A mathematical model of color and orientation processing in V1

Abstract

Orientation processing in the primary visual cortex (V1) has been experimentally investigated in detail and reproduced in models, while color processing remains unclear. Thus, we have constructed a mathematical model of color and orientation processing in V1. The model is mainly based on the following experimental evidence concerning color blobs: A blob contains overlapping neuronal patches activated by different hues, so that each blob represents a full gamut of hue and might be structured with a loop (Xiao et al. in NeuroImage 35:771–786, 2007). The proposed model describes a set of orientation hypercolumns and color blobs, in which color and orientation preferences are represented by the poloidal and toroidal angles of a torus, correspondingly. The model consists of color-insensitive (CI) and color-sensitive (CS) neuronal populations, which are described by a firing-rate model. The set of CI neurons is described by the classical ring model (Ben-Yishai et al. in Proc Natl Acad Sci USA 92:3844–3848, 1995) with recurrent connections in the orientation space; similarly, the set of CS neurons is described in the color space and also receives input from CI neurons of the same orientation preference. The model predictions are as follows: (1) responses to oriented color stimuli are significantly stronger than those to non-oriented color stimuli; (2) the activity of CS neurons in total is higher than that of CI neurons; (3) a random color can be illusorily perceived in the case of gray oriented stimulus; (4) in response to two-color stimulus in the marginal phase, the network chooses either one of the colors or the intermediate color; (5) input to a blob has rather continual representation of a hue than discrete one (with two narrowly tuned opponent signals).

1 Introduction

Visual information is conveyed from the retina through the lateral geniculate nucleus (LGN) to the primary visual cortex (V1), being segregated by features. Some of the most important features processed in V1 are the orientation and color of a stimulus. Orientation processing in V1 was experimentally investigated in detail and reproduced in models, but the mechanisms of color processing in V1 remain unclear. A high percentage of V1 neurons are color sensitive (CS). According to Johnson et al. (2001, 2004, 2008), these make up about 40 % of the whole V1 and up to 60 % of layer 2/3. Moreover, in contrast to CS neurons of LGN, many CS neurons of V1 are orientation selective (Johnson et al. 2001; Economides et al. 2011). Thus, it is reasonable to suggest that V1 carries out the joint processing of color and orientation signals. Available experimental data about color processing are very contradictory and incomplete; therefore, in the present state of our knowledge, it is worth formulating a simple model that might unite the most important facts. Thus, the aim of the present work is to choose a reliable set of experimental data and construct a simple mathematical model of color and orientation processing in V1.

Orientation information is processed in V1 within orientation hypercolumns (Hubel and Wiesel 1962), each of which includes nearby neurons of all different orientation preferences. Does V1 have analogous structures within which color information is processed? Many studies have addressed this question. One of the first discoveries of V1 studies was cytochrome oxidase (CO) blobs in the layer 2/3 with CS neurons within the blobs (Livingstone and Hubel 1984). However, as found in more recent optical imaging observations and microelectrode recordings (Landisman and Ts’o 2002), although CS neurons are mostly located within CO blobs, they are not only found there. Single-unit recording studies (Lennie et al. 1990; Leventhal et al. 1995) showed that there was no evidence of clustering of CS cells in CO blobs. In contrast, Lu and Roe (2008) wrote that V1 had color clusters localized in CO blobs. Xiao et al. (2006, 2007) used optical imaging with high resolution and concluded that color is functionally organized as an array of hue maps, and they also found a high degree of overlap between CO blobs and hue maps. Thus, in spite of contradictory data about the clustering of CS neurons, for the sake of model construction, we assume that color information is processed in V1 within color blobs, each of which includes nearby neurons with all different color preferences (Xiao et al. 2007).

Several other experimental facts are to be noted that generally raise more questions than answers. Color information from R, G, and B cones is transmitted through LGN to V1 via independent channels R $-$ G and B $-$ (R $+$ G) (Derrington et al. 1984). Participating in this transmission neurons is tuned to those cardinal color directions. In contrast, according to Hass and Horwitz (2013), V1 neurons are tuned to a diverse set of color directions and do not segregate naturally into the categories like R $-$ G and B $-$ (R $+$ G) or R, G, and B. Furthermore, in contrast to CS neurons of LGN, CS neurons of V1 are orientation sensitive (Johnson et al. 2001; Economides et al. 2011). Moreover, it was reported that the observed cortical color signal is 5–6 times stronger for the edges than for the inner domain of any presented figure (Friedman et al. 2003), presumably because of joint orientation and color processing. Johnson et al. (2001, 2004, 2008) studied color and orientation specificity of V1 neurons and showed that V1 neurons are separated into double-, single-, and non-opponent types. At first glance, it contradicts the conclusions of Hass and Horwitz (2013). However, Johnson et al.’s (2004) study, where the researchers directly measured the input from three cone types in V1 neurons, led to the conclusion that the double-opponent neurons are not tuned certainly to the cardinal color directions. It is worth noting that while a fraction of V1 neurons responds to the wavelength distribution (Xiao et al. 2007), only V4 cells respond to the perceived color (Zeki 1983).

