Abstract

We present improved strategies to perform photonic information processing using an optoelectronic oscillator with delayed feedback. In particular, we study, via numerical simulations and experiments, the influence of a finite signal-to-noise ratio on the computing performance. We illustrate that the performance degradation induced by noise can be compensated for via multi-level pre-processing masks.

© 2013 OSA

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References

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  2. H. J. Caulfield and S. Dolev, “Why future supercomputing requires optics,” Nat. Photonics 4, 261 (2010).
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  3. W. Maass, T. Natschläger, and H. Markram, “Real-time computing without stable states: a new framework for neural computation based on perturbations,” Neural Comput. 14, 2531–2560 (2002).
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  4. H. Jaeger and H. Haas, “Harnessing nonlinearity: predicting chaotic systems and saving energy in wireless communication,” Science 304, 78–80 (2004).
    [Crossref] [PubMed]
  5. D. Verstraeten, B. Schrauwen, M. D’Haene, and D. Stroobandt, “An experimental unification of reservoir computing methods,” Neural Networks 20, 391–403 (2007).
    [Crossref] [PubMed]
  6. M. Rabinovich, R. Huerta, and G. Laurent, “Transient dynamics of neural processing,” Science 321, 48–50 (2008).
    [Crossref] [PubMed]
  7. W. Maass and H. Markram, “On the computational power of recurrent circuits of spiking neurons,” J. Comput. Syst. Sci. 69, 593–616 (2004).
    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref]
  23. U. Huebner, N. B. Abraham, and C. O. Weiss, “Dimensions and entropies of chaotic intensity pulsations in a single-mode far-infrared NH3 laser”, Phys. Rev. A 40, 6354–6365 (1989).
    [Crossref]
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    [Crossref]

2012 (5)

Y. Paquot, F. Duport, A. Smerieri, J. Dambre, B. Schrauwen, M. Haelterman, and S. Massar, “Optoelectronic reservoir computing,” Sci. Rep. 2, 287 (2012).
[Crossref] [PubMed]

R. Martinenghi, S. Rybalko, M. Jacquot, Y. K. Chembo, and L. Larger, “Photonic nonlinear transient computing with multiple-delay wavelength dynamics,” Phys. Rev Lett. 108, 244101 (2012).
[Crossref] [PubMed]

D. Woods and T. J. Naughton, “Photonic neural networks,” Nature Phys. 8, 257–258 (2012).
[Crossref]

J. Dambre, D. Verstraeten, B. Schrauwen, and S. Massar, “Information processing capacity of dynamical systems,” Sci. Rep. 2, 514 (2012).
[Crossref] [PubMed]

L. Larger, M. C. Soriano, D. Brunner, L. Appeltant, J. M. Gutierrez, L. Pesquera, C. R. Mirasso, and I. Fischer, “Photonic information processing beyond Turing: an optoelectronic implementation of reservoir computing,” Opt. Express 20, 3241–3249 (2012).
[Crossref] [PubMed]

2011 (4)

M. C. Soriano, L. Zunino, L. Larger, I. Fischer, and C. R. Mirasso, “Distinguishing fingerprints of hyperchaotic and stochastic dynamics in optical chaos from a delayed opto-electronic oscillator,” Opt. Lett. 36, 2212–2214 (2011).
[Crossref] [PubMed]

K. Vandoorne, J. Dambre, D. Verstraeten, B. Schrauwen, and P. Bienstman, “Parallel reservoir computing using optical amplifiers,” IEEE Trans. Neural Networks 22, 1469–1481 (2011).
[Crossref]

L. Appeltant, M. C. Soriano, G. Van der Sande, J. Danckaert, S. Massar, J. Dambre, B. Schrauwen, C. R. Mirasso, and I. Fischer, “Information processing using a single dynamical node as complex system,” Nature Commun. 2, 468 (2011).
[Crossref]

