This research applies the method of generalized eigenfunctions to the analysis and description of various biological phenomena. This technique allows the description of a phenomenon in terms of an irreducible collection of intrinsically different standard contributions, called eigenfunctions. One first determines the governing eigenfunctions, then expresses any individual instance of the phenomenon as a linear combination of these eigenfunctions.
Turbulent fluid flow appears to be highly disorganized and chaotic. However, when decomposed into eigenfunctions it shows an unsuspected simplicity of organization. We have analyzed several classical examples of hydrodynamic flow using this technique.
Patterns of activity in the visual cortex of the cat, as determined experimentally using voltage-sensitive fluorescent dyes, are analyzed using the eigenfunction technique. First, we must distinguish the signal (patterns of cortical activity arising from particular visual stimuli) from the noise (a large and rapidly varying range of cortical activities). This makes possible the isolation and study of the patterns associated with the presentation of specific visual stimuli. These patterns are being analyzed and classified, using the eigenfunction method. Although the observed patterns are complex and rapidly varying, only nine eigenfunctions are needed to capture more than 90% of the activity.
Physically distinct visual stimuli may give rise to visually indistinguishable images. We have developed a mathematical description of this phenomenon based on the line spreading that occurs in the visual system. Several classes of edge stimuli have been shown to fall into different equivalence (i.e. indistinguishability) classes, depending on the nature of the line spread function that is active. This analysis placed the phenomenon of distinct but functionally equivalent visual stimuli on a firm quantitative foundation.
The spectral properties of a photopigment often do not correspond to those found spectrophotometrically for any known single pigment. We have proposed that this could occur because the phenomenon did not involve a single pigment, but rather an interaction among several pigments, called a pseudo-pigment. We are developing a mathematical method to determine from a spectral sensitivity function the number and wavelengths of maximum sensitivity of all photopigments involved.
We have developed several new methods to approximately analyze subthreshold solutions of the Hodgkin-Huxley equations. We first considered only the linear portions of these equations, and solved these. This gives a reasonably accurate solution in the subthreshold region. We also developed an asymptotic analysis under two conditions, either small position or long time. Finally, we numerically integrated the full form of these equations on a computer. We showed that all these results were in good agreement with each other and with experiments.