Abstract

Many information processing challenges are difficult to solve with traditional Turing or von Neumann approaches. Implementing unconventional computational methods is therefore essential and optics provides promising opportunities. Here we experimentally demonstrate optical information processing using a nonlinear optoelectronic oscillator subject to delayed feedback. We implement a neuro-inspired concept, called Reservoir Computing, proven to possess universal computational capabilities. We particularly exploit the transient response of a complex dynamical system to an input data stream. We employ spoken digit recognition and time series prediction tasks as benchmarks, achieving competitive processing figures of merit.

© 2012 OSA

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2011 (3)

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,” Nat. Commun. 2, 468 (2011).
[CrossRef] [PubMed]

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

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

2010 (6)

L. Larger and J. M. Dudley, “Optoelectronic chaos,” Nature 465, 41–42 (2010).
[CrossRef] [PubMed]

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]

K. E. Callan, L. Illing, Z. Gao, D. J. Gauthier, and E. Schöll, “Broadband chaos generated by an optoelectronic oscillator,” Phys. Rev. Lett. 104, 113901 (2010).
[CrossRef] [PubMed]

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

R. S. Tucker, “The role of optics in computing,” Nat. Photonics 4, 405 (2010).
[CrossRef]

D. A. B. Miller, “Correspondence to the editor,” Nat. Photonics 4, 406 (2010).
[CrossRef]

2009 (1)

D. V. Buonomano and W. Maass, “State-dependent computations: Spatiotemporal processing in cortical networks,” Nat. Rev. Neurosci. 10, 113–125 (2009).
[CrossRef] [PubMed]

2008 (2)

2007 (1)

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

2005 (1)

D. Verstraeten, B. Schrauwen, D. Stroobandt, and J. Van Campenhout, “Isolated word recognition with the liquid state machine: a case study,” Inf. Process. Lett. 30, 521–528 (2005).
[CrossRef]

2004 (3)

T. Erneux, L. Larger, M. W. Lee, and J. Goedgebuer, “Ikeda hopf bifurcation revisited,” Physica D 194, 49–64 (2004).
[CrossRef]

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

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

2003 (1)

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

1993 (1)

F. T. Arecchi, G. Giacomelli, A. Lapucci, and R. Meucci, “Two–dimensional representation of a delayed dynamical system,” Phys. Rev. A 45, R4225–R4228 (1993).
[CrossRef]

1990 (1)

D. Psaltis, D. Brady, X. G. Gu, and S. Lin, “Holography in artificial neural networks,” Nature 343, 325–330 (1990).
[CrossRef] [PubMed]

1982 (2)

A. Neyer and E. Voges, “Dynamics of electrooptic bistable devices with delayed feedback,” IEEE J. Quantum Electron. 18, 2009–2015 (1982).
[CrossRef]

E. Abraham and S. D. Smith, “Optical bistability and related devices,” Rep. Prog. Phys. 45, 815–885 (1982).
[CrossRef]

1979 (1)

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

1978 (1)

D. A. B. Miller, M. H. Mozolowski, A. Miller, and S. D. Smith, “Nonlinear optical effects in insb with a cw co laser,” Opt. Commun. 27, 133–136 (1978).
[CrossRef]

1970 (1)

A. E. Hoerl and R. W. Kennard, “Ridge Regression: Applications to Nonorthogonal Problems” Technometrics 12, 69–82 (1970).
[CrossRef]

Abraham, E.

E. Abraham and S. D. Smith, “Optical bistability and related devices,” Rep. Prog. Phys. 45, 815–885 (1982).
[CrossRef]

Appeltant, L.

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,” Nat. Commun. 2, 468 (2011).
[CrossRef] [PubMed]

Arecchi, F. T.

F. T. Arecchi, G. Giacomelli, A. Lapucci, and R. Meucci, “Two–dimensional representation of a delayed dynamical system,” Phys. Rev. A 45, R4225–R4228 (1993).
[CrossRef]

Baets, R.

