4.2.0.15

10 Will-be-set-by-IN-TECH

between columns of Layer 2/3 and similarly patchy extra-striate feedback connections to area

It has been argued that the orientation pinwheel comprises a unitary organisational structure or local map in layer 2/3 of area V1 (Hubel & Wiesel, 1974); (Bartfeld & Grinvald, 1992); (Blasdel, 1992). When four pinwheels are reflected about their common borders, a saddle

Fig. 4. Displays a schematic diagram of the pinwheel like structures of visual area V1,

In the macaque, the preferred response properties of V1 neurons can be influenced by activity from a wide extent of the visual field. A review of contextual modulation in the monkey demonstrated contextual modulation in V1 from long-ranges in the visual field (Alexander & Wright, 2006). The review was compiled from a number of experimental paradigms, including visual stimulation with long lines while the neuron's receptive field is occluded (Fiorani et al., 1992), surround only textures (Rossi et al., 2001) and colour patches placed distally to the neuron's receptive field (Wachtler et al., 2003). It was shown that the maximum range of contextual modulation measurable in V1 approaches a large extent of the visual field relative to a neuron's receptive field size or the local cortical magnification factor. Some experimental paradigms, such as the curve tracing effect (Roelfsema & Lamme, 1998); (Khayat et al., 2004), relative luminance (Kinoshita & Komatsu, 2001), and texture defined boundaries (Lee et al., 1998) show excitatory contextual modulation with 'tuning curves' that are flat out to the maximum distance tested. The functional connectivity that underlies this long-range contextual modulation in the monkey is likely to involve cortico-cortical feedback from higher visual areas working in concert with long-range intrinsic patchy connectivity. In the monkey, the feedback connections to V1 from higher visual areas incorporate inputs from a very large

In the analysis that follows, the combination of patchy intrinsic connections and patchy feedback connections are therefore assumed to enable transfer of visual information at ranges approaching the global scale of the visual field. Moreover, we assume that the quantity and distribution of these connections are adequate to deliver the effects of transform *M*10 at the

point arises at the centre of the four pinwheels. See Fig. 4.

extracted from Figure 10 page 43 of Bruce et al. (2003).

extent of the visual field (Angelucci et al., 2002); (Lund et al., 2003).

V1.

4.1.0.12

4.1.0.13

4.1.0.14

scale of the visual field.

In area V1, each neuron makes use of information available to it in real time. There is evidence that contextual information is projected to widespread regions in V1 in an anticipatory manner. Since the spatial changes in the visual field tend to be predictable from previous visual inputs, anticipatory contextual inputs can arrive in time to be integrated in an adaptive manner with ongoing feedforward input. In order to express the properties of widespread contextual integration in a more formal manner, however, we will use the mathematical convenience of assuming that each of the distinct mathematical processes to be described occurs in a step-wise fashion. This more constrained approach allows not only each distinct part of the process to be formulated, but also formulates the inter-relationships between the various sub-processes. Although it is claimed that this approach is appropriate for the purposes of this paper, it must be acknowledged that the question of how such "contextual integration" actually occurs in the neuronal system remains open.

## 4.2.0.16

At the finest scale of connectivity via short-range intrinsic connections, each neuron of a local map is treated as if it were connected to every other neuron minicolumn of that local map. While this is not literally true, considerations of poly-synaptic interactions at this local scale, and the real-time, anticipatory nature of visual processing means that it is a reasonable approximation of the functional connectivity. Consequently, we can assume that each neuron in a local map can sum the outputs of all other neurons in that local map which have been multiplied by unique complex numbers. We call such a collection of parallel computations a local computation. See Fig. 5, which is a schematic diagram of a local computation.

## 4.2.0.17

This paper now needs to discuss three types of local computations, each of which is determined by the interpretation of the input signal and the collection of synaptic strengths which weight the input signal. If the input signal is the spatial domain and the weights are associated with a set of primitive roots of unity, then the resulting local computation is a Fourier transform, denoted *F*. (See Equation 1 for a definition of a Fourier transform.) If the

a local map has been sent to every other neuron in that local map through a synapse whose

<sup>193</sup> Cortical Specification of a Fast Fourier

Although it is commonly accepted that the cortex has a massively parallel architecture, currently there exists no comprehensive model to describe these dynamics. The absence of such a model means that in any particular cortical process, we cannot be sure which aspects of the process are parallel and which are intrinsically sequential. We will employ the following notation to show how the inherently sequential steps of PaSH-FFT can be mapped into neural circuitry. Let the symbol � denote the composition of two local computations as follows: given arbitrary local computations *A* and *B* to operate on a signal in sequence let

Note that the operator is to the right of the input signal it operates on, which is enclosed in

With these concepts in hand, we can now identify the sequential steps of PaSH-FFT. In this special case, the size of the input signal is the square of the size of the local computation and represents two iterations of GOSH-FFT, as described in Section 3. The identification of the sequential steps also suggests the sequence of connections that the input signal must traverse. We now illustrate this with PaSH-FFT, given an input signal *s*, then the application

Given the assumed neural parallelism, a count of the number of components on the right hand side of the equals sign in Equation 5, reveals that a Fourier transform of the entire visual field can be completed by the signal traversing a sequential path connecting five neurons. Likewise,

We now progress to the issue of how convolution could be implemented in cortical architecture. To this end, we describe the various computational constraints imposed by the computational requirements of convolution and argue that the known cortical architecture

The key to the solution of the convolution problem in the neurological domain is provided by the Convolution Theorem, the same one employed by numerous digital signal processing applications. This theorem was discussed in Section 2.2. The importance of the theorem is that the convolution of two functions in the spatial domain can be achieved by the multiplication of the functions in the frequency domain. The implications of this theorem to the cortical convolution conundrum are significant. In our model, the components of the Fourier transform of the function the input signal is to be convolved with are represented by connection weights. Then, once the input signal has been transformed to the frequency

an inverse Fourier transform can be delivered in cortical circuitry as follows:

(*s*)*PaSH* − *FFT* = (*s*)*P* � *F* � *P* � *F* � *P* (5)

(*s*)*inversePaSH* − *FFT* = (*s*)*P* � *I* � *P* � *I* � *P* (6)

strength is associated with the appropriate weight of *F*, *I* or *C* respectively.

Transform Supports a Convolution Model of Visual Perception

4.2.0.20

4.2.0.21

(*s*)*A* � *B* = ((*s*)*A*)*B*.

left and right parenthesis ().

of PaSH-FFT would be expressed as follows:

**4.3 Cortical manifestation of convolution**

satisfies these constraints.

4.3.0.22

Fig. 5. Displays a schematic diagram of a computational unit. The circles represent neurons and the straight lines connecting the circles represent cortical connections. Each neuron depicted at the top of the figure outputs a value *xi*. The neuron depicted at the bottom of the figure inputs the sum of each *xi* multiplied by weight *wi*.

input signal is the frequency domain and the weights are associated with a set of inverse primitive roots of unity, then the resulting local computation is an inverse Fourier transform, denoted *I*. (See Equation 2 for a definition of an inverse Fourier transform.) If the input signal is a frequency domain and the weights represent Fourier components, then the resulting local computation is a convolution in the frequency domain, denoted *C*. (See Equations 3 and 4.) Table 1 provides a summary of this notation.


