**5. Discussion**

14 Will-be-set-by-IN-TECH

domain the required convolutions can be performed by mere multiplications. In cortical terms, each component of the signal, in the frequency domain, must traverse a connection to one more neuron to achieve the desired multiplication. However, the resulting convolution, in the frequency domain, must be transformed back to the spatial domain to complete the convolution. This is achieved with an inverse Fourier transform. Accordingly, the sequence of connections along the path that terminates in the output of a convolved value in the spatial

It is assumed that each component of the input signal traverses parallel paths along the network. Thus, the net time cost to complete a convolution is equivalent to the time required for a component of the input signal to traverse a path connecting 10 neurons. This path is

The plausibility of the cortical model of convolution proposed in this paper is fundamentally predicated on the assumptions made in its formulation. Consequently, we summarise these assumptions along with the arguments offered to justify them before we provide an analysis

1. The number of long-range patchy connections is adequate to achieve a redistribution of the global signal via transform *M*10. This was argued in Section 4.1 and heavily relied on

2. A first order approximation of the number of minicolumns in a local map as 10,000. This

3. The number of short-range intrinsic connections is adequate to consider each local map as being fully connected. This was discussed in Section 4.2 and relied on the work reported

4. A first order approximation of the number of minicolumns in the global map is 10, 0002 = 100 million. This was discussed in Section 2.1 and was based on the work reported in

The first assumption is possibly the most critical as it establishes the fundamental architectural relationship between the local and global maps and is essential to PaSH-FFT. The second two assumptions implied that a Fourier transform of the portion of the signal represented in a local map would be completed by each component of the signal traversing one cortical connection and that on completion of the first iteration of PaSH-FFT, the global signal consists of 10,000 local discrete Fourier transforms each of which is at the scale of a local map. Then, on completion of the second iteration of PaSH-FFT, the 10,000 local Fourier transforms would be transformed into a global Fourier transform of size 10, 0002 = 100 million, which by the fourth assumption represents the size of the global signal. From this we are able to assert that the input spatial signal would be transformed to frequency space at a cost of the signal traversing a path connecting four neurons. With the signal in frequency space, we employed

a conclusion based on a review article reported in (Alexander & Wright, 2006).

was discussed in Section 2.1 and relied on the work reported in (Siegel, 1990).

composed of five short-range intrinsic connections and five long-range connections.

(*s*)*Convolution* = (*s*)*P F P F C P I P I P* (7)

domain is thus given by:

of the model's parameterisation:

in (Siegel, 1990).

4.4.0.24

(Murphy et al., 1998) and Assumption 2.

4.3.0.23

**4.4 Analysis**

The signal processing literature describes many different types of fast Fourier transforms (FFT). Although any one of them represents an alternative candidate to PaSH-FFT, the problem to address is accounting for how they might be implemented within the known constraints of cortical architecture. All fast Fourier transforms need to rearrange components between their intermediate steps of multiply and add. PaSH-FFT derives its rearrangements of components with the transform *M*10 that, as argued, is compatible with the distribution and quantity of long-range cortical connections. If any other FFT could be substituted for PaSH-FFT in the model, one would need to account for the rearrangement phase of that FFT within the known connectivity of area V1.

#### 5.0.0.25

Another issue worthy of some discussion pertains to the Fourier transform and the absence of empirical evidence that would irrefutably demonstrate its cortical implementation. Part of the explanation for this lack of evidence may be provided by the role the Fourier transform plays in the vision process as suggested by this paper. That is, PaSH-FFT was shown to be a means to an end (convolution), not the end itself. Consequently, the question of finding neurons through empirical experimentation that measures response properties of neurons that closely model the profile of a Fourier transform may remain unanswered for some time to come.

