Max Planck Institute for Dynamics and Self-Organization -- Department for Nonlinear Dynamics and Network Dynamics Group
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Pattern formation in the developing visual cortex

The ontogenetic development of the cerebral cortex of the brain is a process of astonishing complexity. In every cubic millimeter of cortical tissue about a million of neurons must be wired appropriately for their respective functions such as the analysis of sensory inputs, the storage of skills and memory, or for motor control. In the brain of an adult mammal, each neuron receives input via about 10000 synapses from neighboring and remote neurons. At the outset of brain development, however, the cortical network is formed only rudimentarily: For instance in the cat’s visual cortex at the day of birth, most neurons have just finished the migration from their birth zone lining the cerebral ventricle to the cortical plate. 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. In the following 2-3 months the cortical circuitry is substantially expanded and reworked and the individual neurons acquire their final specificities in the processing of visual information. Many lines of evidence suggest that during this period the brain in a very fundamental sense “learns to see” [1,2].

Viewed from a dynamical systems perspective, the activity-dependent remodeling of the cortical network during development is a process of dynamical pattern formation. Spontaneous symmetry breaking in the developmental dynamics of the cortical network underlies the initial emergence of cortical selectivities such as orientation preference [3]. After symmetry breaking, ongoing improvement may result in the convergence of the cortical architecture to a stable stationary state constituting an attractor of the visual system’s learning dynamics. Because the development of the visual cortical architecture is in many respects reminiscent of dynamical pattern formation in physical systems far from equilibrium, it is natural to ask whether this process might be modeled by dynamical equations for macroscopic order parameters as often proved successful and instructive in simpler pattern formation problems.

Symmetries in visual cortical development

As a paradigmatic model system our studies are focusing on the development and the spatial structure of the pattern of contour detecting neurons in the visual cortex called the orientation preference map (OPM, see Figure 1b). Functional brain imaging has revealed that the spatial structure of OPMs is aperiodic but roughly repetitive with a characteristic wavelength in the millimeter range (see e.g. [4]) and that OPMs contain numerous singular points called pinwheel centers (see Fig.1b).

In previous work, Wolf and Geisel demonstrated that basic symmetry assumptions imply a universal minimal initial density of pinwheel defects [3]. They also found, however, that the defects generated initially are typically unstable and are decaying by pairwise annihilation in various models of visual cortical development. To solve the problem of pinwheel stability, subsequent studies have raised the hypothesis, that a reduced symmetry of the dynamics of visual cortical pattern formation may underlie the formation of stable pinwheel patterns [5,6]. In the proposed models of reduced symmetry, however, pinwheels generally crystallize in periodic spatial patterns that are clearly distinct from the patterns observed experimentally.

In recent work, we have therefore investigated conditions for the emergence of spatially nonperiodic OPMs theoretically. To this end, Wolf introduced an analytically tractable class of model equations [7], which was shown to possess solutions that qualitatively and quantitatively resemble the experimentally observed patterns. Its construction is based on the requirement that the visual cortex must develop detectors for contours of all orientations. Assuming a supercritical bifurcation of the pattern this turned out to be guaranteed near criticality by a novel permutation symmetry. This symmetry in addition implies the existence of a large number of dynamically degenerate solutions that are qualitatively very similar to OPMs in the visual cortex (Fig.1a). Further analysis revealed that, judged by their pinwheel densities, these patterns even quantitatively resemble the experimentally observed OPMs (see Fig.1c). The stability boundaries of various solutions were calculated in a generalized Swift-Hohenberg model incorporating long-range interactions, a key feature of visual cortical processing. Generically, long-range interactions were found to be essential for the stability of realistic solutions. Their existence and stability, however, turned out to be rather insensitive to full or reduced symmetries of the developmental dynamics [8].


Quantification of cortical column layout

Because these and related results indicate that the apparently complicated layout of visual cortical maps can be quantitatively explained by relatively simple model equations, substantial effort is dedicated to developing methods for the precise quantification of layout parameters from experimentally obtained brain imaging data. The layout of functional cortical maps exhibits a high degree of interindividual variability that may account for individual differences in sensory and cognitive abilities. By quantitatively assessing the interindividual variability of OPMs in the primary visual cortex using a newly developed wavelet based analysis method, we demonstrated that column sizes and shapes as well as a measure of the homogeneity of column sizes across the visual cortex are significantly clustered in genetically related animals and in the two hemispheres of individual brains [9,10]. Taking the developmental timetable of column formation into account, these observations indicate that the essential control parameters of the dynamics of visual cortical pattern formation are tightly controlled genetically.

