Abstract

The operation of an optoelectronic dynamic neural model implementation is extended to higher frequencies. A simplified model of thermal effects in vertical-cavity surface-emitting lasers correctly predicts the qualitative changes in the nonlinear mapping implementation with frequency. Experiments and simulations show the expected resonance properties of this model neuron, along with the possibility of other dynamic effects in addition to the ones observed in the original FitzHugh–Nagumo equations. Results of optical coupling between two similar pulsing artificial neurons are also presented.

© 2007 Optical Society of America

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References

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    [CrossRef]
  3. J. Nagumo, S. Arimoto, and S. Yoshizawa, "An active pulse transmission line simulating nerve axon," Proc. IRE 50, 2061-2070 (1962).
    [CrossRef]
  4. F. Rieke, D. Warland, R. von Steveninck, and W. Bialek, Spikes: Exploring the Neural Code (MIT Press, 1997).
  5. R. van Rullen and S. Thorpe, "Rate coding versus temporal order coding: what the retinal ganglion cells tells the visual cortex," Neural Comput. 13, 1255-1283 (2001).
    [CrossRef] [PubMed]
  6. W. Maass, "Fast sigmoidal networks via spiking neurons," Neural Comput. 9, 279-304 (1997).
    [CrossRef] [PubMed]
  7. J. Hopfield, "Pattern recognition computation using action potential timing for stimulus representation," Nature 376, 33-36 (1995).
    [CrossRef] [PubMed]
  8. R. Sarpeshkar and M. O'Halloran, "Scalable hybrid computation with spikes," Neural Comput. 14, 2003-2038 (2002).
    [CrossRef] [PubMed]
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    [CrossRef]
  10. P. V. Mena, J. J. Morikuni, S.-M. Kang, A. V. Hartron, and K. W. Wyatt, "A simple rate-equation-based thermal VCSEL model," J. Lightwave Technol. 17, 865-872 (1999).
    [CrossRef]
  11. E. Izhikevich, "Neural excitability, spiking and bursting," Int. J. Bifurcation Chaos 10, 1171-1266 (2000).
    [CrossRef]
  12. E. Izhikevich, "Synchronization of elliptic bursters," SIAM Review 43, 315-344 (2001).
    [CrossRef]
  13. E. Izhikevich, "What model to use for cortical spiking neurons?" IEEE Trans. Neural Netw. 15, 1063-1070 (2004).
    [CrossRef] [PubMed]
  14. W. Maass, T. Natschlägger, and H. Markram, "A model for real-time computation in generic neural microcircuits," in Advances in Neural Information Processing Systems, S.Becker, S.Thrun, and K.Obermeyer, eds., vol. 15 (MIT Press, 2003).
  15. K. Johnson, D. McKnight, and I. Underwood, "Smart spatial light modulator," IEEE J. Quantum Electron. 29, 699-714 (1993).
    [CrossRef]
  16. J. Kane, "A smart-pixel-based feedforward neural network," IEEE Trans. Neural Netw. 9, 159-164 (1998).
    [CrossRef]
  17. K. Wagner and T. Slagle, "Optical competitive learning with VLSI liquid-crystal winner-take-all modulators," Appl. Opt. 32, 1408-1435 (1993).
    [CrossRef] [PubMed]
  18. Y. Liu, M. Strzelecka, J. Nohava, M. Hibbs-Brenner, and E. Towe, "Smart-pixel array technology for free-space optical interconnects," Proc. IEEE 88, 764-768 (2000).
    [CrossRef]
  19. H. Chen, D. Francis, T. Nguyen, W. Yuen, G. Li, and C. Chang-Hasnain, "Collimating diode laser beams from a large-area VCSEL-array using microlens array," IEEE Photon. Technol. Lett. 11, 506-508 (1999).
    [CrossRef]
  20. S. McCall, "Instability and regenerative pulsation phenomena in Fabry-Perot nonlinear optic media devices," Appl. Phys. Lett. 32, 284-286 (1978).
    [CrossRef]

