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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  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
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.