The computational characteristics
of each neuron are not independent of the ones of other neurons laying in
the same layer, rather, they are often the consequence of a collective
behavior of neighboring cells.
More specifically, one can assume the
superposition of feed-forward and feedback contributions to investigate how
information is processed through the various stages of the visual pathway.
At each stage, we can identify * components* or * computational
primitives* due only to feed-forward connections (e.g., afferent projections)
and the properties not found in the components of a given stage can emerge
from their organization and their interactions with components of the
subsequent one.
According to such principle, we assume that, at cortical level, simple cell
RF profiles are the consequence of two main factors:
(1) functional projections of LGN afferent neurons and (2)
intracortical recurrent inhibition. Specifically, mechanisms similar
to that proposed by Hubel and Wiesel
[59] or the reported orientation bias of single geniculate neurons
[100,105,117,118] provide to simple cells a weak bias in orientation, but the final profiles of the RFs are molded
by inhibitory processes inside the cortex.

From the point of view of linear operator theory, this means decomposing the operator into a cascade of a feed-forward operator and a feed-back loop through the operator , as schematically depicted in figure 1.

**Figure 1:** *Schematic diagrams for feed-forward (left) and feed-back (right) operators*

Therefore, if

is the initial excitation, the final distribution of excitation will be

where is the identity operator, **a** is the excitatory gain (**a>0**),
that quantifies the impact of feed-forward projections, and **b** is a parameter that
specifies the strength and the sign of the intracortical influence ( operator). Since
we'll focus our attention on the net inhibitory effects, **b** will be
always supposed positive.

The feed-forward operator is defined as

Recent experimental investigations [18,109,110] have shown that simple cell RFs arise from the convergence of ON-center and OFF-center geniculate cells. For the sake of simplicity, in the present study we supposed a direct synaptic excitation from one type of cells only. Functionally, the convergence of feed-forward projections of either ON- or OFF-center cells, can be described through elliptic kernels. Specifically, we shall refer to as a family of positive elongated Gaussians with equal size and with orientations that vary with cortical coordinates, according to a specific orientation map. The ``initial'' RF profile of these cells is thus

where **p** refers to the aspect ratio of the field, and specifies
the RF size.

Also the feed-back operator can be defined in integral term

where is the kernel of intracortical interaction. The two-dimensional profile of the kernel represents the normalized distribution of intracortical spatial coupling, referred to a fixed location on the cortical plane. Its shape models the structural organization of inhibitory circuits and it is thus related to the spread and morphology of dendritic and axon arborizations of inhibitory interneurons.