A. Rodan and P. Tin̂o, “Minimum complexity echo state network,” IEEE Trans. Neural Networks 22, 131–144 (2011).
[Crossref]

2010 (2)

J. P. Crutchfield, L. D. William, and S. Sudeshna, “Introduction to focus issue: intrinsic and designed computation: information processing in dynamical systems beyond the digital hegemony,” Chaos 20, 037101 (2010).
[Crossref] [PubMed]

H. J. Caulfield and S. Dolev, “Why future supercomputing requires optics,” Nat. Photonics 4, 261 (2010).
[Crossref]

2008 (2)

2007 (3)

Y. Kouomou Chembo, L. Larger, H. Tavernier, R. Bendoula, E. Rubiola, and P. Colet, “Dynamic instabilities of microwaves generated with optoelectronic oscillators,” Opt. Lett. 32, 2571–2573 (2007).
[Crossref] [PubMed]

D. Verstraeten, B. Schrauwen, M. D’Haene, and D. Stroobandt, “An experimental unification of reservoir computing methods,” Neural Networks 20, 391–403 (2007).
[Crossref] [PubMed]

J. L. O’Brien, “Optical quantum computing,” Science 7, 1567–1570 (2007).
[Crossref]

2004 (3)

H. Jaeger and H. Haas, “Harnessing nonlinearity: predicting chaotic systems and saving energy in wireless communication,” Science 304, 78–80 (2004).
[Crossref] [PubMed]

W. Maass and H. Markram, “On the computational power of recurrent circuits of spiking neurons,” J. Comput. Syst. Sci. 69, 593–616 (2004).
[Crossref]

L. Larger, J.-P. Goedgebuer, and V. S. Udalsov, “Ikeda–based nonlinear delayed dynamics for application to secure optical transmission systems using chaos,” C.R. de Physique 5, 669–681 (2004).
[Crossref]

2003 (1)

L. Cao, “Support vector machines experts for time series forecasting”, Neurocomputing 51, 321–339 (2003).
[Crossref]

2002 (1)

W. Maass, T. Natschläger, and H. Markram, “Real-time computing without stable states: a new framework for neural computation based on perturbations,” Neural Comput. 14, 2531–2560 (2002).
[Crossref] [PubMed]

1998 (1)

L. Larger, J. P. Goedgebuer, and J. M. Merolla, “Chaotic oscillator in wavelength: a new setup for investigating differential difference equations describing nonlinear dynamics,” IEEE J. Quantum Electron. 34, 594–601 (1998).
[Crossref]

1989 (1)

U. Huebner, N. B. Abraham, and C. O. Weiss, “Dimensions and entropies of chaotic intensity pulsations in a single-mode far-infrared NH3 laser”, Phys. Rev. A 40, 6354–6365 (1989).
[Crossref]

1979 (1)

K. Ikeda, “Multiple-valued stationary state and its instability of the transmitted light by a ring cavity system,” Optics Commun. 30, 257–261 (1979).
[Crossref]

Abraham, N. B.

U. Huebner, N. B. Abraham, and C. O. Weiss, “Dimensions and entropies of chaotic intensity pulsations in a single-mode far-infrared NH3 laser”, Phys. Rev. A 40, 6354–6365 (1989).
[Crossref]

Appeltant, L.

L. Larger, M. C. Soriano, D. Brunner, L. Appeltant, J. M. Gutierrez, L. Pesquera, C. R. Mirasso, and I. Fischer, “Photonic information processing beyond Turing: an optoelectronic implementation of reservoir computing,” Opt. Express 20, 3241–3249 (2012).
[Crossref] [PubMed]

L. Appeltant, M. C. Soriano, G. Van der Sande, J. Danckaert, S. Massar, J. Dambre, B. Schrauwen, C. R. Mirasso, and I. Fischer, “Information processing using a single dynamical node as complex system,” Nature Commun. 2, 468 (2011).
[Crossref]

Baets, R.

Bendoula, R.

