}^{n}, \end{aligned}$$

(51)

where G(x) is often given by the Gaussian. Moreover, we can consider the following time evolution equation with a convolution integral:

$$\begin{aligned} \tau _e \frac{\partial u_e}{\partial t}= a \left( - u_e + (G*I)(x) \right) , \quad t>0, \ x \in {\mathbb {R}}^{n}, \end{aligned}$$

(52)

where $u_e=u_e(t, {x})$ is a blurred output, $\tau _e>0$ stands for the time constant. Here we assume that the shape of spatial-temporal receptive field structure of the photoreceptor cells does not depend on the time t.

First, we show that the stationary solution of the Eq. (7) is equivalent to (51) if we choose a special integral kernel. Let us consider the following integral kernel instead of the Gaussian:

$$\begin{aligned} G_i(x) = \left\{ \begin{array}{ll} \displaystyle \frac{k_i}{2} e^{- k_i |x|}, &{}\quad x\in {\mathbb {R}}^{1}, \\ \displaystyle \frac{k_i^2}{2 \pi } K_0 (k_i |x| ), &{}\quad x\in {\mathbb {R}}^{2} \end{array} \right. \end{aligned}$$

(53)

with the parameter $k_1 = \sqrt{a/d_e}$ as the index $i = 1$, where a and $d_e$ are positive constants and $K_0$ is a modified Bessel function of the second kind of the 0-th order. Then (51) with (53) is equivalent to the Eq. (46). See (A.14) of Appendix in our paper [70], and Lemma 1 of this paper.

Next we compare the numerical solution of the Eq. (7) with the exact solution of the Eq. (52) with $G(x) = G_1(x)$ for $n=1$. Here the initial condition is given by $u_e(0,x) = 0$ for any $x\in {\mathbb {R}}^{1}$, and the input is given by

$$\begin{aligned} I = I(x) = I_1 \quad \mathrm{for} \ x \le 0, \ \mathrm{and} \ I = I(x) = I_2 \quad \mathrm{for} \ x > 0, \end{aligned}$$

(54)

where $I_1$ and $I_2$ are constants. The exact solution of the Eq. (52) with $G(x) = G_1(x)$ is obtained as follows. For $t>0$, we have

$$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle \tau _e \frac{ \partial u_e}{\partial t} = a \left( I_1 - \frac{I_1}{2} e^{k_1 x} + \frac{I_2}{2} e^{k_1 x} - u_e \right) , &{}\quad x \le 0, \\ \displaystyle \tau _e \frac{ \partial u_e}{\partial t} = a \left( I_2 + \frac{I_1}{2} e^{-k_1 x} - \frac{I_2}{2} e^{-k_1 x} - u_e \right) , &{}\quad x >0. \end{array} \right. \end{aligned}$$

Let $\eta = - \frac{a}{\tau _e}$, then we have

$$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle u_e(t,x) = (1 - e^{\eta t} ) \left( I_1 - \frac{I_1}{2} e^{k_1 x} + \frac{I_2}{2} e^{k_1 x} \right) , &{}\quad x \le 0, \\ \displaystyle u_e(t,x) = (1 - e^{\eta t} ) \left( I_2 + \frac{I_1}{2} e^{-k_1 x} - \frac{I_2}{2} e^{-k_1 x} \right) , &{}\quad x >0. \end{array} \right. \end{aligned}$$

From Fig. 13, we recognize that the numerical solution of the Eq. (7) is similar to the exact solution of the Eq. (52) with $G(x) = G_1(x)$.

In the above discussion, we consider for the case of the integral kernel (53). On the other hand, recently, it has been shown in Theorem 2.6 [52] that for one-dimensional case, a finite linear combination of (53) gives an approximation to any even continuous function G(x). See also Corollary 1 [51] for the case of the periodic boundary condition. Therefore, it is reasonable to adopt the Eq. (7) as the model of the first stage.

Fig. 13

a Numerical solution $u_e$ of the Eq. (7). b The solution of the Eq. (52) with $G(x) = G_1(x)$. Here $t=0.1, 0.2, 2.0$, the initial condition is given by $u_e(0,x) = 0$ for any $x\in {\mathbb {R}}^{1}$, and and the input is given by (54)

1.3.2 Stage 2

In the second stage, we focus on the non-local transmission of inhibitory signal at the horizontal cells. It has been often considered non-local interactions as the following formulation by differential equation with convolution term: Let $u_p=u_p(t, {x})$ be an output for the input $u_e(t,x)$ of the second stage at the time $t>0$ and the position $x \in {\mathbb {R}}^{n}$. We consider the equation

$$\begin{aligned} \tau _p \frac{\partial u_p}{\partial t} = b(-u_p + u_e(t, {x})) - \gamma (G * \beta u_p )(x), \quad t>0, \ x\in {\mathbb {R}}^{n}, \end{aligned}$$

(55)

where $\tau _p$ is a time constant and b, $\beta$, $\gamma$ are positive constants. If we specify the integral kernel $G(x)= G_2(x)$ which is (53) with the parameter $k_2 = \sqrt{c/d_h}$ $(c>0, d_h>0)$ as the index $i = 2$, and put $u_h = (G * \beta u_p )(x)$, then the Eq. (55) is written as the following system:

$$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle \tau _p \frac{\partial u_p}{\partial t} = b(-u_p+u_e(t, {x})) - \gamma u_h, &{}\quad t>0, \ x\in {\mathbb {R}}^{n}, \\ \displaystyle 0 = d_h \varDelta u_h + \beta u_p - c u_h, &{}\quad t>0, \ x\in {\mathbb {R}}^{n}, \end{array} \right. \end{aligned}$$

(56)

which implies that the non-local transmission of inhibitory signal exerts instantaneously.

In the subject of neurophysiology, it is known that temporal responses of the S-potential differ in the cases of light adaptation and dark adaptation. Hence we gave a prediction that slow temporal response of the self-control mechanism is an important role for perceptions of afterimage illusions, and we consider the Eq. (8) as governing equations of the second stage.