Abstract

An optical analog of the neural networks involved in sensory processing consists of a dispersive medium with gain in a narrow band of wave numbers, cubic saturation, and a memory nonlinearity that may imprint multiplexed volume holographic gratings. Coupled-mode equations are derived for the time evolution of a wave scattered off these gratings; eigenmodes of the coupling matrix κ saturate preferentially, implementing stable reconstruction of a stored memory from partial input and associative reconstruction of a set of stored memories. Multiple scattering in the volume reconstructs cycles of associations that compete for saturation. Input of a new pattern switches all the energy into the cycle containing a representative of that pattern; the system thus acts as an abstract categorizer with multiple basins of stability. The advantages that an imprintable medium with gain biased near the critical point has over either the holographic or the adaptive matrix associative paradigms are (1) images may be input as noncoherent distributions which nucleate long-range critical modes within the medium, and (2) the interaction matrix κ of critical modes is full, thus implementing the sort of full connectivity needed for associative reconstruction in a physical medium that is only locally connected, such as a nonlinear crystal.

© 1986 Optical Society of America

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References

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  1. H. R. Wilson, J. D. Cowan, “A Mathematical Theory of the Functional Dynamics of Cortical and Thalamic Nervous Tissue,” Kybernetic 13, 55 (1973).
    [CrossRef]
  2. G. B. Ermentrout, J. D. Cowan, “A Mathematical Theory of Visual Hallucination Patterns,” Biol. Cybernet. 34, 137 (1979).
    [CrossRef]
  3. M. S. Cohen, “Interacting Nonlinear Waves in a Neural Continuum Model: Associative Memory and Pattern Recognition,” in Wave Phenomena: Modern Theory and Applications, C. Rogers, T. B. Moodie, Eds. (North-Holland, Amsterdam, 1984).
    [CrossRef]
  4. M. S. Cohen, “Distributed Computation in Neural Networks and Their Optical Analogs,” Proc. Soc. Photo-Opt. Instrum. Eng. 540, 566.
  5. J. A. Anderson, “Cognitive and Psychological Computation with Neural Models,” IEEE Trans. Syst., Man, Cybernet. SMC-13, 799 (1983).
    [CrossRef]
  6. R. Fisher, Ed. Optical Phase Conjugation (Academic, New York, 1983).
  7. M. S. Cohen, “Channeled Bifurcations and Learning in a Neural Network Model,” CRL Memorandum MCSS-86-43, New Mexico State U., Las Cruces, NM (1986).
  8. B. D. Hassard, N. D. Kazarinoff, Y-H. Wan, Theory and Applications of Hopf Bifurcation, London Mathematical Society Lecture Note Series 41 (Cambridge U. Pr., London, 1981).
  9. A. Newell, “Bifurcation and Nonlinear Focusing,” in Pattern Formation and Pattern Recognition, Vol. 5 (Springer-Verlag, New York, 1980), pp. 244–265.
    [CrossRef]
  10. J. Feinberg, “Self-Pumped, Continuous Wave Phase Conjugator Using Internal Reflection,” Opt. Lett. 7, 486 (1979).
    [CrossRef]
  11. T. Kohonen, Self Organization and Associative Memory (Springer-Verlag, New York, 1984).
  12. J. Hopfield, “Neural Networks and Physical Systems with Emergent Collective Abilities,” Proc. Natl. Acad. Sci. USA 79, 2554 (1982).
    [CrossRef] [PubMed]

1983

J. A. Anderson, “Cognitive and Psychological Computation with Neural Models,” IEEE Trans. Syst., Man, Cybernet. SMC-13, 799 (1983).
[CrossRef]

1982

J. Hopfield, “Neural Networks and Physical Systems with Emergent Collective Abilities,” Proc. Natl. Acad. Sci. USA 79, 2554 (1982).
[CrossRef] [PubMed]

1979

J. Feinberg, “Self-Pumped, Continuous Wave Phase Conjugator Using Internal Reflection,” Opt. Lett. 7, 486 (1979).
[CrossRef]

G. B. Ermentrout, J. D. Cowan, “A Mathematical Theory of Visual Hallucination Patterns,” Biol. Cybernet. 34, 137 (1979).
[CrossRef]

1973

H. R. Wilson, J. D. Cowan, “A Mathematical Theory of the Functional Dynamics of Cortical and Thalamic Nervous Tissue,” Kybernetic 13, 55 (1973).
[CrossRef]

Anderson, J. A.

J. A. Anderson, “Cognitive and Psychological Computation with Neural Models,” IEEE Trans. Syst., Man, Cybernet. SMC-13, 799 (1983).
[CrossRef]

Cohen, M. S.