Bienstman, P.

K. Vandoorne, J. Dambre, D. Verstraeten, B. Schrauwen, and P. Bienstman, “Parallel reservoir computing using optical amplifiers,” IEEE Trans. Neural Networks 22, 1469–1481 (2011).
[Crossref]

K. Vandoorne, W. Dierckx, B. Schrauwen, D. Verstraeten, R. Baets, P. Bienstman, and J. Campenhout, “Towards optical signal processing using photonic reservoir computing,” Opt. Express 16, 11182—11192 (2008).
[Crossref] [PubMed]

Brunner, D.

Campenhout, J.

Cao, L.

L. Cao, “Support vector machines experts for time series forecasting”, Neurocomputing 51, 321–339 (2003).
[Crossref]

Caulfield, H. J.

H. J. Caulfield and S. Dolev, “Why future supercomputing requires optics,” Nat. Photonics 4, 261 (2010).
[Crossref]

Chembo, Y. K.

R. Martinenghi, S. Rybalko, M. Jacquot, Y. K. Chembo, and L. Larger, “Photonic nonlinear transient computing with multiple-delay wavelength dynamics,” Phys. Rev Lett. 108, 244101 (2012).
[Crossref] [PubMed]

Colet, P.

Crutchfield, J. P.

J. P. Crutchfield, L. D. William, and S. Sudeshna, “Introduction to focus issue: intrinsic and designed computation: information processing in dynamical systems beyond the digital hegemony,” Chaos 20, 037101 (2010).
[Crossref] [PubMed]

D’Haene, M.

D. Verstraeten, B. Schrauwen, M. D’Haene, and D. Stroobandt, “An experimental unification of reservoir computing methods,” Neural Networks 20, 391–403 (2007).
[Crossref] [PubMed]

Dambre, J.

J. Dambre, D. Verstraeten, B. Schrauwen, and S. Massar, “Information processing capacity of dynamical systems,” Sci. Rep. 2, 514 (2012).
[Crossref] [PubMed]

Y. Paquot, F. Duport, A. Smerieri, J. Dambre, B. Schrauwen, M. Haelterman, and S. Massar, “Optoelectronic reservoir computing,” Sci. Rep. 2, 287 (2012).
[Crossref] [PubMed]

L. Appeltant, M. C. Soriano, G. Van der Sande, J. Danckaert, S. Massar, J. Dambre, B. Schrauwen, C. R. Mirasso, and I. Fischer, “Information processing using a single dynamical node as complex system,” Nature Commun. 2, 468 (2011).
[Crossref]

K. Vandoorne, J. Dambre, D. Verstraeten, B. Schrauwen, and P. Bienstman, “Parallel reservoir computing using optical amplifiers,” IEEE Trans. Neural Networks 22, 1469–1481 (2011).
[Crossref]

Danckaert, J.

L. Appeltant, M. C. Soriano, G. Van der Sande, J. Danckaert, S. Massar, J. Dambre, B. Schrauwen, C. R. Mirasso, and I. Fischer, “Information processing using a single dynamical node as complex system,” Nature Commun. 2, 468 (2011).
[Crossref]

Dierckx, W.

Dolev, S.

H. J. Caulfield and S. Dolev, “Why future supercomputing requires optics,” Nat. Photonics 4, 261 (2010).
[Crossref]

Duport, F.

Y. Paquot, F. Duport, A. Smerieri, J. Dambre, B. Schrauwen, M. Haelterman, and S. Massar, “Optoelectronic reservoir computing,” Sci. Rep. 2, 287 (2012).
[Crossref] [PubMed]

Fischer, I.

Goedgebuer, J. P.

L. Larger, J. P. Goedgebuer, and J. M. Merolla, “Chaotic oscillator in wavelength: a new setup for investigating differential difference equations describing nonlinear dynamics,” IEEE J. Quantum Electron. 34, 594–601 (1998).
[Crossref]

Goedgebuer, J.-P.