Based on the literature, our conceptual view is as follows: There are CS and color-insensitive (CI) neurons in V1. CI neurons correspond to non-opponent cells described by Johnson et al. (2008). CS neurons are separated into the two following subtypes:

• $\hbox {CS}_\mathrm{I}$ neurons, which are distributed uniformly in layers $4\hbox {C}\upbeta$, 4A, and 2/3 (Conway et al. 2010); that is, they do not form clusters. They receive a direct input from LGN neurons. $\hbox {CS}_\mathrm{I}$ neurons are tuned to the cardinal color directions (R $-$ G opponency in layer $4\hbox {C}\upbeta$ and B $-$ (R $+$ G) in layers 2/3 and 4A; Conway et al. 2010) and are not orientation selective (i.e., single opponent); and

• $\hbox {CS}_\mathrm{II}$ neurons, which form clusters located in layer 2/3 in “color blobs” (we do not focus on CO blobs, but imply that most color blobs overlap with CO blobs). The number of color blobs in V1 is approximately equal to the number of orientation hypercolumns (OHs); that is, ordinarily only one blob is located within an OH. Generally, a blob locates within one or a few nearby orientation columns (see Lu and Roe 2008); thus, each blob presumably contains neurons of close orientation preferences. Most $\hbox {CS}_\mathrm{II}$ neurons are double opponent (there might also be some percentage of single-opponent ones), and they are tuned for a diverse set of color directions. $\hbox {CS}_\mathrm{II}$ neurons tuned to opposite hues inhibit each other, where the “opposite” hue is determined in the isoluminant slice of the Derrington–Krauskopf–Lennie (DKL) color space (Derrington et al. 1984; see more details below) as an angle differing in $\uppi$ from the preferred hue. We suppose that the distribution of inputs from $\hbox {CS}_\mathrm{I}$, the close location to one another, and consequently, stronger recurrent interactions between neighbor neurons provide a property such that close $\hbox {CS}_\mathrm{II}$ neurons prefer similar hues (Xiao et al. 2007).

We restrict our consideration to $\hbox {CS}_\mathrm{II}$ neurons determining the color output from V1, implying that the input is formed by the opponent (thalamic and $\hbox {CS}_{\mathrm{I}}$) neurons (below, the subscript for $\hbox {CS}_\mathrm{II}$ neurons will be omitted). After the description of the working hypothesis underlying our model, let us consider some known models related to the visual cortex.

Do we know of any models of color processing? Land and McCann (1971) formulated the retinex theory to simulate how the human visual system processes color. Following this, several retinex algorithms have been developed. These simulate the color constancy effect, modifying the R, G, and B values at each pixel to give an estimate of the physical color independent of the shading. The mechanism of the effect is complicated and involves the whole visual system, whereas the role of V1 is minimal or absent (Foster 2011); for this reason, retinex does not reflect the details of V1 functioning. For the sake of comparability with models of the neocortex, the retinex was implemented using Wilson–Cowan equations (Bertalmío and Cowan 2009). Like other retinex algorithms, the Wilson–Cowan equation-based variant treats R, G, and B signals separately; hence, it does not reflect the abovementioned experimental data. On the other hand, some of the retinex algorithms reproduce another important effect of color contrast (Rizzi et al. 2004; Bertalmío and Cowan 2009). The color contrast effect involves spatially inhomogeneous stimuli; this is why it is out of the scope of our present consideration.

Among models that describe specific functions of V1, orientation-selectivity models are the most developed. A ring network is probably the simplest model that can describe an OH. Such a model consists of neuronal populations, each implying an infinite number of neurons receiving a common input (with the same orientation) and individual noise. In the study of Ben-Yishai et al. (1995), the OH was described by the ring model based on a first-order differential equation for the firing rate. This model is useful for mathematical analysis and analytically tractable under certain conditions (Hansel and Sompolinsky 1998). It has been applied to explain a number of effects in V1, such as contrast invariance, virtual rotation, or symmetry breaking. Some structural properties of V1 are not significant for observation of the mentioned effects; thus, the simple ring model does not cover them. For instance, in this canonical ring model, a layered structure of the cortex is reduced, implying that the main functional role belongs to recurrent connections. This is in contrast to physiological data (Thomson and Lamy 2007) showing that the probability of connections between neurons of different layers (e.g., input layers 2/3 to 5) might be larger than or comparable to the probability of the recurrent connections. A model that takes into account the role of connections between layers was presented by Battaglia and Hansel (2011). This work proposed an extension of the canonical ring model of an OH, thus considering two interacting rings separately representing layers 1–4 and 5–6. Each ring comprised excitatory and inhibitory single-compartmental exponential integrate-and-fire neurons.

Biophysically detailed models are on the other extreme of the description of V1. Such models take into account the kinetics of different neuronal ionic channels and so on, as proposed, by Chizhov (2014), for instance. Based on the conductance-based refractory density (CBRD) approach, this model describes several nearby OHs comprising interacting excitatory and inhibitory populations of two-compartmental neurons distributed in two cortical layers. Although probabilistic models provide accuracy for transient regime simulations, they are too complex for analysis of most of the effects observed in V1. An alternative detailed models are network based (Schummers et al. 2007; Basalyga et al. 2012; Rangan et al. 2009) and population rate based (Symes and Wennekers 2009).

