Theory and implementation of infomax filters for the retina

In the first part of this paper we discuss a technical visual sensory system, which--in analogy with the retina--includes some preprocessing of visual information. In so doing, we use an information-theoretic criterion, the infomax ansatz, to optimize the response of the sensory system. In particular, it is shown that the lattice structure of the photoreceptor array has to be taken into account. By a discrete Fourier transformation on a triangular lattice we derive the frequency response of the infomax filter within the first Brillouin zone. To illustrate the response properties, infomax filters adapted to different noise levels are applied to images with different signal-to-noise ratios. This clearly demonstrates the necessity of adaptation of the filter properties to the given noise level. Furthermore, it is shown how to efficiently implement infomax-like filters by simple networks with only nearest-neighbour interactions. A two-layered network topology proves to be very advantageous in implementing the desired high-pass or low-pass properties. The network topology allows for adaption of the network to low and high noise levels by simply adjusting the nearest-neighbour couplings. In the second part of this paper, we compare the previously described information-theoretic requirements on a visual sensory system with biological facts known from the vertebrate retina. The substantial physiological response properties of the vertebrate retina are in agreement with the main features of the infomax filter. Since available experimental data lacks information which is necessary for a more quantitative comparison, we present suggestions for future experiments. Some key anatomical features of the retina of many vertebrates compare well with our two-layered implementation of the infomax filter. The analogy, in particular, concerns the adaption mechanism. To illustrate this point, we summarize some recent experiments which demonstrate that in the retina of some species adaption is based on the release of the neuromodulator dopamine by the interplexiform cells. This causes the horizontal cells to decouple. On the basis of recently gained understanding of the outer plexiform layer of the retina some further hypotheses about the functionality of the retina become obvious and possible future experiments to verify or refute them are suggested. Finally, we discuss the infomax approach from a more general point of view. In particular, we show that redundancy is essential to obtaining noise robustness of an internal representation of the environment as it is produced by a sensory system such as the retina.