**2. Background**

This section reviews neurophysiological and mathematical concepts pertinent to the understanding of GOSH-FFT and its cortical implementation.

#### **2.1 Physiological background**

The primary visual cortex (V1) has a distinctive layer structure. Inputs from the lateral geniculate nucleus (LGN) of the thalamus arrive chiefly to layer 4 in the monkey (Fitzpatrick et al., 1985). Layers 2/3 of monkey V1 contain the first neurons in the feedforward hierarchy that are strongly responsive to orientated stimuli (Schiller et al., 1976); (De Valois et al., 1979); (Parker & Hawken, 1988); (Leventhal et al., 1995). Central to understanding response properties of neurons in V1 is the notion of the receptive field (Hubel & Wiesel, 1962). The receptive field of a neuron is that region of its visual field to which it responds most strongly. Neurons within layers 2/3 V1 have receptive fields that are oriented. That is, they prefer stimuli that are lines of a certain orientation, or oriented texture elements (Hubel & Wiesel, 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 energy (Basole et al., 2003); (Basole et al., 2006).

## 2.1.0.4

2 Will-be-set-by-IN-TECH

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

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

• GOSH-FFT has a natural implementation in the cortical architecture of visual area V1, and • Its implementation provides a plausible cortical mechanism to account for the

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

This section reviews neurophysiological and mathematical concepts pertinent to the

The primary visual cortex (V1) has a distinctive layer structure. Inputs from the lateral geniculate nucleus (LGN) of the thalamus arrive chiefly to layer 4 in the monkey (Fitzpatrick et al., 1985). Layers 2/3 of monkey V1 contain the first neurons in the feedforward hierarchy that are strongly responsive to orientated stimuli (Schiller et al., 1976); (De Valois et al., 1979); (Parker & Hawken, 1988); (Leventhal et al., 1995). Central to understanding response properties of neurons in V1 is the notion of the receptive field (Hubel & Wiesel, 1962). The receptive field of a neuron is that region of its visual field to which it responds most strongly. Neurons within layers 2/3 V1 have receptive fields that are oriented. That is, they prefer stimuli that are lines of a certain orientation, or oriented texture elements (Hubel & Wiesel,

convolutions implied by long-range contextual modulation.

understanding of GOSH-FFT and its cortical implementation.

for in cortical architecture?

Transform (GOSH-FFT) and argue:

1.0.0.3

GOSH-FFT.

**2. Background**

**2.1 Physiological background**

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 10<sup>4</sup> (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 & Blasdel, 1993).
