Exact solution of the nonlinear dynamics of recurrent neural mechanisms for direction selectivity

Di erent theoretical models have tried to investigate the feasibility of recurrent neural mechanisms for achieving direction selectivity in the visual cortex. The mathematical analysis of such models has been restricted so far to the case of purely linear networks. We present an exact analytical solution of the nonlinear dynamics of a class of direction selective recurrent neural models with threshold nonlinearity. Our mathematical analysis shows that such networks have form-stable stimulus-locked traveling pulse solutions that are appropriate for modeling the responses of direction selective cortical neurons. Our analysis shows also that the stability of such solutions can break down giving raise to a di erent class of solutions ("lurching activity waves") that are characterized by a speci c spatio-temporal periodicity. These solutions can not arise in models for direction selectivity with purely linear spatio-temporal ltering. This report describes research done within the Center for Biological and Computational Learning in the Department of Brain and Cognitive Sciences and in the Arti cial Intelligence Laboratory at the Massachusetts Institute of Technology. This research is sponsored by a grant from OÆce of Naval Research under contract No. N00014-93-1-3085, OÆce of Naval Research under contract No. N00014-95-1-0600, National Science Foundation under contract No. IIS-9800032, and National Science Foundation under contract No. DMS-9872936. Additional support is provided by: MIT UROP Program, AT&T, Central Research Institute of Electric Power Industry, Eastman Kodak Company, Daimler-Benz AG, Digital Equipment Corporation, Honda R&D Co., Ltd., NEC Fund, Nippon Telegraph & Telephone, and Siemens Corporate Research, Inc. M. Giese was supported by DFG, Honda Americas, Inc., and the Deutsche Volkswagen Foundation. X. Xie was suppoted by a a Chun Doug Shiah Memorial fellowship. To appear in Neurocomputing, 2001.