L. Larger, J.-P. Goedgebuer, and V. S. Udalsov, “Ikeda–based nonlinear delayed dynamics for application to secure optical transmission systems using chaos,” C.R. de Physique 5, 669–681 (2004).
[Crossref]

Gutierrez, J. M.

Haas, H.

H. Jaeger and H. Haas, “Harnessing nonlinearity: predicting chaotic systems and saving energy in wireless communication,” Science 304, 78–80 (2004).
[Crossref] [PubMed]

Haelterman, M.

Y. Paquot, F. Duport, A. Smerieri, J. Dambre, B. Schrauwen, M. Haelterman, and S. Massar, “Optoelectronic reservoir computing,” Sci. Rep. 2, 287 (2012).
[Crossref] [PubMed]

Huebner, U.

U. Huebner, N. B. Abraham, and C. O. Weiss, “Dimensions and entropies of chaotic intensity pulsations in a single-mode far-infrared NH3 laser”, Phys. Rev. A 40, 6354–6365 (1989).
[Crossref]

Huerta, R.

M. Rabinovich, R. Huerta, and G. Laurent, “Transient dynamics of neural processing,” Science 321, 48–50 (2008).
[Crossref] [PubMed]

Ikeda, K.

K. Ikeda, “Multiple-valued stationary state and its instability of the transmitted light by a ring cavity system,” Optics Commun. 30, 257–261 (1979).
[Crossref]

Jacquot, M.

R. Martinenghi, S. Rybalko, M. Jacquot, Y. K. Chembo, and L. Larger, “Photonic nonlinear transient computing with multiple-delay wavelength dynamics,” Phys. Rev Lett. 108, 244101 (2012).
[Crossref] [PubMed]

Jaeger, H.

H. Jaeger and H. Haas, “Harnessing nonlinearity: predicting chaotic systems and saving energy in wireless communication,” Science 304, 78–80 (2004).
[Crossref] [PubMed]

Kouomou Chembo, Y.

Larger, L.

R. Martinenghi, S. Rybalko, M. Jacquot, Y. K. Chembo, and L. Larger, “Photonic nonlinear transient computing with multiple-delay wavelength dynamics,” Phys. Rev Lett. 108, 244101 (2012).
[Crossref] [PubMed]

L. Larger, M. C. Soriano, D. Brunner, L. Appeltant, J. M. Gutierrez, L. Pesquera, C. R. Mirasso, and I. Fischer, “Photonic information processing beyond Turing: an optoelectronic implementation of reservoir computing,” Opt. Express 20, 3241–3249 (2012).
[Crossref] [PubMed]

M. C. Soriano, L. Zunino, L. Larger, I. Fischer, and C. R. Mirasso, “Distinguishing fingerprints of hyperchaotic and stochastic dynamics in optical chaos from a delayed opto-electronic oscillator,” Opt. Lett. 36, 2212–2214 (2011).
[Crossref] [PubMed]

Y. Kouomou Chembo, L. Larger, H. Tavernier, R. Bendoula, E. Rubiola, and P. Colet, “Dynamic instabilities of microwaves generated with optoelectronic oscillators,” Opt. Lett. 32, 2571–2573 (2007).
[Crossref] [PubMed]

L. Larger, J.-P. Goedgebuer, and V. S. Udalsov, “Ikeda–based nonlinear delayed dynamics for application to secure optical transmission systems using chaos,” C.R. de Physique 5, 669–681 (2004).
[Crossref]

L. Larger, J. P. Goedgebuer, and J. M. Merolla, “Chaotic oscillator in wavelength: a new setup for investigating differential difference equations describing nonlinear dynamics,” IEEE J. Quantum Electron. 34, 594–601 (1998).
[Crossref]

Laurent, G.

M. Rabinovich, R. Huerta, and G. Laurent, “Transient dynamics of neural processing,” Science 321, 48–50 (2008).
[Crossref] [PubMed]

Maass, W.

