Course 12 - Symmetry Breaking and Pattern Selection in Visual Cortical Development

The ontogenetic development of the cerebral cortex of the brain is a process of astonishing complexity. In every cubic millimeter of cortical tissue in the order of 10⁶ neurons must be wired appropriately for their respective functions such as the analysis of sensory inputs, the storage of skills and memory, or motor control [1]. In the brain of an adult animal, each neuron receives input via about 10⁴ synapses from neighboring and remote cortical neurons and from subcortical inputs [1]. At the outset of postnatal development, the network is formed only rudimentarily: For instance in the cat's visual cortex, most neurons have just finished the migration from their birth zone lining the cerebral ventricle to the cortical plate at the day of birth [2]. The number of synapses in the tissue is then only 10% and at the time of eye-opening, about two weeks later, only 25% of its adult value [3]. In the following 2-3 months the cortical circuitry is substantially expanded and reworked and the individual neurons aquire their final specificities in the processing of visual information [4].

It is a very attractive but still controversial hypothesis that in the ontogenetic development of the brain the emerging cortical organization is constructed by learning mechanisms which are similar to those that enable us to aquire skills and knowledge in later life [[5], [6], [7]]. Several lines of evidence strongly suggest that the brain in a very fundamental sense learns to see. First, visual experience is very important for the normal development of sight. If the use of the visual sense is prevented early in life vision becomes irreversibly impaired [4]. Since this is not due to a malformation of the eye or of peripheral stages of the visual pathway, it suggests that in development visual input it used to improve the processing capabilities of the visual cortical networks. In addition, the performance of the developing visual system responds very sensitively to visual experience. In human babies, for instance, already a few hours of visual experience lead to a marked improvement of visual acuity [8]. Second, the synaptic organization of the visual cortex is highly plastic and responds with profound and fast functional and structural reorganization to appropriate experimental manipulations of visual experience [9, 10]. These and similar observations suggest that the main origin of perceptual improvement in early development is due to an activity-dependent and thus use-dependent refinement of the cortical network. In the course of this neuronal activity patterns that arise in the processing of visual information in turn guide the further refinement of the cortical network until a mature configuration is reached. Whereas, theoretically, this hypothesis is very attractive, it is, experimentally, still controversial, whether neural activity actually plays such an instructive role (for discussion see [[11], [12], [13]]).

Viewed from a dynamical systems perspective, the activity-dependent remodeling of the of the cortical network described above is a process of dynamical pattern formation. In this picture, spontaneous symmetry breaking in the developmental dynamics of the cortical network underlies the emergence of cortical selectivities such as orientation preference [14]. The subsequent convergence of the cortical circuitry towards a mature pattern of selectivities can be viewed as the development towards an attractor of the developmental dynamics [15]. In this set of lectures, I will discuss universal dynamical properties of a paradigmatic process in visual cortical development: the development of orientation columns and the formation of so called orientation pinwheels.

In the visual cortex, as in most areas of the cerebral cortex information is processed in a 2-dimensional (2D) array of functional modules, called cortical columns [16, 17]. Individual columns are groups of neurons extending vertically throughout the entire cortical thickness that share many functional properties. Orientation columns in the visual cortex are composed of neurons preferentially responding to visual contours of a particular stimulus orientation [ 18]. In a plane parallel to the cortical surface, neuronal selectivities vary systematically, so that columns of similar functional properties form highly organized 2D patterns, known as functional cortical maps. In the case of orientation columns, this 2D organization is characterized by so called pinwheels, regions in which columns preferring all possible orientations are organized around a common center in a radial fashion [19,20] (see Figure 1).

Experimental evidence suggests that the formation of orientation columns is a dynamical process guided by neuronal activity and sensitive to visual experience. In normal development, orientation columns first form at about the time of eye opening [[21], [22], [23]]. Comparison of this process to the development under conditions of modified visual experience demonstrates that adequate visual experience is essential for the complete maturation of orientation columns and that impaired visual experience, as with experimentally closed eye-lids can suppress or impair the formation of orientation columns [23] (see Figure 2). Most intriguingly, when visual inputs are experimentally redirected to drive what would normally become primary auditory cortex, orientation selective neurons and a pattern of orientation columns even forms in this brain region that would normally not at all be involved in the processing of visual information ([24] see Figure 3). In particular the latter observation strongly suggests that the capability to form a system of orientation columns is intrinsic to the learning dynamics of the cerebral cortex given appropriate inputs. Taken together these lines of evidence mark the formation of orientation columns as a paradigmatic problem in the dynamics of cortical development and plasticity.

Owing to the large number of degrees of freedom of any realistic scale microscopic model of visual cortical development, the description of the development of the pattern of columns by equations for the synaptic connections between the LGN and cortex is very complicated. On the order of 10⁶ synaptic strengths would be required to realistically describe, for example, the pattern of orientation preference in a 4x4mm² piece of the visual cortex. This complexity and the presently very incomplete knowledge about the nature of realistic equations for the dynamics of visual cortical development demand that theoretical analyzes concentrate on aspects that are relatively independent of the exact form of the equations and are representative for a large class of models. An appropriate framework for this is provided by models in which the emerging cortical architecture is described by order parameter fields and its development by a dynamics of such fields [15, [25], [26], [27], [28], [29], [30], [31]]. A few years ago, Theo Geisel and the author discovered that experimentally accessible signatures of an activity-dependent refinement of the cortical network are predicted by universal properties of this very general class of models for the development of visual cortical orientation preference maps [15]. We could demonstrate that that if the pattern of orientation preferences is set up by learning mechanisms, then the number of pinwheels generated early in development exhibits a universal minimal value that depends only on general symmetry properties of the cortical network. This implies that in species exhibiting a lower number of pinwheels in the adult, pinwheels must move and annihilate in pairs during the refinement of the cortical circuitry. Verification of this intriguing prediction would provide striking evidence for the activity-dependent generation of the basic visual cortical processing architecture. In the initial sections of this chapter (2. The pattern of orientation preference columns, 3. Symmetries in the development of orientation columns, 4. , 5. Generation and motion of pinwheels), I will present a self-contained treatment of the mathematical origin of this kind of universal behavior. In particular, I will discuss the description of the development of orientation preference columns in terms of a dynamics of abstract order parameter fields, connect this description to the theory of Gaussian random fields, and show how the theory of Gaussian random fields can be used to obtain quantitative information on the generation and motion of pinwheels, in the two dimensional pattern of visual cortical orientation columns. I will then extended the symmetry based approach used to derive this prediction to study also the kind of patterns to which the map will asymptotically converge and the interactions essential for the stabilization of different kinds of solutions (Section 6). In Section 7, I will provide an exposition of an appropriate perturbation method called weakly nonlinear analysis for the problem of orientation column formation. Using this method, a class of generalized Swift-Hohenberg models for the formation of patterns of contour detecting neurons during visual cortical development is constructed in Section 8. In this model class, a permutation symmetry of the model equations satisfies the requirement that the visual cortex develops selectivity for all contour orientations. By this symmetry a large number of dynamically degenerate solutions exist that quantitatively reproduce the experimentally observed patterns. Long-range interactions are found to be essential for the stability of realistic solutions.