# Eulerian Numbers Weigths in Distributed Computing Nets

## Abstract

We explore the possibilities of Eulerian numbers to define weights in layered networks and model distributed computation at the level of neurons receptive fields. These networks are then compared to those defined by binomial coefficients (Newton filters). Their potential as structures for signals convergence, divergence and overlapping is also established.

## 1 Introduction

### 1.1 Convergent-Divergent Layered Nets

The existence of a convergent-divergent path in the structure of the neurons system of vertebrates is a fact that has been long demonstrated in both peripheral (sensory) and central parts [1, 2].

Also, the retina as well as cortex, show a conspicuous layered anatomy, with interconnections among the units (neurons) of each layer and the following layers, always exist overlapping of input (or sensory) “fields” (zones), and thus, a double process of convergence and divergence of the input information and the outputs from the layers also exists.

In our representations, it is a type of distributed computation by a layered network of almost equally functional computing agents. This is illustrated in Fig. 1 for one dimensional \((x=x_1,x_2,\ldots ,x_{20})\) input field. The value of the discrete inputs are \(x_i\). Note that this representation is highly paradigmatic and that there is not a one-to-one correspondence between each of them and a real nervous structure.

The outputs of the nets are \(\varOmega _1,\ldots ,\varOmega _j\). Overlapping, convergence and divergence of signals are evident.

*m*layers, in which each computing agent have just two input lines. It is no difficult to show that the equivalent nets always exist for linear networks, that is for nets in which:

### 1.2 Newton-Hermite Filters

*m*are given by the row of the

*m*binomial (or Newton) coefficients, that is the coefficients of the polynomials:

*m*and order

*k*, have weights which are the coefficients of the polynomial:

*k*corresponds then to the number of layers in which the weights (+1,+1) are changed to (+1,-1).

*m*inputs, that is:

*m*layers, each layer having the weights \((+1,e_i)\), where \(e_i\) are the roots of the polynomial:

*m*and order

*k*can be arranged in a triangular array of numbers, starting \(m=1\) in the vertex. For \(N_m(0)\) it corresponds to the Pascal Triangle. This is illustrated in Fig. 2(a) for \(N_7(2)\), which is the so called “inverted Mexican hat” filter. Figure 2(b) shows the layered net for the filter having two “inhibitory layers”. Notice that it is irrelevant where the “inhibitory layers” are, but their number.

*k*, tend to Hermite functions of order

*k*, \(H(x,k)=\displaystyle \frac{d^k\;(e^{-x^2})}{dx^k}\). The resulting kernels in the continuous formulation will be:

Notice that the basis for representing a Newton array into an equivalent layered computing structure is its triangular nature, when they are generated. Also, any arbitrary discrete linear filter of real numbers have an equivalent layered network, so it finally admit a triangular array equivalent representation. Notice that also this applies to Eulerian Networks (next section).

## 2 Eulerian Numbers, Eulerian Networks and Eulerian Filters

*x*, taking the coefficients of the polynomials of the resulting numerators and putting them into a triangular array, it results the Euler triangle.

The array of the first 6 Eulerian numbers is shown in Fig. 3(a), with a convenient superimposed coordinate system. Notice that both *x* and *y* are integers starting in 1 (not in 0).

*E*(

*x*,

*y*) is the Eulerian number in position (

*x*,

*y*).

*x*,

*y*), the weight of the input coming from the left \((\lambda )\) to each computing unit is the coordinate

*y*, and that of the inputs coming from the right \((\rho )\) is the coordinate

*x*. This is illustrated in Fig. 3(b).

*l*is then:

By appropriate re-scaling the (integer) indexes of the input lines [4], normalized Eulerian weights (numbers), approaches a gaussian (just as binomial coefficients). We shall consider re-scaling later, for the continuous filters. Figure 4 shows the plot of Newton and Euler weights for zero order filters of length 51, where the approaching to gaussian is already apparent.

As it is also apparent, Euler filters of order 0 have a narrower receptive field extension, which correspond to a higher spatial frequency averaging effect. Also, higher order filters will have a more sharp contrast detection effect and provide for narrower harmonic representation on input signals.

Higher order Euler filters \(\varOmega _l(k)\) should be obtained by analogy to the Newton filters, that is, changing the sign of the inputs left to right to each computing unit in *k* layers. However in this case, the change does not correspond to a *k*th discrete derivatives, because the weights are no longer all +1. What is more, the resulting filter weights depend on what layers we select to change signs, that is, in Newton Filter’s terminology, what layers are inhibitory. This, obviously, means a much lower reliability to changes of the place of the inhibitions for Eulerian filters.

## 3 Higher Order Euler Filters and Hermitian Euler Formulation

*l*, one place to the right (or left), change the sign of the output and add a final new computing unit. This means that by every discrete derivative, we increase the supposed resolution by one, ending in effect in a filter of order

*k*, but length \(k+l\). Or viceversa, to obtain a filter of order

*k*, and length

*l*, we shall start with

*k*Eulerian filters of order \(l-k\) and add

*k*inhibitory layers. For

*l*large and

*k*not very large (first derivatives), the effects are not different from the ones for Newton filters.

Figure 5(a) shows the filter profiles for \(N_{51}(1)\) and \(\varOmega _{52}(1)\) (first derivatives) and Fig. 5(b) the corresponding Mexican hats. All weights have been normalized.

Again, the parallelism between Newton and Eulerian filters is very apparent. The receptive field for Eulerian filters is “narrower” than for Newton’s filters, which again means more sensitivity to higher spatial frequencies.

*k*can be approximated by \(k=2\sqrt{2}=\sqrt{8}\). That is, an homothecy in the continuous input field of coordinate

*x*that transforms Newton filters into Euler filters. The corresponding Hermitian filters (Newton’s and Euler) of order 2 are shown in Fig. 6.

## 4 Conclusion

Newton and Euler filters have qualitatively the same “shape”, for any order of the filter. However, Eulerian discrete filter distribution of different weights per layer permits to experiment their behavior when introducing inhibitory layers. This has to be explored.

It is expected that, from the signal processing point of view, Newton and Euler filters will have practically the same effect as changing the resolution when going from one representation to the other.

In any case, potential possible uses of these neuron-like convergent-divergent networks is still under study by computer simulation of larger networks.

## Notes

### Acknowledgments

This work has been supported, in part, by Spanish Ministry of Science projects MTM2011-28983-CO3-03 and MTM2014-56949-C3-2-R.

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