The role of V1 is not limited by orientation processing. V1 comprises neurons selective to spatial frequency, direction of motion, color, and other characteristics of the stimulus. Bressloff and Cowan (2002) considered two modalities, namely orientation and spatial frequency. The authors proposed a spherical topology of the network in the features’ space so that coordinates of a neuron define its preferred orientation and spatial frequency. The neuron with the coordinates of one pole of the sphere is located in the center of a low-frequency OH, whereas the neuron with the coordinates of the other pole is located in the center of a high-frequency OH. Their model, describing several interacting OHs, demonstrates the role of local (within an OH) and long-range horizontal (between OHs) connections in orientation and spatial frequency tuning. The goal of the present study is also to describe two modalities, that is why we chose the topology of a surface in three-dimensional space for our representation of orientation and color.

Because both variables—orientation and color—may be considered periodical, we propose a toroidal topology for our model, where small rings of CS neurons with a hue as a coordinate are distributed along a big ring with an orientation angle as a coordinate (Fig. 1). Each small ring represents a single color blob. CI neurons are distributed on an orientation ring.

Fig. 1

Schematic of the torus model. The big ring is the system of CI neuronal populations. The small rings represent the system of CS neuronal populations with similar orientation preferences

Why might the summation of CS neurons of a blob be considered as a ring? It was shown experimentally that information about color is supplied from the retina through LGN to V1 by two parallel channels, R $-$ G and B $-$ (R $+$ G). In contrast to this separation of LGN outputs into those categories, CS neurons of layers 2/3 of V1 (Hass and Horwitz 2013) may be tuned to any visible color hue (Xiao et al. 2007). As shown experimentally (Xiao et al. 2007), a uniform color stimulus activates an array of patches in layers 2/3 of V1; patches activated by different hues partially overlap so that nearby peaks (locations of maximal responses in patches) represent close hues. Each blob represents a full gamut of hue (Xiao et al. 2007) and might be structured with a loop, corresponding to an isochromic isoluminant line of the DKL color space (Derrington et al. 1984). There is a tendency for neurons with similar functions to interact more strongly with each other. That tendency is reflected in the architecture of the ring: Close neurons interact more strongly and have similar preferences. It also proves that the simplest blob’s network topology is a ring in color space. We have chosen a closed loop structure also because any opened network structure has strong boundary effects. As Suppl. 1 shows, because of the boundary effect a change of chroma of a color stimulus results in a change of the chosen color hue.

The profile of connections for CI neurons will be borrowed from the classical ring model (Ben-Yishai et al. 1995). Because evidence about any sufficiently strong influence of a color stimulus on CI neurons is absent, we suppose that CI neurons do not have input from CS neurons.

In what form does the information about color come into a blob? It is not a sum of independent R, G, and B signals, because even at the level of ganglion cells (retina), independent signals from cones are transformed to opponent signals R $-$ G and B $-$ (R $+$ G), and such opponent center-surround receptive fields are typical for neurons of LGN and for single-opponent neurons of V1. As for CS neurons in blobs, they are tuned for a diverse set of color directions; thus, we characterize their preferred color hue by polar angle at the isoluminant slice of the DKL color space proposed by Derrington et al. (1984). Derrington et al. (1984) suggested characterizing the perceived colors by the following three coordinates (according to three types of receptive fields of LGN neurons) inside a sphere:

• (1) L is the conditional luminance that is calculated as the sum of retinal signals from the long- and middle-wave sensitive cones. The luminance gradually changes from the black pole to the white;

• (2) RG is the difference in retinal signals from the long- and middle-wave sensitive cones; and

• (3) BY (Blue $-$ (Red $+$ Green)) is the difference in retinal signals from the short-wave sensitive cones and the sum of long- and middle-wave sensitive cones.

We use the DKL space to characterize color by polar coordinates, where $\theta$ is the angle characterizing the color hue, the radial coordinate is the color chroma, and the vertical coordinate is the luminance. However, note that in order to save the representation of the classic ring model, we use the non-normalized parameters of an input background $I_0$ and an input color tuning $I_2$ instead of the luminance and the chroma, correspondingly (so, $I_2=0$ in the case of a gray stimulus).

It is not clear whether the input to a blob is an analog signal or a discrete one representing a sum of opponent RG and BY signals. Arguments in favor of an analog representation will be given in Sect. 3.4.

Section 2 describes the model, including its mathematical and numerical realization. In Sect. 3, first, the responses to color oriented and color non-oriented stimuli are simulated. Then, the responses to untuned (gray and non-oriented) stimuli are considered. Finally, we examine a two-color stimulus and analyze difference of the discrete and analog color representations.