W. Maass and H. Markram, “On the computational power of recurrent circuits of spiking neurons,” J. Comput. Syst. Sci. 69, 593–616 (2004).
[Crossref]

W. Maass, T. Natschläger, and H. Markram, “Real-time computing without stable states: a new framework for neural computation based on perturbations,” Neural Comput. 14, 2531–2560 (2002).
[Crossref] [PubMed]

Markram, H.

W. Maass and H. Markram, “On the computational power of recurrent circuits of spiking neurons,” J. Comput. Syst. Sci. 69, 593–616 (2004).
[Crossref]

W. Maass, T. Natschläger, and H. Markram, “Real-time computing without stable states: a new framework for neural computation based on perturbations,” Neural Comput. 14, 2531–2560 (2002).
[Crossref] [PubMed]

Martinenghi, R.

R. Martinenghi, S. Rybalko, M. Jacquot, Y. K. Chembo, and L. Larger, “Photonic nonlinear transient computing with multiple-delay wavelength dynamics,” Phys. Rev Lett. 108, 244101 (2012).
[Crossref] [PubMed]

Massar, S.

Y. Paquot, F. Duport, A. Smerieri, J. Dambre, B. Schrauwen, M. Haelterman, and S. Massar, “Optoelectronic reservoir computing,” Sci. Rep. 2, 287 (2012).
[Crossref] [PubMed]

J. Dambre, D. Verstraeten, B. Schrauwen, and S. Massar, “Information processing capacity of dynamical systems,” Sci. Rep. 2, 514 (2012).
[Crossref] [PubMed]

L. Appeltant, M. C. Soriano, G. Van der Sande, J. Danckaert, S. Massar, J. Dambre, B. Schrauwen, C. R. Mirasso, and I. Fischer, “Information processing using a single dynamical node as complex system,” Nature Commun. 2, 468 (2011).
[Crossref]

Merolla, J. M.

L. Larger, J. P. Goedgebuer, and J. M. Merolla, “Chaotic oscillator in wavelength: a new setup for investigating differential difference equations describing nonlinear dynamics,” IEEE J. Quantum Electron. 34, 594–601 (1998).
[Crossref]

Mirasso, C. R.

Natschläger, T.

W. Maass, T. Natschläger, and H. Markram, “Real-time computing without stable states: a new framework for neural computation based on perturbations,” Neural Comput. 14, 2531–2560 (2002).
[Crossref] [PubMed]

Naughton, T. J.

D. Woods and T. J. Naughton, “Photonic neural networks,” Nature Phys. 8, 257–258 (2012).
[Crossref]

O’Brien, J. L.

J. L. O’Brien, “Optical quantum computing,” Science 7, 1567–1570 (2007).
[Crossref]

Paquot, Y.

Y. Paquot, F. Duport, A. Smerieri, J. Dambre, B. Schrauwen, M. Haelterman, and S. Massar, “Optoelectronic reservoir computing,” Sci. Rep. 2, 287 (2012).
[Crossref] [PubMed]

Pesquera, L.

Rabinovich, M.

M. Rabinovich, R. Huerta, and G. Laurent, “Transient dynamics of neural processing,” Science 321, 48–50 (2008).
[Crossref] [PubMed]

Rodan, A.

A. Rodan and P. Tin̂o, “Minimum complexity echo state network,” IEEE Trans. Neural Networks 22, 131–144 (2011).
[Crossref]

Rubiola, E.

Rybalko, S.

R. Martinenghi, S. Rybalko, M. Jacquot, Y. K. Chembo, and L. Larger, “Photonic nonlinear transient computing with multiple-delay wavelength dynamics,” Phys. Rev Lett. 108, 244101 (2012).
[Crossref] [PubMed]

Schrauwen, B.

