From receptive profiles to a metric model of V1

Abstract

In this work we show how to construct connectivity kernels induced by the receptive profiles of simple cells of the primary visual cortex (V1). These kernels are directly defined by the shape of such profiles: this provides a metric model for the functional architecture of V1, whose global geometry is determined by the reciprocal interactions between local elements. Our construction adapts to any bank of filters chosen to represent a set of receptive profiles, since it does not require any structure on the parameterization of the family. The connectivity kernel that we define carries a geometrical structure consistent with the wellknown properties of long-range horizontal connections in V1, and it is compatible with the perceptual rules synthesized by the concept of association field. These characteristics are still present when the kernel is constructed from a bank of filters arising from an unsupervised learning algorithm.

KerCNNs: biologically inspired lateral connections for classification of corrupted images

“…The geometrical structure encoded in this kernel is shown in Montobbio et al (2019a) to be compatible with the properties of V1 horizontal connections, and with the perceptual principles synthesized by association fields. Results are also shown for a bank of filters arising from an unsupervised learning algorithm: this shows that meaningful information of the geometry of horizontal connections can be recovered from numerically known filters, thus motivating the present work.…”

“…A kernel model for lateral connectivity. A different connectivity kernel, induced by a structure of metric space associated to the RPs of simple cells, has been introduced in Montobbio et al (2019a). The core of the model is the definition of a kernel describing the interactions between local elements, which determines a metric structure directly induced by the shape of the RPs of V1 simple cells.…”

“…We conclude this section with an important remark on the action of the correlation kernel of Montobbio et al (2019a) as an operator acting on functions defined on G. Given a function…”

“…where N is the same normalization operator introduced by Coifman and Lafon (2006) and appearing in Montobbio et al (2019a), see Eq. ( 4).…”

Reconstructing Group Wavelet Transform From Feature Maps With a Reproducing Kernel Iteration

“…A possible interpretation of the proposed iteration with a kernel defined by the SE(2) group as a neural computation in V1 comes from the modeling of the neural connectivity as a kernel operation (Wilson and Cowan, 1972;Ermentrout and Cowan, 1980;Citti and Sarti, 2015;Montobbio et al, 2018), especially if considered in the framework of a neural system that aims to learn group invariant representations of visual stimuli (Anselmi and Poggio, 2014;Anselmi et al, 2020). A direct comparison of the proposed technique with kernel techniques recently introduced with radically different purposes in Montobbio et al (2018) and Montobbio et al (2019) shows, however, two main differences at the level of the kernel that is used: here, we need the dual wavelet to build the projection kernel, and the iteration kernel effectively contains the feature maps. On the other hand, a possible application is the inclusion of a solvability condition such as Equation ( 14) as iterative steps within learning frameworks such as those of Anselmi et al (2019) and Anselmi et al (2020).…”