2 Methods

The model representation of a region that includes neurons of all color and orientation preferences is shown schematically in Fig. 1. All neurons are orientation selective and divided into two classes: CI neurons (big ring in Fig. 1) and CS (small rings in Fig. 1). Every CI population is parameterized by its preferred orientation $\varphi$, whereas every CS population is parameterized by preferred orientation $\varphi$ and preferred color hue $\theta$. In the case of orientation, the underlying period is $\pi$. Thus, the system of CI neurons is structured with a ring; the position in the ring of the population preferring orientation $\varphi$ is specified by polar angle $2\varphi$. In addition, the system of CS neurons with the same preferred orientation $\varphi$ (neurons of one color blob located in the orientation column with the preferred orientation $\varphi$ of one OH) is also structured with a ring. This ring represents the full gamut of saturated hues as in the isoluminant slice of the DKL color space. Thus, in the case of color hue, the period is $2\pi$.

CS rings are assembled in a torus. The location of a CS population on the torus is given by coordinates $\{\varphi ,\theta \}$, so that color hue and orientation preferences are represented by poloidal and toroidal angles of the torus, correspondingly. The full network corresponds to a set of OHs and color blobs, the range of preferred orientations of all the blobs covering the interval from 0 to $\uppi$. However, the spatial distribution of either neuronal populations within the cortex or stimuli on the screen is not taken into account. Synaptic weights of the connections between CI populations depend only on an orientation angular distance between them. CS populations that prefer the same orientation are connected with synaptic weights that depend only on the color angular distance between them.

For simplicity, the evolution of the population activities is described by a relatively simple rate model, as in Hansel and Sompolinsky (1998), in which the highly nonlinear population dynamics was expressed by simple ordinary differential equations with a threshold-linear activation function. Let $\nu (\varphi ,\theta ,t)$ denote the activity at the time t of a CS population and $\mu (\varphi ,t)$ that of a CI population. The equations are as follows:

$$\begin{aligned} \tau \frac{\partial \mu }{\partial t}= & {} -\mu \nonumber \\&+\,\left[ \int \limits _0^\pi {\left( {J_0^\mu +J_1 {\hbox {Cos}} \left( 2\left( \varphi -\varphi ^{\prime }\right) \right) } \right) \;\mu \left( \varphi ^{\prime },t\right) } \frac{d\varphi ^{\prime }}{\pi }\right. \nonumber \\&\left. +\,I_0 +I_1 {\hbox {Cos}}\left( 2\left( \varphi -\varphi _0\right) \right) +\eta \left( \varphi ,t\right) \right] _+ \end{aligned}$$

(1)

$$\begin{aligned} \tau \frac{\partial \nu }{\partial t}= & {} - \nu \nonumber \\&+\,\left[ \int \limits _0^{2\pi } {\left( {J_0^\nu +J_2 {\hbox {Cos}}\left( \theta -\theta ^{\prime }\right) } \right) \;\nu \left( \varphi ,\theta ^{\prime },t\right) } \frac{d\theta ^{\prime }}{2\pi }\right. \nonumber \\&\left. +\,J_3 \mu \left( \varphi ,t\right) +I\left( \varphi ,\theta ,t\right) \right] _+ , \end{aligned}$$

(2)

where

$$\begin{aligned} I\left( \varphi ,\theta ,t\right)= & {} I_0 +I_1 {\hbox {Cos}} \left( 2\left( \varphi -\varphi _0\right) \right) \nonumber \\&+\,I_2 {\hbox {Cos}} \left( \theta -\theta _0\right) +\eta \left( \varphi ,\theta ,t\right) , \end{aligned}$$

(3)

$\tau$ is the relaxation time constant, $\eta$ is the white Gaussian noise with zero mean and variance $\sigma$, $\langle \eta (t) \eta ({t}^{\prime })\rangle _t\,=\,\sigma ^{2}\delta (t-{t}^{\prime })$, $\langle \eta \rangle _t\,=\,0$. The terms in square brackets represent synaptic currents. The parameters of the input and recurrent connections are explained below. The F-I curve (the dependence of firing rate F on input I) is implied to be approximated by the linear-threshold function:

$$\begin{aligned}{}[x]_+ =\left\{ {\begin{array}{l} x,\quad \mathrm{if}\;x\ge 0 \\ 0,\quad \mathrm{if}\;x<0 \\ \end{array}} \right. \;. \end{aligned}$$

(4)

For simplicity, the threshold is zero.

The network processes angle features and thus periodic boundary conditions are appropriate here. Then, the profiles of recurrent connections can be decomposed into sums of Fourier series. For numerical simulations, we approximate each of them by the first two Fourier components, as in Hansel and Sompolinsky (1998), namely $J_0^\mu +J_1 {\hbox {Cos}}(2\varphi )$ and $J_0^\nu +J_2 {\hbox {Cos}}\theta$. Such approximations provide stronger connections between neurons with similar preferences. Similarly, the functions of external signals were $I_0 +I_1 {\hbox {Cos}}(2\varphi )$ and $I_0 +I_1 {\hbox {Cos}}(2\varphi )+I_2 {\hbox {Cos}}\theta$. The influence of CI neurons on CS neurons was characterized by the parameter $J_3$.

