responses in the absence of LGN input, evoking color hallucinations via a Turing-like mech-anism of spontaneous pattern formation in DKL space. MODEL In light of the patchy distribution of color-sensitive cells reported in Landisman and Ts 'o (2002b), Li et al. (2022),Livingstone and Hubel (1984), and Salzmann et al. (2012), we model the color map of V1 as a set of neuronal networks, each encoding the chromaticity of its corresponding region of the visual field. This organization brings to mind the hypercolumnar structure of ori-entation preference within V1 ( Hubel & Wiesel, 1974 ), which, on the basis of its feature-based connectivity properties, allows for the representation of network activity as a function of a local-ized feature space. Here, we assume a mean hue-dependent activity a(θ,t) where θrepresents a direction in the DKL stimulus space, a strictly physiological conception of “hue”from the hues categorizing color perception, as explained above. In drawing this distinction, and in agreementwith Wool et al. (2015) and Li et al. (2022), we give no special status to V1 cells tuned to the DKL directions associated with the unique hues of color-opponent theory, while simultaneously emphasizing the cone-opponent nature of feedforward afferents from the LGN. The resulting activity a(θ,t) of a network of hue-preferring cells, expressed as a firing rate in units of spikes/second, is dominated by the membrane properties of its constituent cells, whose potential variations occur on the order of the membrane time constant τ 0, taken to be 10 msec (Ben-Yishai et al., 1995 ;Carandini & Ringach, 1997 ;Izhikevich, 2010 ). In the vein of previous neural mean field models of feature detection ( Bressloff & Cowan, 2002,2003b ;Bressloff, Cowan, Golubitsky, Thomas, & Wiener, 2001 ;Dayan, Abbott, & Labahn, 2001 ;Ermentrout, 1998 ;Gutkin, Pinto, & Ermentrout, 2003 ), and in close analogy to the ring model of orien-tation tuning ( Ben-Yishai et al., 1995 ;Hansel & Sompolinksy, 1998 ), we let a(θ,t) evolve according to the single-population firing-rate formulation of the Wilson-Cowan equations: τ0daθ;tð Þ dt¼-aθ;tð Þ þ ghθ;tð Þ1/2/C138 ; (1) where h(θ,t), the synaptic input, takes into account both excitatory and inhibitory afferents into a population of cells preferring hue θ, and g(h) is an activation function, as described below. To analyze the relationships between feedforward and recurrent processing and to distin-guish between their respective effects on a(θ,t), we write h(θ,t) as a sum of the stimulus-driven synaptic input from the LGN and the intracortical input resulting from the hue-dependentnetwork connectivity within V1: hθ;tð Þ ¼ h ctxθ;tð Þ þ hlgnθð Þ: (2) We express the input as the average effect of the net synaptic current on the membrane potential of a cell, following the conventions of Ermentrout (1998) and Carandini and Ringach (1997). Thus, h(θ,t) has units of m V and can take on both positive and negative values, chosen here so that a(θ,t) typically ranges from 0 to 60 spikes/sec, consistent with electrophysiological experiments penetrating individual color-responsive cells ( Conway, 2001 ;Johnson et al., 2001 ;Landisman & Ts 'o, 2002a ;Wachtler et al., 2003 ). The input is converted to a firing rate according to the nonlinear activation function ghð Þ ¼ β⋅h-T ð Þ ⋅Hh-Tð Þ ; (3) where H(x) is the Heaviside step function defined as H(x) = 1 for x> 0 and zero for x≤0. Note that in the context of machine learning, this form of activation is also known as the rectified Spontaneous pattern formation: A system 's ability to self-generate new symmetries in the absence ofexternal input. Activation function: A function mapping the afferentinput into a population of neurons(expressed as a current or membranepotential) to the population 's firing rate or probability of firing. Network Neuroscience 682Pattern forming mechanisms of color vision Downloaded from http://direct. mit. edu/netn/article-pdf/7/2/679/2118501/netn_a_00294. pdf by guest on 09 August 2024 