Y. Paquot, F. Duport, A. Smerieri, J. Dambre, B. Schrauwen, M. Haelterman, and S. Massar, “Optoelectronic reservoir computing,” Sci. Rep. 2, 287 (2012).
[Crossref] [PubMed]

J. Dambre, D. Verstraeten, B. Schrauwen, and S. Massar, “Information processing capacity of dynamical systems,” Sci. Rep. 2, 514 (2012).
[Crossref] [PubMed]

L. Appeltant, M. C. Soriano, G. Van der Sande, J. Danckaert, S. Massar, J. Dambre, B. Schrauwen, C. R. Mirasso, and I. Fischer, “Information processing using a single dynamical node as complex system,” Nature Commun. 2, 468 (2011).
[Crossref]

K. Vandoorne, J. Dambre, D. Verstraeten, B. Schrauwen, and P. Bienstman, “Parallel reservoir computing using optical amplifiers,” IEEE Trans. Neural Networks 22, 1469–1481 (2011).
[Crossref]

K. Vandoorne, W. Dierckx, B. Schrauwen, D. Verstraeten, R. Baets, P. Bienstman, and J. Campenhout, “Towards optical signal processing using photonic reservoir computing,” Opt. Express 16, 11182—11192 (2008).
[Crossref] [PubMed]

D. Verstraeten, B. Schrauwen, M. D’Haene, and D. Stroobandt, “An experimental unification of reservoir computing methods,” Neural Networks 20, 391–403 (2007).
[Crossref] [PubMed]

Smerieri, A.

Y. Paquot, F. Duport, A. Smerieri, J. Dambre, B. Schrauwen, M. Haelterman, and S. Massar, “Optoelectronic reservoir computing,” Sci. Rep. 2, 287 (2012).
[Crossref] [PubMed]

Soriano, M. C.

Stroobandt, D.

D. Verstraeten, B. Schrauwen, M. D’Haene, and D. Stroobandt, “An experimental unification of reservoir computing methods,” Neural Networks 20, 391–403 (2007).
[Crossref] [PubMed]

Sudeshna, S.

J. P. Crutchfield, L. D. William, and S. Sudeshna, “Introduction to focus issue: intrinsic and designed computation: information processing in dynamical systems beyond the digital hegemony,” Chaos 20, 037101 (2010).
[Crossref] [PubMed]

Tavernier, H.

Tin^o, P.

A. Rodan and P. Tin̂o, “Minimum complexity echo state network,” IEEE Trans. Neural Networks 22, 131–144 (2011).
[Crossref]

Udalsov, V. S.

L. Larger, J.-P. Goedgebuer, and V. S. Udalsov, “Ikeda–based nonlinear delayed dynamics for application to secure optical transmission systems using chaos,” C.R. de Physique 5, 669–681 (2004).
[Crossref]

Van der Sande, G.

L. Appeltant, M. C. Soriano, G. Van der Sande, J. Danckaert, S. Massar, J. Dambre, B. Schrauwen, C. R. Mirasso, and I. Fischer, “Information processing using a single dynamical node as complex system,” Nature Commun. 2, 468 (2011).
[Crossref]

Vandoorne, K.

K. Vandoorne, J. Dambre, D. Verstraeten, B. Schrauwen, and P. Bienstman, “Parallel reservoir computing using optical amplifiers,” IEEE Trans. Neural Networks 22, 1469–1481 (2011).
[Crossref]

K. Vandoorne, W. Dierckx, B. Schrauwen, D. Verstraeten, R. Baets, P. Bienstman, and J. Campenhout, “Towards optical signal processing using photonic reservoir computing,” Opt. Express 16, 11182—11192 (2008).
[Crossref] [PubMed]

Verstraeten, D.