The feedforward signal to CS populations includes a background component $I_0$ (which can be treated as luminance), an oriented signal with the amplitude $I_1$, and a color signal with the amplitude $I_2$ (which might be treated as color saturation, chroma). The non-cortical signal to CI neurons includes only the background and orientation signals. In the numerical simulations, we considered several types of stimuli, as follows: oriented $(I_1 >0)$ or non-oriented $(I_1 =0)$, color $(I_2 >0)$, gray $(I_2 =0)$ or two-color stimuli. The simplest two-color external signal was represented by a sum of two rectified functions as follows: $I_2 \left( {[{\hbox {Cos}}(\theta -\theta _1)]_+ +[{\hbox {Cos}}(\theta -\theta _2)]_+ } \right)$. Rectification, but not just harmonic function as for other stimulus cases, was applied here in order to avoid zeroing of the stimulus in the case of two “orthogonal” color hues.

In numerical simulations, the torus was discretized with a grid of $M\times N$ sites so that $\Delta \theta =2 \pi /M\; rad$, $\Delta \varphi = \pi /N \;rad$, $M=40$, and $N=40$. To solve the differential equations numerically, we used the Euler method.

3 Results

3.1 Model response to oriented and non-oriented color stimuli

Figure 2a shows the stationary solution of the torus model in response to a non-oriented color stimulus. The activity profile of the CI ring is plotted at the top of the figure, and the activity distribution of the CS rings in the $(\varphi ,\theta )$-plane is at the bottom. Similarly, Fig. 2b presents the response to oriented color stimulus. Both stimuli generate a hill-shaped profile in color space, centered on the stimulus color $\theta _{ 0} =1.8 \;rad$. If a non-oriented stimulus is presented, then the network has a flat activity profile in the orientation space; if the stimulus is oriented, then the network has a hill-shaped profile in the orientation space, centered on the stimulus orientation $\varphi _{ 0} =0.9 \;rad$. It is noteworthy that the response of CS neurons to a color oriented stimulus is significantly stronger than to a non-oriented one. This is consistent with experimental data from the work by Friedman et al. (2003).

Fig. 2

Activity distribution of CS populations (at the bottom) in the $(\varphi ,\theta )$ plane and the activity profile in orientation space of CI populations (at the top) in response to a non-oriented color stimulus $I_1 =0$; b oriented color stimulus $I_1 =3, \varphi _{0} =0.9\;rad$. Other parameters are the same as in a and b and have the following values: $J_0^\mu =0$, $J_0^\nu =0$, $J_1 =2, J_2 =3, J_3 =2, I_0 =1, I_2 =0.1,$ and $\theta _{0} =1.8 \;rad$

3.2 Model response to non-oriented gray stimulus

Due to the proposed structure of the model, its analysis can be largely borrowed from the classical ring model (Hansel and Sompolinsky 1998) and supplemented by the results of numerical analysis. We think that the most important aspect is to analyze types of solutions in response to non-oriented stimuli. Further, we intend to (1) reproduce the results obtained in the classical ring model for CI neurons; (2) use an analogy between the equations for CS and CI populations in order to provide an analysis for the CS rings; and (3) demonstrate examples of different types of solutions for CS populations.

3.2.1 Types of solution for the CI ring

In the work by Ben-Yishai et al. (1995), where the ring model was used to describe an OH, it was shown that orientation preference may arise without stimulus anisotropy. Depending on the parameters of recurrent connections, the canonical ring model (Ben-Yishai et al. 1995; Hansel and Sompolinsky 1998) reveals one of three types of solutions, namely homogeneous, marginal (emergence of a hill-shaped activity distribution in orientation space), and amplitude instability (unlimited increase of activity), as seen in the form of the phase diagram in $(J_0^\mu ,J_1)$ plane in Fig. 3a. In this work, we also use the equations of the ring model for CI populations, and therefore, the phase diagram for the CI ring is identical to that for the ring model (Fig. 3a). Homogeneous and marginal states are depicted at the top of Fig. 5 (homogeneous in Fig. 5a and marginal in Fig. 5b, c).

Fig. 3

a Phase diagram in the $(J_0 ,J_1 )$ plane for the stability of the various states of the ring model in the presence of homogeneous stimulus $I_1 =0$, adapted from Hansel and Sompolinsky (1998). Here $J_0$ relates to $J_0^\mu$ in the torus model; b Dependence of maximal activity of CI neurons in response to weakly tuned stimulus $(I_1 =0.1)$ on the parameter $J_1$ characterizing the tuning of recurrent interactions between CI neurons. The parameters are as follows: $J_0^\mu =-2$, $I_0 =1$, $\sigma =0.03$

Figure 3b demonstrates how the amplitude of the modulation of the recurrent interactions affects maximal firing rate in the model: For the fixed value of $J_0^\mu$, the increase of $J_1$ leads to a nonlinear increase of firing rate. Values of $J_1$ greater than about 6.8 correspond to the amplitude instability.

