2.1.0.6

The finest scale of axonal projections within V1 are the short-range intrinsic connections that provide connectivity between neurons up to the range approximated by an ocular dominance column width, or 400 *μ*m. Within V1, long-range patchy connections extend for 3 mm within the supra-granular layers (Stettler et al., 2002) and long-range connections within the infra-granular layers extend for up to 6 mm (Rockland & Knutson, 2001). V1 also receives feedback from at least nine extra-striate areas (Rockland & Vanhoesen, 1994). Extra-striate feedback is considered by most researchers to be the primary source of long-range horizontal interactions measured in V1 (Alexander & Wright, 2006). These feedback connections are fast conducting myelinated cortico-cortical fibres, and while they traverse distances of up to 10 cm in the monkey, the transmission delays are of the same order as intrinsic short and long-range axons within V1 (Bringuier et al., 1999); (Girard et al., 2001). These feedback connections are often in register with the intrinsic patchy system within V1, depending on the area of origin (Angelucci et al., 2002); (Lund et al., 2003). The middle temporal (MT) visual area will serve here as brief illustration of the role of extra-striate feedback in V1. Receptive field sizes in MT are about 10 times larger than in V1 at all eccentricities (Albright & Desimone, 1987). Small focal injections of tracer into V1 indicate that the sizes of the feedback fields from MT to V1 are 21-fold larger than the aggregate receptive size of the V1 injection sites (Angelucci et al., 2002). These feedback connections are an obvious substrate for the integration of global signals into V1 (Bullier, 2001). The local-global map hypothesis (Alexander et al., 2004) of V1 posits a non-local influence on the structure of local maps in V1. This hypothesis states that the global visual map in V1 is remapped to the local map scale in V1 in the form of a map of response properties, e.g. orientation, and in the case of the monkey, spatial frequency preference and colour selectivity. These local maps tile the surface of V1 and each receive inputs from a large extent of the visual field. *So rather than the local map being simply a map of primitive visual features that apply to a point in visual space, the local map is a map of primitive visual features as they arise in the organisation of the visual field and become relevant to a location in visual space.* As the maximum range of contextual modulation in V1 approaches the size of the visual field (Alexander & Wright, 2006), the local organisation of response properties can be influenced by the functional properties of the global visual field.

#### **2.2 Mathematical background**

Fundamental to the Fourier transform proposed in this paper is the Spiral Honeycomb Image Algebra (SHIA). This is a data structure that embodies important properties of the natural visual constraints imposed by the primate eye (Sheridan et al., 2000). In particular, SHIA has a discrete, finite and bounded domain which mimics the distribution of photo receptors on the retinal field. The underlying geometry of the SHIA is a hexagonal or rectangular lattice. In the former case, each hexagon has a designated positive integer address expressed in base seven. The numbered hexagons form clusters of super-hexagons of size 7*n*. These 4 Will-be-set-by-IN-TECH

of V1 occurs via a number of anatomical routes, apart from the well described feedforward connections from layer 4 (Fitzpatrick et al., 1985). Other routes of information transfer include extra-striate feedback (Rockland et al., 1994); (Rockland & Vanhoesen, 1994); (Angelucci et al., 2002), long-range intrinsic fibres within V1 (Blasdel et al., 1985), as well as feedback from V1 to the lateral geniculate nucleus (Marrocco et al., 1982); (Briggs & Usrey, 2007), and diffusion

The finest scale of axonal projections within V1 are the short-range intrinsic connections that provide connectivity between neurons up to the range approximated by an ocular dominance column width, or 400 *μ*m. Within V1, long-range patchy connections extend for 3 mm within the supra-granular layers (Stettler et al., 2002) and long-range connections within the infra-granular layers extend for up to 6 mm (Rockland & Knutson, 2001). V1 also receives feedback from at least nine extra-striate areas (Rockland & Vanhoesen, 1994). Extra-striate feedback is considered by most researchers to be the primary source of long-range horizontal interactions measured in V1 (Alexander & Wright, 2006). These feedback connections are fast conducting myelinated cortico-cortical fibres, and while they traverse distances of up to 10 cm in the monkey, the transmission delays are of the same order as intrinsic short and long-range axons within V1 (Bringuier et al., 1999); (Girard et al., 2001). These feedback connections are often in register with the intrinsic patchy system within V1, depending on the area of origin (Angelucci et al., 2002); (Lund et al., 2003). The middle temporal (MT) visual area will serve here as brief illustration of the role of extra-striate feedback in V1. Receptive field sizes in MT are about 10 times larger than in V1 at all eccentricities (Albright & Desimone, 1987). Small focal injections of tracer into V1 indicate that the sizes of the feedback fields from MT to V1 are 21-fold larger than the aggregate receptive size of the V1 injection sites (Angelucci et al., 2002). These feedback connections are an obvious substrate for the integration of global signals into V1 (Bullier, 2001). The local-global map hypothesis (Alexander et al., 2004) of V1 posits a non-local influence on the structure of local maps in V1. This hypothesis states that the global visual map in V1 is remapped to the local map scale in V1 in the form of a map of response properties, e.g. orientation, and in the case of the monkey, spatial frequency preference and colour selectivity. These local maps tile the surface of V1 and each receive inputs from a large extent of the visual field. *So rather than the local map being simply a map of primitive visual features that apply to a point in visual space, the local map is a map of primitive visual features as they arise in the organisation of the visual field and become relevant to a location in visual space.* As the maximum range of contextual modulation in V1 approaches the size of the visual field (Alexander & Wright, 2006), the local organisation of response properties can be

of visual signal in the retina (Kruger et al., 1975); (Berry et al., 1999).

influenced by the functional properties of the global visual field.

Fundamental to the Fourier transform proposed in this paper is the Spiral Honeycomb Image Algebra (SHIA). This is a data structure that embodies important properties of the natural visual constraints imposed by the primate eye (Sheridan et al., 2000). In particular, SHIA has a discrete, finite and bounded domain which mimics the distribution of photo receptors on the retinal field. The underlying geometry of the SHIA is a hexagonal or rectangular lattice. In the former case, each hexagon has a designated positive integer address expressed in base seven. The numbered hexagons form clusters of super-hexagons of size 7*n*. These

**2.2 Mathematical background**

2.1.0.6

self-similar super-hexagons tile the plane in a recursively modular manner. As an example, a super-hexagon of size 7<sup>2</sup> = 49 and its concomitant addressing scheme is displayed in Fig. 1 (a).

#### (a) Hexagonal SHIA


(b) Rectangular SHIA

Fig. 1. Displays the two-level addressing scheme of SHIA: (a) Hexagonal and (b) Rectangular.

In the latter case, each rectangle has a designated positive integer address expressed in base five. An example of this addressing scheme is displayed in Fig. 1 (b).

(a) Four Level SHIA

<sup>187</sup> Cortical Specification of a Fast Fourier

Transform Supports a Convolution Model of Visual Perception

(b) M10

*f*(*x*)*g*(*x* − *a*) (3)

Fig. 2. Displays (a) an image of a duck represented on a four-level SHIA; (b) the result of applying SHIA transform M10 twice to the image displayed in (a). There are four observable effects: 1) multiple near copies of the input image (a), 2) each copy is rotated by the same angle, 3) each copy is scaled by the same amount, 4) applying *M*10 twice to the image

*<sup>f</sup>*(*x*) <sup>∗</sup> *<sup>g</sup>*(*x*) = <sup>∑</sup>*<sup>a</sup>*∈*<sup>g</sup>*

displayed in (b) results in the image displayed in (a).

its discrete definition is
