# Flexibility of Boolean Network Reservoir Computers in Approximating Arbitrary Recursive and Non-Recursive Binary Filters

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Reservoir Computer

#### 2.2. Reservoir

#### 2.3. Input

#### 2.4. Output

#### 2.5. Objective Functions

- Non-recursive functions, defined as ${y}^{t}=g\left({u}^{t-\tau},{u}^{t-\tau -1},\dots ,{u}^{t-\tau -\left(M-1\right)}\right)$
- Recursive functions, defined as ${y}^{t}=g\left({u}^{t-\tau},{u}^{t-\tau -1},\dots ,{u}^{t-\tau -\left(M-2\right)},{y}^{t-1}\right)$

#### 2.6. Training and Testing Algorithm

#### 2.7. Overall Strategy

## 3. Results

#### 3.1. Benchmark Functions: Median and Parity

#### 3.2. Median

#### 3.3. Parity

#### 3.4. Estimating a Range of Functions

#### 3.5. Reservoir Flexibility

#### 3.6. Determinants of Difficulty

## 4. Discussion

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Reservoir computer layout. The reservoir computer (RC) is composed of a binary input node, a Boolean network reservoir, and a binary output node.

**Figure 2.**Mean accuracy, ${\overline{a}}_{j}$, vs. $L$ for the three-bit median (

**A**) and parity (

**B**) functions for different $\overline{K}$-valued reservoirs with $N$ = 500.

**Figure 3.**Mean accuracy, ${\overline{a}}_{j},$ vs. $L$ for the five-bit median (

**A**) and parity (

**B**) functions for different $\overline{K}$-valued reservoirs with $N$ = 500.

**Figure 4.**Mean accuracy, ${\overline{a}}_{j}$, vs. $L$ for the recursive three-bit median (

**A**) and parity (

**B**) functions for different $\overline{K}$-valued reservoirs with $N$ = 500.

**Figure 5.**Mean accuracy, $\overline{a}$, vs. $L$ for all of the three-bit functions for different $\overline{K}$-valued reservoirs. Different sizes of reservoirs are shown: $N$ = 10 (

**A**); $N$ = 50 (

**B**); $N$ = 100 (

**C**); and $N$ = 500 (

**D**).

**Figure 6.**Mean accuracy, $\overline{a}$, vs. $L$ for five-bit functions for different $\overline{K}$-valued reservoirs. Different sizes of reservoirs are shown: $N$ = 10 (

**A**); $N$ = 50 (

**B**); $N$ = 100 (

**C**); and $N$ = 500 (

**D**).

**Figure 7.**Mean accuracy, $\overline{a}$, vs. $L$ for all of the recursive three-bit functions for different $\overline{K}$-valued reservoirs. Different sizes of reservoirs are shown: $N$ = 10 (

**A**); $N$ = 50 (

**B**); $N$ = 100 (

**C**) and $N$ = 500 (

**D**).

**Figure 8.**Histogram of ${\Phi}_{i}$ across all 100 reservoirs for each $N,L$ with $\overline{K}$ = 2 for three-bit functions. Each subplot represents the density for all of the reservoirs with one $N$ and $L$, with the x-axis being $\Phi $, and the y-axis being the number of reservoirs [0, 256].

**Figure 9.**Mean accuracy, ${\overline{a}}_{s}$, vs. function average sensitivity, ${\overline{s}}_{g}$. Three-bit functions are shown in blue, and recursive three-bit functions shown in red with $L$ = 10 (stars), 50 (circles), and 100 (diamonds). Four different values for $N$ are shown: (

**A**) $N$ = 10; (

**B**) $N$ = 50; (

**C**) $N$ = 100; and (

**D**) $N$ = 500. Only reservoirs with $\overline{K}$ = 2 are shown. See Supplementary Materials for $\overline{K}$ = 1 and $\overline{K}$ = 3 (Figure S6).

**Figure 10.**Mean accuracy, ${\overline{a}}_{s}$, vs. function average sensitivity, ${\overline{s}}_{g}$ for five-bit functions. $L$ = 0.1 (stars), 0.5 (circles), and 1 (diamonds) are given in each plot. Four different values for $N$ are shown: (

**A**) $N$ = 10; (

**B**) $N$ = 50; (

**C**) $N$ = 100; and (

**D**) $N$ = 500. Only reservoirs with $\overline{K}$ = 2 are shown.

**Figure 11.**Example of mean function accuracy vs. average sensitivity with activities of each variable displayed. Data shown for three-bit functions: $N$ = 10, $L$ = 0.1, and $\overline{K}$ = 2. (

**A**) Each of 256 functions is visualized as a horizontal triplet of circles, where each circle corresponds to a variable (left to right, ${u}^{t-\tau}$, ${u}^{t-\tau -1}$, ${u}^{t-\tau -2}$), as colored by its activity (inset). For example, the parity function can be seen at ${\overline{s}}_{g}$ = 3, ${\overline{a}}_{j}\approx $ 0.5, and ${A}_{g}$ = [1, 1, 1]. (

**B**) In order to more clearly see the relationship between the distribution of activity and accuracy, functions are plotted by ranked accuracy rather than absolute accuracy. Here, the height of the columns is a result of the number of functions with a given ${\overline{s}}_{g}$, and does not reflect absolute accuracy.

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**MDPI and ACS Style**

Echlin, M.; Aguilar, B.; Notarangelo, M.; Gibbs, D.L.; Shmulevich, I. Flexibility of Boolean Network Reservoir Computers in Approximating Arbitrary Recursive and Non-Recursive Binary Filters. *Entropy* **2018**, *20*, 954.
https://doi.org/10.3390/e20120954

**AMA Style**

Echlin M, Aguilar B, Notarangelo M, Gibbs DL, Shmulevich I. Flexibility of Boolean Network Reservoir Computers in Approximating Arbitrary Recursive and Non-Recursive Binary Filters. *Entropy*. 2018; 20(12):954.
https://doi.org/10.3390/e20120954

**Chicago/Turabian Style**

Echlin, Moriah, Boris Aguilar, Max Notarangelo, David L. Gibbs, and Ilya Shmulevich. 2018. "Flexibility of Boolean Network Reservoir Computers in Approximating Arbitrary Recursive and Non-Recursive Binary Filters" *Entropy* 20, no. 12: 954.
https://doi.org/10.3390/e20120954