3.2.2 The influence of background stimulation on the model’s behavior

The amplitude of a background stimulation $I_0$ does not affect the type of response in our torus model, nor does it do so in the canonical ring model; that is, the boundaries of the homogeneous, marginal, and amplitude instability zones do not depend on $I_0$. In the torus model, each CS ring receives an additional signal $I_1 {\hbox {Cos}}(2(\varphi -\varphi _0 ))+J_3 \mu (\varphi )$. Therefore, effective background stimulation is obtained as $I_{eff} =I_0 +I_1 {\hbox {Cos}}(2(\varphi -\varphi _0 ))+J_3 \mu (\varphi )$, and, like $I_0$, this does not affect the type of response of CS neurons.

It was also found that the maximal response amplitude of CS neurons depends linearly on the parameter $J_3$. The maximal firing rate $\nu _{\max }$ at $J_3 =0$ and the gain of $\nu _{\max } (J_3 )$ depend on the parameters $J_0^\nu$, $J_0^\mu$, $J_1$, and $J_2$. Different combinations of the solution types corresponding to marginal and homogeneous phases for CS and CI populations were considered. The results showed that the largest gain of the dependence (solid line) was observed in the case of the marginal phase for both the $(J_0^\nu ,J_2 )$ and $(J_0^\mu , J_1)$ planes (Fig. 4).

Fig. 4

Dependence of maximal activity of CS neurons on the parameter $J_3$, characterizing the synaptic influence of CI neurons on CS neurons of the same orientation preference, as a function of $J_1$ and $J_2 ; J_0^\mu =0$, $J_0^\nu =0$, $I_0 =1, I_1 =0$, and $I_2 =0$

3.2.3 Types of solutions of a CS ring

The phase diagram in the $(J_0^\nu , J_2)$ plane for every CS ring is identical to the phase diagram in the $(J_0^\mu , J_1)$ plane (Fig. 3a) because the equation for a CS population is similar to the equation for a CI population with the variable $\theta$ instead of $\varphi$ and $I_{eff}$ instead of $I_0$. As was shown in Sect. 3.2.2, a shift of $I_0$ does not change the type of solution.

With $J_0^\mu , J_1$ from the zone of homogeneous solutions and $J_0^\nu , J_2$ from the zone of marginal solutions, the activity demonstrates a hill-shaped profile for every CS ring in color space, independent for each ring and having the same amplitude (Fig. 5a). Figure 5b shows a marginal state for $J_3 =0$. The model demonstrates a color-selective (with random choice of color) response for every CS ring, but with a smaller amplitude than in the case of color stimulus. For the total system, on average, there is no color preference in these conditions. Note that the considered case of $J_3 =0$ is not physiologically relevant, because it corresponds to an isolation of CS neurons from CI, which contradicts experimental observations.

Fig. 5

Distribution of CS population activity (at the bottom) in the $(\varphi ,\theta )$ plane and the activity profile in orientation space for CI populations (at the top) in response to a gray non-oriented stimulus, as follows: a $J_1 =2, J_3 =2$; b $J_1 =3, J_3 =0$; c $J_1 =3, J_3 =2$. Other parameters in a, b, and c are the same and have the following values: $J_0^\mu =0$, $J_0^\mu =0$, $J_2 =3$, $I_0 =1$, $I_1 =0$, and $I_2 =0$. The sets of parameters correspond to the marginal phase for CS population in a–c, homogeneous phase for CI in a and marginal phase for CI in b–c

In case of a nonzero input from CI populations, the amplitude of the response of CS neurons is stronger (Fig. 5c). The model predicts an appearance of color preference in response to a gray stimulus in the condition of narrow orientation tuning. In this case, essential anisotropy in color space is observed in only a few CS rings; thus, it is probable that the whole system, on average, might select an arbitrary color.

3.3 The model behavior in the case of input signals containing two colors

According to (Xiao et al. 2007), cortical zones activated by different colors are partially overlapped within a blob. Thus, it is an open question whether one blob discriminates between two simultaneously presented colors. Let’s consider the model behavior when the two-color input is broadly tuned and represented by the function $I_2 \left( [{\hbox {Cos}}(\theta -\theta _1 )]_+\right. \left. +[{\hbox {Cos}}(\theta -\theta _2 )]_+ \right)$. It was found that the model behavior in the case of two-color input signal at a fixed value of $I_2$ qualitatively depends on $J_0^\nu$, $J_2$ and the coordinate difference between colors $\theta _1$ and $\theta _2$. In the homogeneous phase, the response qualitatively reflects the stimulus; i.e., the network has a two-hill profile in the color space. In the marginal phase, the network has one-hill profile with the peak either between $\theta _1$ and $\theta _2$ (Fig. 6a) or centered at the stimulus, in the particular case of Fig. 6b, it is at $\theta _2$. Thus, in the marginal phase the model fails to discriminate between the two colors.

