Theory for the alignment of cortical feature maps during development

We present a developmental model of ocular dominance column formation that takes into account the existence of an array of intrinsically specified cytochrome oxidase blobs. We assume that there is some molecular substrate for the blobs early in development, which generates a spatially periodic modulation of experience-dependent plasticity. We determine the effects of such a modulation on a competitive Hebbian mechanism for the modification of the feedforward afferents from the left and right eyes. We show how alternating left and right eye dominated columns can develop, in which the blobs are aligned with the centers of the ocular dominance columns and receive a greater density of feedforward connections, thus becoming defined extrinsically. More generally, our results suggest that the presence of periodically distributed anatomical markers early in development could provide a mechanism for the alignment of cortical feature maps.

Figure 1

(a) Difference of Gaussians lateral interaction function $w\left(r\right)$ showing short-range excitation and long-range inhibition. (b) Dispersion curve $\lambda \left(k\right)=-\mu +\widehat{W}\left(k\right)$. The shaded regions indicate the band of unstable eigenmodes around the critical wave number $k_c$ where $\widehat{W}\left(k_c\right)={\text{max}}_k\left\{\widehat{W}\left(k\right)\right\}$. Black: unstable modes for $\mu >0$. $\text{Black}+\text{gray}$: unstable modes for $\mu =0$.

Figure 2

Even and odd eigenmodes localized around blob and interblob regions.

Figure 3

(a) If $k=\left|\mathbf{k}-\mathbf{Q}\right|$ then $\mathbf{k}$ must lie on the bisector of the reciprocal lattice vector $\mathbf{Q}$. (b) and (c) Construction of the first Brillouin zone (shaded region) in the reciprocal square and hexagonal lattices with ${\mathbf{Q}}_j=2\pi {\widehat{\ell }}_j/d$.

Figure 4

(Color online) One-dimensional pinning of OD columns to a periodic array of CO blobs. (a) Space-time plot of the density $n_{-}\left(x,t\right)$ obtained by numerically solving a one-dimensional version of Eq. (4) for $N=1+0.5\kappa \left[1+\text{cos}\left(2\pi x/d\right)\right]$ with $\kappa =0.4$, $\mu =0.08\approx {\mu }_c$, and a network of size $\Lambda =8d$. The parameters of the Mexican hat function (2) are chosen to be $A=1.8$, $B=1.0$, ${\sigma }_E=0.29d$, and ${\sigma }_I=0.72d$, so that $k_c\approx \pi /d$. The initial binocular state (gray) evolves into alternating left and right eye ocular dominance columns (colored white and black, respectively) whose centers align with the centers of the CO blobs (indicated by red horizontal lines). (b) Corresponding plot of $n_{+}\left(x,t\right)$ showing the emergence of a periodic variation in the total density of feedforward afferents (white represents a high density of synaptic drive).

Figure 5

(a) Plot of degree of pinning $\chi$ as a function of the modulation strength $\kappa$ for $k_c=\pi /d$ and $\mu =0.08$ taken over 45 trials with white noise about a binocular state. (b) One-dimensional OD pattern with a spatial phase defect. The steady-state density $n_{-}\left(x\right)$ is plotted as a function of $x$ for $\Lambda =16d$. Other parameter values are as in Fig. 4. It can be seen that pinning occurs except within a transition region (shaded in gray) where the OD columns are gradually shifted by an additional spatial phase equal to a single blob spacing. The variation in the CO blob density $u\left(x\right)$ is shown by the alternating light and dark dashed curve. The maxima of the density $n_{-}\left(x\right)$ shift from the dark to the light peaks of $u\left(x\right)$ as $x$ increases.

Figure 6

(a) Plot of degree of pinning $\chi$ as a function of the modulation strength $\kappa$ for $k_c=\pi /d$. (b) Plot of degree of pinning $\chi$ as a function of the wavelength mismatch ${\Delta }_k=d{k}_c/\pi -1$ for $\kappa =1.0$. Black dots show pinning with respect to OD centers. Weight parameters are as in Fig. 4 except that $\mu =0$. Averages were performed over 40 trials.

Figure 7

(a) Plot of degree of pinning $\chi$ versus degree of irregularity $\Gamma$ in distribution of CO blobs as given by Eq. (20). Here, $\kappa =1.0$, $\mu =0$, and $k_c=\pi /d$. (b) One-dimensional OD pattern (black curve) pinning to an irregular distribution $\left(\Gamma =0.7\right)$ of CO blobs (solid gray curve). The corresponding regular distribution of spacing $d$ is shown by the dashed gray curve. Same parameter values as (a) with $\Gamma =0.7$.

Figure 8

Two-dimensional pinning of OD columns to a square lattice of CO blobs with $N=1+\kappa u\left(x,y\right)$ and $u\left(x,y\right)=0.25\left[2+\text{cos}\left(2\pi x/d\right)+\text{cos}\left(2\pi y/d\right)\right]$. We take $\mu =0$ and $\kappa =1$ and consider a radial weight distribution (2) with $A=3.8$, $B=3.3$, ${\sigma }_e=0.51d$, and ${\sigma }_i=0.64d$ such that $k_c\approx \pi /d$, where $d$ is the blob spacing. The two examples correspond to different initial conditions obtained by introducing small random perturbations of the uniform binocular state. For visualization purposes, the color scale was chosen so that white corresponds to left eye dominance and gray corresponds to right eye dominance, with the lattice shown in black.

Figure 9

Two-dimensional pinning of OD columns to an irregular lattice of CO blobs. Distribution of CO blobs is given by Eq. (22) with $\Gamma =0.3$.

Figure 10

Examples of pinning to the CO blob lattice in the cases of MD and “catlike” conditions, shown in (a) and (b), respectively. In both (a) and (b), the CO blob lattice and the primary weight distribution are the same as in Fig. 8. In (a), we simulate MD conditions by reducing the excitation level of the deprived eye to 60% of its original value. The area that the nondeprived eye then innervates is 67% of V1. In (b), since the contralateral visual input is dominant in early cat development, in order to simulate catlike conditions, both excitation and inhibition levels of the ipsilateral drive are reduced to 60% of their original values for the initial phase of development (here, $\kappa$ is taken to be $\kappa =0.1$). For visualization purposes, the color scale was chosen so that white corresponds to left eye dominance and gray corresponds to right eye dominance, with the lattice shown in black.