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Boolean networks (BNs) are nonlinear systems and each BN has a simple structure, thus it is easy to construct large networks. The BNs are becoming increasingly important as they have been widely used in many fields like random number generation, gene regulation, and reservoir computing. In recent years, autonomous Boolean networks (ABNs) have been proposed and realized by actual digital logic circuit. The BNs each have a clock or selection device to determine the update time of each node. Unlike BNs, ABNs have no device to control the update mechanism, and the update of each node is determined by response characteristics of the logic gate that make up the node, which leads to continuous and complicated outputs. Time series with different complexities including periodic and chaotic sequences can be generated by the ABNs, which is very meaningful in different applications.Research on the regulation of ABNs’ output is of big significance. Non-ideal response characteristics of the logic gates and time delay on the link are two major factors which can regulate the output state. Many studies focus on time delay on the link and indicate that the large delay inconsistency leads to complex outputs.In this paper, in order to study the regulation of ABNs’ output, it is demonstrated that the response characteristics of the logic gate can be continuously adjusted by the parameters in the ABNs’ equations. Then the effects of logic gates’ response characteristics on ABNs’ outputs are studied by simulation. The simulation results indicate that the ABNs’ outputs can transform between periodic and chaotic state with the change of logic gates’ response characteristics. Moreover, the interrelationship between logic gates’ response characteristics and propagation delays along the links is reinvestigated. The results show that the high complexity series space is extended by the fast logic gates’ response characteristics. Also the effects of different logic gates’ response characteristics on the ABNs’ output are compared, and the results indicate that node 2 has a good performance on the regulation of ABNs’ output while node 1 and node 3 show small effect on the ABNs’ output.It is concluded that the complexity of the ABNs’ output can be regulated by the logic gates’ response characteristics, and the high complexity series’ generation can be promoted by the fast logic gates’ response characteristics. This conclusion is conducive to the logic gates’ selection in random number generation, gene regulation, reservoir computing and other applications.
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Keywords:
- autonomous Boolean network /
- chaos /
- logic gates’ response characteristics /
- permutation entropy
[1] Gaucherel C, Thero H, Puiseux A, Bonhomme V 2017 Ecol. Complex 31 104114
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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图 1 异或非逻辑门示意图
Figure 1. The schematic illustration of the XNOR logic gate.
图 2 异或非逻辑门输入输出响应波形图 (a) 完全响应波形图; (b) 不完全响应波形图; (c) 非正确响应波形图
Figure 2. I/O response waveform of XNOR logic gate: (a) Full response waveform; (b) incomplete response waveform; (c) incorrect response waveform.
图 3 异或非逻辑门输入相同, τlp = 0.03, 0.15, 0.28, 0.40时输出波形 (a)输入波形; (b)输出波形
Figure 3. Output waveforms of XNOR logic gate for τlp = 0.03, 0.15, 0.28, 0.40 when inputs are the same: (a) Input waveform; (b) output waveform.
图 4 异或非逻辑门输出脉冲幅值和宽度随τlp变化曲线 (a) 输入波形图; (b) 输出脉冲幅值ymax随τlp变化曲线; (c) 输出脉冲宽度∆tY随τlp变化曲线
Figure 4. Output pulse amplitude and width as a function of τlp : (a) Input waveform; (b) output pulse amplitude as a function of τlp; (c) output pulse width as a function of τlp.
图 5 自治布尔网络示意图
Figure 5. Schematic illustration of autonomous Boolean network.
图 6 自治布尔网络分岔图
Figure 6. Bifurcation diagram of the autonomous Boolean network.
图 7 自治布尔网络在τlp = 0.630, 0.305, 0.050 ns的模拟结果 (a1)−(a3) 时序; (b1)−(b3) 频谱; (c1)−(c3) ∆tLDP序列相图
Figure 7. Simulation results of the autonomous Boolean network for τlp = 0.630, 0.305, 0.050 ns: (a1)−(a3) Time-evolution; (b1)−(b3) power spectra; (c1)−(c3) phase diagrams of ∆tLDP series.
图 8 排序熵值在二维参数空间τlp和τij上的分布图 (a) τij = τ12; (b) τij = τ13; (c) τij = τ22; (d) τij = τ21; (e) τij = τ31; (f) τij = τ33
Figure 8. Two dimensional maps of H in the parameter space of τlp and τij: (a) τij = τ12; (b) τij = τ13; (c) τij = τ22; (d) τij = τ21; (e) τij = τ31; (f) τij = τ33.