Fig. 6

Distribution of population activity (solid lines) in the $(\varphi ,\theta )$ plane in response to a two-color stimulus (dashed lines). a The external input with close color coordinates $\theta _1 =0\;rad$ and $\theta _2 =2.3\;rad$; b The external input with distant coordinates, $\theta _1 =0\;rad$ and $\theta _2 =3.0\;rad$. Other parameters for the solutions in a and b are as follows: $J_0^\nu =-2$, $J_1 =3$, $J_2 =3$, $J_3 =1, I_0 =1, I_1 =0$, and $I_2 =0.5$

Then, let’s study the activity profile in the color space in response to narrowly tuned input containing two colors. Such input was represented by the function $-I_2 +\frac{I_2 }{2}\left( \exp \left( {-\;\frac{\left| {\theta -\theta _1 } \right| }{0.2}} \right) +\exp \left( {-\;\frac{\left| {\theta -2\pi -\theta _1 } \right| }{0.2}} \right) +\exp \left( {-\;\frac{\left| {\theta -\theta _2 } \right| }{0.2}} \right) \right. \left. +\exp \left( {-\;\frac{\left| {\theta -2\pi -\theta _2 } \right| }{0.2}} \right) \right)$. It was found that the types of behavior were the same; in particular, in the marginal phase the model fails to discriminate between two colors. The ring model of an OH with narrowly tuned inhibition (Carandini and Ringach 1997; Mundel et al. 1997) in response to stimulus containing two orientations shows repulsion (illusion when the angle between perceived orientations is bigger than the angle between stimulus orientations) or attraction. Carandini and Ringach supposed that these illusions are caused by the intracortical feedback through inhibition with a center-surround weighting function. In our model with global inhibition, we also found attraction (Fig. 6a), but in contrast to their finding of repulsion, we observed selection. The signal containing two colors may appear, for example, if an observer expects to see one color (a signal sent by the associative areas of the brain cortex) but receives a stimulus of another color.

3.4 Comparison of “analog” and “discrete” inputs to a blob

The above consideration relates to analog representation of the color input. Let’s now check the plausibility of discrete representation of stimulus hue by RG, BY signals. For this we can reinterpret the responses to two-color stimulus. The discrete input means that RG and BY opponent signals come to a blob without mixing. Let’s compare responses of the model to analog and discrete inputs both coding the violet hue. In the analog representation, the violet hue has one coordinate $\theta _0 =\pi /4$, whereas in the discrete representation the violet hue is a sum of the positive RG $(\theta _1 =0\;rad)$ signal with the intensity 153 / 255 and the BY $(\theta _2 =\pi /2)$ signal with the intensity 205 / 255. The “discrete input” is given by the function $-I_2 + \frac{I_2 }{2}\left( \frac{153}{255}\left( {\exp \left( {-\;\frac{\theta }{0.2}} \right) +\exp \left( {-\;\frac{\left| {\theta -2\pi } \right| }{0.2}} \right) } \right) \right. \left. +\frac{205}{255}\left( {\exp \left( {-\;\frac{\left| {\theta -\pi /2} \right| }{0.2}} \right) +\exp \left( {-\;\frac{\left| {\theta -5\pi /2} \right| }{0.2}} \right) } \right) \right)$The “analog input” is given by the function $-I_2 +I_2 \left( \exp \left( {-\;\frac{\left| {\theta -\pi /4} \right| }{0.2}} \right) \right. \left. +\exp \left( {-\;\frac{\left| {\theta -9\pi /4} \right| }{0.2}} \right) \right)$.

Figure 7a demonstrates distribution of CS population activity in the $(\varphi , \theta )$ plane in the case of “analog input signal.” In the color space, there is only one peak in the coordinate $\theta _0 =\pi /4$, meaning that the network perceives violet hue. Figure 7b shows distribution of activity in the case of “discrete input signal.” In contrast to the response to “analog signal,” there are two peaks in the color space. Such two-hill profile contradicts to the experiments by Hass and Horwitz (2013) and Xiao et al. (2007), showing that every hue has an individual location in a blob. It should be also noted that the difference in intensities changes the type of response from two- to one-hill profile (compare Figs. 7b and 6a). The profile in response to the analog input, shown in Fig. 7a, is more consistent with the experimental data. We conclude that input to a blob is analog.

Fig. 7

Distribution of population activity in the $(\varphi ,\theta )$ plane in response to the narrowly tuned two-color stimulus. a The activity profile in response to the “analog input” with the color coordinate $\theta _0 =\pi /4$; b the activity profile in response to the “discrete input” with the intensity 153 / 255 on the RG $(\theta _1 =0\;rad)$ and with the intensity 205 / 255 on the BY $(\theta _2 =\pi /2)$; other parameters for the solutions in a and b are as follows: $J_0^\nu =-2$, $J_0^\mu =-2$, $J_1 =3$, $J_2 =3$, $J_3 =1, I_0 =1, I_1 =0,$ and $I_2 =0.5$

We also considered a generalization of the model based on Eqs. 1–4 (Model 1) that takes into account the synaptic kinetics and non-stationary behavior of the population using conductance-based population model (Model 2, see Suppl. 2). Model 2 reproduced the response of Model 1 to the two-color stimulus, the effect of amplification of the response to color by orientation. In response to non-oriented gray stimulus, Model 2 gives homogeneous and marginal solutions. The domain of amplitude instability of the simple model splits to homogeneous and marginal subdomains. In contrast to the marginal solution of Model 1 in a form of a standing wave, the marginal solution of Model 2 is a traveling wave.

