## Photonic information processing beyond Turing: an optoelectronic implementation of reservoir computing |

Optics Express, Vol. 20, Issue 3, pp. 3241-3249 (2012)

http://dx.doi.org/10.1364/OE.20.003241

Acrobat PDF (1125 KB)

### 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

## 1. Introduction

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]

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

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

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

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

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

9. 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]

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

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

^{2}to 10

^{3}) of randomly connected nonlinear dynamical nodes receiving the information to be processed via input signals. These input signals are injected from

*l*input channels into

*m*reservoir nodes, with random weights

*j*via a linear weighted sum of

*k*node states, with coefficients

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

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

10. 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]

**304**, 78–80 (2004). [CrossRef] [PubMed]

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

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

10. 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]

12. 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]

13. 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]

10. 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]

**2**, 468 (2011). [CrossRef] [PubMed]

**2**, 468 (2011). [CrossRef] [PubMed]

## 2. Experimental setup

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

16. 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]

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

15. 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]

17. 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]

_{3}) provides an electro-optic nonlinear modulation transfer function (sin

^{2}–function). A long optical fiber implements the delayed feedback loop and a photodiode is employed for optical detection. An electronic feedback circuit closes the nonlinear delay loop, connecting its output to the MZM input electrode. This circuit serves several purposes. It acts as a low pass filter, with a characteristic response time

*T*

*. It allows to add the input information*

_{R}*u*

*(*

_{I}*t*) to the delayed signal

*x*(

*t*), and amplifies this signal before it is applied to the MZM to allow for sufficient nonlinear operation. In addition, it provides the data output

*w*(

*t*).

*β*and the offset phase of the MZM Φ

_{0}, enabling easy tunability of nonlinearity and dynamical behaviors. Parameter

*β*is controlled via the laser diode power, while Φ

_{0}is controlled by the DC bias input of the MZM. In the absence of input signal, the system is set to operate in a steady (fixed point) state by keeping

*β*at a sufficiently low value. By setting the system in the steady state, a consistent response of the device to the same input signal is guaranteed.

*ρ*is the relative weight of the input information compared to the feedback signal

*x*and

*μ*corresponds to the feedback scaling. Parameter

*ε*=

*T*

*/*

_{R}*τ*

*is the oscillator response time normalized to the delay and*

_{D}*s*=

*t*/

*τ*

*is the normalized time. Setting*

_{D}*ρ*= 0, the system performs the well known Ikeda dynamics [18

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

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

*β*< 1). Dynamical complexity occurs during the transient response of the nonlinear delay system when it is excited by the input information.

20. 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]

*τ*

*, realized by 4.2 km optical fiber, into subintervals of length*

_{D}*θ*[10

**2**, 468 (2011). [CrossRef] [PubMed]

*T*

*) of the nonlinear delayed feedback loop through its impulse response. The longer (shorter)*

_{R}*T*

*is relative to the separation*

_{R}*θ*, the more (less) consecutive virtual nodes are connected. Temporal separations

*θ*slightly smaller than

*T*

*were found to yield the best RC performance [10*

_{R}**2**, 468 (2011). [CrossRef] [PubMed]

13. 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]

*N*= 400 virtual nodes [10

_{N}**2**, 468 (2011). [CrossRef] [PubMed]

*τ*= 20.87

_{D}*μ*s, i.e.

*θ*=

*τ*/

_{D}*N*= 52.18 ns. With the internal system timescale of

_{N}*T*= 240 ns, we calculate a ratio of

_{R}*T*/

_{R}*θ*≃ 4.6 between the system response time and node width. It is worth mentioning that other values of

*N*and

_{N}*τ*yield similar results, as long as the indicated relative scaling is fulfilled. This is of particular relevance when the proposed setup has to be extended to an ultra-fast version involving standard high speed telecom components.

_{D}## 3. Benchmark tests for evaluating computational power

21. 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]

*M*(dimension

_{l}*N*x

_{f}*N*) constructed with the Lyon’s Cochlear ear model consists of the corresponding

_{s}*N*=86 frequency channels and a maximum of

_{f}*N*=130 samples in time.