Bienstman, P.

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

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

Brady, D.

D. Psaltis, D. Brady, X. G. Gu, and S. Lin, “Holography in artificial neural networks,” Nature 343, 325–330 (1990).
[CrossRef] [PubMed]

Buonomano, D. V.

D. V. Buonomano and W. Maass, “State-dependent computations: Spatiotemporal processing in cortical networks,” Nat. Rev. Neurosci. 10, 113–125 (2009).
[CrossRef] [PubMed]

Callan, K. E.

K. E. Callan, L. Illing, Z. Gao, D. J. Gauthier, and E. Schöll, “Broadband chaos generated by an optoelectronic oscillator,” Phys. Rev. Lett. 104, 113901 (2010).
[CrossRef] [PubMed]

Campenhout, J.

Cao, L. J.

L. J. 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]

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]

Dambre, J.

K. Vandoorne, J. Dambre, D. Verstraeten, B. Schrauwen, and P. Bienstman, “Parallel reservoir computing using optical amplifiers,” IEEE Trans. Neural Netw. 22, 1469–1481 (2011).
[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,” Nat. Commun. 2, 468 (2011).
[CrossRef] [PubMed]

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,” Nat. Commun. 2, 468 (2011).
[CrossRef] [PubMed]

Dierckx, W.

Dolev, S.

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

Dudley, J. M.

L. Larger and J. M. Dudley, “Optoelectronic chaos,” Nature 465, 41–42 (2010).
[CrossRef] [PubMed]

Erneux, T.

T. Erneux, L. Larger, M. W. Lee, and J. Goedgebuer, “Ikeda hopf bifurcation revisited,” Physica D 194, 49–64 (2004).
[CrossRef]

Fischer, I.

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,” Nat. Commun. 2, 468 (2011).
[CrossRef] [PubMed]

Gao, Z.

K. E. Callan, L. Illing, Z. Gao, D. J. Gauthier, and E. Schöll, “Broadband chaos generated by an optoelectronic oscillator,” Phys. Rev. Lett. 104, 113901 (2010).
[CrossRef] [PubMed]

Gauthier, D. J.

K. E. Callan, L. Illing, Z. Gao, D. J. Gauthier, and E. Schöll, “Broadband chaos generated by an optoelectronic oscillator,” Phys. Rev. Lett. 104, 113901 (2010).
[CrossRef] [PubMed]

Giacomelli, G.

F. T. Arecchi, G. Giacomelli, A. Lapucci, and R. Meucci, “Two–dimensional representation of a delayed dynamical system,” Phys. Rev. A 45, R4225–R4228 (1993).
[CrossRef]

Goedgebuer, J.

T. Erneux, L. Larger, M. W. Lee, and J. Goedgebuer, “Ikeda hopf bifurcation revisited,” Physica D 194, 49–64 (2004).
[CrossRef]

Goedgebuer, J.-P.

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

Gu, X. G.

D. Psaltis, D. Brady, X. G. Gu, and S. Lin, “Holography in artificial neural networks,” Nature 343, 325–330 (1990).
[CrossRef] [PubMed]

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]

Hoerl, A. E.

A. E. Hoerl and R. W. Kennard, “Ridge Regression: Applications to Nonorthogonal Problems” Technometrics 12, 69–82 (1970).
[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,” Opt. Commun. 30, 257–261 (1979).
[CrossRef]

Illing, L.

K. E. Callan, L. Illing, Z. Gao, D. J. Gauthier, and E. Schöll, “Broadband chaos generated by an optoelectronic oscillator,” Phys. Rev. Lett. 104, 113901 (2010).
[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]

Kennard, R. W.

A. E. Hoerl and R. W. Kennard, “Ridge Regression: Applications to Nonorthogonal Problems” Technometrics 12, 69–82 (1970).
[CrossRef]

Lapucci, A.

