Perspectives and Problems in Nonlinear Science

Ehud Kaplan, Jerry Marsden, Kartik Sreenivasan

Research output: Contribution to journalArticlepeer-review


Functions of a single variable that are simultaneously band and space-limited are useful for spectral estimation and have also been proposed as reasonable models for the sensitivity profiles of receptive fields of neurons in the primary visual cortex. Here we consider the two-dimensional extension of these ideas. Functions that are simultaneously space- and band-limited in circular regions form a natural set of families, parameterized by the “hardness” of the space- and band- limits. For a Gaussian (“soft”) limit, these functions are the two-dimensional Hermite functions, with a modified Gaussian envelope. For abrupt space and spatial frequency limits,
these functions are the two-dimensional analog of the Slepian (prolate spheroidal) functions (Slepian and Pollack [1961]; Slepian [1964]). Between these limiting cases, these families of functions may be regarded as points
along a 1-parameter continuum. These families and their associated operators have certain algebraic properties in common. The Hermite functions
play a central role, for two reasons. They are good asymptotic approximations of the functions in the other families. Moreover, they can be decomposed both in polar coordinates and in Cartesian coordinates. This joint
decomposition provides a way to construct profiles with circular symmetries
from the superposition of one-dimensional profiles. This result is approximately
universal: it holds exactly in the “soft” (Gaussian) limit and in good approximation across the one-parameter continuum to the “hard” (Slepian)
limit. These properties lead us to speculate that such two-dimensional profiles will play an important role in the understanding of visual processing
in cortical areas beyond the primary visual cortex. A comparison with published
experimental results lends support to this conjecture.
Original languageEnglish (US)
JournalSpringer Climate
StatePublished - 2003


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