J. Dambre, D. Verstraeten, B. Schrauwen, and S. Massar, “Information processing capacity of dynamical systems,” Sci. Rep. 2, 514 (2012).
[Crossref] [PubMed]

K. Vandoorne, J. Dambre, D. Verstraeten, B. Schrauwen, and P. Bienstman, “Parallel reservoir computing using optical amplifiers,” IEEE Trans. Neural Networks 22, 1469–1481 (2011).
[Crossref]

K. Vandoorne, W. Dierckx, B. Schrauwen, D. Verstraeten, R. Baets, P. Bienstman, and J. Campenhout, “Towards optical signal processing using photonic reservoir computing,” Opt. Express 16, 11182—11192 (2008).
[Crossref] [PubMed]

D. Verstraeten, B. Schrauwen, M. D’Haene, and D. Stroobandt, “An experimental unification of reservoir computing methods,” Neural Networks 20, 391–403 (2007).
[Crossref] [PubMed]

Weiss, C. O.

U. Huebner, N. B. Abraham, and C. O. Weiss, “Dimensions and entropies of chaotic intensity pulsations in a single-mode far-infrared NH3 laser”, Phys. Rev. A 40, 6354–6365 (1989).
[Crossref]

William, L. D.

J. P. Crutchfield, L. D. William, and S. Sudeshna, “Introduction to focus issue: intrinsic and designed computation: information processing in dynamical systems beyond the digital hegemony,” Chaos 20, 037101 (2010).
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Woods, D.

D. Woods and T. J. Naughton, “Photonic neural networks,” Nature Phys. 8, 257–258 (2012).
[Crossref]

Zunino, L.

C.R. de Physique (1)

L. Larger, J.-P. Goedgebuer, and V. S. Udalsov, “Ikeda–based nonlinear delayed dynamics for application to secure optical transmission systems using chaos,” C.R. de Physique 5, 669–681 (2004).
[Crossref]

Chaos (1)

J. P. Crutchfield, L. D. William, and S. Sudeshna, “Introduction to focus issue: intrinsic and designed computation: information processing in dynamical systems beyond the digital hegemony,” Chaos 20, 037101 (2010).
[Crossref] [PubMed]

IEEE J. Quantum Electron. (1)

L. Larger, J. P. Goedgebuer, and J. M. Merolla, “Chaotic oscillator in wavelength: a new setup for investigating differential difference equations describing nonlinear dynamics,” IEEE J. Quantum Electron. 34, 594–601 (1998).
[Crossref]

IEEE Trans. Neural Networks (2)

A. Rodan and P. Tin̂o, “Minimum complexity echo state network,” IEEE Trans. Neural Networks 22, 131–144 (2011).
[Crossref]

K. Vandoorne, J. Dambre, D. Verstraeten, B. Schrauwen, and P. Bienstman, “Parallel reservoir computing using optical amplifiers,” IEEE Trans. Neural Networks 22, 1469–1481 (2011).
[Crossref]

J. Comput. Syst. Sci. (1)

W. Maass and H. Markram, “On the computational power of recurrent circuits of spiking neurons,” J. Comput. Syst. Sci. 69, 593–616 (2004).
[Crossref]

Nat. Photonics (1)

H. J. Caulfield and S. Dolev, “Why future supercomputing requires optics,” Nat. Photonics 4, 261 (2010).
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Nature Commun. (1)

L. Appeltant, M. C. Soriano, G. Van der Sande, J. Danckaert, S. Massar, J. Dambre, B. Schrauwen, C. R. Mirasso, and I. Fischer, “Information processing using a single dynamical node as complex system,” Nature Commun. 2, 468 (2011).
[Crossref]

Nature Phys. (1)

D. Woods and T. J. Naughton, “Photonic neural networks,” Nature Phys. 8, 257–258 (2012).
[Crossref]

Neural Comput. (1)

W. Maass, T. Natschläger, and H. Markram, “Real-time computing without stable states: a new framework for neural computation based on perturbations,” Neural Comput. 14, 2531–2560 (2002).
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Neural Networks (1)

D. Verstraeten, B. Schrauwen, M. D’Haene, and D. Stroobandt, “An experimental unification of reservoir computing methods,” Neural Networks 20, 391–403 (2007).
[Crossref] [PubMed]