_{s}*M*is multiplied with the input connectivity matrix

_{l}*W*(dimension

_{i}*N*x

_{N}*N*,

_{f}*N*=400 being the number of virtual nodes in the delay line), creating the data input

_{N}*M*for the reservoir. Most of the elements

_{i}*W*are set to zero, realizing a sparse and random connectivity between the input layer and the reservoir. The remaining elements are chosen randomly from two discrete mask values, keeping the system in a transient state for the duration of the spoken digit, while also breaking the symmetry between the

_{i}*N*nodes. The elements of the connectivity matrix remain constant for the duration of the node separation

_{n}*θ*. For training the output weights we have randomly chosen 475 spoken digits among a data set of 500, leaving 25 for testing. The read-out weights

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

*W*, which is expected to provide the identification of the spoken digit in the form of a so-called target function. The entire training and test procedure is repeated 20 times with different, non-overlapping fragmentations of the 500 speech samples. By following this approach, we minimize the influence of individual speakers and spoken digits on our results, as well as providing statistical information.

_{r}*β*,Φ

_{0})–plane. Part (c) of the same Figure provides the Φ

_{0}-dependence for a constant

*β*, while the transmission function of the MZM as a function of Φ

_{0}is shown in part (d). As demonstrated by the nonlinear transfer function of the MZM, depicted in Fig. 4(d), and by Eq. (1), we can experimentally realize a variety of different nonlinear response properties to data input. These can be directly tuned by scanning the (

*β*,Φ

_{0})–plane, allowing to control magnitude and sign of the linear, as well as nonlinear response. We can choose to work with settings for different sign and magnitude of slope as well as curvature. Accordingly, our experiment represents not only a powerful electro-optical realization of RC, but at the same time it allows for studying the influence of nonlinearity and dynamical properties on the RC performance. A strong dependence in classification capability of the reservoir is found, with the WER ranging from (7.24±0.79) % down to only (0.04±0.017) %. The systematic dependence of the WER on Φ

_{0}shows the importance of the nonlinearity for the classification performance. We find the lowest WER always to be at points close, but not equal, to the local extrema of the nonlinear response. Around these points the nonlinearity can be approximated by a quadratic function. The optimal operational point has a tendency to be shifted from the local extrema towards the side with a negative slope in the response function. Corresponding points, sharing the same nonlinearity, differ in stability properties of the fixed point for a change in sign of the slope [19

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

*β*, we find the optimal operational conditions for intermediate values. As soon as

*β*is sufficiently large (

*β*>0.1) the performance does not critically depend on

*β*, as long as Φ

_{0}is kept optimized. An increase in

*β*, however, results in a growing sensitivity on Φ

_{0}. In the absence of feedback (

*μ*=0), the system’s performance significantly degrades, with the best classification yielding a WER of 1.84 %. Removing the delayed feedback strips the system of its memory, which is thus proven to be beneficial for successful spoken digit classification using our setup. Figure 4(c) shows the WER and margin as a function of Φ

_{0}for

*β*= 0.3 and

*ρ*≃

*π*in more detail. Error bars are extracted from three independent measurements, repeated under identical experimental conditions. It can be seen that good performance is not limited to a single point, with a WER remaining below 0.5% for the range 0.75

*π*≤ Φ

_{0}≤ 0.95

*π*.

24. A. S. Weigend and N. A. Gershenfeld, “Time series prediction: Forecasting the future and understanding the past,” ftp://ftp.santafe.edu/pub/Time-Series/Competition (1993).