#### 5.0.0.26

Underpinning the proposed implementation of PaSH-FFT in cortical architecture is a highly simplistic model of the parallelism inherent in the cortex. The model employed did not take into account at least two well accepted features of this parallel architecture. First, the system itself somehow synchronises the flow of the signal. Second, the cortex does not

these long-range connections facilitated the transformation of the signal into and out of the frequency domain via a new fast Fourier transform named PaSH-FFT. A mathematical proof of the most general form of this FFT, GOSH-FFT, was provided in the appendix along with MatLab-like pseudo-code to facilitate the implementation of GOSH-FFT in computer

<sup>197</sup> Cortical Specification of a Fast Fourier

It was shown that, to a first order approximation, a cortical implementation of PaSH-FFT could account for the large scale convolution implied by known models of contextual modulation.

• represents a plausible cortical mechanism to account for long-range contextual

• suggests a theoretical explanation of how the brain might be wired to achieve large scale

• opens up the possibility of explaining other cortical processes via frequency space

It is the conclusion of this paper that the processing of the visual signal in the frequency

In this appendix, we present the pseudo-code for GOSH-FFT and a mathematical proof of

domain via a fast Fourier transform plays a fundamental role in primate vision.

This section presents a formal statement of GOSH-FFT in MatLab-like pseudo-code.

software. 6.0.0.29

modulation;

Fourier analysis;

computations.

**7. Appendix**

GOSH-FFT.

Notation:

02 *y* ← *x*

09 *r* ← *w* 10 end 11 end

**7.1 Appendix A**

The significance of PaSH-FFT is that it:

Transform Supports a Convolution Model of Visual Perception

*x* = Complex array specifying the input signal *base* = 7*α*, where *α* is an integer greater than zero *n* = 7*β*, where *β*/*α* , is an integer greater than zero.

⊗ denotes scalar multiplication in SHIA

01 function y = *GOSH* − *FFT*(*x*, *n*, *base*, *α*)

07 *F* ← *localDFFT*(*f* , *w*, *level*, *n*, *base*) 08 *y* ← *localM*10*Inverse*(*F*, *level*, *n*, *base*, *α*)

03 *f orlevel* = 1 : *logbase*(*n* 04 *f* ← *m*10(*y*, *α*)

06 *w* ← *m*10(*r*, *α*)

<sup>05</sup> *<sup>r</sup>* <sup>←</sup> *rootsO f Unity*(*baselevel*)

need computation-like synchronisation or state update. Synchronisation can be provided by considering "Small World" relationships. (Gao et al., 2001) have shown that a "Small World" network needs only a small fraction of long-range couplings to obtain a great improvement in both stochastic resonance and synchronisation in network connectivity of bistable oscillators. We suggest that the known topology of the visual cortex (Zeki, 1993) if considered as a "Small World" network can provide the foregoing benefits. They would be consistent with the long-range and short-range connectivities of V1 to retinal neurons which have the required bistable oscillator condition provided by on-centre or off-centre neurons responses to light and dark and including those with colour opponency properties. The long and short-range selectivity for connections can be dynamic based on the neuron threshold levels and spatial frequency channels (Dudkin, 1992). The system updates a neuronal state only when new information indicates a change in the input signal.

## 5.0.0.27

The cortical implementation of PaSH-FFT was discussed in Section 4.2 where it was argued that the known connectivity of area V1 was sufficient to support its cortical implementation. It was then argued that this implementation could deliver the required convolution in a 'small' number of sequential steps. However, the argument did not rule out the possibility of an alternative mechanism that would deliver the required convolution in fewer steps than PaSH-FFT. It would appear that without a sufficiently developed model of the brain's parallelism, it is unlikely that a mathematical proof of a lower bound for the minimum number of sequential steps could be produced. Currently, the only bound that we can be sure of is that the required convolution could not be completed in one step. The question of determining the minimum lower bound remains an open question.

#### 5.0.0.28

The role of the frequency domain was at the heart of the solution to the cortical convolution conundrum proposed in this paper. However, the possibility of performing the convolution in the spatial domain without resorting to the frequency domain cannot be ruled out by any argument presented in this paper. Although it is unclear how this could be accomplished without resorting to a highly asymmetric model of the distribution of the connectivity of long-range connections. In any case, the search for an explanation of how the dynamic reconfiguration implied by the analysis of this paper is actually accomplished is likely to provide many different conjectures along the way. One possible avenue in this endevour might be provided by further tracer experiments such as those reported in (Angelucci et al., 2002).