4 Discussion

Based on known experimental data, we formulated a conceptual view of the structure and functioning of V1 which provides color and orientation processing. The concept was expressed in the form of a minimal V1 model describing this functionality. Below we list the experimental facts incorporated into the model and then its predictions. The following experimental findings have been incorporated into or reproduced by the model:

• 1. V1 contains CS and CI neurons. Most neurons in layer 2/3 of V1 are orientation selective (Johnson et al. 2001; Economides et al. 2011).

• 2. Most of CS neurons in layer 2/3 of V1 are clustered (i.e., localized within blobs) and tuned to a diverse set of color directions (Hass and Horwitz 2013; Xiao et al. 2007). The clustering of CS neurons promotes mapping of CS neurons in V1 with a continual range of hues (Xiao et al. 2007);

• 3. The number of color blobs is comparable with the number of OHs. A single color blob is considerably smaller than an OH (Xiao et al. 2007) and, generally, locates within one or a few nearby orientation columns (Lu and Roe 2008). On the basis of these experimental findings, it can be concluded that the effect of CI neurons on CS is essential whereas the feedback effect is negligible. In turn, this is in agreement with the conclusion of Friedman et al. (2003) that orientation selectivity in V1 is independent of color;

• 4. Firing activity is higher within color blobs, as shown by Economides et al. (2011). In the model, the total input signal to CS neurons of a blob can be much greater than that to CI neurons, because CS neurons within a blob of V1 have strong recurrent connections and receive input from CI neurons and the input formed by the opponent (thalamic and $\hbox {CS}_\mathrm{I}$, see Sect. 1) neurons.

An important feature of the model is the periodic structure for the network of CS neurons within a blob. The periodic structure helps to avoid undue boundary effects (see Suppl. 1), particularly, a dependence of hue perception on the stimulus chroma.

Another important feature is the representation of a color input signal. With the help of the DKL color space the three-component signal at the level of the cones transforms into the continual signal at the level of color blobs, implicitly taking into account the transmission of the three-component signal through the opponent color pathways at the level of ganglion, LGN and $\hbox {CS}_\mathrm{I}$ cells. Color and orientation signals are mixed only in color blobs.

The effect of color constancy (Land and McCann 1971) is reproduced by the model by construction, because the color hue and luminance are independent in the model, and therefore, luminance does not affect color hue processing.

It should be noted that the suggested model does not take into account spatial effects. To construct a distributed model, the structure of connections between CS neurons of a color blob has to be known but such data are not yet available. The proposed model is at least valid to study spatially uniform stimuli processing. In the experiments we based on (Xiao et al. 2007), the structure of color blobs was visualized only in response to uniform color stimuli. We suppose that the same functional architecture of the network would be revealed in response to distributed stimuli.

The mathematical analysis of the proposed model can be largely borrowed from the classical ring model. The equations for CS populations in our model differ in input signals depending on orientation, which, as shown in Sect. 3.2.2, does not lead to qualitative change of solutions, nevertheless, has some interesting interpretations. We reinterpreted the ring model in terms of color and used known results of its mathematical analysis. Moreover, the influence of orientation on color processing in the model has been analyzed in simulations.

The proposed model allows some predictions to be made, which are as follows:

• 1. Input to a blob is not discrete, consisting of two narrowly tuned opponent (RG and BY) signals (see Sect. 3.4). Instead, it is analog. As shown in (Hass and Horwitz 2013; Xiao et al. 2007), a CS neuron of a blob responds to the wavelength distribution, while the maximal activity corresponds to the preferred color;

• 2. The response of a blob to a color oriented stimulus is largely higher than the response to a color non-oriented stimulus (Sect. 3.1). It was also predicted experimentally (Friedman et al. 2003);

• 3. In response to two-color stimulus (Sect. 3.3), three different types of solutions are possible: For the parameters from the homogeneous phase, the activity profile is qualitatively similar to the input (there are two hills with the peaks corresponding to the color coordinates of stimulus), for the parameters from the marginal phase the network tunes to a single color either between the ones of the stimulus or any of them;

• 4. The model reveals color and orientation tunings in response to a gray non-oriented stimulus, if the coupling parameters correspond to the marginal phase. The activity profile is hill-shaped with the peak at a random color coordinate for every blob. In the case of a gray oriented stimulus, the activity of a blob preferring orientation of the stimulus is enhanced, and therefore, the whole network chooses one color.

We also constructed a model (Model 2) which takes into account the synaptic kinetics and unbalanced population dynamics. Model 2 confirmed the effect of two-color stimulus and the enhancing of color response by oriented stimulus. In response to a non-oriented gray stimulus, the Model 2 has one of two possible solutions, marginal or homogeneous. The phase space is divided on two regions, and there is no phase of amplitude instability. In contrast to Model 1, Model 2 reveals traveling waves in the marginal phase.

Given the above findings, the proposed torus model links a number of experimental observations on color and orientation processing in V1. We expect further development of the model, first involving its extension to the spatially distributed case.