M. S. Cohen, “Interacting Nonlinear Waves in a Neural Continuum Model: Associative Memory and Pattern Recognition,” in Wave Phenomena: Modern Theory and Applications, C. Rogers, T. B. Moodie, Eds. (North-Holland, Amsterdam, 1984).
[CrossRef]

M. S. Cohen, “Distributed Computation in Neural Networks and Their Optical Analogs,” Proc. Soc. Photo-Opt. Instrum. Eng. 540, 566.

M. S. Cohen, “Channeled Bifurcations and Learning in a Neural Network Model,” CRL Memorandum MCSS-86-43, New Mexico State U., Las Cruces, NM (1986).

Cowan, J. D.

G. B. Ermentrout, J. D. Cowan, “A Mathematical Theory of Visual Hallucination Patterns,” Biol. Cybernet. 34, 137 (1979).
[CrossRef]

H. R. Wilson, J. D. Cowan, “A Mathematical Theory of the Functional Dynamics of Cortical and Thalamic Nervous Tissue,” Kybernetic 13, 55 (1973).
[CrossRef]

Ermentrout, G. B.

G. B. Ermentrout, J. D. Cowan, “A Mathematical Theory of Visual Hallucination Patterns,” Biol. Cybernet. 34, 137 (1979).
[CrossRef]

Feinberg, J.

Hassard, B. D.

B. D. Hassard, N. D. Kazarinoff, Y-H. Wan, Theory and Applications of Hopf Bifurcation, London Mathematical Society Lecture Note Series 41 (Cambridge U. Pr., London, 1981).

Hopfield, J.

J. Hopfield, “Neural Networks and Physical Systems with Emergent Collective Abilities,” Proc. Natl. Acad. Sci. USA 79, 2554 (1982).
[CrossRef] [PubMed]

Kazarinoff, N. D.

B. D. Hassard, N. D. Kazarinoff, Y-H. Wan, Theory and Applications of Hopf Bifurcation, London Mathematical Society Lecture Note Series 41 (Cambridge U. Pr., London, 1981).

Kohonen, T.

T. Kohonen, Self Organization and Associative Memory (Springer-Verlag, New York, 1984).

Newell, A.

A. Newell, “Bifurcation and Nonlinear Focusing,” in Pattern Formation and Pattern Recognition, Vol. 5 (Springer-Verlag, New York, 1980), pp. 244–265.
[CrossRef]

Wan, Y-H.

B. D. Hassard, N. D. Kazarinoff, Y-H. Wan, Theory and Applications of Hopf Bifurcation, London Mathematical Society Lecture Note Series 41 (Cambridge U. Pr., London, 1981).

Wilson, H. R.

H. R. Wilson, J. D. Cowan, “A Mathematical Theory of the Functional Dynamics of Cortical and Thalamic Nervous Tissue,” Kybernetic 13, 55 (1973).
[CrossRef]

Biol. Cybernet.

G. B. Ermentrout, J. D. Cowan, “A Mathematical Theory of Visual Hallucination Patterns,” Biol. Cybernet. 34, 137 (1979).
[CrossRef]

IEEE Trans. Syst., Man, Cybernet.

J. A. Anderson, “Cognitive and Psychological Computation with Neural Models,” IEEE Trans. Syst., Man, Cybernet. SMC-13, 799 (1983).
[CrossRef]

Kybernetic

H. R. Wilson, J. D. Cowan, “A Mathematical Theory of the Functional Dynamics of Cortical and Thalamic Nervous Tissue,” Kybernetic 13, 55 (1973).
[CrossRef]

Opt. Lett.

Proc. Natl. Acad. Sci. USA

J. Hopfield, “Neural Networks and Physical Systems with Emergent Collective Abilities,” Proc. Natl. Acad. Sci. USA 79, 2554 (1982).
[CrossRef] [PubMed]

Proc. Soc. Photo-Opt. Instrum. Eng.

M. S. Cohen, “Distributed Computation in Neural Networks and Their Optical Analogs,” Proc. Soc. Photo-Opt. Instrum. Eng. 540, 566.

Other

M. S. Cohen, “Interacting Nonlinear Waves in a Neural Continuum Model: Associative Memory and Pattern Recognition,” in Wave Phenomena: Modern Theory and Applications, C. Rogers, T. B. Moodie, Eds. (North-Holland, Amsterdam, 1984).
[CrossRef]

R. Fisher, Ed. Optical Phase Conjugation (Academic, New York, 1983).

M. S. Cohen, “Channeled Bifurcations and Learning in a Neural Network Model,” CRL Memorandum MCSS-86-43, New Mexico State U., Las Cruces, NM (1986).