_{0}and therefore on the characteristics of the nonlinearity. For Φ

_{0}= 0.1

*π*we obtain the lowest prediction error with a NMSE= 0.124 ±4 ×10

^{−4}. For the task of time series prediction the system’s performance is optimized for Φ

_{0}being shifted further away from the local extrema in the response function, closer towards the inflection point. In addition, the system’s performance significantly degrades for these values of Φ

_{0}corresponding to the local extrema. This is different to the behavior obtained in the spoken digit recognition task, where at these values of Φ

_{0}the performance was not optimal, still the loss in performance was far less significant. We interpret this as a manifestation of the importance of the memory for the one-time-step prediction task, however, a small amount of nonlinearity is still required for obtaining good performance. To provide evidence that the performance indeed stems from the interplay of high-dimensional mapping and nonlinearity and not from the nonlinearity alone, we in addition plot the data obtained when disconnecting the feedback line (red points,

*μ*= 0). The lower performance without feedback loop (i.e. memory) is clearly visible. Data presented for

*β*= 0.2 shows consistently better optimal performance for Φ

_{0}<0.5

*π*, where the slope of Eq. (1) is positive. For the case of zero feedback the performance is almost symmetric around Φ

_{0}=0.5

*π*, again indicating that this effect might be connected to properties of the system’s memory. Timeseries prediction based on numerical methods achieved even lower prediction errors (below 1 % using echo state networks [25

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

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

## 4. Conclusion

## Acknowledgments

## References and links

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. |

2. | E. Abraham and S. D. Smith, “Optical bistability and related devices,” Rep. Prog. Phys. |

3. | J. L. O’Brien, “Optical quantum computing,” Science |

4. | H. J. Caulfield and S. Dolev, “Why future supercomputing requires optics,” Nat. Photonics |

5. | R. S. Tucker, “The role of optics in computing,” Nat. Photonics |

6. | D. A. B. Miller, “Correspondence to the editor,” Nat. Photonics |

7. | H. Jaeger and H. Haas, “Harnessing nonlinearity: predicting chaotic systems and saving energy in wireless communication,” Science |

8. | D. V. Buonomano and W. Maass, “State-dependent computations: Spatiotemporal processing in cortical networks,” Nat. Rev. Neurosci. |

9. | 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 |

10. | 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. |

11. | M. Rabinovich, R. Huerta, and G. Laurent, “Transient dynamics of neural processing,” Science |

12. | 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 |

13. | K. Vandoorne, J. Dambre, D. Verstraeten, B. Schrauwen, and P. Bienstman, “Parallel reservoir computing using optical amplifiers,” IEEE Trans. Neural Netw. |

14. | A. Neyer and E. Voges, “Dynamics of electrooptic bistable devices with delayed feedback,” IEEE J. Quantum Electron. |

15. | 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. |

16. | K. E. Callan, L. Illing, Z. Gao, D. J. Gauthier, and E. Schöll, “Broadband chaos generated by an optoelectronic oscillator,” Phys. Rev. Lett. |

17. | K. Ikeda, “Multiple-valued stationary state and its instability of the transmitted light by a ring cavity system,” Opt. Commun. |

18. | L. Larger and J. M. Dudley, “Optoelectronic chaos,” Nature |

19. | T. Erneux, L. Larger, M. W. Lee, and J. Goedgebuer, “Ikeda hopf bifurcation revisited,” Physica D |

20. | F. T. Arecchi, G. Giacomelli, A. Lapucci, and R. Meucci, “Two–dimensional representation of a delayed dynamical system,” Phys. Rev. A |

21. | D. Verstraeten, B. Schrauwen, D. Stroobandt, and J. Van Campenhout, “Isolated word recognition with the liquid state machine: a case study,” Inf. Process. Lett. |

22. | 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). |

23. | A. E. Hoerl and R. W. Kennard, “Ridge Regression: Applications to Nonorthogonal Problems” Technometrics |

24. | A. S. Weigend and N. A. Gershenfeld, “Time series prediction: Forecasting the future and understanding the past,” ftp://ftp.santafe.edu/pub/Time-Series/Competition (1993). |

25. | A. Rodan and P. Tino, “Minimum complexity echo state network,” IEEE Trans. Neural Netw. |

26. | L. J. Cao, “Support vector machines experts for time series forecasting,” Neurocomputing |

