A Computational Framework for Modeling Emergence of Color Vision in the Human Brain

## Abstract

Abstract It is a mystery how the brain decodes color vision purely from the optic nerve signals it receives, with a core inferential challenge being how it disentangles internal perception with the correct color dimensionality from the unknown encoding properties of the eye.
In this paper, we introduce a computational framework for modeling this emergence of human color vision by simulating both the eye and the cortex. Existing research often overlooks how the cortex develops color vision or represents color space internally, assuming that the color dimensionality is known a priori; however, we argue that the visual cortex has the capability and the challenge of inferring the color dimensionality purely from fluctuations in the optic nerve signals. To validate our theory, we introduce a simulation engine for biological eyes based on established vision science and generate optic nerve signals resulting from looking at natural images. Further, we propose a model of cortical learning based on self-supervised principle and show that this model naturally learns to generate color vision by disentangling retinal invariants from the sensory signals. When the retina contains N 𝑁 N italic_N types of color photoreceptors, our simulation shows that N 𝑁 N italic_N -dimensional color vision naturally emerges, verified through formal colorimetry. Using this framework, we also present the first simulation work that successfully boosts the color dimensionality, as observed in gene therapy on squirrel monkeys, and demonstrates the possibility of enhancing human color vision from 3D to 4D.

### 1 Introduction

“Color is the place where our brain and the universe meet.” – Paul Klee

We experience colors in everyday life so effortlessly that it is easy to take the underlying neural computations for granted. In fact, the sensory signals exiting our eye, called optic nerve signals (ONS), are nothing like our color vision (see Fig. 1 and Video 5:32). For example, ONS are spatially warped, akin to an image taken with a fish-eye lens, due to varying densities of photoreceptor cells in the retina [curcio1990human]. ONS does not come in color either – colors of the scene are spectrally sampled by different types of color sensitive cells (cone cells) in the retina, appearing as a layer of spatial noise in the ONS. Furthermore, other processes, such as lateral inhibition and action potentials, render the image structure barely recognizable, in gradient domain where spatiotemporal “edges” dominate. Now, the question is: how are we still seeing colors?

Specifically, this paper introduces a novel computational framework for modeling the emergence of human color vision by simulating the eye and the cortex.
For the eye, we present a biophysically accurate implementation of a textbook scientific model of the retinal neural circuitry.
For the brain, we hypothesize a low-level, self-supervised learning mechanism in the cortex that operates purely on the optic nerve signal stream.
For color representation in the brain, we propose modeling color in the brain as a high-dimensional vector, rather than assuming any specific color dimensionality and show that the correct color dimensionality emerges naturally through the proposed learning.

For eye simulation, our goal is to create a computational engine that takes in any spectral image of the world and outputs the corresponding optic nerve signal stream, based on established vision science. Our model captures how scene form, color, and motion become spatiotemporally entangled in the optic nerve signal stream.
Our simulation is based on the “textbook” model of vision science [rodieck1998first] for midget, private-line visual pathways, detailed in Section 3, and comprising fixational eye motion, spectral encoding by cone cells, foveation, and lateral inhibition.

Figure: Figure 1: Overview of our proposed framework for modeling the emergence of human color vision. Our simulation engine of biological eyes converts a scene stimulus to a stream of optic nerve signals (Section 3 & Video 5:32). We simulate cortical learning purely from these optic nerve signals (Section 4) and show the emergence of color vision. We show how to analyze the emegent neural color qualitatively with $\mathit{NS}$ and quantitatively with $\mathit{CMF\textrm{-}SIM}$ (Section 5).
Refer to caption: x1.png

For brain simulation, we hypothesize that the cortex could disentangle the optic nerve signal from the invariant retinal properties to generate color vision – purely through a self-supervised learning process that aims to predict the constant fluctuations in cellular-level activations of the optic nerve signal during small eye movements. The neural conditions for such self-supervised learning are biologically plausible in the sense that the cortex continuously receives optic nerve signals under the tiny gaze movements of fixational eye drift.The theoretical intuition for why such learning might succeed is that eye motion repeatedly draws a static scene image across the invariant spectral and spatial sampling properties of the retina, potentially enabling the retinal properties and scene images to be mutually filtered out of the optic nerve signal where they are entangled. We show that this simple learning mechanism succeeds at discovering color vision with the correct dimensionality.

But first, a somewhat esoteric yet technically critical feature of the modeling framework needs discussion: color representation in the brain.
Existing research often overlooks how the cortex develops color vision or represents color space internally, assuming that the color dimensionality is known a priori, e.g.
RGB.
We argue that the cortex has the capability and the task of inferring color dimensionality, purely from fluctuations in the optic nerve signals. Therefore, we propose representing color in the cortex as a high-dimensional vector in $\mathbb{R}^{N}$ and find that the correct color space and color dimensionality emerge naturally as geometric properties of the hypothesized learning mechanism. We show how to formally quantify and visualize the emergent color space (Section 5).

Remarkably, the hypothesized learning mechanism results in clear color vision of the correct color dimensionality. When the retina contains 1, 2, 3 or 4 types of color photoreceptors (cones), correct color dimensionality emerges: respectively, 1D mono-, 2D di-, 3D tri- or 4D tetra-chromatic vision. In fact, it is an esoteric but well-known fact in vision science that the three types of 2D dichromacy result in highly specific color-spaces (blue-yellow, blue-orange, and teal-pink), which we see emerge naturally.
Even more, the simulation presents a model of how color dimensionality boosting occurred in the squirrel monkeys [mancuso2009gene]. We simulate injection of the third cone pigment virus into the dichromat retina, which results in boosting from dichromacy to trichromacy.
Intriguingly, the model also shows the possibility that normal human trichromacy could be boosted to tetrachromacy.

In sum, our proposed framework formulates the emergence of human color vision in a computational manner. The simulation engine of realistic optic nerve signals generates training data to this problem, and an intentional and challenging constraint is that the cortical model must strictly learn only from the optic nerve signal with no auxiliary information. This paper presents the first, simple and yet complete existence proof that such cortical inference is possible, and we show that this learning simulation is consistent with various surprising and unexplained vision science phenomena.

### 2 Related Work

#### 2.1 Vision Science on Optic Nerve Signal Encoding

In the “textbook” model, the retina transforms light into electrical signals through three primary functions: color sampling via cone cell spectral response functions, lateral inhibition via horizontal neural connections, and spatial sampling via cell positioning (Fig. 2). Under daylight, this process begins when light activates cone cells, which are of three types for most humans [young1801ii, young1802ii, von1867handbuch], each sensitive to different wavelengths [stockman2000spectral]. These signals are then modulated by horizontal cells that enhance visual contrast through lateral inhibition [rodieck1965quantitative, dacey1996horizontal, verweij2003surround]. The signals continue to bipolar and retinal ganglion cells (RGCs), with a direct connection in the fovea via the midget private-line pathway [dowling1966organization, mcmahon2000fine, wool2018nonselective], vital for high-resolution color vision. Notably, the cone cell density varies, reaching its peak in the fovea [osterberg1935topography, curcio1990human]. The axons of the ganglion cells bundle to form optic nerve signals, maintaining their spatial arrangement, which results in a retinotopy in the visual cortex [holmes1918disturbances, tootell1982deoxyglucose, dougherty2003visual]. Fixational eye movements cause dynamic photoreceptor activation at all times by constantly shifting the gaze, even when focusing on static objects [rucci2015unsteady, young2021emulated, martinez2013impact].

Recordings of real optic nerve signals exist, but cannot be used for our cortical simulations because they are orders of magnitude too low resolution (from only a few thousand cells) and only in response to grayscale rather than color imagery
[litke2004does, brackbill2020reconstruction, marre2017multi, liu2022simple].
To overcome these data limitations, we present a simulation engine for the midget private-line pathway in the fovea in Section 3.

#### 2.2 Color Representation in Computational Neuroscience

It is an open question how to meaningfully model neural representations of color in simulations of visual perception.
Most computational neuroscience sidesteps this issue, “hard-coding” a dimensionality of $3$, representing and constraining cortical color to tristimulus values, such as RGB [parthasarathy2017neural, botella2018nonlinear, brackbill2020reconstruction, kim2021nonlinear, zhang2022image, wu2022maximum], LMS [young1802ii, von1867handbuch] or cone-opponent space [derrington1984chromatic, macleod1979chromaticity].
Constraining cortical models to such a ceiling of three for dimensionality clearly conflicts with the study of a functional tetrachromat observer with 4D color vision [jordan2010dimensionality, rezeanu2021explaining]. Instead, we model the brain with the capability and the challenge of deducing the inherent color dimensionality encoded in the optic nerve signals.

#### 2.3 Theory on How Vision Emerges in the Brain

In computational vision science modeling, it is often overlooked that the cortex relies solely on a stream of optic nerve signals to discover color vision, with no access to a teacher signal or perceptual ground truth.
Rather, various efforts have been made to reconstruct visual stimuli from neural responses by giving the cortical model access to ground truth stimulus image [naselaris2009bayesian, nishimoto2011reconstructing, parthasarathy2017neural, botella2018nonlinear, brackbill2020reconstruction, kim2021nonlinear], but the neural reality is that the brain never has direct access to the visual scene.
This characteristic makes the human visual system a quintessential example of self-supervised learning, which is a growing field in computer vision [de1993learning, chen2020simple, he2022masked]. In this work, we propose a learning principle in the cortex which aims to predict the cellular-level fluctuations in activation that occur during small eye movements, which is associated with the idea of temporal prediction [palmer2015predictive, lotter2016deep, singer2018sensory, singer2023hierarchical], as well as the broader concept of predictive coding [rao1999predictive, srinivasan1982predictive]. It is also closely linked to the sensorimotor contingency theory from cognitive science [o2001sensorimotor] and slow feature analysis [hinton1990connectionist, foldiak1991learning] which suggests that the brain learns to anticipate the sensory outcomes of motor actions (e.g. eye movements) and utilizes these anticipations to filter out invariant (slow) features.

#### 2.4 Theory on How Brains Infer Color Dimensionality

Previous studies has invistigated inference of invariant retinal properties from sensory signals. For instance, research has demonstrated the ability to deduce cone cell types [wachtler2007cone, brainard2008trichromatic, benson2014unsupervised] and the positions of photoreceptors [maloney_ahumada, brainard2008trichromatic] via statistical methods from sensory signals. [brainard2008trichromatic] hinted at analyzing sensory inputs from different time points during fixational drift as a way to reveal retinal features, and in this paper we provide a computational realization of this idea with a specific learning mechanism that achieves complete disentanglement of retinal invariants from optic nerve signals.