2007

2004

E. Izhikevich, "What model to use for cortical spiking neurons?" IEEE Trans. Neural Netw. 15, 1063-1070 (2004).
[CrossRef] [PubMed]

2002

R. Sarpeshkar and M. O'Halloran, "Scalable hybrid computation with spikes," Neural Comput. 14, 2003-2038 (2002).
[CrossRef] [PubMed]

J. Gustavsson, J. Vukusic, J. Bengtsson, and A. Larson, "A comprehensive model for the modal dynamics of vertical-cavity surface-emitting lasers," IEEE J. Quantum Electron. 38, 203-212 (2002).
[CrossRef]

2001

R. van Rullen and S. Thorpe, "Rate coding versus temporal order coding: what the retinal ganglion cells tells the visual cortex," Neural Comput. 13, 1255-1283 (2001).
[CrossRef] [PubMed]

E. Izhikevich, "Synchronization of elliptic bursters," SIAM Review 43, 315-344 (2001).
[CrossRef]

2000

Y. Liu, M. Strzelecka, J. Nohava, M. Hibbs-Brenner, and E. Towe, "Smart-pixel array technology for free-space optical interconnects," Proc. IEEE 88, 764-768 (2000).
[CrossRef]

E. Izhikevich, "Neural excitability, spiking and bursting," Int. J. Bifurcation Chaos 10, 1171-1266 (2000).
[CrossRef]

1999

P. V. Mena, J. J. Morikuni, S.-M. Kang, A. V. Hartron, and K. W. Wyatt, "A simple rate-equation-based thermal VCSEL model," J. Lightwave Technol. 17, 865-872 (1999).
[CrossRef]

H. Chen, D. Francis, T. Nguyen, W. Yuen, G. Li, and C. Chang-Hasnain, "Collimating diode laser beams from a large-area VCSEL-array using microlens array," IEEE Photon. Technol. Lett. 11, 506-508 (1999).
[CrossRef]

1998

J. Kane, "A smart-pixel-based feedforward neural network," IEEE Trans. Neural Netw. 9, 159-164 (1998).
[CrossRef]

1997

W. Maass, "Fast sigmoidal networks via spiking neurons," Neural Comput. 9, 279-304 (1997).
[CrossRef] [PubMed]

1995

J. Hopfield, "Pattern recognition computation using action potential timing for stimulus representation," Nature 376, 33-36 (1995).
[CrossRef] [PubMed]

1993

K. Wagner and T. Slagle, "Optical competitive learning with VLSI liquid-crystal winner-take-all modulators," Appl. Opt. 32, 1408-1435 (1993).
[CrossRef] [PubMed]

K. Johnson, D. McKnight, and I. Underwood, "Smart spatial light modulator," IEEE J. Quantum Electron. 29, 699-714 (1993).
[CrossRef]

1978

S. McCall, "Instability and regenerative pulsation phenomena in Fabry-Perot nonlinear optic media devices," Appl. Phys. Lett. 32, 284-286 (1978).
[CrossRef]

1962

J. Nagumo, S. Arimoto, and S. Yoshizawa, "An active pulse transmission line simulating nerve axon," Proc. IRE 50, 2061-2070 (1962).
[CrossRef]

1961

R. FitzHugh, "Impulses and physiological states in theoretical models of nerve membrane," Biophy. J. 1, 445-466 (1961).
[CrossRef]

Appl. Opt.

Appl. Phys. Lett.

S. McCall, "Instability and regenerative pulsation phenomena in Fabry-Perot nonlinear optic media devices," Appl. Phys. Lett. 32, 284-286 (1978).
[CrossRef]

Biophy. J.

R. FitzHugh, "Impulses and physiological states in theoretical models of nerve membrane," Biophy. J. 1, 445-466 (1961).
[CrossRef]

IEEE J. Quantum Electron.