27. | D. Psaltis, D. Brady, X. G. Gu, and S. Lin, “Holography in artificial neural networks,” Nature |

28. | Y. Paquot, F. Duport, A. Smerieri, J. Dambre, B. Schrauwen, M. Haelterman, and S. Massar, “Optoelectronic Reservoir Computing,” http://arxiv.org/abs/1111.7219 |

**OCIS Codes**

(190.3100) Nonlinear optics : Instabilities and chaos

(200.3050) Optics in computing : Information processing

(250.4745) Optoelectronics : Optical processing devices

**ToC Category:**

Optics in Computing

**History**

Original Manuscript: October 24, 2011

Revised Manuscript: January 13, 2012

Manuscript Accepted: January 16, 2012

Published: January 27, 2012

**Citation**

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)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-3-3241

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### References

- 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]
- E. Abraham and S. D. Smith, “Optical bistability and related devices,” Rep. Prog. Phys.45, 815–885 (1982). [CrossRef]
- J. L. O’Brien, “Optical quantum computing,” Science7, 1567–1570 (2007). [CrossRef]
- H. J. Caulfield and S. Dolev, “Why future supercomputing requires optics,” Nat. Photonics4, 261 (2010). [CrossRef]
- R. S. Tucker, “The role of optics in computing,” Nat. Photonics4, 405 (2010). [CrossRef]
- D. A. B. Miller, “Correspondence to the editor,” Nat. Photonics4, 406 (2010). [CrossRef]
- H. Jaeger and H. Haas, “Harnessing nonlinearity: predicting chaotic systems and saving energy in wireless communication,” Science304, 78–80 (2004). [CrossRef] [PubMed]
- D. V. Buonomano and W. Maass, “State-dependent computations: Spatiotemporal processing in cortical networks,” Nat. Rev. Neurosci.10, 113–125 (2009). [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,” Chaos20, 037101 (2010). [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]
- M. Rabinovich, R. Huerta, and G. Laurent, “Transient dynamics of neural processing,” Science321, 48–50 (2008). [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. Express16, 11182–11192 (2008). [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. Neyer and E. Voges, “Dynamics of electrooptic bistable devices with delayed feedback,” IEEE J. Quantum Electron.18, 2009–2015 (1982). [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]
- 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]
- 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]
- L. Larger and J. M. Dudley, “Optoelectronic chaos,” Nature465, 41–42 (2010). [CrossRef] [PubMed]
- T. Erneux, L. Larger, M. W. Lee, and J. Goedgebuer, “Ikeda hopf bifurcation revisited,” Physica D194, 49–64 (2004). [CrossRef]
- F. T. Arecchi, G. Giacomelli, A. Lapucci, and R. Meucci, “Two–dimensional representation of a delayed dynamical system,” Phys. Rev. A45, R4225–R4228 (1993). [CrossRef]
- 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]
- 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).
- A. E. Hoerl and R. W. Kennard, “Ridge Regression: Applications to Nonorthogonal Problems” Technometrics12, 69–82 (1970). [CrossRef]
- A. S. Weigend and N. A. Gershenfeld, “Time series prediction: Forecasting the future and understanding the past,” ftp://ftp.santafe.edu/pub/Time-Series/Competition (1993).
- A. Rodan and P. Tino, “Minimum complexity echo state network,” IEEE Trans. Neural Netw.22, 131–144 (2011). [CrossRef]
- L. J. Cao, “Support vector machines experts for time series forecasting,” Neurocomputing51, 321–339 (2003). [CrossRef]
- D. Psaltis, D. Brady, X. G. Gu, and S. Lin, “Holography in artificial neural networks,” Nature343, 325–330 (1990). [CrossRef] [PubMed]
- Y. Paquot, F. Duport, A. Smerieri, J. Dambre, B. Schrauwen, M. Haelterman, and S. Massar, “Optoelectronic Reservoir Computing,” http://arxiv.org/abs/1111.7219

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