More recently, application of these methods led to the first demonstration of a dynamical rearrangement of cortical orientation columns during visual development. In a study comparing the layout of orientation columns in two different visual cortical areas in kittens and cats we found that maps in the two areas exhibit matched column sizes at retinotopically corresponding positions in adult cats. In addition the data revealed that in kittens of various ages, column sizes progressively became better matched in the two areas over the course of visual cortical development [10]. Based on these methodological advances, we also started to characterize the statistics of pinwheel defects in collaboration with Len E. White (Duke University). Here it turned out that virtually all basic statistics of pinwheels in orientation maps are universal in a wide range of animals [11]. These findings are very surprising and beginning to shed a completely new light on the formation of cortical columns. Current model studies are addressing the conditions under which dynamical pattern formation can account qualitatively and quantitatively for the observed universal statistics.


[1] M.C. Crair, Curr. Opin. Neurobiol. 9:88 (1999)

[2] L.C. Katz, J.C. Crowley, Nat. Rev. Neurosci. 3:34 (2002)

[3] F. Wolf, T. Geisel, Nature 395:73 (1998)

[4] M. Kaschube, F. Wolf, T. Geisel, S. Löwel, J. Neurosci. 22:7206 (2002). [5] H.Y. Lee, M. Yahyanejad, M.Kardar, Proc. Natl. Acad. Sci. 100:16036 (2003)

[6] P.J. Thomas, J.D. Cowan. Phys. Rev. Lett. 92:188101 (2004)

[7] F. Wolf, Phys. Rev. Lett. 95:208701 (2005)

[8] M. Schnabel, M. Kaschube, S. Löwel, F. Wolf (in prep.)

[9] M. Kaschube, F. Wolf, M. Puhlmann, S. Rathjen, K.F. Schmidt, T. Geisel, S. Löwel, Eur. J. Neurosci. 22, 7206 (2003)

[10] M. Kaschube, M. Schnabel, S. Löwel, F. Wolf, (in prep.) [11] M. Kaschube, M. Schnabel, S. Löwel, D. Coppola, L. White, F. Wolf, (in prep.)

Contact:  Fred Wolf 

Members working within this Project:

 Rainer Engelken 
 Manuel Schottdorf 
 Juan Daniel Flórez Weidinger 

Former Members:

 Dmitry Tsigankov 
 Wolfgang Keil 
 Min Huang 
 Matthias Kaschube 
 Lars Reichl 
 Michael Schnabel 

Selected Publications:

M. Schottdorf, W. Keil, D. Coppola, L.E. White, and F. Wolf (2015).
Random Wiring, Ganglion Cell Mosaics, and the Functional Architecture of the Visual Cortex
PLoS Computational Biology 11(11): e1004602.

M. Schottdorf, S.J. Eglen, F. Wolf, and W. Keil (2014).
Can Retinal Ganglion Cell Dipoles Seed Iso-Orientation Domains in the Visual Cortex?
PLoS ONE 9(1): e86139.

W. Keil, and F. Wolf (2011).
Coverage, Continuity and Visual Cortical Architecture
arXiv:1104.1946v1 [q-bio.NC]. download file

D. Tsigankov, and A. Koulakov (2010).
Sperry versus Hebb: Topographic mapping in Isl2/EphA3 mutant mice
BMC Neuroscience 11(155).

L. Reichl (2010).
Symmetry Breaking and Pattern Selection in Models of Visual Development
Phd thesis, University of Göttingen, Germany. download file

L. Reichl, S. Loewel, and F. Wolf (2009).
Pinwheel Stabilization by Ocular Dominance Segregation
Phys. Rev. Lett. 102:208101. download file

F. Wolf, L. Reichl, and S. Loewel (2008).
Contralateral eye dominance induces pinwheel crystallization in models of visual cortical development
In: . Society for Neuroscience Abstracts Vol 34, 326.18.

M. Kaschube, M. Schnabel, and F. Wolf (2008).
Self-Organization and the Selection of Pinwheel Density in Visual Cortical Development
New Journal of Physics 10:015009 (20pp). download file

M. Schnabel (2008).
A Symmetry of the Visual World in the Architecture of the Visual Cortex
Phd thesis, University of Göttingen, Germany. download file

L. Reichl, D. Heide, S. Loewel, and F. Wolf (2006).
Transition to disordered ocular dominance maps by inter-map coupling
In: . Society for Neuroscience, Abstracts Vol 32, 619.7 .

F. Wolf (2005).
École d'Été de Physique des Houches, 2003, Methods and Models in Neurophysics
In: . Elsevier, Amsterdam, chapter Symmetry Breaking and Pattern Selection in Visual Cortical Development, pages 575-639.

F. Wolf (2005).
Symmetry, Multistability, and Long-Range Interactions in Brain Development
PRL 95:208701.

M. Kaschube, F. Wolf, T. Geisel, and S. Löwel (2002).
Genetic Influence on Quantitative Features of Neocortical Architecture
Journal of Neuroscience 22(16):7206-7217.

M. Kaschube, F. Wolf, T. Geisel, and S. Löwel (2000).
Quantifying the Variability of Patterns of Orientation Domains in the Visual Cortex of Cats
Neurocomputing 32-33:415-423.

F. Wolf, and T. Geisel (1998).
Spontaneous pinwheel annihilation during visual development
Nature 395:73-78.