Figure: Figure 2: Overview of our simulation engine of biological eyes. 1. This engine takes a scene stimulus as an input, processes it through a “textbook” model of eye motion and retinal neural circuitry, to generate a stream of optic nerve signals. 2. This engine accepts custom eye and retina parameters. 3. It allows visualization of neural signals in steps (A-F), illustrating the progressive entanglement of scene imagery with retinal properties. 4. Visualization of changing one of the input parameters, the number of cone cell types – showing that the signals become noisier as the number increases.
Refer to caption: x2.png

### 3 Simulation Engine for Biological Eyes and Optic Nerve Signals

We model the three primary functions of the human retina based on the known science(^1^11https://color-vision.github.io/).
The scene imagery is sampled from a dataset of hyperspectral images [arad2022ntire] (Fig. 2.A). A model of fixational eye drifts [rucci2015unsteady, young2021emulated, martinez2013impact] generates a sequence of frames that sample different parts of the image. Each frame is projected on the retina, stimulating a randomized array of cone cells, according to known cell density variation as a function of eccentricity (distance from the fovea) [curcio1990human] and known statistics of different cone cell types [carroll2000flicker] (Fig. 2.B). Each cone cell converts the scene light into photoreceptor activations based on cone type spectral response functions [stockman_spectral_1999, stockman2000spectral] (Fig. 2.C), followed by lateral inhibition from horizontal cells [wool2018nonselective, rodieck1965quantitative] (Fig. 2.D). These signals are transformed into spike trains (Fig. 2.E), then bundled into optic nerve signals (ONS), resulting in spatial distortion (Fig. 2.F), turning a color scene stimulus into a noisy, spatially distorted ONS. Additional details are in Appendix S1 and Video 1:30.

One observation to make is that there are three invariant properties of the simulated retina (i.e. retinal invariants): cell positions, cell types, and horizontal cell connections. These properties are held constant during generation of ONS for a particular eye, but we use the engine with different values for these properties to generate ONS datasets for a diverse set of eyes. For example, we create datasets with different cone types, including monochromatic (L, M, S), dichromatic (LM, LS, MS), trichromatic (LMS) and tetrachromatic (LMSQ) configurations (Fig. 2.4). The generated ONS becomes spatially noisier as the number of cone types increases, and we investigate how the cortex infers the inherent color dimensionality of these eyes purely from the encoded differences in their respective ONS. We will release the dataset and code for generating ONS upon acceptance.

Figure: Figure 3: Overview of our hypothesized cortical learning mechanism and exclusive study of the learning behavior of the cortical model with trichromatic retina. 1. Given the stream of optic nerve signals as the only input data, the cortical model aims to predict the next ONS from the current one with 3 learnable functions, decoder $\Phi$, translation operator $\Omega$ and re-encoder $\Psi$. 2. Prediction error decreases as learning progresses, converging after 100K learning steps. 3. During learning, the color dimensionality of the internal percepts transition from 1D, 2D to 3D, formally measured by $\mathit{CMF\textrm{-}SIM}$ and visualized by $\mathit{NS}$ (Section 5). 4. The cortical model infers the retinal invariant properties during learning: cell positions $\mathrm{P}$ (higher density in fovea), cone cell types $\mathrm{C}$, and lateral inhibition weights $\mathrm{W}$ (center-surround receptive field).
Refer to caption: x3.png

### 4 Simulating Cortical Learning and Emergence of Color Vision

Our cortical model is structured to learn three functions in a pipeline (Figure 3.1): 1. $\mathrm{\Phi}$ that decodes the optic nerve signal at time $t$ into its internal percept; 2. $\mathrm{\Omega}$ that translates this percept according to eye motions inferred from the signal over a short time interval, $dt$; and 3. $\mathrm{\Psi}$ that re-encodes the translated percept back into a predicted optic nerve signal at time $t+dt$, which is compared against the real signal received at that time. Mathematically, given the optic nerve signal ${O_{t}}$ at time $t$;

$$ ${\hat{O}_{t+dt}}=\mathrm{\Psi}(\mathrm{\Omega}(\mathrm{\Phi}({O_{t}})))$ (1) $$

where ${\hat{O}_{t+dt}}$ is the predicted ONS at time $t+dt$.
Here the task of decoder $\mathrm{\Phi}$ is analogous to inverting the retinal processes to transform ${O_{t}}$ into the visual percept image ${V_{t}}$ (i.e. ${V_{t}}\leftarrow\mathrm{\Phi}({O_{t}})$). Likewise, the re-encoder function $\mathrm{\Psi}$ resembles the retinal processes, as it aims to reproduce an optic nerve signal from the visual percept (i.e. ${O_{t}}\leftarrow\mathrm{\Psi}({V_{t}})$). Therefore, $\mathrm{\Phi}$ and $\mathrm{\Psi}$ are pseudo-inverses.

The learning objective is to minimize the prediction error $E_{\mathrm{prediction}}$, the difference between predicted and real optic nerve images at time $t+dt$, such that:

$$ $E_{\mathrm{prediction}}=\|{O_{t+dt}}-{\hat{O}_{t+dt}}\|_{2}^{2}=\|{O_{t+dt}}- \mathrm{\Psi}(\mathrm{\Omega}(\mathrm{\Phi}({O_{t}})))\|_{2}^{2}$ (2) $$

where ${O_{t+dt}}$ is the real observed ONS at time $t+dt$.

Three main ideas drove the evolution of our selected model features. First, we reasoned that it would be a big step towards successful decoding and re-encoding if the cortex could infer the key encoding properties of each cone cell. We gave the cortical model sets of learnable parameters, which we call “neural buckets”, in which to store and update guesses of these properties during learning. The buckets contain the following information for each cone cell: 2D position in visual space ($\mathrm{P}$), cone spectral identity ($\mathrm{C}$), and lateral inhibition weights ($\mathrm{W}$) to neighboring cones.

The second main idea was the observation that decoder $\mathrm{\Phi}$ and encoder $\mathrm{\Psi}$ are mathematically factorizable into a pipeline of subfunctions, such that:

$$ $\displaystyle\mathrm{\Phi}$ $\displaystyle=\mathrm{\Phi}_{\mathrm{P}}\circ\mathrm{\Phi}_{\mathrm{C}}\circ \mathrm{\Phi}_{\mathrm{W}}$ $\displaystyle\mathrm{\Psi}$ $\displaystyle=\mathrm{\Psi}_{\mathrm{W}}\circ\mathrm{\Psi}_{\mathrm{C}}\circ \mathrm{\Psi}_{\mathrm{P}}.$ $$

Each sub-function is an operator conditioned on its corresponding neural bucket, cell positions $\mathrm{P}$, cone spectral types $\mathrm{C}$, and lateral inhibition weights to neighboring cells $\mathrm{W}$.
In case of decoder $\mathrm{\Phi}$, it first executes inversion of lateral inhibition using $\mathrm{\Phi}_{\mathrm{W}}$ in order to estimate the activations of each photoreceptor associated with an optic nerve axon; second, projects scalar cone activations into $\mathbb{R}^{N}$ by $\mathrm{\Phi}_{\mathrm{C}}$ using inferred spectral identities in $\mathrm{C}$ (and interpolating color across space – the third main idea below); and finally inverting the spatial distortion of foveation via $\mathrm{\Phi}_{\mathrm{P}}$. The re-encoding function is a pipeline of analogous subfunctions in reverse order: re-applying spatial warping with $\mathrm{\Psi}_{\mathrm{P}}$; re-projecting color into scalar photoreceptor activations with $\mathrm{\Psi}_{\mathrm{C}}$; and re-applying lateral inhibition with $\mathrm{\Psi}_{\mathrm{W}}$. Further implementation details are provided in Appendix S2.

The third main idea was that, in order to accurately re-encode after translation, the model needs to learn to interpolate color information spatially, because there is only one cone type at each point on the retina.
This need to interpolate is analogous to the situation in cameras with image sensors [bayer1976color, fossum1997cmos, kimmel1999demosaicing] that physically sample only one of the R,G,B channels at each pixel, and fundamentally require demosaicking algorithms to interpolate full R,G,B values at all pixels. The required interpolation function in the cortical model is more complex because the spectral sampling pattern is random, so learning this function is entangled with correctly resolving $\mathrm{C}$. We enable and force the cortical model to learn the color interpolation function by representing it as a convolutional neural network with neural bucket parameters $\mathrm{D}$ (Appendix S2.2.2).

An important detail is the handling of eye motion in the simulation. In one experiment we show that the model can learn a subfunction that estimates the eye motion translation between times $t$ and $t+dt$ purely from the optic nerve signals at those times.
The main computational challenge here is the spatial warp in optic nerve signals. The uniform translation in stimulus space corresponds to a non-uniform translation in the optic nerve signal space, which makes the prediction of the eye motion dependent on the inference of cell positions $\mathrm{P}$. We find that the cortical model iteratively updates the neural bucket $\mathrm{P}$ from imperfect initial eye motion estimate, which helps to improve the prediction of eye motion, and vice versa. This inference converges to the correct eye motion estimate, as the model learns to minimize prediction error (Appendix S2.3).

With this pipeline of subfunctions and associated neural buckets, the hypothesized learning is equivalent to parallel numerical optimization of all neural buckets in striving to minimize prediction error. We simulate learning using stochastic gradient descent [kingma2014adam].

