Figure 1: Steerable and shiftable filters (receptive fields). The properties of steerability and shiftability allow a continuum of orientation preferences to be obtained from a small number of discretely tuned orientation-selective basis RFs (the Shaping the RFs section).
Several models for the formation of the original set of discrete orientations by projections from the lateral geniculate nucleus (LGN) to the striate cortex have been offered in the past [18,47]. In particular, it has been argued that the receptive fields at the output of the LGN are already broadly tuned for a small number of discrete orientations (possibly just horizontal and vertical), and that at the cortical level the entire spectrum of orientations is generated from the discrete set present in the geniculate projection . In the present model, the size of the original discrete orientation columns is determined by the minimal cortical separation of cells with non-overlapping RFs, called the point image . Thus, the model incorporates a network of orientation columns, whose size is determined by the diameter of their constituent RFs. Each column is tuned to a specific angle, and located at an approximately constant distance from another column with the same orientation tuning. The RFs of adjacent units with the same orientation preference are overlapping, with the amount of overlap determined by the number of RFs built into the network. The preferred orientations are distributed uniformly in the range . Each RF is modeled by a product of a 2D Gaussian , with center at , and an orientation selective filter , with optimal angle : . This model for a receptive field is equivalent to a directional derivative of a 2D Gaussian (cf. ).
According to the theory of shiftable/steerable filters [14,40], an RF located at and tuned to the orientation can be obtained by a linear combination of basis RFs, as illustrated in Figure 1. The numbers and denote the steering and shifting coefficients, respectively; because orientation and localization are independent parameters, the 's can be calculated separately from the 's. The number of steering coefficients depends on the polar Fourier bandwidth of the basis RF, while the number of steering filters is inversely proportional to the basis RF size. Details regarding this scheme and its performance may be found in .
Figure 2: Lateral connections in V1. The construction of the observed wide variety of orientation tuning curves from steerable/shiftable basis RFs may be implemented by the lateral connections in the primary visual area.
The mathematical properties of shiftable/steerable filters outlined above suggest that the columnar architecture in V1 provides a basis for creating a continuum of RF properties. Computationally, this requires that the input to a V1 neuron be a linear combination of outputs of several RFs, as in Figure 1. Is this assumption warranted by anatomical and physiological data regarding cortical interconnection patterns, and, in particular, patterns of lateral connections? Horseradish peroxidase (HRP) labeling studies  have shown that lateral connections of orientation columns extend to a range of . In other studies that used 2DG autoradiography and retrograde labeling, connectivity patterns were superimposed on functional maps . The results showed that cells tended to connect to cells of like orientation preference. The relationship between functionally defined columns and patchy connections was studied by . They used optical imaging techniques to construct functional maps of orientation columns, then targeted injections of biocytin tracer to selected functional domains. Their results show that long-range connections, extending or more, tend to link cells with like orientation preference. In the short range, up to from the injection site, connections were made to cells of diverse orientation preferences. The selectivity of the short-range connections is markedly disrupted, probably because dendritic arbors and axonal connections freely cross orientation column borders .
These findings suggest that the long-range connections, which link cells of like orientation preference, may provide the inputs necessary to shift the position of the desired RF, while the short-range connections, linking cells of diverse orientation preference, may provide the substrate for steering the RF to an arbitrary angle (Figure 2). Note that the model requires both excitatory and inhibitory connections; biological data suggest that this requirement is not unreasonable. According to , the majority of long range horizontal connections are excitatory and link pyramidal cells. Cross-correlation studies  support this observation. Inhibitory connections come from two sources: first, a small proportion of postsynaptic cells (as high as ) may be inhibitory interneurons; second, it is possible that orientation-biased cells within cytochrome oxidase-rich blobs in primates (where high GABA-decarboxylase activity indicates probable presence of inhibitory synapses) provide inhibitory inputs to the sharply tuned orientation selective cells .