The arbors of most interneurons are confined to laterally restricted circular areas of 200:300 [73,104], thus leading to an inhibitory action which is strongly local and uniformly distributed on the horizontal plane. Such local inhibitory circuits are mainly involved in the regulation of the gain of excitatory input [32,33]. Much more interesting, for its functional implications, is long-range inhibition, evoked, often along preferred directions, by clustered locations on the cortical surface. The origin of this kind of inhibition can be related to disynaptic activation of locally confined inhibitory interneurons through long-range clustered branches of pyramidal cells [81,94], and to the monosynaptic inhibitory action of basket cells which also have relatively long clustered axon collaterals, elongated in particular directions and running tangentially for up to a millimeter from the soma [63,64,65,67,104]. To model the consequences of this long-range inhibition, the kernel presents significant values only at a certain distance from their centers, and along a direction forming a specific angle with the horizontal axis.
In the simplest case, we model such kernel as the sum of two weighted offset circular Gaussians, that are displaced symmetrically with respect to the location on the cortical plane they are referred to. For an horizontal displacement () the kernel will be
where the g's are Gaussian functions
and the w's are weights. For a displacement () along a direction the kernel will be
with the - sign for and the + sign for .
The width of the Gaussians and their offset d are such that no connections are established between two cells when their intracortical distance is either too large or too small. Moreover, the lateral spreading of interconnections can be characterized by the angular bandwidth related to the half peak magnitude extension of the kernel
This relation implies that we cannot change the angular spread of inhibition without affecting also its spatial localization. To overcome this limitation, we define generalized kernels by summing up equal circular Gaussians whose centers are uniformly spaced on the arc of circumference with radius d under the span angle (see figure 2).
In this way, the inhibitory kernel becomes:
where , . In this case the angular bandwidth results:
Since we are mainly interested in the organization of the RFs along their cross-orientation axis (i.e., the axis orthogonal to the RF orientation), we dwell mainly upon those axon fields that extend for longer distances perpendicular to the orientation axis of the selected cell than along its orientation axis. Hence, we'll consider (see figure 2b), even though, in the following, the effects of intracortical kernels with orientations slightly different from the orthogonal one will be also investigated.