### 5 Neural Representation of Color Space and Its Dimensionality

Figure: Figure 4: Overview of $\mathit{CMF\textrm{-}SIM}$. 1. The goal of $\mathit{CMF\textrm{-}SIM}$ is to find the minimum number of primary colors that can be matched to any target colors. A match color is a weighted sum of primary colors, and coefficients are iteratively updated to minimize the perceptual error until convergence. 2. Example output for the cortical model with trichromat retina (3 cone cell types). The matching errors for all target spectral light, ranging from 400nm to 700nm, first converge to zero error with 3 primaries, formally indicating that the model has 3D color vision.
Refer to caption: x4.png

We model color in the cortex as a vector in high-dimensional space, $\mathbb{R}^{N}$. This decision represents our view that the brain has both the freedom and the challenge of somehow inferring the intrinsic color dimensionality of the visual signals it is receiving along the optic nerve.
Specifically, we define a cortical decoding function $\mathrm{\Phi}$ that takes the optic nerve signal ${O_{t}}$ at time $t$ and transforms it into its internal visual percept ${V_{t}}$ (Figure 3), such that:

$$ ${V_{t}}=\mathrm{\Phi}({O_{t}})$ (3) $$

where each pixel in ${V_{t}}$ is a $N$-dimensional vector. Inside $\mathbb{R}^{N}$, we assume that cortical color space emerges as a $K$-dimensional manifold. To measure this intrinsic dimensionality $K$, we introduce two methods: formal, numerical $\mathit{CMF\textrm{-}SIM}$; and intuitive, visual $\mathit{NS}$.

Our quantitative analysis method $\mathit{CMF\textrm{-}SIM}$ treats the retina model plus cortical model as a black-box color observer. That is, we limit ourselves to showing the model two patches of scene color and obtaining as feedback only a scalar score of the difference in color appearance (with zero being a color match). This interface is intentionally limited and identical to color matching experiments with human subjects. Then, resting on the formal, technical bedrock for colorimetry established by maxwell1856theory and grassmann1853theorie, we exhaustively probe to determine the minimum number of color primaries needed to match any test color through linear combination (Figure 4 & Appendix S3). For example, Maxwell found that most humans require $3$ primaries and are formally trichromatic, but red-green colorblind persons require only $2$ primaries and are dichromatic. We model diverse color observers and measure the color dimensionality of their emergent vision.

Building on this, we address the theoretical question of whether cortical inference is essential for color vision. We posit that it is indeed crucial, comparing to a baseline of representing optic nerve signals directly as internal percepts (i.e. no cortical inference) and showing that this baseline fails to pass color matching tests. $\mathit{CMF\textrm{-}SIM}$ reveals the color dimensionality is $1$-D for observers with $3$ cone types. This underscores the necessity of cortical processing in color perception (Appendix S4).

We further provide a qualitative analysis tool, Neural Scope ($\mathit{NS}$) which maps the emergent cortical color manifold in $\mathbb{R}^{N}$ linearly to conventional RGB color, in order to enable visual inspection of the emergent color vision.
$\mathit{NS}$ provides striking color visualizations and complementary intuition visually, which are fully consistent with formal $\mathit{CMF\textrm{-}SIM}$ results.
For example, $\mathit{NS}$ allows us to compute and contrast the color palettes of the three different types of human dichromacy (Figure 5).

In sum, we model cortical color as vectors in $\mathbb{R}^{N}$, allow color space to emerge naturally through the hypothesized learning mechanism, and measure the emergent color space’s dimensionality quantitatively with $\mathit{CMF\textrm{-}SIM}$ and analyze it visually with $\mathit{NS}$.

### 6 Simulation Results - Emergence of Color Vision

Figure 3.2 begins with a model of typical human retina containing L, M and S cones, and illustrates the time-varying behavior of the cortical model as it learns color vision. The visualized prediction error decreases as the training progresses, and the internal percept converges to $3$-D color vision both formally ($\mathit{CMF\textrm{-}SIM}$) and visually ($\mathit{NS}$ imagery), shown in Figure 3.3. Notably, the visual timeline highlights that the cortical model learns color vision one dimension at a time: achieving monochromacy at 700 learning steps, dichromacy at 1,200 steps, and converging to trichromacy after $10^{6}$ steps.
At convergence, the cortical model has accurately inferred all retinal properties: spectral identity of each cell, cell positions and lateral inhibition neighbor weights (Figure 3.4).

The remainder of this section shows $\mathit{CMF\textrm{-}SIM}$ and $\mathit{NS}$ results at convergence for a diversity of simulation scenarios. Figure 4.2 dissects the $\mathit{CMF\textrm{-}SIM}$ analysis for the case where the retina contains $3$ cone types. The graphs show color matching function results using optimal sets of color primaries from 0 0 to $4$ primaries, along with the residual perceptual error as a function of wavelength.
The area under the curve (AUC) for each error graph suggests that the error falls to near-zero only with at least 3 primaries – this formally proves that the color dimensionality is $3$.

Figure 5 presents results of the hypothesized cortical learning in a diversity of color observers where the retina contains different numbers of cone cell types.
This shows that the model learns $K$-dimensional color vision when the retina contains $K$ cone types. That is, when the retina contains 1, 2, 3 or 4 cell types, the cortical model converges on mono, di, tri, or tetrachromat color vision, formally quantified with $\mathit{CMF\textrm{-}SIM}$ (further analysis of tetrachromat models in Appendix S6). And qualitatively, we observe that the $\mathit{NS}$ images are grayscale for $K=1$, colorblind with only shades of blue and yellow for $K=2$, and full color with all trichromatic hues only with $K=3$ (Figure 5.1).
Use of $\mathit{NS}$ is limited to color dimensionality up to 3, and is to not applicable to $K=4$ case, but $\mathit{CMF\textrm{-}SIM}$ formally confirms that 4-dimensional color emerges there.
All variants of retinas with two cone types (protanopia, deuteranopia and tritanopia), converge to 2D color vision. But more striking, $\mathit{NS}$ reveals hue shifts among these models that are consistent with CVD studies [brettel1997computerized] (yellow-blue hues for deuteranopia / protanopia [judd1948color, graham1959studies] and
teal-pink hues for tritanopia [alpern1983trit]) (Figure 5.2).

Figure: Figure 5: Results of simulating emergence of color vision from various retinas, with analysis of learned color dimensionality using qualitative visualization ($\mathit{NS}$) and formal methods ($\mathit{CMF\textrm{-}SIM}$). 1. Cortical models trained with dataset generated with retinas containing 1, 2, 3, 4 cone types result in mono-, di-, tri, tetrachromatic color vision, respectively. 2. Qualitative color of dichromat variants is consistent with known vision science on color vision deficiency.
Refer to caption: x5.png

Figure 6 simulates boosting of color dimensionality in adulthood by gene therapy. Previous genetic studies [jacobs2007emergence, mancuso2009gene, zhang2017gene] demonstrated that the introduction of an additional class of photopigments in the mammalian cone mosaic, even in adulthood via gene therapy, resulted in a new dimension of chromatic sensory experience. Here, we simulate the experiment performed in squirrel monkeys [mancuso2009gene], finding simulation results consistent with the noted boost in color dimensionality (Figure 6.1).
First, we model the vision of an adult male squirrel monkey with a protanopic retina (M and S cones only) and a cortical model that has converged. Next, we simulate the effects of gene therapy, by modifying the retina model so that a random subset of cones begin to express L opsin. Immediately after this retinal change, $\mathit{NS}$ continues to show blue-yellow dichromatic vision. However, if we allow the cortical model to continue the hypothesized self-supervised learning, vision re-converges to boosted, 3-dimensional color (Figure 6.1). As an aside, the real gene therapy experiment [mancuso2009gene] resulted in expression of both M and L photopigment in affected cones, but the relative amounts are difficult to ascertain.
We additionally model scenarios in which equal M and L expression occurs or a variable amount of L relative to M at each cell, and both scenarios converge to trichromacy as well (Appendix S5). These result indicate that the hypothesized self-supervised learning can explain experimental boosting of color dimensionality in adulthood consistent with mancuso2009gene, if the hypothesized learning is assumed to occur continuously even in adulthood.

Figure 6.B further demonstrates that simulations of boosting color dimensionality succeed even in 3D$\rightarrow$4D cases by the addition of a fourth cone
between S and M cones. This simulation results present the first theoretical work that highlights the possibility of boosting humans with trichromatic vision to tetrachromat one by addition of a fourth photopigment. Further details, including the study of human tetrachromat observer model, are described in Appendix S6.

Figure: Figure 6: Simulated experiments for boosting color dimensionality via gene therapy. 1. A 2D dichromat color vision is boosted to 3D trichromacy by mutating some M cones to express L opsins, with cortical learning re-converging to 3D color. 2. Similarly, a 3D trichromat model is boosted to 4D tetrachromacy by adding a fourth cone type between M and L cones, with cortical learning re-converging to 4D color.
Refer to caption: x6.png

### 7 Conclusion

In this work, we presented a framework for modeling the emergence of color vision in the human brain. We introduced a simulation engine for optic nerve signals, simulated cortical learning purely from such input signals, and measured the emergent color dimensionality qualitatively with $\mathit{NS}$ and quantitatively with $\mathit{CMF\textrm{-}SIM}$.

We believe that the critical contribution of this work actually lies in the computational formulation of the problem in human perception itself – given access to only a stream of optic nerve signals, we probe how to meaningfully model the emergence of color vision in the cortex via simulated learning. Once the problem is formulated this way, it may not be entirely surprising to machine learning researchers that color with the correct dimensionality can be inferred; however, from the vision science perspective, this new approach is in many ways a foreign way to formulate the problem, because it is more common to think color comes from hardwired neural circuits (Appendix S4).

The connection to camera imaging systems is noteworthy as well. The proposed computational framework is akin to a camera processing pipeline attached to an unknown, random color filter array pattern, where even the number of different color filters is a mystery. This is related to a branch of computational imaging called “auto calibration” that jointly solves for the scene and unknown system calibration parameters from measurements. One could imagine a new class of engineered sensing systems that have general processing units along the lines of the learning mechanism in this paper, that enables perceptual inference from a broader range of sensory streams that do not require the precision manufacturing common to many current sensors and cameras.

The learning notion of perception emerging from a process of disentangling it from encoded sensory streams is an interesting intellectual view of how perception may generally emerge in the brain.
Any sensory stream going to the brain encodes information from the world entangled with sensing organ characteristics. The self-supervised learning mechanism proposed in this paper might be abstracted into a general neural process of learning to predict fluctuations in sensory stream due to ego perturbation,
with perception for that sense emerging neurally as the optimal internal representation and associated decoder/encoder pair that enables accurate prediction.

To conclude, we invite the research community to build on the computational framework proposed in this paper, by improving the visual system components (e.g. even more accurately modeling details in the eye model, or replacing back-propagation in the cortical model with more bio-plausible learning rules [lillicrap2020backpropagation, hinton2022forward]), or applying the framework to other sensory modalities.