F. T. Arecchi, G. Giacomelli, A. Lapucci, and R. Meucci, “Two–dimensional representation of a delayed dynamical system,” Phys. Rev. A 45, R4225–R4228 (1993).
[CrossRef]

Larger, L.

L. Larger and J. M. Dudley, “Optoelectronic chaos,” Nature 465, 41–42 (2010).
[CrossRef] [PubMed]

T. Erneux, L. Larger, M. W. Lee, and J. Goedgebuer, “Ikeda hopf bifurcation revisited,” Physica D 194, 49–64 (2004).
[CrossRef]

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

Laurent, G.

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

Lee, M. W.

T. Erneux, L. Larger, M. W. Lee, and J. Goedgebuer, “Ikeda hopf bifurcation revisited,” Physica D 194, 49–64 (2004).
[CrossRef]

Lin, S.

D. Psaltis, D. Brady, X. G. Gu, and S. Lin, “Holography in artificial neural networks,” Nature 343, 325–330 (1990).
[CrossRef] [PubMed]

Lyon, R. F.

R. F. Lyon, “A computational model of filtering, detection, and compression in the cochlea,” Proc. of the IEEE Int. Conf. Acoust., Speech, Signal Processing (1982).

Maass, W.

D. V. Buonomano and W. Maass, “State-dependent computations: Spatiotemporal processing in cortical networks,” Nat. Rev. Neurosci. 10, 113–125 (2009).
[CrossRef] [PubMed]

Massar, S.

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,” Nat. Commun. 2, 468 (2011).
[CrossRef] [PubMed]

Meucci, R.

F. T. Arecchi, G. Giacomelli, A. Lapucci, and R. Meucci, “Two–dimensional representation of a delayed dynamical system,” Phys. Rev. A 45, R4225–R4228 (1993).
[CrossRef]

Miller, A.

D. A. B. Miller, M. H. Mozolowski, A. Miller, and S. D. Smith, “Nonlinear optical effects in insb with a cw co laser,” Opt. Commun. 27, 133–136 (1978).
[CrossRef]

Miller, D. A. B.

D. A. B. Miller, “Correspondence to the editor,” Nat. Photonics 4, 406 (2010).
[CrossRef]

D. A. B. Miller, M. H. Mozolowski, A. Miller, and S. D. Smith, “Nonlinear optical effects in insb with a cw co laser,” Opt. Commun. 27, 133–136 (1978).
[CrossRef]

Mirasso, C. R.

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,” Nat. Commun. 2, 468 (2011).
[CrossRef] [PubMed]

Mozolowski, M. H.

D. A. B. Miller, M. H. Mozolowski, A. Miller, and S. D. Smith, “Nonlinear optical effects in insb with a cw co laser,” Opt. Commun. 27, 133–136 (1978).
[CrossRef]

Neyer, A.

A. Neyer and E. Voges, “Dynamics of electrooptic bistable devices with delayed feedback,” IEEE J. Quantum Electron. 18, 2009–2015 (1982).
[CrossRef]

O’Brien, J. L.

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

Psaltis, D.

D. Psaltis, D. Brady, X. G. Gu, and S. Lin, “Holography in artificial neural networks,” Nature 343, 325–330 (1990).
[CrossRef] [PubMed]

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. Tino, “Minimum complexity echo state network,” IEEE Trans. Neural Netw. 22, 131–144 (2011).
[CrossRef]

Schöll, E.

K. E. Callan, L. Illing, Z. Gao, D. J. Gauthier, and E. Schöll, “Broadband chaos generated by an optoelectronic oscillator,” Phys. Rev. Lett. 104, 113901 (2010).
[CrossRef] [PubMed]

Schrauwen, B.

