**1. Introduction**

180 Fourier Transform Applications

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*Medicine*, Vol. 82, No,1B, pp. 34-40, ISSN 1555-7162 (Electronic).

*Applied Physiology*, Vol. 101, pp. 590 –597, ISSN 0021-8987 (Print).

Currently, the full extent of the role Fourier analysis plays in biological vision is unclear. Although we have examples of sensory organs that perform Fourier transforms, e.g. the lens of the eye and the cochlear, to date there is no direct empirical evidence for its implementation in cortical architecture. However, there does exist intriguing theoretical evidence that suggests a role for the Fourier transform in a primate's primary visual cortex (area V1) which emerges from recent developments in our knowledge of contextual modulation. This paper proposes a new Fourier transform and a specification of how this transform has a natural implementation in cortical architecture. The significance of this new Fourier transform and its specification in neural circuitry is that it provides a plausible explanation for previously unexplained observable properties of the primate vision system.

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The spatial response properties, such as orientation tuning and spatial frequency tuning, of neurons in area V1 have been known for some time (Schiller et al., 1976). For a while, it was generally accepted that these tuning functions of receptive fields are largely context-independent (De Valois et al., 1979). However, later research has demonstrated contextual influences from the region close to the receptive field (Sceniak et al., 2001); (Cavanaugh et al., 2002); (Bair & Movshen, 2004). Moreover, it has been found that this near surround region of a receptive field can modify receptive field responses through suppression (Blakemore & Tobin, 1972) and by cross-orientation facilitation effects (Sillito & Jones, 1996); (Cavanaugh et al., 2002); (Kimura & Ohzawa, 2009). It has also been demonstrated that long-range contextual modulation is as robust a feature of neural function in area V1 as the extensively studied receptive field properties of this area (Lamme, 1995). Since that time, the evidence for long-range contextual modulation continues to grow, e.g. (Zipser et al., 1996); (Lamme et al., 1998); (Lee et al., 1998).

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Concurrent with this research establishing the empirical evidence for contextual modulation has been research aimed at developing functional models of V1 that are consistent with the empirical evidence. In the early 1980's the concept of convolution was employed by David Marr (Marr & Hildreth, 1980) as a model that accounted for considerable observable

1974). The spatial and temporal frequency tuning preferences of neurons in V1 can also be measured. The neuron's response properties measured via the receptive fields resemble spatially localized filters with a preferred orientation and spatial frequency (Schiller et al., 1976); (Foster et al., 1985); (Mikami et al., 1986); (Edwards et al., 1995) or spatio-temporal

<sup>183</sup> Cortical Specification of a Fast Fourier

The orientation preference of neurons can be mapped using optical imaging techniques and neurological studies, which show good agreement with single cell measurements (Blasdel, 1992); and groups of neurons which act as a single unit. It has been experimentally shown that this single unit activity of large groups of single cells are composed of 104 (first order approximation) interconnected cells even in one local V1 column (Siegel, 1990). The advantages of modeling large scale neuron activity which exhibit cohort macroscopic organisation was shown by (Sirovitch et al., 1996). No model was presented but organising principles for analyzing and viewing data were presented. These techniques have revealed an intricate structure to the orientation preference map in layers 2/3. A critical feature of these structures is the orientation pinwheel (local map), in which the orientation preference of the neuronal population changes through the entire range of 180 degrees of orientations over the 360 degrees of polar range of the circular pinwheel. At the centre of the pinwheel is the singularity, which is the point at which lines of iso-orientation preference meet (Obermayer &

The cortex is often called the iso-cortex because of the repeated structures of which it is comprised (Douglas & Martin, 1991). The smallest scale of structure is the minicolumn which, in the monkey, consists of 30 adjacent pyramidal cell shafts in layers 2/3 packed within a diameter of 23 *μ*m (Peters & Sethares, 1996). There are approximately 20 cell bodies within a minicolumn in layers 2/3. The next largest physical scale in V1 at which repeated structures occur is the cortical column (Lund et al., 2003). The cortical column is 200 *μ*m in diameter and is the scale at which long-range patchy connections terminate. A number of anatomical and functional markers repeat at a larger scale of 400 *μ*m. These include the distance between CO blobs, the approximate periodicity of the orientation preference map, and the spatial scale of a single ocular dominance band (Lund et al., 2003). Orientation pinwheels are also of approximately this spatial scale. Each of these functional markers has been shown to be closely related to the system of patchy connections, in which like response preference connects to like, and the inter-patch distance in V1 has this same periodicity of 400 *μ*m (Bartfeld & Grinvald, 1992); (Malach et al., 1993); (Bosking et al., 1997). The largest spatial scale is V1 itself, which is some 4 cm wide in the monkey. There are of the order of 10,000 CO blobs in layers 2/3 of V1 (Murphy et al., 1998), and ocular dominance bands of 120 in number (Horton & Hocking, 1998), suggesting that the multiple response property maps with periodicity of 400 *μ*m repeat around 10,000 times over layers 2/3 of V1. The input connections from the LGN arborize at a range of scales within layer 4C of V1. These inputs are arranged in block-like structures at the approximate scale of an ocular dominance band in layer 4C, but at a finer scale of approximately one column in layer 4C (Fitzpatrick et al., 1985). Further fine scale arborizations occur at approximately the scale of one minicolumn in layer 4A. At the global scale of the cortex, inputs from the LGN are organized into a retinotopic mapping of the visual field (Rolls and Cowey 1970; (Tootell et al., 1988). Connectivity into the layers 2/3

energy (Basole et al., 2003); (Basole et al., 2006).

Transform Supports a Convolution Model of Visual Perception

2.1.0.4

Blasdel, 1993).

2.1.0.5

properties of the human vision system. Since that time further theoretical and empirical evidence has been mounting that supports such a model. In particular, it has been shown that response properties of neurons in area V1 are modeled by convolution of the input image with a family of Gabor functions (Sanger, 1988). Further research has demonstrated that the upper layers of area V1 are modeled well by a bank of Gabor filters (Grigorescu et al., 2003); (Huang et al., 2008); (Lee & Choe, 2003); (Ursine et al., 2004); (Tang et al., 2007). A related, but alternative, approach to the Gabor response functions to model simple and complex cells of V1 is the use of Gaussian derivatives (Huang et al., 2009). The common denominator of these contextual modulation models is long-range convolution. However, the issue of accepting these state of the art computational models of contextual modulation as plausible functional models of Layer 2/3 of V1 thus becomes one of addressing the *cortical convolution conundrum*, more specifically: how are the large scale convolutions required by such models accounted for in cortical architecture?

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This paper's goal is to address the cortical convolution conundrum. In the process, we will propose a new fast Fourier transform, named Generalised Overarching SHIA Fast Fourier Transform (GOSH-FFT) and argue:


The rest of this paper is organised as follows: Section 2 provides a description of key neurophysiological and mathematical concepts underpinning the main thrust of this paper. Section 3 describes the Generalised Overarching SHIA Fast Fourier Transform (GOSH-FFT). Section 4 proposes a new interpretation of the physiology of long-range intrinsic connections and reinterprets previously introduced physiological concepts to propose a plausible cortical implementation of GOSH-FFT. Section 5 discusses various implications of the novel material of this paper. Section 6 summarises and concludes the paper. Section 7 is an appendix that contains a MatLab-like pseudo-code description of GOSH-FFT and a mathematical proof of GOSH-FFT.