## Supplementary Materials

In this supplementary section, we provide additional details on our computational framework for modeling the emergence of human color vision. Section S1 covers the simulation of optic nerve signals from hyperspectral scene images and our modeling of retina circuitry. Section S2 details the design of our simulated cortical model. Section S3 provides a step-by-step description of how we simulate formal color matching function tests of color dimensionality, $\mathit{CMF\textrm{-}SIM}$. In Section S4, we probe whether the emergence of color vision depends on cortical learning, by introducing comparative, baseline cortical models.
Section S5 simulates variant cell expression phenotypes after gene therapy in squirrel monkeys [mancuso2009gene], showing robustness of the simulation result that addition of a new cell opsin gene enhances color dimensionality. Finally, Section S6 discusses our tetrachromacy simulation, highlighting the differences between the human Q cone (between M and L cones) and the pigeon Q cone (between S and M cones).

Upon acceptance, we will make our entire codebase available along with trained model weights and comprehensive documentation.

**Table 1: Glossary of terms used in our framework**
| Scalars |  |
| --- | --- |
| $t,x,y$ | Time, world view coordinates |
| $u,v$ | Optic nerve image coordinates |
| $N$ | $N$-dimensional vector represents cortical color |

### S1 Simulation Engine of Biological Eyes

We simulate the optic nerve signals from hyperspectral scene images, simulate eye gaze movements, and model the retina circuitry. We describe the details of these simulations in the following.

#### S1.1 Database of Hyperspectral Scene Images

We model the scene viewed by the eye using image databases comprising $900$ photographs of everyday scenes shot with a hyperspectral camera [arad2022ntire] where each pixel is represented by spectral power distributions (“spectra” below).

#### S1.2 Eye Gaze Movements – Fixational Drift

We simulate small eye movement as fixational eye drifts with a random walk [young2021emulated]. This produces a stream of the current scene image translating across the retina. Over a time interval of duration $dt$, we sample the change in eye motion from a uniform probability distribution $(dX,dY)\sim\mathcal{U}\{(-15,15)\times(-15,15)\}$ where $15$ corresponds to $15$ pixels in the scene image $S$.

#### S1.3 Retina Circuitry Model

Our model of light detection and signal encoding in the retina is the textbook model [rodieck1998first] of the photopic, midget, “private-line” pathway (see Fig. 2). Details of our implementation follow.

##### S1.3.1 Cone Mosaic and Spectral Sampling

In our simulation, we model $256\times 256$ cone cells, with the random distribution of L, M and S cone cells on the retina with a relative probability of $0.63$, $0.32$ and $0.05$ from sabesan2015characterizing. We model L, M and S spectral response using the template response function from carroll2000flicker, with spectral peaks of $560$, $530$, $419$ nm, respectively.
We model photon-shot noise in the photoreceptor activation values, simulating a signal-to-noise ratio of approximately $100$. The simulation omits foveal tritanopia [williams1981foveal].

##### S1.3.2 Foveation and Cone Positions

We model cone cell spatial density on the retina in accordance with curcio1990human, while enforcing retinotopy. Starting from a regular grid of cells, we computationally perturb positions until the target spatial distribution is stochastically achieved, while constraining each cell to be always confined by its neighbors. We model biological variation in cell locations, which represents a challenge for cortical learning, by adding multi-resolution positional randomness [perlin1985image].

##### S1.3.3 Center-Surround Lateral Inhibition

We mathematically model textbook lateral inhibition in retinal signals as a Difference-of-Gaussians (DoG) convolution kernel [enroth1983spatio], with parameters fitted from electrophysiology [wool2018nonselective]. Specifically our DoG kernel has standard deviations of $0.15$ and $0.9$ cone diameters, respectively, for the positive center and negative surround Gaussians. Significantly, the relative amplitudes are such that the kernel has a non-zero mean of $0.09$ [lennie1991design, wool2018nonselective].
We convolve the array of cone activation values by this DoG kernel to compute outputs to bipolar cells.

##### S1.3.4 On/Off Pathways and Optic Nerve Spiking

We model textbook on- and off- connections from cone activations to bipolar cells, forwarding to on- and off- retinal ganglion cells (RGC)s. The on- activation is modeled by a rectified linear unit activation function, $\mathrm{ReLU}(x)=\mathrm{max}(0,x)$, where $x$ is the laterally-inhibited cone output. The off- activation is modeled as $\mathrm{ReLU}(-x)$.
Optic nerve spikes are the outputs at RGC axons, with action potentials are generated with Leaky Integrate and Fire model [lapicque1907recherches, abbott1999lapicque, arbib2003handbook].

### S2 Cortical Model

#### S2.1 Time-Averaged Optic Nerve Signals

We model the first step of cortical processing as re-combination of the on- and off- RGC action potentials followed by time-averaging. This results in an image stream ${O_{t}}$ with real-valued pixel values that are proportional to the laterally-inhibited output from cone cells. ${O_{t}}$ is the sole input to the remainder of the cortical model.

#### S2.2 Prediction Pipeline and Learning Objective

Figure 7 shows an elaboration of the cortical prediction pipeline introduced in Figure 3.1 and summarized in Setion 4. This pipeline represents an existence-proof simulation that the hypothesized self-supervised learning on ${O_{t}}$ can successfully produce: (1) emergence of vision with the correct color dimensionality, and (2) inference of invariant retinal properties, including learned neural buckets for cone spectral identities $\mathrm{C}$, cell positions $\mathrm{P}$ and lateral inhibition weights $\mathrm{W}$.
Our results show that both can be simultaneously achieved by learning to minimize the error in predicting the fluctuations in optic nerve signal values.

To re-summarize Section 4, the prediction is made by applying the learned functions for decoding $\mathrm{\Phi}$, translation $\mathrm{\Omega}$ and re-encoding $\mathrm{\Psi}$ to the optic nerve signal ${O_{t}}$, to obtain a predicted optic nerve signal ${\hat{O}_{t+dt}}=\mathrm{\Psi}(\mathrm{\Omega}(\mathrm{\Phi}({O_{t}})))$. The learning objective function is to minimize prediction error, the difference between predicted and real optic nerve images at time $t+dt$: $E_{\mathrm{prediction}}=\|{O_{t+dt}}-{\hat{O}_{t+dt}}\|_{2}^{2}=\|{O_{t+dt}}-
\mathrm{\Psi}(\mathrm{\Omega}(\mathrm{\Phi}({O_{t}})))\|_{2}^{2}.$
The decoder $\mathrm{\Phi}$ and re-encoder $\mathrm{\Psi}$ functions are each factorized into a pipeline, such that:
$\mathrm{\Phi}=\mathrm{\Phi}_{\mathrm{P}}\circ\mathrm{\Phi}_{\mathrm{C}}\circ
\mathrm{\Phi}_{\mathrm{W}}$ and $\mathrm{\Psi}=\mathrm{\Psi}_{\mathrm{W}}\circ\mathrm{\Psi}_{\mathrm{C}}\circ
\mathrm{\Psi}_{\mathrm{P}}$, where each sub-function is an operator conditioned on its corresponding neural bucket, $\mathrm{C}$, $\mathrm{P}$ and $\mathrm{W}$. The following sections describe the implementation of each learnable neural bucket and its associated cortical sub-function.

Figure: Figure 7: Pipeline of decoding, translation and re-encoding cortical functions. The optic nerve signal image ${O_{t}}$ is decoded into visual percept image ${V_{t}}$. The percept is translated in accordance with eye motion over time $dt$, then re-encoded back into a prediction of the optic nerve signal ${\hat{O}_{t+dt}}$ at a short time in the future.
Refer to caption: x7.png

##### S2.2.1 Learning Lateral Inhibition Neighbor Weights W W \mathrm{W} roman_W

In the factorized function pipeline, sub-decoder $\mathrm{\Phi}_{\mathrm{W}}$ conceptually inverts lateral inhibition to transform optic nerve image ${O_{t}}$ into cone activation image ${A_{t}}$. Conversely, sub-encoder $\mathrm{\Psi}_{\mathrm{W}}$ reproduces lateral inhibition to transform ${A_{t}}$ into ${O_{t}}$. These operators are modeled, respectively, as inverse and forward convolution operators, implemented by pixel-wise division and multiplication in the Fourier domain. The learnable parameters for this operator are neural bucket $\mathrm{W}$, representing the Fourier transform image pixels of the lateral inhibition DoG kernel’s Fourier transform. The resolution of $\mathrm{W}$ is set equal to the image resolution of ${O_{t}}$.

##### S2.2.2 Learning Color Types C C \mathrm{C} roman_C and Interpolation D D \mathrm{D} roman_D

In the factorized function pipeline, sub-decoder $\mathrm{\Phi}_{\mathrm{C}}$ conceptually transforms cone activation image ${A_{t}}$ (scalar-valued pixels) into a full-color visual percept image ${\tilde{V}_{t}}$ ($N$-dimensional vector pixels). Conversely, $\mathrm{\Psi}_{\mathrm{C}}$ re-encodes ${\tilde{V}_{t}}$ into ${A_{t}}$. These functions depend on two neural buckets of learnable parameters, $\mathrm{C}$ and $\mathrm{D}$. $\mathrm{C}$ is an image of $N$-dimensional vectors, in which the cortical model learns to represent the spectral type of the source cone associated with each pixel in ${A_{t}}$. $\mathrm{D}$ are the parameter weights of a learnable function $\phi_{\mathrm{D}}$ that spatially interpolates color across the image, which we implement as a convolutional neural network based on the U-Net [ronneberger2015u] architecture.

Mathematically, $\mathrm{\Phi}_{\mathrm{C}}$ and $\mathrm{\Psi}_{\mathrm{C}}$ are defined as $\mathrm{\Phi}_{\mathrm{C}}({A_{t}})=\phi_{\mathrm{D}}({A_{t}}\otimes\mathrm{C})$,
where $\otimes$ denotes element-wise multiplication; $\mathrm{\Psi}_{\mathrm{C}}({\tilde{V}_{t}})={\tilde{V}_{t}}\otimes\mathrm{C}$, where $\odot$ denotes element-wise dot product. To maintain $\mathrm{\Phi}_{\mathrm{C}}$ and $\mathrm{\Psi}_{\mathrm{C}}$ as pseudo-inverses, we constrain $\mathrm{C}\odot\mathrm{C}$ to be an image with all pixels equal to 1.

