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Synergetic analysis of the Häussler‐von der Malsburg equations for manifolds of arbitrary geometry
Author(s) -
Güßmann M.,
Pelster A.,
Wunner G.
Publication year - 2007
Publication title -
annalen der physik
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.009
H-Index - 68
eISSN - 1521-3889
pISSN - 0003-3804
DOI - 10.1002/andp.200610243
Subject(s) - connection (principal bundle) , instability , synergetics (haken) , nonlinear system , physics , basis (linear algebra) , geometry , dimension (graph theory) , classical mechanics , statistical physics , mathematics , pure mathematics , mechanics , quantum mechanics
We generalize a model of Häussler and von der Malsburg which describes the self‐organized generation of retinotopic projections between two one‐dimensional discrete cell arrays on the basis of cooperative and competitive interactions of the individual synaptic contacts. Our generalized model is independent of the special geometry of the cell arrays and describes the temporal evolution of the connection weights between cells on different manifolds. By linearizing the equations of evolution around the stationary uniform state we determine the critical global growth rate for synapses onto the tectum where an instability arises. Within a nonlinear analysis we use then the methods of synergetics to adiabatically eliminate the stable modes near the instability. The resulting order parameter equations describe the emergence of retinotopic projections from initially undifferentiated mappings independent of dimension and geometry.