Orientation maps from pattern formation

Turing patterns in biology

 

Alan Turing first proposed in 1952 that simple interacting chemicals, which he called morphogens, can create complex patterns that bear resemblance to ones found in biology. For example, he postulated one type of morphogen could result in the darkening of hair (called an activator), whereas another type could bleach it (called an inhibitor); as these morphogens diffuse across the animal's coat, typically at different rates, their reactions with each other can create intricate pigmentation patterns. The reaction–diffusion equations that he developed have been used to replicate the spots on a leopard's coat, the stripes on a zebra's hide, and can even account for how these patterns change as an animal ages.

One intriguing possibility, which I explore in my thesis, is that a reaction–diffusion process creates a scaffold during cortical development. For example, the striped pattern below shows a striking resemblance to ocular dominance bands. A similar idea is that spot-like patterns could lead to an orientation map; that is, one configuration of spots could correspond to a particular orientation preference, and shifted spot patterns could correspond to different preferences. This idea would explain why orientation maps are smooth (because the spots have spatial extent), and why orientation domains repeat at regular intervals (because the spots are periodic)—both features being natural consequences of reaction–diffusion interactions.

A Stripe Pattern that Resembles Ocular Dominance Columns in Visual Cortex

The notion that the functional architecture of the cortex could be specified through a similar reaction–diffusion process is a compelling idea. For one, it taps into an extremely efficient design strategy: All that is required to create the scaffold is the production of particular chemicals at the appropriate developmental stages. Second, these chemical patterns (or their imprints) can anchor features, which could account for the remarkable resilience of orientation maps to manipulations. Finally, the reaction–diffusion process can unfold in the presence of varied environmental constraints (different size and shape V1 regions), and is therefore robust.

 

Electrical pattern formation

In 2001, Ernst et al. proposed a model of V1 that uses electrical pattern formation to form a preferred orientation map. Their model substituted activator–inhibitor chemicals with excitatory–inhibitory neural activity, and implemented spatial interactions (loosely analogous to chemical diffusion) using local excitatory and distal inhibitory connections. The result is a two-dimensional recurrent network that can form bumps of neural activity; not surprisingly, these bumps closely resemble the spot-like patterns generated by a simple reaction–diffusion model. Note that the inputs to the network are isotropic and remain fixed throughout the simulation.

There are two requirements to achieve orientation selectivity from bumps: First, different oriented inputs must elicit different configurations of bumps. And second, repeated presentations of the same inputs must yield roughly the same configurations regardless of the initial state of the network. From the point of view of a single cell, these two requirements ensure that its tuning curve is not flat (i.e., different inputs elicit different responses), and the shape is statistically significant (i.e., repeatable for independent trials).

A simulation of electrical pattern formation The recurrent network produces bumps when driven by drifting gratings (top half). Each cell's preferred is plotted using a color code (bottom half). Similar to animal maps, the preferences change smoothly and repeat regularly, direct results of the bump patterns.

A perfect recurrent network (i.e., the properties of each cell are identical, and the connections between them are also identical) does not satsify the second requirement. Even though this perfect network can have non-flat tuning curves over individual trials, the shapes of these curves will not be consistent across trials for two reasons: First, the locations of bumps are extremely sensitive to initial conditions. Second, the movements of individual bumps are unbiased (i.e., they have an equal likelihood of moving in any direction). In this situation, cells are not really selective; they become active simply by chance occurrences.

How can we achieve robust selectivity so that neurons respond in a consistent manner? One possibility is that intracortical connections could provide an innate bias that constrains network activity. For example, bumps could be tethered to certain regions regardless of initial conditions; in addition, these pinned bumps might prefer to move in particular directions, resulting in consistent interactions with different orientations. Ernst et al. introduced such a bias in their network by adding jitter to the strengths of the recurrent connections (see movie).

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