##### S2.2.3 Learning Cell Positions P P \mathrm{P} roman_P

As shown in Fig. 2.3.F, optic nerve images are spatially distorted due to foveation, whereas visual perception is apparently undistorted.
In the factorized function pipeline, sub-decoder $\mathrm{\Phi}_{\mathrm{P}}$ conceptually transforms ${\tilde{V}_{t}}(u,v)$, which is foveated and spatially distorted, into the final visual percept image ${V_{t}}(x,y)$, which is Euclidean and undistorted. Conversely, $\mathrm{\Psi}_{\mathrm{P}}$ re-warps ${V_{t}}(x,y)$ into ${\tilde{V}_{t}}(u,v)$.

We define a learnable spatial warping function $\phi_{\mathrm{P}}(u,v)=(x,y)$, implemented using normalizing flow [rezende2015variational, dinh2016density] that is an invertible function. Neural bucket $\mathrm{P}$ comprises the learnable parameters of this normalizing flow network. Then $\mathrm{\Phi}_{\mathrm{P}}({\tilde{V}_{t}})(x,y)={\tilde{V}_{t}}(\phi_{\mathrm
{P}}^{-1}(x,y))$, and $\mathrm{\Psi}_{\mathrm{P}}({V_{t}})(u,v)={\tilde{V}_{t}}(\phi_{\mathrm{P}}(u,v))$.

#### S2.3 Learning Eye Motion Estimation M M \mathrm{M} roman_M

In the learning loop, cortical sub-function $\mathrm{\Omega}$ translates the visual percept image according to the spatial motion $(dx,dy)$ that occurs during the brief period between $t$ and $t+dt$. Where does $(dx,dy)$ come from? Here, we describe how the cortex can learn a helper cortical function $\mathrm{\mu}_{\mathrm{M}}({O_{t}},{O_{t+dt}})$ that estimates $(dx,dy)$ directly from the optic nerve stream values at times $t$ and $t+dt$, maintaining the strict operation of the cortical learning model purely from optic nerve signals.

We model learning of $\mathrm{\mu}_{\mathrm{M}}({O_{t}},{O_{t+dt}})$ with neural bucket $\mathrm{M}$ by optimizing:

$$ $\displaystyle\mathrm{M}=\underset{\mathrm{M},\mathrm{P}}{\mathrm{argmin}}\| \mathrm{B}({O_{t+dt}})-\mathrm{\Psi}_{\mathrm{P}}(\mathrm{\Omega}_{\mathrm{\mu }_{\mathrm{M}}({O_{t}},{O_{t+dt}})}(\mathrm{\Phi}_{\mathrm{P}}(\mathrm{B}({O_{ t}}))))\|_{2}^{2},$ $$

where $\mathrm{B}$ is a convolution operator that low-pass filters the optic nerve images, effectively halving the highest frequencies; $\mathrm{\Phi}_{\mathrm{P}}$ aims to dewarp the optic nerve image into Euclidean coordinates, and $\mathrm{\Psi}_{\mathrm{P}}$ is the inverse operator that rewarps the image. Details are discussed also in Video 11:30.

#### S2.4 Efficient Learning Implementation & Alternative Representation of Internal Percepts

We simulate the self-supervised learning by parallel numerical optimization of all neural buckets
by applying the stochastic gradient descent algorithm to minimize the prediction error metric. In practice, this is implemented by repeating the following step thousands of times until convergence: pick a batch of different scene images from the database; generate the optic nerve signals at time $t$ with the retina encoding model; send this into the cortical model and execute the decode / translate / re-encode functions with current neural bucket parameters to estimate the signal at $t+dt$; compute the actual optic nerve signal at $t+dt$ with the retina encoding model; compute the prediction error by taking the difference between prediction and actual signal; backpropagate the prediction error to update neural bucket parameters.

For efficient processing of cortical functions, we avoid computing full-resolution undistorted visual percept images ${V_{t}}(x,y)$ during the learning simulation. Instead, we directly compute ${\tilde{V}_{t+dt}}(u,v)$ by optical flow of the pixels in ${\tilde{V}_{t}}(u,v)$, where the optical flow map is defined by $(\mathrm{\Psi}_{\mathrm{P}}\circ\mathrm{\Omega}_{\mathrm{\mu}_{\mathrm{M}}({O_
{t}},{O_{t+dt}})}\circ\mathrm{\Phi}_{\mathrm{P}})$. This is mathematically equivalent to the learning implementation described in Methods S2.2,
and the cortical model continues to learn cell positions $\mathrm{P}$, and therefore decoder $\mathrm{\Phi}$ can still be applied to compute undistorted visual percepts when desired.

Figure: Algorithm 1 Pseudocode for $\mathit{CMF\textrm{-}SIM}$

### S3 Color Matching Function Tests & 𝐶𝑀𝐹 ⁢ - ⁢ 𝑆𝐼𝑀 𝐶𝑀𝐹 - 𝑆𝐼𝑀 \mathit{CMF\textrm{-}SIM} italic_CMF - italic_SIM

In the classical color matching theory of Maxwell [maxwell1856theory] and Grassman [grassmann1853theorie], an observer compares a test color spectrum, $t$, and a set of $K$ primary color spectra, $p_{i}$ for $i=1,...,K$ and tunes real weights $\alpha_{i}$ for each primary until a color match is achieved. If a particular weight has a negative value, primary light of that magnitude is added to the test color $t$ rather than to the other primary colors. Mathematically, a color match is achieved when the matching spectrum $\sum_{i=1}^{K}\mathrm{\alpha}_{i}^{+}p_{i}$ and test spectrum $t+\sum_{i=1}^{K}(-\alpha_{i})^{+}p_{i}$ appear identical. Here, $\alpha_{i}^{+}$ is defined as $\max(\alpha_{i},0)$.

We simulate such classical color matching by formulating the retina model plus learned cortical model as a black-box color observer in the colorimetric sense. That is, we take the test and matching spectra, apply the retina encoding model $\mathrm{\Upsilon}$ and the learned cortical decoder $\mathrm{\Phi}$ to each separately, and iteratively update the weights $\alpha_{i}$ until there is no further improvement in the computed perceptual difference $E=\|\mathrm{\Phi}(\mathrm{\Upsilon}(\sum_{i=1}^{K}\mathrm{\alpha}_{i}^{+}p_{i}
))-\mathrm{\Phi}(\mathrm{\Upsilon}(t+\sum_{i=1}^{K}(-\alpha_{i})^{+}p_{i}))\|_
{2}^{2}$. Note that the optimal $E$ will not be close to zero if the primaries cannot match test color $t$.

The color dimensionality of an observer is formally equal to the minimum number of primary colors required to match any test color. $\mathit{CMF\textrm{-}SIM}$ simulates thousands of color matches to rigorously compute this dimensionality for any given retina model plus cortical model. In $\mathit{CMF\textrm{-}SIM}$, we simulate classical color matching function (CMF) tests of exhaustively attempting to match test colors equal to each monochromatic wavelength (100 samples from 400 to 700 nm), with a linear combination of $K$ primary colors. For the primary colors, we use distinct, monochromatic spectra. We start CMF tests with a single primary, and add primaries one-by-one until all test wavelengths in the CMF can be matched successfully. The pseudocode for $\mathit{CMF\textrm{-}SIM}$ is described in Algorithm 1.

For example, for our cortical model to be formally measured as trichromat, we must show that there do not exist any 1 or 2 primaries that can pass the CMF tests, and show 3 primaries that succeed. A brute-force approach would be to exhaustively test all possible sets of $K$ primaries; for efficiency, we instead perform stratified sampling to randomly generate a large number (e.g. 500) distinct sets of primaries for each choice of $K$. We require that all of these possible primary sets fail, with the aggregate perceptual error across all wavelengths being greater than a threshold $\epsilon$, before incrementing the CMF tests to $N$+1 primaries.

Figure: Figure 8: Comparison $\mathit{CMF\textrm{-}SIM}$ results of our baseline cortical models with limited or no inferential processing. In all of these models, the input optic nerve stream comes from a retina with 3 cone cells as in regular human trichromacy. 1. The first baseline model represents no cortical learning involvement in human visual perception, by setting the internal percept directly equal to the input optic nerve signal. The resulting base error, the perceptual error between the same color patch at two different retinal locations, results in higher errors, compared to the energy of the internal percept. This means that this cortical model fails to recognize the same color consistently across different patches of the retina, so it is procedurally impossible to even try color matching function tests against reference match colors. Formally then, this model fails resolve color vision. 2. In the second baseline model, we treat cone cell activations as internal percepts, omitting lateral inhibition from the eye simulation; $\mathit{CMF\textrm{-}SIM}$ measures the resulting internal percept as 1D color vision, failing to resolve trichromacy. 3. The third baseline model is an ablation model in which we omit the demosaicking network $\mathrm{D}$ from our proposed model of Section S2; $\mathit{CMF\textrm{-}SIM}$ also measures the resulting internal percept as 1D color. Red base errors are superimposed on energy / 1 primary error plots for visual comparison, and 150% of these base errors are applied as threshold determination.
Refer to caption: x8.png

### S4 Baseline Results – Testing if Cortical Inference is Required for Color Vision

A fundamental question about color perception is whether cortical inference is necessary, or if human color perception arises directly from hardwired neural circuits. The main paper presents analysis an existence proof that cortical inference can indeed result in color vision of the correct dimensionality. Here, we add evidence that cortical inference of some kind is necessary, by modeling and testing three baseline cortical models with limited or no cortical processing.

For our first baseline, we assume an extreme case where no cortical learning occurs, treating raw optic nerve signals directly as internal percepts (i.e., the cortical decoder function $\Phi$ is an identity function). We apply $\mathit{CMF\textrm{-}SIM}$ directly to the optic nerve signals generated by a trichromat retina. Figure 8.1 presents analysis that this model fails to generate vision that can be recognized as color consistent. The analysis is to perform baseline color matching experiments where the test and match spectral functions are identical. This “base perceptual error” is larger than the energy of each color percept itself, meaning that the emergent vision fails to recognize a patch as the same color across different parts of the retina.