K. Vandoorne, J. Dambre, D. Verstraeten, B. Schrauwen, and P. Bienstman, “Parallel reservoir computing using optical amplifiers,” IEEE Trans. Neural Netw. 22, 1469–1481 (2011).
[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,” Nat. Commun. 2, 468 (2011).
[CrossRef] [PubMed]

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

D. Verstraeten, B. Schrauwen, D. Stroobandt, and J. Van Campenhout, “Isolated word recognition with the liquid state machine: a case study,” Inf. Process. Lett. 30, 521–528 (2005).
[CrossRef]

Smith, S. D.

E. Abraham and S. D. Smith, “Optical bistability and related devices,” Rep. Prog. Phys. 45, 815–885 (1982).
[CrossRef]

D. A. B. Miller, M. H. Mozolowski, A. Miller, and S. D. Smith, “Nonlinear optical effects in insb with a cw co laser,” Opt. Commun. 27, 133–136 (1978).
[CrossRef]

Soriano, M. C.

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,” Nat. Commun. 2, 468 (2011).
[CrossRef] [PubMed]

Stroobandt, D.

D. Verstraeten, B. Schrauwen, D. Stroobandt, and J. Van Campenhout, “Isolated word recognition with the liquid state machine: a case study,” Inf. Process. Lett. 30, 521–528 (2005).
[CrossRef]

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]

Tino, P.

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

Tucker, R. S.

R. S. Tucker, “The role of optics in computing,” Nat. Photonics 4, 405 (2010).
[CrossRef]

Udaltsov, V. S.

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

Van Campenhout, J.

D. Verstraeten, B. Schrauwen, D. Stroobandt, and J. Van Campenhout, “Isolated word recognition with the liquid state machine: a case study,” Inf. Process. Lett. 30, 521–528 (2005).
[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,” Nat. Commun. 2, 468 (2011).
[CrossRef] [PubMed]

Vandoorne, K.

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

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

Verstraeten, D.

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

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

D. Verstraeten, B. Schrauwen, D. Stroobandt, and J. Van Campenhout, “Isolated word recognition with the liquid state machine: a case study,” Inf. Process. Lett. 30, 521–528 (2005).
[CrossRef]

Voges, E.

A. Neyer and E. Voges, “Dynamics of electrooptic bistable devices with delayed feedback,” IEEE J. Quantum Electron. 18, 2009–2015 (1982).
[CrossRef]

William, L. D.

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Figures (5)

Fig. 1
Fig. 1

Schematic representation of RC based on (a) a complex network of nonlinear nodes or (b) a single nonlinear element subject to delayed feedback via time multiplexing, where f (x) stands for the the system’s nonlinear transformation and h(t) denotes the system’s impulse response, respectively.

Fig. 2
Fig. 2

Optoelectronic implementation of RC with a single nonlinear element subject to delayed feedback.

Fig. 3
Fig. 3

Injection of a spoken digit into the reservoir showing the input connectivity matrix (left), a Cochleagram of a spoken digit (middle) and the resulting input data of the network (right). In the connect matrix the color code presents the magnitude of the input scaling factors w l m i, in the Cochleagram and the Network input data the color encodes the amplitudes of the signals, with red (blue) corresponding to large (small) values.

Fig. 4
Fig. 4

(a) and (b) show the WER and Margin for spoken digit recognition in the (β0)–plane (bifurcation parameter vs. MZM phase). The two figures of merit show a similar dependency on both parameters, with excellent performance at β = 0.3 and Φ0 = 0.89π. (c) Detailed dependence of the RC performance on the MZM phase at β = 0.3. (d) MZM transmission function as a function of phase Φ0.

Fig. 5
Fig. 5

MZM phase dependence of the RC performance in a time series prediction task, using the Santa-Fe data set. Best performance for β = 0.2 is found around Φ0 = 0.1π0 = 0.5π0 = 0.7π and Φ0 = 0.85π phase values in the vicinity of local extrema of the transfer function of the MZM (see Figs. 4(d), 1(a), and 1(b)).

Equations (1)

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ε x ˙ ( s ) + x ( s ) = β sin 2 [ μ x ( s 1 ) + ρ u I ( s 1 ) + Φ 0 ] ,

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