Neurocomputing (1)

L. Cao, “Support vector machines experts for time series forecasting”, Neurocomputing 51, 321–339 (2003).
[Crossref]

Opt. Express (2)

Opt. Lett. (2)

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K. Ikeda, “Multiple-valued stationary state and its instability of the transmitted light by a ring cavity system,” Optics Commun. 30, 257–261 (1979).
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Phys. Rev Lett. (1)

R. Martinenghi, S. Rybalko, M. Jacquot, Y. K. Chembo, and L. Larger, “Photonic nonlinear transient computing with multiple-delay wavelength dynamics,” Phys. Rev Lett. 108, 244101 (2012).
[Crossref] [PubMed]

Phys. Rev. A (1)

U. Huebner, N. B. Abraham, and C. O. Weiss, “Dimensions and entropies of chaotic intensity pulsations in a single-mode far-infrared NH3 laser”, Phys. Rev. A 40, 6354–6365 (1989).
[Crossref]

Sci. Rep. (2)

J. Dambre, D. Verstraeten, B. Schrauwen, and S. Massar, “Information processing capacity of dynamical systems,” Sci. Rep. 2, 514 (2012).
[Crossref] [PubMed]

Y. Paquot, F. Duport, A. Smerieri, J. Dambre, B. Schrauwen, M. Haelterman, and S. Massar, “Optoelectronic reservoir computing,” Sci. Rep. 2, 287 (2012).
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Science (3)

M. Rabinovich, R. Huerta, and G. Laurent, “Transient dynamics of neural processing,” Science 321, 48–50 (2008).
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H. Jaeger and H. Haas, “Harnessing nonlinearity: predicting chaotic systems and saving energy in wireless communication,” Science 304, 78–80 (2004).
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Figures (6)

Fig. 1
Fig. 1

(a) Sketch of a traditional RC setup. (b) RC based on a single nonlinear element subject to delayed feedback.

Fig. 2
Fig. 2

Experimental implementation of the optoelectronic oscillator with delay.

Fig. 3
Fig. 3

Example of pre-processing for input samples {0.5, 0.25, 0.75} expanded for a delay line with 10 virtual nodes in the case of (a) a two-valued mask and (b) a six-valued mask. The vertical lines indicate the start of a new input sample.

Fig. 4
Fig. 4

(a) NMSE test prediction error in the Santa Fe laser time-series prediction task for β = 0.8 and γ = 0.45 as a function of the offset phase Φ. The red (blue) line corresponds to a binary mask in the presence (absence) of 10 bits quantization noise. The black line corresponds to a six-valued mask in the presence of 10 bits quantization noise. (b) Colour-coded NMSE prediction error in the βγ plane for Φ = −0.65π and a six-valued input mask.

Fig. 5
Fig. 5

Minimum NMSE test prediction error in the Santa Fe laser time-series prediction task for β = 0.8 and γ = 0.45 as a function of the number of quantization bits. The red (black) line corresponds to a binary (six-valued) mask. The error bars correspond to ten different random realizations of the masks.

Fig. 6
Fig. 6

(a) Experimentally recorded nonlinearity (dotted line) and operating point (solid line with triangles) as a function of the Mach-Zehnder offset phase for β = 0.8. (b) Test prediction error (NMSE) for the Santa Fe laser time-series prediction task with 400 virtual nodes for two-valued (dashed line) and six-valued (solid line) input masks (γ = 0.45). (c) NMSE of the predicted time-series for an improved detection with 5:1 oversampling and subsequent averaging obtained for the six-valued mask.

Equations (2)

Equations on this page are rendered with MathJax. Learn more.

x ˙ ( s ) = x ( s ) + β ( sin 2 [ x ( s τ ) + γ u I ( s τ ) + Φ ] 0.5 ) ,
β = I I t h I β 1 I t h ,

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