The second baseline model is a variant of the first, where we change the optic nerve stream to directly transfer cone activation values, omitting the encoding complications of foveated warping and lateral inhibition. This is intended as a far simpler encoding, to pressure test whether color vision can be detected in the spectrally encoded cone values without further processing. However, $\mathit{CMF\textrm{-}SIM}$ formally measures that this model results in 1D color (Fig.8.2), instead of the expected 3D color from such a retina. This result shows that cortical processing is required even on cone activations to produce color vision of the correct color dimensionality.

The third baseline model adds another perspective by taking the full-featured model detailed in Section S2 but omitting the demosaicking network $\phi_{\mathrm{D}}$ (as defined in Supplement S2.2.2). As shown in Fig.8.3, $\mathit{CMF\textrm{-}SIM}$ measures the resulting percepts of this model also as 1D, providing evidence of the importance of demosaicking process in the emergence of color vision.

### S5 Variations in Cell Expression after Gene Therapy

As described in Section 6, we simulate experiments aimed at boosting color dimensionality [mancuso2009gene]. In our simulation, approximately 60% of M cones are affected by gene therapy, with three possible scenarios for their modification. First, affected M cones are completely transformed into pure L cones (Fig.9.1). Second, affected M cones equally express M and L opsins (Fig.9.2). Third, affected M cones express M and L opsins in random ratios, that is $\alpha$L $+$ $\beta$M, where $\alpha$ and $\beta$ spatially vary across the retina (Fig.9.3). In all cases, we confirm that our cortical model acquires 3D color vision after adaptation (i.e. re-learning), formally measured by $\mathit{CMF\textrm{-}SIM}$.
This makes intuitive sense if we consider the emergent vision from linear systems theory. Each different opsin can be interpreted as a basis vector for the resulting linear color space. In this view, cells containing a mixture of two cones (M and L in this case) can be interpreted as linear combinations of two basis vectors, which cannot increase the dimensionality of the linear space.

Figure: Figure 9: Three scenarios of cell expression after gene therapy adding a third cone type: 1. affected M cones transform into L cones, 2. affected M cones express 50:50 M and L opsins, and 3. affected M cones express M and L opsins at spatially-varying, random ratios (i.e. $\alpha,\beta\in[0,1],\alpha+\beta=1$). $\mathit{CMF\textrm{-}SIM}$ shows that all cases converge to 3D color vision after post-therapy re-learning.
Refer to caption: x9.png

### S6 Probing Effect of Spectral Response and Environmental Colors in Acquiring Tetrachromatic Vision

In Section 6, tests of tetrachromatic eyes assumed the fourth cone type was a pigeon Q cone, with peak sensitivity at 506 nm, between the S and M cone peak sensitivities (419 nm and 530 nm, respectively). In this section, we report additional experiments using instead a human Q cone, modeled with peak sensitivity at 545 nm between M and L peak sensitivities (530 nm and 560 nm, respectively). As shown in the spectral graphs of Figure 10, the pigeon Q cone is more decorrelated from L, M, S, which in principle may more easily support emergence of 4D color vision.

Further, we probe the effect of visual environment on the emergence of tetrachromacy, simulating the learning of color vision with three different input scene image datasets. The first environment is the set of real hyperspectral photographs of everyday scenes [arad2022ntire] as described in Section S1.1. However, natural hyperspectral images are thought to be lacking in human tetrachromatic colors [leeTetrachromacy2024], so in the second and third datasets we augment the 900 hyperspectral images with 100 tetrachromatic color patches. In the second dataset, the colors are sampled from the tetrachromatic hue sphere as computed using recently developed $N$-dimensional color theory for tetrachromacy [leeTetrachromacy2024], customized to each of the observers here (i.e. with human Q or pigeon Q, respectively). In the third dataset, the colors are sampled from an idealized tetrachromatic color space in which each of the four cone channels is allowed to take any value, allowing theoretically maximum levels of chromatic contrast. Since natural hyperspectral images have been found lacking in human tetrachromatic colors [leeTetrachromacy2024], in principle we may expect the likelihood that 4D color vision emerges to increase across these three simulated environments.

Indeed, in line with theoretical intuition, Figure 10 shows that 4D color vision emerges, as measured formally by $\mathit{CMF\textrm{-}SIM}$, only for the third visual environment in the case of the human Q cone, but 4D color vision emerges with any of the visual environments for the pigeon Q cone.
Our simulations suggest strong genetic and environmental effects on emergence of tetrachromatic color vision.

Figure: Figure 10: Results of expanded experiments of boosting color dimensionality from 3D to 4D, by addition of a Q cone to trichromatic L, M, S cones. Panel 1 models addition of a Q cone from natural human tetrachromacy, and panel 2 is for a Q cone from pigeon vision; spectral responses are shown. Each panel compares emergence of color vision with three different visual environments: real hyperspectal images [arad2022ntire]; synthetic images containing tetrachromatic colors sampled from the hue sphere customized to that color observer [leeTetrachromacy2024]; synthetic images containing idealized tetrachromatic colors with chromatic contrast between cone channels at the theoretical maximum. The simulation results show that 4D color vision only emerges for the human Q cone with the third environment, while it emerges for the pigeon Q cone under all three environments.
Refer to caption: x10.png

## Peer Reviews

confidence: 3
contribution: 2
initial_rating: 8
presentation: 3
questions: A few additional questions: 
 
1. What is $N$ in the cortical learning model? What would happen if the model directly set $N=3$? 
 
2. Does the $NX3$ matrix (neural scope) hold constant in all the shown examples (like Figure 3.3) or is this matrix refitted every single time given the different internal percepts? 
 
3. Would the authors comment on why it takes additional 99K steps to learn the 3rd color? Would this (#steps) be different if there were only 2 colors? 
 
4. If I read the manuscript correctly, the visual percepts are the output of U-net? 
 
5. U-net (the backbone for demosaicking here) contains substantial inductive bias in its architecture. Is the separate translation operator a model merely a biologically inspired model choice? It seems that the model is trained on pairs of data (the input ONS is from time t and the output ONS is from $t+\Delta t$) instead of sequences. If U-net absorbs the learning of $\Omega$, would it change the performance? I understand eye movements shift ONS, but is it necessary to map it into a separate subfunction?
soundness: 3
strengths: The authors claim that the main novelty of the paper is the framing of the emergence of color vision as a learning problem. I agree the machine learning framework includes a couple biologically inspired constraints. I would appreciate some clarifications to better understand the significance of the cortical learning model.
summary: The authors claimed that they reverse engineered how color vision may emerge in a biologically inspired cortical learning model. They use simulated optic nerve signals (ONS) as input and trained neural networks to individually approximate three invariant properties of the retina: cell positions, cone spectral type, and magnitudes of lateral inhibition. Their model optimizes the error between the predicted ONS signal at the next sampling step and the real simulated ONS signal. They showed that during the minimization of error between inferred ONS and real ONS, the internal perception of their model gradually adds individual primary colors. To show the aggregation of primary colors, they probe the learned embedding with another separately trained model; they call this separate color matching model CMK-SIM. This CMK-SIM is based on text book colorimetry. They use the observation that CMK-SIM found three primary colors in their trained model (instead of the raw optic nerve signal as the model input) as the success of recovering color vision.
weaknesses: The technical details of this paper are a bit hard to follow (those missing ?? reference in supplementary do not help). For example, I would appreciate further clarification on how the authors choose their baseline models. The authors merely took the entire cortical learning model out (baseline I) or simplified their ONS signal (Baseline II). I guess color/cone spectral type information is present in the ONS signal? Can the author verify this (although CMK-SIM choose to only use 1 color)? 
 
Now another possibility is that the U-net architecture the authors use for the C and D gets way overfit (for example, [1] showed that U-net can generate natural image from white noise inputs and its overfitting occurs between 2400 and 50K steps. The authors here use 100K steps) to recover 3 colors. Considering how much inductive bias is there in the deep architecture, this large number of iterations makes it unclear *where* exactly the color dimensionality emerges. Although the authors postulated the decoder and encoder into factorizable components. There are no ablation studies showing how different learnable components contribute to the emergence of color vision (or if this is not possible given how intricate these components are, would the authors please clarify?)

[1] Ulyanov et al., Deep Image Prior https://arxiv.org/abs/1711.10925

In fact, recent applications of U-net [2] [3] rarely exceeds 5000 steps and they take advantage of its inductive bias on natural images. My main hesitation to fully agree with the results is that the model seems to operate in the "over-fitting" regime to recover the 3rd color. 

[2] Falck et al., A Multi-Resolution Framework for U-Nets with Applications to Hierarchical VAEs NeurIPS 2022

[3] Wang, et al., Learning low-dimensional generalizable natural features from retina using a U-net NeurIPS 2022

---

confidence: 4
contribution: 3
initial_rating: 5
presentation: 3
questions: * Authors claim the emergence of color vision. Please tone down / reformulate this contribution. What you get is the dimensionality of the color space. 

* The limitation of the 3-d result should be acknowledged and discussed in the work. (My guess) is that richer training and more general testing (e.g. asymmetric matching) would lead to larger dimensionality.

* Cite alternative principles that can be used to derive the number of necessary sensors to encode spectral (or chromatic) information.

* How do you reconcile these two apparently contradictory sentences? (line 23-25): "When the retina contains N types of color photoreceptors, our simulation shows that N-dimensional color vision naturally emerges". (line 352): "CMF-SIM reveals the color dimensionality is 1-D for observers with 3 cone types." 

* The Color Matching Function Simulation in the main paper is only explained in FIg.4 and this may not be enough for the average ICLR reader.

* I missed the explanation of the Neural Scope. Note that the generation of images that simulate the perception of dichromats is not trivial (Brettel et al. JOSA 97, Capilla et al. JOSA 04).

* Please do not refer to supplementary videos to explain concepts (as in line 64 of fig.1 or as in line 201 of the supplementary material). If extra explanations are required, please do it through additional sections in the supplementary material. 

