4.2.0.18

At the next higher scale of connectivity each local map is assumed to have access to ongoing activity of every other local map via long-range patchy connections and striate-extrastriate interactions. These provide the means by which the results of local computation can be transported to another local map as input for a further local computation. We denote such a projection as *P* to represent the class of transformations (as described in Section 4.1).

## 4.2.0.19

All of the components of PaSH-FFT can be considered as being built of sequences of *P* projections followed by a local computation. Likewise, when we employ the term local computation in conjunction with the symbols *F*, *I* or *C*, it is implied that the signal within a local map has been sent to every other neuron in that local map through a synapse whose strength is associated with the appropriate weight of *F*, *I* or *C* respectively.

## 4.2.0.20

12 Will-be-set-by-IN-TECH

m

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

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.)

> *F* Fourier transform spatial domain primitive roots of unity *I* Inverse Fourier frequency domain primitive roots of unity *C* Convolution frequency domain Fourier components

At the next higher scale of connectivity each local map is assumed to have access to ongoing activity of every other local map via long-range patchy connections and striate-extrastriate interactions. These provide the means by which the results of local computation can be transported to another local map as input for a further local computation. We denote such a projection as *P* to represent the class of transformations (as described in Section 4.1).

All of the components of PaSH-FFT can be considered as being built of sequences of *P* projections followed by a local computation. Likewise, when we employ the term local computation in conjunction with the symbols *F*, *I* or *C*, it is implied that the signal within

**Symbol Local Computation Input Signal Weights**

Table 1. Summary of notation and definitions of local computations

 wixi i=1

w3

wm

 N1 N2 N3 Nm x1 x2 x3 xm

w2

w1

figure inputs the sum of each *xi* multiplied by weight *wi*.

Table 1 provides a summary of this notation.

4.2.0.18

4.2.0.19

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 (*s*)*A* � *B* = ((*s*)*A*)*B*.

Note that the operator is to the right of the input signal it operates on, which is enclosed in left and right parenthesis ().

## 4.2.0.21

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 of PaSH-FFT would be expressed as follows:

$$(\text{s})PaSH-FFT=\text{(s)}P\odot F\odot P\odot F\odot P\tag{5}$$

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

$$\mathbf{r}(\mathbf{s}) \\ \text{inverse} \\ \text{PaSH} - \text{FFT} = \mathbf{(s} ) \\ \mathbf{P} \odot \mathbf{I} \odot \mathbf{P} \odot \mathbf{I} \odot \mathbf{P} \tag{6}$$
