The relationship between RFs and inhibition schemata can be further investigated by deriving analytical expressions for their dependence. To this purpose, to make the problem tractable we remove the dependence of the kernels on the orientation map, or, in other terms, we consider an uniform distribution of orientations, const, e.g., . Under this assumption, analytical expressions for the resulting RFs can be easily written, from which to draw valuable predictions on their properties, and, in general, on the modus operandi of recurrent inhibition. Due to the constant orientation map, the structure of the interaction fields, obtained by the repetition of the same pattern of connections, is highly regular, so that the effect of recurrent inhibition in the formation of periodic structures is particularly strong. However, the results should not to be considered applicable on a narrower scale, compared to those obtained with a non-constant map. Indeed, a constant distribution of orientation preferences over the small areas involved in inhibition is just the real situation within iso-orientation domains, and is a good approximation over regions with a low orientation drift rate.
Under these conditions, and assuming visual field and cortical plane extending indefinitely, (9) and (10) can be rewritten as
where symbolizes the spatial convolution operator and denotes both retinal and cortical coordinates, since the hypothesis of a one-to-one projection between the visual field and the cortical surface () still holds.
After convolution theorem, we have
where upper case letters refer to Fourier transforms and is the spatial frequency vector. Rearranging this equation yields
According to our linear setting, represents the spectral response profile of the resultant RF and can be readily calculated from the Fourier transforms of and . For the kernels defined by (12) and (16) and assuming , is given by
where and are the components of spatial frequency vector . First, let's infer from (26) a limiting condition on b to avoid unstable behavior, in relation with the geometrical parameters of the inhibitory kernel:
Then, we can use this explicit form of to understand how the resultant RF depends on the geometrical parameters (, , and d) of the kernels and on the strength of inhibition b. The response properties of a simple cell can be characterized in frequency domain by comparing the resulting spectrum of its RF with that of a Gabor RF. To this purpose, let us consider the frequency tuning curve (i.e., the frequency sensitivity curve of the simple cell along the axis orthogonal to the RF orientation), and characterize it by two parameters: the optimal spatial frequency (i.e., the value at which gets its peak value) and the absolute bandwidth (i.e., the difference between the upper () and lower () spatial frequencies at which the response amplitude drops to of its maximum). Figure 7 shows a typical dependence of spatial frequency tuning curves on the parameter b. An increase of the inhibitory strength b results in a rise of the response at and in a narrowing of the tuning. For low values of inhibition strength, the spectrum of maintain a low-pass structure; for higher values of b two peaks start growing, symmetrically displaced from the origin. While the center frequency, , of the peaks remains practically constant, the bandwidth, , decreases with b.
Figure 7: Frequency tuning curves for increasing values of the inhibition strength b. Only positive frequencies are shown.
This behavior can be analyzed by considering in detail how inhibition strength b influences and , in dependence on the geometrical parameters of the RF () and of the inhibitory kernel ( and d). To analyze the effects of the various parameters, it is convenient to introduce the normalized frequency tuning curve:
where , , and .
In figure 8a-b, we have plotted, for different values of and a typical (fixed) value of , the behavior of the normalized optimal spatial frequency and of the normalized absolute bandwidth versus the inhibition strength, respectively. Similarly, in figure 8c-d, the same quantities are represented for different values of and a typical (fixed) value of . We observe a large influence of for low values of b, while increasing the value of b, (i) the normalized optimal spatial frequency first increases suddenly and afterwards flattens toward an asymptotic value independent of , but dependent on , and (ii) the absolute bandwidth, eventually after a slight increase, quickly drops to a small fraction of the initial value, independently of , and almost of .
Not a terminal
This behavior of can be summarized in the parameter space as shown in figure 9a-b (see caption).
A closer consideration on the specific role of the parameters and d is presented in figure 10a-b, where the optimal spatial frequency and absolute bandwidth are plotted versus the distance (d) of the clusters for different values of the initial RF size () and for a fixed value of the inhibition strength (). We can observe that, increasing the distance d, both and first rise up and then drop, approaching an asymptotic value which is independent of the size of the initial excitatory contribution from LGN projections.
Let's now consider, how the resulting RF compares with the Gabor RFs in the spatial domain. Rearranging (25) yields
where and are the Fourier transforms of the equivalent inhibitory kernel in the feed-forward form, and of the kernel of the over-all open-loop operator, respectively. figure 11 illustrates the meaning of these relations with a block-diagram representation.
Though the inverse Fourier transform of is not easy to evaluate, some comments can be evidenced by its Taylor expansion. If , one obtains
Since is a low pass filter, its powers in (30) lead to a decrease in bandwidth and so to an increasing width of the coupling function . Indeed, back in the space domain the powers of result in a sum of shifted Gaussians with alternating signs, with a considerable increase of the effective coupling function. Specifically, as the power grows the number of Gaussians increases as well as their width, while their contribution on the coupling decreases:
The expression for the open loop kernel well evidences the regular distribution of the additional induced coupling and impacts on the spatial structure of simple cell RFs. Specifically, by combining (31) and (32) with (29), it is easy to obtain a series expression for the resultant RF after inhibition:
We observe that the resulting RF appears as a sum of displaced elongated Gaussians which resembles the simple cell models based on differences of offset Gaussians (DOOG) [52,53,106]. Summing up the Gaussians, for typical values of the parameters (a sum limited to l=3 is sufficient to have an error smaller than for ), we obtain a well-defined Gabor-like profile, as predicted by DOOG models. More precisely, the progressive enlarging of the Gaussians with their offset distance, predicts that the successive sidebands in the resulting RFs are increasingly widely spaced out, as predicted by the Gaussian derivative model . From the examination of the spatial separation of the RF subregions, and by their length in the direction parallel to the axis of orientation we can make some considerations on the effects of the model parameters on the orientation specificity of the resulting RF. To this end, we have evaluated from (33) and (34) the aspect ratio relative to the central subregion, determined at of the peak height, for different initial orientation biases (p) and for different values of . The results, for a fixed value of the inhibition strength (), are plotted in figure 12 versus the distance of the inhibitory clusters d.
Figure 12: Aspect ratio relative to the central subregion of the resulting RF, evaluated at of its peak value. Each group of curves is related to a particular value of the initial orientation bias p (i.e., the aspect ratio of the initial excitatory contribution from LGN).
In all cases, the plots are bell-shaped, evidencing that for each value of the size of the excitatory contribution , there exists a value of d for which we obtain the best improvement in orientation specificity. Increasing the amount of the initial orientation bias, the effect of inhibition in improving orientation tuning becomes much more stronger, anyway, the couples of values corresponding to the maxima of the aspect ratios seem to be independent of the initial orientation bias. Moreover, it is worthy to note that a slight improvement in orientation can be observed also without any initial bias (i.e., p=1). Hence, the kind of intracortical inhibition proposed in our model is able to break circular symmetries from LGN projections and therefore provides cues for the intracortical origin of orientation selectivity.