J. Gustavsson, J. Vukusic, J. Bengtsson, and A. Larson, "A comprehensive model for the modal dynamics of vertical-cavity surface-emitting lasers," IEEE J. Quantum Electron. 38, 203-212 (2002).
[CrossRef]

K. Johnson, D. McKnight, and I. Underwood, "Smart spatial light modulator," IEEE J. Quantum Electron. 29, 699-714 (1993).
[CrossRef]

IEEE Photon. Technol. Lett.

H. Chen, D. Francis, T. Nguyen, W. Yuen, G. Li, and C. Chang-Hasnain, "Collimating diode laser beams from a large-area VCSEL-array using microlens array," IEEE Photon. Technol. Lett. 11, 506-508 (1999).
[CrossRef]

IEEE Trans. Neural Netw.

J. Kane, "A smart-pixel-based feedforward neural network," IEEE Trans. Neural Netw. 9, 159-164 (1998).
[CrossRef]

E. Izhikevich, "What model to use for cortical spiking neurons?" IEEE Trans. Neural Netw. 15, 1063-1070 (2004).
[CrossRef] [PubMed]

Int. J. Bifurcation Chaos

E. Izhikevich, "Neural excitability, spiking and bursting," Int. J. Bifurcation Chaos 10, 1171-1266 (2000).
[CrossRef]

J. Lightwave Technol.

Nature

J. Hopfield, "Pattern recognition computation using action potential timing for stimulus representation," Nature 376, 33-36 (1995).
[CrossRef] [PubMed]

Neural Comput.

R. Sarpeshkar and M. O'Halloran, "Scalable hybrid computation with spikes," Neural Comput. 14, 2003-2038 (2002).
[CrossRef] [PubMed]

R. van Rullen and S. Thorpe, "Rate coding versus temporal order coding: what the retinal ganglion cells tells the visual cortex," Neural Comput. 13, 1255-1283 (2001).
[CrossRef] [PubMed]

W. Maass, "Fast sigmoidal networks via spiking neurons," Neural Comput. 9, 279-304 (1997).
[CrossRef] [PubMed]

Proc. IEEE

Y. Liu, M. Strzelecka, J. Nohava, M. Hibbs-Brenner, and E. Towe, "Smart-pixel array technology for free-space optical interconnects," Proc. IEEE 88, 764-768 (2000).
[CrossRef]

Proc. IRE

J. Nagumo, S. Arimoto, and S. Yoshizawa, "An active pulse transmission line simulating nerve axon," Proc. IRE 50, 2061-2070 (1962).
[CrossRef]

SIAM Review

E. Izhikevich, "Synchronization of elliptic bursters," SIAM Review 43, 315-344 (2001).
[CrossRef]

Other

W. Maass, T. Natschlägger, and H. Markram, "A model for real-time computation in generic neural microcircuits," in Advances in Neural Information Processing Systems, S.Becker, S.Thrun, and K.Obermeyer, eds., vol. 15 (MIT Press, 2003).

F. Rieke, D. Warland, R. von Steveninck, and W. Bialek, Spikes: Exploring the Neural Code (MIT Press, 1997).

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Figures (13)

Fig. 1
Fig. 1

Experimental setup for the neural model. The RLC circuit implements the linear part of the dynamics, while a birefringence filter provides a nonlinear map from driving voltage to detected optical signal. VCSEL: LaserMate VCT-A85A42-S. PBS: Halbo air-spaced 90° Glan-Taylor. Double-passed birefringent crystal: 3   cm × 1   cm × 0 .5   cm LiNbO 3 . Photodetector: Thorlabs, Newton, New Jersey, PDA155. In most experiments reported here, R 1 = 47 Ω , R 2 = 10 Ω , C = 3.9   nF , L = 5.6   μH .

Fig. 2
Fig. 2

Schematic view of the simple thermal model for representing the dynamics of the wavelength modulation, necessary to predict the system behavior at frequencies close to or above the thermal cut-off frequency. A simple electrical model of the diode is used, along with linear electro-optic conversion and first order thermal integration.

Fig. 3
Fig. 3

Comparison of simple thermal model and experiment at different frequencies. A large sinusoidal signal was applied to the system in open loop. Parameters as in Table 1. Low-frequency ( 1   kHz ) results used to adjust some parameters. See text for details.

Fig. 4
Fig. 4

(a) Simulation result for higher frequency pulsing. τ v = 0.15   μs , τ w = 0.6   μs . (b) Experimental waveform. Time scale 0 .5   μs / div .

Fig. 5
Fig. 5

Bursting behavior obtained with the faster electronic feedback, for a bias value slightly above the VCSEL threshold. τ v = 0.15   μs , τ w = 1   μs , A = 4 , bias tee low frequency cutoff at 100   kHz , remaining parameters as in Table 1.

Fig. 6
Fig. 6

Simulations showing bursts of activity, with a 100   kHz high-pass filter in the feedback loop and additive Gaussian noise, σ = 1   mV .

Fig. 7
Fig. 7

Bursting in response to a long optical pulse. (a) Simulation. (b) Experimental results. The detected signal is a sum of the internally-generated optical signal and the external optical pulse. External optical pulse peak power 0 .7   mW .

Fig. 8
Fig. 8

Verification of the resonant behavior of the system. (a) Weak input pulses too far apart ( 5   μs ) to be integrated. (b) Same pulse shape, with the separation reduced to 0.5   μs . Strong responses to the last few pulses in the sequence are clearly seen.

Fig. 9
Fig. 9

Another test of resonant behavior. The driving waveform (repeated three times) has three closely spaced pulses ( 0 .1   μs separation), with little effect on output, followed by three pulses at resonant separation, as in Fig. 8, producing a strong response to the last pulse in the triplet.

Fig. 10
Fig. 10

Setup for the simultaneous implementation of the nonlinear transfer function for two separate sources. Lenses L3 and L4 reduce the separation between the collimated beams from the VCSELs. Remaining setup is the same as for the case of a single source. L1, L2: Kodak A414 molded glass aspheric, EFL = 3.3   mm , NA = 0.47 . L3: EFL = 250   mm , D = 50   mm . L4: EFL = 50   mm , D = 25   mm .

Fig. 11
Fig. 11

Results for the simultaneous implementation of the nonlinear map for two separate sources. Slow 1   kHz sinusoidal driving term applied in parallel to both VCSELs.

Fig. 12
Fig. 12

Effect of coupling with the receiving circuit bias adjusted to the the point of sporadic activity in the absence of coupling. (a) No coupling. (b) Coupling induces a larger number of spikes, but not phase-locking.

Fig. 13
Fig. 13

Effect of coupling with both circuits biased to instability. Optical signals were attenuated to get more uniform spiking. Time scale increased, in comparison with the other figures, to investigate phase-locking in the waveform detail. (a) Coupling off. (b) Coupling on. Receiving neuron waveform is much more regular. A form of antiphase locking seems to be occurring, with the peaks of the sending neuron waveform matching the troughs in the receiving circuit waveform.

Tables (1)

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Table 1 Parameters for the Thermal Model

Equations (14)

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V det = f [ V drv ] = g [ V drv ] sin 2 ( π V drv V T ϕ 0 ) .
τ v v ˙ ( t ) = f [ v ( t ) ] v ( t ) w ( t ) + u ( t ) ,
τ w w ˙ ( t ) = A v ( t ) w ( t ) B ,
i = H [ v V D ] ( V V D ) R S + R D ,
P Diode = i 2 R S ,
P Opt ( t ) = H [ i ( t ) i Th ] η ( i ( t ) i Th ) .
V Det ( t ) = f [ v , θ ] = G ( v ) sin 2 ( π θ θ T ϕ ) ,
τ θ θ ˙ ( t ) = R Thermal P Thermal ( t ) θ ( t ) ,
τ v v ˙ = v + g ( v ) f ( θ ) w + u
τ w w ˙ = w + A v
τ θ θ ˙ = θ + R Thermal h ( v ) .
C v ˙ = v + u + f [ v ] R 1 i i ,
L i i = R 2 i i + v V B ,
τ v = R 1 C , τ w = L R 2 , A = R 1 R 2 ,

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