图 9 排序熵值在二维参数空间(τlp,i, τlp,j)上的分布图 (a1)−(a3) τlp,1 = 0.1, 0.3, 0.5 ns且(τlp,i, τlp,j) = (τlp,2, τlp,3); (b1)−(b3) τlp,2 = 0.1, 0.3, 0.5 ns且(τlp,i, τlp,j) = (τlp,1, τlp,3); (c1)−(c3) τlp,3 = 0.1, 0.3, 0.5 ns且(τlp,i, τlp, j) = (τlp,1, τlp,2)
Figure 9. Two dimensional maps of H in the parameter space of (τlp,i, τlp,j): (a1)−(a3) τlp,1 = 0.1, 0.3, 0.5 ns and (τlp,i, τlp,j) = (τlp,2, τlp,3); (b1)−(b3) τlp,2 = 0.1, 0.3, 0.5 ns and (τlp,i, τlp,j) = (τlp,1, τlp,3); (c1)−(c3) τlp,3 = 0.1, 0.3, 0.5 ns and (τlp,i, τlp,j) = (τlp,1, τlp,2).
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[1] Gaucherel C, Thero H, Puiseux A, Bonhomme V 2017 Ecol. Complex 31 104114
[2] Albert R, Barabasi A L 2000 Phys. Rev. Lett. 84 5660
Google Scholar
[3] Ghil M, Zaliapin I, Coluzzi B 2008 Physica D 237 2967
Google Scholar
[4] Kauffman S A 1969 J. Theor. Biol. 22 437
Google Scholar
[5] Bornholdt S 2008 Jr. Soc. Nterface 5 S85
Google Scholar
[6] Tran V, Mccall M N, Mcmurray H R, Almudevar A 2013 Front. Genet. 4 263
[7] Chaves M, Albert R, Sontag E D 2005 J. Theor. Biol. 235 431
Google Scholar
[8] Farrow C L, Heidel J, Maloney J, Rogers J 2004 IEEE T. Neural. Networ. 15 348
Google Scholar
[9] Darby M S, Mysak L A 1993 Clim. Dyn. 8 241
Google Scholar
[10] Zhang R, Cavalcante H L D D S, Gao Z, Gauthier D J, Socolar J E S, Adams M M, Lathrop D P 2009 Phys. Rev. E 80 045202
[11] Rosin D P, Rontani D, Gauthier D J 2013 Phys. Rev. E 87 040902
Google Scholar
[12] Dong LH, Yang H, Zeng Y 2017 13th International Conference on Computational Intelligence and Security, Hong Kong, China, DEC 15
[13] Park M, Rodgers J C, Lathrop D P 2015 Microelectr. J. 46 1364
Google Scholar
[14] 马荔, 张建国, 李璞, 徐航, 王云才 2018 中南大学学报 49 888
Ma L, Zhang J G, Li P, Xu H, Wang Y C 2018 J. Cent. South. Univ. (Sci. Tech.) 49 888
[15] 张琪琪, 张建国, 李璞, 郭龑强, 王云才 2019 通信学报 40 2019014
Google Scholar
Zhang Q Q, Zhang J G, Li P, Guo Y Q, Wang Y C 2019 J. Commun. 40 2019014
Google Scholar
[16] Canaday D, Griffith A, Gauthier D J 2018 Chaos 28 123119
Google Scholar
[17] Haynes N D, Soriano M C, Rosin D P, Fischer I, Gauthier D J 2014 Phys. Rev. E 91 020801
[18] Cheng X R, Sun M Y, Socolar J E S 2012 J. R. Soc. Interface 10 20120574
[19] Sun M Y 2013 Ph. D. Dissertation (Berlin: Duke University).
[20] Charlot N, Canaday D, Pomerance A, Gauthier D J 2020 arXiv: 1907.12542 v2 [cs. CR.]
[21] Cavalcante H L D D S, Gauthier D J, Socolar J E S, Zhang R 2010 Phil. Trans. R. Soc. A 368 495
[22] Rosin D P, Rontani D, Gauthier D J 2014 Phys. Rev. E 89 042907
Google Scholar
[23] Rosin D P, Rontani D, Haynes N D, Scholl E, Gauthier D J 2014 Phys. Rev. E 90 030902
Google Scholar
[24] D'Huys O, Lohmann J, Haynes N D, Gauthier D J 2016 Chaos 26 094810
Google Scholar
[25] 龚利爽, 侯二林, 刘海芳, 李凯凯, 王云才 2019 通信学报 40 2019048
Gong L S, Hou E L, Liu H F, Li K K, Wang Y C 2019 J. Commun. 40 2019048
[26] Rosin D 2014 Ph. D. Dissertation (Berlin: Duke University).
[27] Xiang S Y, Pan W, Li N Q, Zhang L Y, Zhu H N 2013 Opt. Commun. 311 294
Google Scholar
[28] Toker D, Sommer F T, D'Esposito M 2020 Commun. Biol. 3 11
Google Scholar
[29] Ghil M, Mullhaupt A 1985 J. Stat. Phys. 41 125
Google Scholar
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