* Please define the acronyms the first time they appear. Pay special attention to CMF-SIM and NS. I think CVD is not even defined.
soundness: 3
strengths: The paper is original in proposing predictive coding as a way to fit the elements of color coding. 
Another originality is bringing to the machine learning community a concept from vision science: using a response matching experiment to define the dimensionality of the encoding space for certain concept (in this case, the color percept).
However some of the results obtained from this original view seem obvious (see below).
summary: The authors propose that realistic (because of being 3-dimensional) encoding of color percepts emerge assuming a certain number of spectral sensors and temporal predictive coding. They explore this idea with different number of spectral sensors (or kinds of cones).
In their proposal, first, they learn the different elements of the encoding function by maximizing the consistency (minimizing the error) between the signal captured at time t+dt and the prediction computed from the signal at time t and the displacement information between saccades. Then, the dimensionality of the color percept is defined as in classical colorimetry by looking for the minimal number of independent sources that need to be changed to make a color match (to make the encoded responses the same).
weaknesses: * Authors claim the emergence of color vision (e.g. titles of the section 4 and 6), however what they reproduce is just the dimensionality of the color space, or the number of illuminants required for response matching. This dimensionality (even if correct) is not a full description of color vision, and should not be reported as that. For example, the dimension does not explain the positive and negative lobes of spectral sensitivity of spectral sensors [Wiesel&Hubel J.Neurophysiol.66], it does not explain their adaptivity under change of illuminant [Von Kries 1902], nor the nonlinearity of their response [Krauskopf & Gegenfurtner JOSA 92], nor the non-Euclidean nature of subjective distances between colors [MacAdam JOSA 42], to cite a few facts. 

* Moreover authors refer that the dimension of color space is 3 when assuming 3 different spectral sensors. That is consistent with the result of classical *symmetric* color matching experiments with colors seen in *isolation*, the paradigm of classical tristimulus colorimetry [Wyszeki & Stiles Wiley 82]. However, when seing colors *in context*, the independent dimensions are not 3 but at least 5: not only brightness, hue and colorfulness, but also lightness (not the same as brightness), and chroma (not the same as colorfulness) [see Fairchild, Color Appearanace Models, Wiley 2013]. Therefore, the chromatic content of the spatial context (not only the number of sensors) is critical for the emergence of this 5-dimensional nature, and this is not mentioned in the work. This 3d limitation may come from training limited to regular (white / daylight) illumination, and testing using symmetric color matching with simple (isolated flat) samples, as in the CIE1931 colorimetry. This limitation should be acknowledged and discussed in the work.

* The authors say that previous literature always assumed the dimensionality of the color space. However, there is literature, based on different principles, that can be used to derive the minimal number of sensors and their spectral sensitivity. For instance, [Singh et al. Proc. 3rd Int’l Workshop on Statis. Comp. Theories of Vision 03, Jimenez & Malo IEEE Trans.Geosci.Rem.Sens.14] develop expressions for the accuracy of the recovery of the reflectance of the surfaces and spectrum of the illuminant depending on the use of spatial information and on the number and spectral sensitivity of the sensors. These represent a different principle (inference of object reflectance with color constancy) that can be used to set the dimensionality of the color encoding in a different way as that proposed by the authors.

* The proposed method gives obvious results at some points and these are contradicted at some other points. An example of obvious result is what is said at the abstract (line 23-25): "When the retina contains N types of color photoreceptors, our simulation shows that N-dimensional color vision naturally emerges". However, a later statement seems to contradict the above: in line 352: "CMF-SIM reveals the color dimensionality is 1-D for observers with 3 cone types." 

* Methods as the Color Matching Function Simulation or the Neural Scope are not clearly explained in the paper.

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confidence: 5
contribution: 3
initial_rating: 8
presentation: 4
questions: 1. In part 1 of Figure 2, fixational eye movements are modeled as small drifts controlled by a uniform probability distribution (*dX*, *dY* ) ∼ *U*{(−15, 15) × (−15, 15)}. I watched the supplementary vedio about the model, it appears (if I'm correct) that the drift pattern displays frequent small adjustments with occasional larger movements. Could you clarify how the uniform probability distribution achieves this effect, given that it should theoretically produce equally likely shifts across the specified range? And why do you choose 15-pixel as the range of movements?
2. I am also curious about the drift pattern moving from the eyes to the mouth. How do the fixational eye movements capture both the eyes and the mouth?
3. I suggest authors review the use of LaTeX \ref{} commands, as there are missing figure numbers in several instances (e.g., lines 71, 141, and 180) in the Supplementary Materials.
4. In your model, could you explain the process used to convert the "scene stimulus" into "incoming spectra"? Specifically, I'm interested in whether you apply any spectral sampling or decomposition steps and if there are transformations based on the eye’s optical properties to simulate the spectra reaching the retina.
5. Could you clarify how cone cell activation is calculated based on the received light? I am wondering whether you apply spectral sensitivity functions for each cone type (L, M, S) or any other specific computational approach to simulate how cones respond to incoming light.
6. ON/OFF pathways in bipolar cells and the center-surround organization in RGCs work together to enhance contrast and edge/contour detection, but in your Figure 2 part 3, I don't see the edge or contour section of the scene has been enhanced?
7. Could you clarify how lateral inhibition works among cone cells in your model? If I understand correctly, this mechanism balances the activation of different cone cells and adjusts their response to light, and then enhances edge and contour. Additionally, could you provide more details on how cone cells, bipolar cells, and RGCs are connected, especially in terms of how these connections contribute to spike generation? I believe these details would be helpful to discuss.
8. In Part 1 of the simulation engine of biological eyes, are there any specific learning rules or algorithms used to adjust parameters or process visual information? 
9. Why do you choose the standard deviations of 0.15 and 0.9 in DoG kernel?
10. Could you elaborate on the mechanisms that contribute to the emergence of color representation in your model? From an intuitive perspective, I’m curious how the self-supervised learning and prediction error minimization guide the model toward developing a stable and structured color space, especially as it expands beyond initial color dimensions.
11. Concerning Questions point 10, the model predicts a 4D color space (LMQS) from 3D RGB color images. Since we don’t have four types of cones, we can’t truly envision or confirm what 4D images would look like. Have you tried using 2D colorful images to predict a 3D color space? This approach could help test and validate the model’s predictions.
12. According to Weaknesses point 3. I'm interested in understanding whether the mechanisms in your model might have implications for curing color blindness, particularly in primates such as squirrel monkeys. If the color representation depends on ventral pathway of V1-V2-V4, do you think the your model could potentially inform approaches to treat or "cure" color blindness?  I’d appreciate any insights you could share more information with me.
13. A typo "investigated" on line 156 should be corrected.
14. I don't see the code releases.
soundness: 4
strengths: 1. The paper proposes a new model that can simulate the emergence of color vision through a cortical learning mechanism. This model uses self-supervised learning to develop color representation from optic nerve input, demonstrating how color perception emerges naturally from visual data without labeled learning.
2. By simulating color vision without explicit biological pathways, the authors propose a new theory of how color vision could emerge in the brain, suggesting that specific neural pathways may not be strictly necessary for color representation.
3. The model’s insights could inform research into treating or understanding color blindness, possibly providing theoretical guidance for therapeutic approaches.
summary: This paper presents a novel computational framework for understanding the emergence of color vision in the brain, centered on a self-supervised learning model that interprets optic nerve signals without predetermined color dimensions. The framework includes a simulation engine that processes spectral images through realistic eye mechanisms, including eye motion and cone cell sampling, to generate optic nerve signals. The model demonstrates how different types of color vision (monochromatic, dichromatic, trichromatic, or tetrachromatic) can naturally emerge based on the number of cone types in the retina without requiring predefined color encoding. Notably, the framework successfully models enhancement scenarios, such as the conversion from dichromatic to trichromatic vision demonstrated in squirrel monkey gene therapy experiments. It suggests the possibility of enhancing human color vision from trichromatic to tetrachromatic.
weaknesses: 1. The paper proposes a cortical learning mechanism for color representation without explicitly mimicking the V1-V2-V4 pathway commonly associated with color processing in biological systems. Does this imply that, in your view, the V1-V2-V4 pathway may not be strictly necessary for achieving color representation? I’m curious if your findings suggest that color perception could emerge through other mechanisms or pathways outside of this traditional hierarchy.
2. Does the model in your paper learn directly from RGB format images? If so, how does it expand from the RGB color space to a 4D color space within the learning process? I’m curious about the mechanisms that allow the model to infer or represent a 4D color space based on RGB input.
3. While your model provides an explanation for the emergence of color representation, I find it challenging to accept that the V1-V2-V4 pathway might not play an essential role in color perception. For instance, the study by Li, Liu, Juusola, and Tang (2014) (see below) on perceptual color mapping in macaque visual area V4 underscores the significance of this pathway in color processing. Could you clarify how your model accounts for or relates to this traditional neural pathway?

> Li, M., Liu, F., Juusola, M., & Tang, S. (2014). Perceptual color map in macaque visual area V4. _Journal of Neuroscience, 34_(1), 202–217. https://doi.org/10.1523/JNEUROSCI.4549-12.2014

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confidence: 5
contribution: 4
initial_rating: 8
presentation: 3
questions: Below is a list of not only questions but also general comments:

### Optic Nerve Signals
- I think the depiction of optic nerve signals in Figure 1 and the supplementary video should include the idea of distinct photoreceptor cells, which create separate pathways, like parvocellular and magnocellular pathways.
- Following the previous point, in the simulation, if I understand correctly, the optic nerve signal (ONS) is modelled as a 2D matrix, which closely resembles raw images from cameras without further processing. However, since photoreceptor positions are known, a more accurate simulation might use a 3D matrix with the third dimension representing photoreceptor channels. This would mean at each spatial location only one channel contains information (light absorption values). I’d be interested in your thoughts on this.
- In the statement, "[...] colors of the scene are spectrally sampled by different types of color sensitive cells (cone cells) in the retina, appearing as a layer of spatial noise in the ONS [...]", I would argue that raw photoreceptor signals, while noisy, are not spatial noise but spatially meaningful. This is reflected in Figure 3, "True ONS at time t". I think Figure 1 should display a similar visualisation rather than the highly noisy image currently shown.

### Colour Decoding or Encoding?
Throughout the manuscript and in the supplementary video, the authors describe human colour vision as decoding colour. For instance, the abstract begins: "It is a mystery how the brain decodes color vision purely from the optic nerve signals it receives, with a core inferential challenge being how it disentangles internal perception with the correct colour dimensionality from the unknown encoding properties of the eye."

I’d suggest that we do not decode colour vision but rather construct it (in a sense, we encode colour to interpret the world more easily). It is not only from optic nerve signals; our colour perception is heavily influenced by semantic and cognitive processes. I don't mean to play with linguistic terms but raising a conceptual difference. For instance, in my view, the function Phi in Figure 3 "encodes" the internal percept and the CMF-SIM block shows different potential decoding of the internal percept. Currently, Phi is based solely on ONS, but a more complete model would incorporate input from cognitive processes that impact how we encode colour information of a scene.

