Synaptic Formation Rate as a Control Parameter in a Model for the Ontogenesis of Retinotopy

We study the shape of patterns formed under different values of a control parameter in a model system for the ontogenesis of retinotopy proposed by Haussler and von der Malsburg. Guided by linear modes, their eigenvalues and nonlinear interactions, a few deciding values of the synaptic formation rate i¾?are chosen, under which final states are obtained using computer simulations of the full dynamics. We find that a precise topographic mapping can only be developed under a very narrow range of i¾?close to its critical value. The allowed range of i¾?is relaxed if the system is equipped with a proper structure, presumably by evolution.

[1]  L. Luo,et al.  Axon retraction and degeneration in development and disease. , 2005, Annual review of neuroscience.

[2]  H. Haken.  Synergetics: an Introduction, Nonequilibrium Phase Transitions and Self-organization in Physics, Chemistry, and Biology , 1977 .

[3]  A. Pelster,et al.  Synergetic analysis of the Häussler‐von der Malsburg equations for manifolds of arbitrary geometry , 2006, physics/0607253.

[4]  M. Cross,et al.  Pattern formation outside of equilibrium , 1993 .

[5]  J. Cowan,et al.  A mathematical theory of visual hallucination patterns , 1979, Biological Cybernetics.

[6]  Christoph von der Malsburg,et al.  Derivation of nonlinear amplitude equations for the normal modes of a self-organizing system , 2007, ESANN.

[7]  K. Obermayer,et al.  Statistical-mechanical analysis of self-organization and pattern formation during the development of visual maps. , 1992, Physical review. A, Atomic, molecular, and optical physics.

[8]  B. Ermentrout.  Stripes or spots? Nonlinear effects in bifurcation of reaction—diffusion equations on the square , 1991, Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences.

[9]  D J Willshaw,et al.  A marker induction mechanism for the establishment of ordered neural mappings: its application to the retinotectal problem. , 1979, Philosophical transactions of the Royal Society of London. Series B, Biological sciences.

[10]  N. Swindale,et al.  A model for the formation of orientation columns , 1982, Proceedings of the Royal Society of London. Series B. Biological Sciences.

[11]  Fernanda Botelho,et al.  Fluids and Waves , 2007 .

[12]  H. Meinhardt.  Models of biological pattern formation , 1982 .

[13]  A. Turing.  The chemical basis of morphogenesis , 1952, Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences.