Abstract

An optoelectronic implementation of a modified FitzHugh–Nagumo neuron model is proposed, analyzed, and experimentally demonstrated. The setup uses linear optics and linear electronics for implementing an optical wavelength-domain nonlinearity. The system attains instability through a bifurcation mechanism present in a class of neuron models, a fact that is shown analytically. The implementation exhibits basic features of neural dynamics including threshold, production of short pulses (or spikes), and refractoriness.

© 2007 Optical Society of America

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References

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  41. T. Takizawa, Y. Liu, and J. Ohtsubo, "Chaos in a feedback Fabry-Perot Interferometer," IEEE J. Quantum Electron. 30, 334-338 (1994).
    [CrossRef]
  42. J. Goedgebuer, L. Larger, and H. Porte, "Chaos in wavelength with a feedback tunable laser diode," Phys. Rev. E 57, 2795-2798 (1998).
    [CrossRef]
  43. J. N. Blakely, L. Illing, and D. J. Gauthier, "High-speed chaos in an optical feedback system with flexible timescales," IEEE J. Quantum Electron. 40, 299-305 (2004).
    [CrossRef]
  44. E. Mos, J. Hoppenbrouwers, M. Hill, M. Blüm, J. Schleipen, and H. de Waardt, "Optical neuron by use of a laser diode with injection seeding and external optical feedback," IEEE Trans. Neural Netw. 11, 968-976 (2000).
    [CrossRef]
  45. H. Porte and J. Goedgebuer, "Bistability in wavelength using an electro-optically tuned dye laser," Opt. Commun. 51, 331-336 (1984).
    [CrossRef]
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    [CrossRef]
  47. 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]
  48. C. Degen, I. Fischer, and W. Elsäßer, "Transverse modes in thermally detuned oxide-confined vertical-cavity surface-emitting lasers," Phys. Rev. A 63, 023817 (2001).
    [CrossRef]
  49. P. Mena, J. Morikuni, S. Kang, A. Hartron, and K. Wyatt, "A simple rate-equation-based thermal VCSEL model," J. Lightwave Technol. 17, 865-871 (1999).
    [CrossRef]
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    [CrossRef]
  51. Y. Kuznetsov, Elements of Applied Bifurcation Theory (Springer-Verlag, 1995).
  52. B. Linares-Barranco, E. Sánchez-Sinencio, A. Rodríguez-Vázquez, and J. Huertas, "A CMOS implementation of FitzHugh-Nagumo neuron model," IEEE J. Solid-State Circuits 26, 956-965 (1991).
    [CrossRef]

2007

A. Romariz and K. Wagner, "Tunable vertical cavity surface-emitting laser with feedback to implement a pulsed neural model. 2. High-frequency effects and optical coupling," Appl. Opt. 46,4746-4753 (2007).
[CrossRef] [PubMed]

2006

S. Lecoeuche and D. Tsaptsinos, "Novel applications of neural networks in engineering," Eng. Applic. Artif. Intell. 19, 719-720 (2006).
[CrossRef]

2004

J. N. Blakely, L. Illing, and D. J. Gauthier, "High-speed chaos in an optical feedback system with flexible timescales," IEEE J. Quantum Electron. 40, 299-305 (2004).
[CrossRef]

2002

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]

S. Draghici, "On the capabilities of neural networks using limited precision weights," Neural Networks 15, 395-414 (2002).
[CrossRef] [PubMed]

J. L. Johnson, "All-optical pulse generators for optical computing," in Proceedings of the International Topical Meeting on Optics in Computing, 195-197, Taipei, Taiwan, 2002.

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

P. Silveira, G. Pati, and K. Wagner, "Optical finite impulse response neural networks," Appl. Opt. 41, 4162-4180 (2002).
[CrossRef] [PubMed]

2001

Y. Frauel, G. Pauliat, A. Villing, and G. Roosen, "High-capacity photorefractive neural network implementing a Kohonen topological map," Appl. Opt. 40, 5162-5169 (2001).
[CrossRef]

S. Tang and J. Liu, "Chaotic pulsing and quasi-periodic route to chaos in a semiconductor laser with delayed optoelectronic feedback," IEEE J. Quantum Electron. 37, 329-336 (2001).
[CrossRef]

C. Degen, I. Fischer, and W. Elsäßer, "Transverse modes in thermally detuned oxide-confined vertical-cavity surface-emitting lasers," Phys. Rev. A 63, 023817 (2001).
[CrossRef]

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

2000

E. Mos, J. Hoppenbrouwers, M. Hill, M. Blüm, J. Schleipen, and H. de Waardt, "Optical neuron by use of a laser diode with injection seeding and external optical feedback," IEEE Trans. Neural Netw. 11, 968-976 (2000).
[CrossRef]

R. H. Hahnloser, R. Sarpeshkar, M. Mahowald, R. Douglas, and H. S. Seung, "Digital selection and analogue amplification coexist in a cortex-inspired silicon circuit," Nature 405, 947-951 (2000).
[CrossRef] [PubMed]

A. Giaquinta, M. Argentina, and M. Velarde, "A simple generalized excitability model mimicking salient features of neuron dynamics," Journal of Statistical Physics 101, 665-678 (2000).
[CrossRef]

1999

D. V. Buonomano and M. Merzenich, "A neural network model of temporal code generation and position-invariant pattern recognition," Neural Comput. 11, 103-116 (1999).
[CrossRef] [PubMed]

P. Mena, J. Morikuni, S. Kang, A. Hartron, and K. Wyatt, "A simple rate-equation-based thermal VCSEL model," J. Lightwave Technol. 17, 865-871 (1999).
[CrossRef]

1998

P. Moerland, E. Fiesler, and I. Saxena, "Discrete all-positive multilayer perceptrons for optical implementation," Opt. Eng. 37, 1305-1315 (1998).
[CrossRef]

J. Goedgebuer, L. Larger, and H. Porte, "Chaos in wavelength with a feedback tunable laser diode," Phys. Rev. E 57, 2795-2798 (1998).
[CrossRef]

1997

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

M. V. Tsodyks and H. Markram, "The neural code between neocortical pyramidal neurons depends on neurotransmitter release probability," Proc. Natl. Acad. Sci. U.S.A. 94, 719-723 (1997).
[CrossRef] [PubMed]

1996

W. Nakwaski, "Thermal aspects of efficient operation of vertical-cavity surface-emitting lasers," Opt. Quantum Electron. 28, 335-352 (1996).
[CrossRef]

1995

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

1994

1993

1991

B. Linares-Barranco, E. Sánchez-Sinencio, A. Rodríguez-Vázquez, and J. Huertas, "A CMOS implementation of FitzHugh-Nagumo neuron model," IEEE J. Solid-State Circuits 26, 956-965 (1991).
[CrossRef]

1990

R. Eckhorn, H. Reitboeck, M. Arndt, and P. Dicke, "Feature linking via synchronization among distributed assemblies: simulations of results from cat visual cortex," Neural Comput. 2, 293-307 (1990).
[CrossRef]

C. Peterson, S. Redfield, J. Keeler, and E. Hartman, "Optoelectronic implementation of multilayer neural networks in a single photorefractive crystal," Opt. Eng. 29, 359-368 (1990).
[CrossRef]

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

1988

D. Psaltis, D. Brady, and K. Wagner, "Adaptive optical networks using photorefractive crystals," Appl. Opt. 27, 334-341 (1988).

1987

1985

1984

H. Porte and J. Goedgebuer, "Bistability in wavelength using an electro-optically tuned dye laser," Opt. Commun. 51, 331-336 (1984).
[CrossRef]

1981

C. Morris and H. Lecar, "Voltage oscillations in the barnacle giant muscle fiber," Biophys. J. 35, 193-213 (1981).
[CrossRef] [PubMed]

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 models of nerve membrane," Biophys. J. 1, 445-466 (1961).
[CrossRef] [PubMed]

1952

A. L. Hodgkin and A. F. Huxley, "A quantitative description of ion currents and its applications to conduction and excitation in nerve membranes," J. Physiol. 117, 500-544 (1952).
[PubMed]

Appl. Opt.

Biophys. J.

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

C. Morris and H. Lecar, "Voltage oscillations in the barnacle giant muscle fiber," Biophys. J. 35, 193-213 (1981).
[CrossRef] [PubMed]

Eng. Applic. Artif. Intell.

S. Lecoeuche and D. Tsaptsinos, "Novel applications of neural networks in engineering," Eng. Applic. Artif. Intell. 19, 719-720 (2006).
[CrossRef]

IEEE J. Quantum Electron.

J. N. Blakely, L. Illing, and D. J. Gauthier, "High-speed chaos in an optical feedback system with flexible timescales," IEEE J. Quantum Electron. 40, 299-305 (2004).
[CrossRef]

S. Tang and J. Liu, "Chaotic pulsing and quasi-periodic route to chaos in a semiconductor laser with delayed optoelectronic feedback," IEEE J. Quantum Electron. 37, 329-336 (2001).
[CrossRef]

T. Takizawa, Y. Liu, and J. Ohtsubo, "Chaos in a feedback Fabry-Perot Interferometer," IEEE J. Quantum Electron. 30, 334-338 (1994).
[CrossRef]

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]

IEEE J. Solid-State Circuits

B. Linares-Barranco, E. Sánchez-Sinencio, A. Rodríguez-Vázquez, and J. Huertas, "A CMOS implementation of FitzHugh-Nagumo neuron model," IEEE J. Solid-State Circuits 26, 956-965 (1991).
[CrossRef]

IEEE Trans. Neural Netw.

E. Mos, J. Hoppenbrouwers, M. Hill, M. Blüm, J. Schleipen, and H. de Waardt, "Optical neuron by use of a laser diode with injection seeding and external optical feedback," IEEE Trans. Neural Netw. 11, 968-976 (2000).
[CrossRef]

J. Lightwave Technol.

J. Physiol.

A. L. Hodgkin and A. F. Huxley, "A quantitative description of ion currents and its applications to conduction and excitation in nerve membranes," J. Physiol. 117, 500-544 (1952).
[PubMed]

Journal of Statistical Physics

A. Giaquinta, M. Argentina, and M. Velarde, "A simple generalized excitability model mimicking salient features of neuron dynamics," Journal of Statistical Physics 101, 665-678 (2000).
[CrossRef]

Nature

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

R. H. Hahnloser, R. Sarpeshkar, M. Mahowald, R. Douglas, and H. S. Seung, "Digital selection and analogue amplification coexist in a cortex-inspired silicon circuit," Nature 405, 947-951 (2000).
[CrossRef] [PubMed]

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

Neural Comput.

D. V. Buonomano and M. Merzenich, "A neural network model of temporal code generation and position-invariant pattern recognition," Neural Comput. 11, 103-116 (1999).
[CrossRef] [PubMed]

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

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

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

R. Eckhorn, H. Reitboeck, M. Arndt, and P. Dicke, "Feature linking via synchronization among distributed assemblies: simulations of results from cat visual cortex," Neural Comput. 2, 293-307 (1990).
[CrossRef]

Neural Networks

S. Draghici, "On the capabilities of neural networks using limited precision weights," Neural Networks 15, 395-414 (2002).
[CrossRef] [PubMed]

Opt. Commun.

H. Porte and J. Goedgebuer, "Bistability in wavelength using an electro-optically tuned dye laser," Opt. Commun. 51, 331-336 (1984).
[CrossRef]

Opt. Eng.

P. Moerland, E. Fiesler, and I. Saxena, "Discrete all-positive multilayer perceptrons for optical implementation," Opt. Eng. 37, 1305-1315 (1998).
[CrossRef]

C. Peterson, S. Redfield, J. Keeler, and E. Hartman, "Optoelectronic implementation of multilayer neural networks in a single photorefractive crystal," Opt. Eng. 29, 359-368 (1990).
[CrossRef]

Opt. Lett.

Opt. Quantum Electron.

W. Nakwaski, "Thermal aspects of efficient operation of vertical-cavity surface-emitting lasers," Opt. Quantum Electron. 28, 335-352 (1996).
[CrossRef]

Phys. Rev. A

C. Degen, I. Fischer, and W. Elsäßer, "Transverse modes in thermally detuned oxide-confined vertical-cavity surface-emitting lasers," Phys. Rev. A 63, 023817 (2001).
[CrossRef]

Phys. Rev. E

J. Goedgebuer, L. Larger, and H. Porte, "Chaos in wavelength with a feedback tunable laser diode," Phys. Rev. E 57, 2795-2798 (1998).
[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]

Proc. Natl. Acad. Sci. U.S.A.

M. V. Tsodyks and H. Markram, "The neural code between neocortical pyramidal neurons depends on neurotransmitter release probability," Proc. Natl. Acad. Sci. U.S.A. 94, 719-723 (1997).
[CrossRef] [PubMed]

Other

M. Recce, "Encoding information in neuronal activity," in Pulsed Neural Networks, W. Maass and C. Bishop, eds., pp. 111-126 (MIT Press, 1999).

T. Sejnowski, "Neural pulse coding," in Pulsed Neural Networks, W. Maass and C. M. Bishop, eds. (MIT Press, 1999).

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

J. Rinzel and G. Ermentrout, "Analysis of neural excitability and oscillations," in Methods in Neuronal Modeling: From Synapses to Networks, I. Segev and J. Fleshman, eds., 135-169 (MIT Press, 1989).

C. Mead, Analog VLSI and Neural Systems (Addison-Wesley, 1989).
[CrossRef]

A. Murray, "Pulse-based computation in VLSI neural networks," in Pulsed Neural Networks, W. Maass and C. Bishop, eds., 87-107 (MIT Press, 1999).

L. Abbott and T. Kepler, "Model neurons: from Hodgkin-Huxley to Hopfield," in Statistical Mechanics of Neural Networks, L. Garrido, ed., 5-18 (Springer, 1990).
[CrossRef]

J. L. Johnson, "All-optical pulse generators for optical computing," in Proceedings of the International Topical Meeting on Optics in Computing, 195-197, Taipei, Taiwan, 2002.

H. Gibbs, Optical Bistability: Controlling Light with Light (Academic Press, 1985).

F. Hoppensteadt and E. Izhikevich, Weakly Connected Neural Networks (Springer, 1997).
[CrossRef]

Y. Kuznetsov, Elements of Applied Bifurcation Theory (Springer-Verlag, 1995).

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Figures (7)

Fig. 1
Fig. 1

(Color online) Basic idea of wavelength nonlinearity implementation. A birefringent material placed between crossed polarizers implements a nonlinear function of wavelength, in its turn a function of input in a tunable source.

Fig. 2
Fig. 2

Nonlinear mapping from driver voltage to detected signal at low frequency, obtained with single-mode VCSEL VCT-A85A42-S.

Fig. 3
Fig. 3

(a) Typical time evolution of dynamical variables in the FN model under constant input. (b) Corresponding trajectory in the state plane. Continuous line: trajectory described by the system. Dotted lines: fast variable (v) nullcline. Dashed line: w nullcline. Nonlinear term f [ v ] = v ( 1 v ) ( v 0.5 ) . Constant input u = 0.2 .

Fig. 4
Fig. 4

Diagram showing features of the equilibrium points in parameter space, for a fixed ratio of time constants K = 10 . The parabola divides the region of complex eigenvalues from that of real eigenvalues. The vertical dashed line defines the transitions from stable to unstable behavior. With the exception of the small region for A 0 , all transitions to instability occur inside the region of complex eigenvalues, a signature of the Andronov–Hopf bifurcation, which is the bifurcation of the original FN system.

Fig. 5
Fig. 5

Overall experimental setup with electronic feedback. VCSEL: LaserMate VCT-A85A42-S. PBS: Halbo air-spaced 90° Glan-Taylor. Reflective double-pass through 3   cm × 1   cm × 0 .5 cm LiNbO 3 crystal (slightly misaligned to avoid direct optical feedback). Photodetector: Thor Labs PDA155.

Fig. 6
Fig. 6

(Color online) (a) Simulation results showing sustained periodic oscillations for constant input u = 0.12 . Time in units of τ v . Recovery gain A = 10 . Recovery bias B = 1.2 . Recovery time constant τ w = 10 τ v , threshold V Th = 0.123   V , sinusoidal map period V T = 0.054   V , sinusoidal map amplitude G C = 0.075 . (b) Experimental results. From top to bottom: driver voltage, recovery variable and detected optical power through birefringent wavelength discriminating filter. Time scale 0.5   ms / div . Parameters: τ v = 0.1   ms , τ w = 1   ms .

Fig. 7
Fig. 7

(Color online) (a) Simulated response to a train of pulses. Time in units of τ v . Parameters: τ w = 10 τ v , A = 10 , B = 1.2 . (b) Experimental results. Parameters: τ v = 0.1   ms , τ w = 1   ms . Time scale: 1   ms / div .

Equations (88)

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

P det P = sin 2 ( π Δ n L λ ) = sin 2 ( π Δ n L ν c ) ,
Δ n ( λ )
P ( i )
λ ( i )
V det = R I det = R P ( i ) sin 2 ( π Δ n L λ ) = R P ( i ) sin 2 ( π Δ n L ν c ) ,
λ 0
λ ( i ) = λ 0 + α i
ν = ν 0 + α i
V det = f [ i ] = G P sin 2 ( π Δ n L α i λ 0 2 ϕ 0 ) ,
ϕ 0 = π Δ n L / λ 0
G = R
i ( t )
P ( i )
E g = E g 0 + E g 1 ( T T 0 )   + ⋯ 
P ( i )
1   μs
V det = f [ V drv ] = g [ V drv ] sin 2 ( π V drv V T ϕ 0 ) ,
g [ V drv ] = G H [ V drv V Th ] ( V drv V Th ) ,
H [ ]
V T
g [ V drv ]
850 nm
0 .51   nm / mA
LiNbO 3
f [ v ]
τ v v ˙ ( t ) = f [ v ( t ) ] w ( t ) + u ( t ) ,
τ w w ˙ ( t ) = A v ( t ) B w ( t ) ,
u ( t )
τ v
τ w
τ v τ w
f [ v ]
f [ v ] = v ( 1 v ) ( v a )
0 < a < 1
f [ v ]
f [ v ]
v ˙ = 0
w ˙ = 0
v ˙ = 0
f [ v ]
w ˙ = 0
f [ v ]
τ v v ˙ ( t ) = f [ v ( t ) ] v ( t ) w ( t ) + u ( t ) ,
τ w w ˙ ( t ) = A v ( t ) w ( t ) B ,
V drv
f [ v ]
v ( t )
f [ v ] = 0
J = [ v ˙ v v ˙ w w ˙ v w ˙ w ] = [ f ( v ) 1 τ v 1 τ v A τ w 1 τ w ] ,
A > [ 1 + K ( 1 f ( v 0 ) ) ] 2 4 K + f ( v 0 ) 1 ,
K = τ w / τ v
f ( v 0 )
v 0
v ˙ = 0
f ( v 0 ) > 1 K + 1 ,
f ( v 0 )
( v 0 , w 0 )
f ( v 0 )
K = 10
A 0
τ v = 0.1   ms
τ w = 1   ms
A = 10
f [ v ]
f [ v ] = v ( 1 v ) ( v 0.5 )
u = 0.2
K = 10
A 0
3   cm × 1   cm × 0 .5 cm
LiNbO 3
u = 0.12
τ v
A = 10
B = 1.2
τ w = 10 τ v
V Th = 0.123   V
V T = 0.054   V
G C = 0.075
0.5   ms / div
τ v = 0.1   ms
τ w = 1   ms
τ v
τ w = 10 τ v
A = 10
B = 1.2
τ v = 0.1   ms
τ w = 1   ms
1   ms / div

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