### Measurement of Colour Dimensionality
I agree that colour is represented in the cortex as a high-dimensional space, $R^N$, but I think $N$ includes more than mono-, di-, tri-, and tetrachromacy. It also encompasses other aspects of colour appearance that are independent of photoreceptor dimensionality, such as transparency, glossiness, etc. In essence, properties of colour perception often relate to objects rather than individual pixels (e.g., see "Colour appearance and the end of Hering’s Opponent-Colors Theory", *Trends in Cognitive Sciences*, 2023, and "Colour perception: objects, constancy, and categories", *Annual Review of Vision Science*, 2018). I’m curious if you’ve analysed different dimensions of the internal percept in the cortical model, as hypothesised, and whether similar concepts emerged there.

### Why Did Human Colour Vision Emerge?
One of the manuscript’s interesting findings is that monochromacy, dichromacy, trichromacy, and tetrachromacy emerge in sequence during the learning steps. This suggests that colour space dimensionality increases evolutionarily without altering the ecological conditions of the system (i.e., the system’s interaction with its input remains the same through the learning process). In contrast, the colour vision dimensionality in biological organisms varies based on their environmental needs. For example, human trichromacy is thought to have evolved for tasks like distinguishing ripe fruit and foliage or recognising emotional states in faces. In the manuscript’s unsupervised learning model, the system’s needs remain constant, yet the colour vision dimensionality expands as learning progresses. I have a few questions about this:
   - What drives this change in dimensionality within the network’s colour vision?
   - How consistent is this emergence? If you repeated the process 100 times, how much variation would appear in the resulting colour vision systems?
   - Why does this evolution stop at four dimensions rather than expanding to five or more?

### Minor Points
- What does a "private-line" visual pathway refer to?
- In Figure 2, for consistency, I suggest using the same balloon image as Figure 1 to clarify the model’s detailed processes.
- In the context of this statement, "[...] random distribution of L, M and S cone cells on the retina with a relative probability of 0.63, 0.32 and 0.05 [...]", it might be helpful to mention the large individual variation in these probabilities, which significantly impacts individual colour perception. Have you experimented with different distributions, and could this framework contribute to understanding this variation?
- In the context of this statement "Specifically our DoG kernel has standard deviations of 0.15 and 0.9 cone diameters, respectively, for the positive center and negative surround Gaussians [...]", the authors might consider modelling dynamic, contrast-dependent centre-surround modulation of lateral inhibition (e.g., see physiological studies by Alessandra Angelucci). This could enhance the efficiency of DoG models (e.g., "Colour constancy beyond the classical receptive field," *PAMI*, 2017, and "Feedback and surround modulated boundary detection," *IJCV*, 2018).
soundness: 4
strengths: Overall, I found the article very interesting with several fascinating ideas. Its greatest strength lies in its innovative thinking, presenting a fresh computational approach—sensory-agnostic perception—that stands apart from prior models.
summary: The manuscript introduces a new approach to computational modelling of colour vision, combining established knowledge of human visual systems with modern artificial neural networks. The proposed framework suggests that the cortex (the agent’s processing unit) lacks direct information about the retina (the agent’s sensory hardware) and investigates the types of colour perception that could arise in such a setup. The model is trained using a biologically realistic principle: predicting future optic nerve signals, aligning with predictive coding theory, a successful approach to unsupervised learning in vision research. To examine the internal representations created by this model, the study employs psychophysical tests commonly used in human colour-matching experiments. The results provide interesting insights into the dimensionality of colour perception at the single-pixel level, offering promising avenues for future exploration into the possibility of tetrachromacy in humans.
weaknesses: I would summarise the current weaknesses of the manuscript into three areas:

1. **Presentation**: While generally well-written and easy to follow, some parts of the manuscript require considerable effort from readers to fully understand. For example, abbreviations like "CMF-SIM" and "NS" are not immediately clear. Later, NS is explained as "Neural Scope," but CMF-SIM remains unclear. In the supplementary materials, the "colour matching function" is mentioned, but what about SIM? This ambiguity demands extra effort from readers. Also, there are multiple unresolved cross-references ("??") in the supplementary materials.

2. **Results**: The results section could benefit from being more thorough. Specifically, a broader set of analyses across multiple training instances would strengthen the rigour of the findings and clarify them for readers.

3. **Related Work**: Although the manuscript introduces a unique approach that differs from previous models, a brief section in Related Work could discuss other methods using artificial neural networks to model aspects of human colour perception, such as colour categories, contrast sensitivity in chromatic channels, and colour opponency.

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confidence: 3
contribution: 2
initial_rating: 6
presentation: 3
questions: 1 There are many parameters in the model, especially in the cortical learning layers. How were these parameters selected, and are they based on biological references or empirical tuning?
2 The experiments are mostly theoretical, focusing on emergent color dimensions in simplified scenarios. Have the authors considered testing their model on more complex datasets or tasks, such as object recognition with varying lighting conditions?
3  The model relies on self-supervised learning for color perception emergence. Biological systems often use feedback and interaction between cortical areas, which may influence color perception development. How do the authors justify the exclusion of such feedback mechanisms?
4 The paper mentions using divisive normalization for edge detection in the model but does not detail why this specific normalization was chosen over others. Could the authors elaborate on why divisive normalization aligns best with their framework, or if they explored alternatives?
soundness: 2
strengths: 1. The originality stems from the integration of self-supervised learning with a computational framework inspired by biological color perception. By combining a retinal model with a cortical learning mechanism, the authors attempt to create a biologically plausible framework for emergent color and shape perception. This interdisciplinary approach bridges machine learning with neurobiological modeling, which is a valuable exploration in color perception studies.
2 The paper’s methodology reflects a reasonable approach to simulating early visual processing stages. The authors carefully design the retinal model to simulate color cone responses and introduce predictive coding for the cortical learning mechanism. Though some limitations exist, the study provides valuable insights into the use of predictive coding in visual processing models, highlighting how cortical networks could feasibly learn color information.
3 The paper is generally well-written, providing a clear overview of the methods, experiment design, and results. Figures and visualizations are well-crafted, aiding in understanding the experimental findings. The authors successfully explain complex processes, such as predictive coding and color perception, in an accessible way.
4 This study is significant within the niche area of computational neuroscience focusing on color perception. It addresses the unsolved question of how color perception might emerge in a self-learning cortical network, which is a meaningful contribution to understanding visual processing mechanisms. The attempt to simulate gene therapy’s effect on color perception (such as enhancing dichromatic to trichromatic vision) offers a unique perspective that could inspire further studies in perceptual enhancement.
summary: This paper presents a computational framework to explore how color perception might emerge in the cortex through a self-supervised learning model inspired by biological vision. The authors aim to simulate the early visual processing stages by building a model that integrates a retinal layer with different types of cone cells (sensitive to various wavelengths) and a cortical layer using predictive coding. The model learns to extract color and shape information from visual inputs without explicit color labels, reflecting a form of emergent perception.

A significant contribution is the model's ability to simulate the transition from dichromatic to trichromatic vision, akin to genetic interventions shown in experimental biology. The framework demonstrates that under different cone configurations, the model can adapt to various perceptual dimensions, which provides insights into color processing mechanisms in the brain. This interdisciplinary approach contributes to our understanding of self-learning in color perception and offers a basis for future studies on adaptive vision systems in neuroscience and AI.
weaknesses: Biological Plausibility of Input Data:
The model uses natural RGB images as input, which already contain color information. This approach diverges from biological realism, as the retina does not process pre-encoded color data but rather interprets light of various wavelengths. To improve biological plausibility, the model could use spectrally-encoded images or raw spectral data as inputs to better reflect how retinal cones process different wavelengths. This adjustment would align the model closer to real visual processing and allow more meaningful exploration of emergent color dimensions.

Validation against Biological Data:
While the paper presents a computational framework, it lacks direct validation with biological data, which is essential to support its claims of modeling emergent color perception. Adding quantitative comparisons with known biological data on cortical color responses, such as V1 or V4 neuron responses in primates, would strengthen the study’s credibility. Incorporating benchmarks like psychophysical data on color matching could also make the findings more robust and biologically grounded.

Ambiguity in Learning Mechanisms:
The model’s reliance on predictive coding for learning color dimensions in the cortex is interesting but raises questions about biological accuracy, as it assumes self-supervised learning can fully capture color processing. However, in biological systems, factors like feedback from other cortical regions and adaptive changes play significant roles. Expanding the model to incorporate feedback loops or other cortical interactions could offer a more comprehensive view and make the learning process more aligned with biological color perception theories.

Overstated Novelty in Emergent Color Dimensions:
While the paper claims novelty in emergent color dimensions, related work on unsupervised learning of color and perceptual features is well-established. For example, studies on hierarchical models of color vision (e.g., Komatsu, 2006; Conway et al., 2023) and other unsupervised learning models in color processing (Zhou & Mel, 2008) address similar concepts. The authors could enhance clarity by explicitly discussing these foundational works, explaining how their approach extends or diverges from these models, and adjusting novelty claims accordingly.

Komatsu H. The neural mechanisms of perceptual filling-in. Nature reviews neuroscience. 2006 Mar 1;7(3):220-31.
Zhou C, Mel BW. Cue combination and color edge detection in natural scenes. Journal of vision. 2008 Apr 1;8(4):4-.
Conway BR, Malik-Moraleda S, Gibson E. Color appearance and the end of Hering’s Opponent-Colors Theory. Trends in Cognitive Sciences. 2023 Sep 1.

Meta-review: This paper introduces a novel computational framework to model the emergence of color vision in the human brain. Using biologically realistic simulations of optic nerve signals and a self-supervised learning mechanism, the study demonstrates how the brain can infer color dimensionality without predefined assumptions. The work effectively bridges neuroscience and machine learning, addressing a fundamental mystery in vision science. The authors validate their approach through rigorous experiments, showcasing applications to both normal and enhanced vision scenarios. While the paper could benefit from a deeper discussion of perceptual complexities and prior related works, its innovative contributions and interdisciplinary significance strongly support the acceptance of this paper.

Decision: Accept (Oral)