B. D. Hassard, N. D. Kazarinoff, Y-H. Wan, Theory and Applications of Hopf Bifurcation, London Mathematical Society Lecture Note Series 41 (Cambridge U. Pr., London, 1981).

A. Newell, “Bifurcation and Nonlinear Focusing,” in Pattern Formation and Pattern Recognition, Vol. 5 (Springer-Verlag, New York, 1980), pp. 244–265.
[CrossRef]

T. Kohonen, Self Organization and Associative Memory (Springer-Verlag, New York, 1984).

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Figures (3)

Fig. 1
Fig. 1

Dispersion relation k2(μ,). kc2k2(μc,c) and [(∂k2)/(∂ω)]c are both real. (Decaying waves broken, growing waves solid lines.)

Fig. 2
Fig. 2

Critical modes ϕ1 and ϕ2 excited by a projected image η consist of spherical waves diverging from image points p1 and p2. They form a hyperboloidal interference grating (broken lines) with foci p1,p2. A spherical wave from p1 is then incident on the grating producing a reflected wave apparently diverging from p2. The Bragg conditions (klkm) + kj = kp hold locally: grating vector + incident wave vector = reflected wave vector.

Fig. 3
Fig. 3

Dynamical behavior of the system. The system has been trained by exposure to the modal vectors V 1 H = [ 1 / 18 ] ( 1 , 4 , 1 , 0 , 0 , 0 ) (dark) and V 2 H = [ 1 / 18 ] ( 0 , 0 , 0 , 4 , 1 , 1 ) (light) imprinting the modal autocorrelation matrix κ = (V1V1H + V2V2H) (C = 1). The state vector V(T) = Aj(T)ϕj evolves according to the coupled-mode equations (12) with g = 0.1. Input patterns η set the initial components of V by 〈η,ϕj〉 = Aj(0). The simulation shows the evolution of the 6-D state vector V projected into 3-D space; each dot represents a time step. For the parameter value λ = (1 − M)C = −1, basins of attraction of the full nonlinear system (saturated output states) lie along the eigenvectors V1 and V2 of κ, V ˜ = αV1 or αV2, |α|2 = C/g = 10. Initial states V(0) evolve to the output state VI for which 〈V(0),VI〉 was maximal corresponding to recognition of an input pattern as belonging either to class {V1} or {V2}.

Equations (34)

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i 2 ( k 2 ω ) v T = [ 2 + k 2 ( μ , i ω c ) ] v - h v 2 v ,
d k 2 d T = c v 2 - β [ k 2 - k 2 ( μ ) ] ,
D { v 2 ; T } = c exp ( - β T ) - T exp ( β T ) v ( T ) 2 d T
k 2 ( μ ) k 2 ( μ c ) + ( μ - μ c ) ( k 2 μ ) c k c 2 + 2 μ 2 ( k 2 μ ) c
i 2 ( k 2 ω ) c v T = 2 v + [ k c 2 + 2 μ 2 ( k 2 μ ) c ] v + D { v 2 ; T } v - h v 2 v
v ( x , T ) = v 1 ( x , T ) + 2 v 2 ( x , T ) + 3 v 3 ( x , T ) +
2 v 1 + k c 2 v 1 = 0 ,
v 1 ( x , T ) = j = 1 N A j ( T ) ϕ j ( x ) ,
( 2 + k c 2 ) v 3 = i ( k 2 ω ) c j = 1 N d A j d T exp ( i k j · x ) - μ 2 ( k 2 μ ) c j A j exp ( i k j · x ) + h a = 1 N b = 1 N p = 1 N A a A ¯ b A p exp i ( k a - k b + k p ) · x + l = 1 n m = 1 n K l m ( T ) exp [ i ( k l - k m ) · x ] × p = 1 N A p exp ( i k p · x )
K l m ( T ) = c exp ( - β T ) - T exp ( - β T ) A l ( T ) A ¯ m ( T ) d T
k l - k m + k p = k j ,             l , m , = 1 n ,             p , j = 1 , , N ,
k a - k b + k p = k j ,             a , b , p , j = 1 , , N .
k t = k m and k j = k p , or k p = k m and k j = k l
k a = k b and k j = k p , or k p = k b and k j = k a
d A l d T = ( λ + m = 1 n κ m m ) A l + m = 1 n κ l m A m - 2 g A l a = 1 N A a 2 + g A l 2 A l - κ l l A l             for l = 1 , , n ,
d A α d T = λ A α - 2 g A α a = 1 n A a 2 + g A α 2 A α - g ( nonplanar A a A ¯ b A β ) + κ l m A β             for α = n + 1 , , N .
d A j d T = ( λ + m = 1 n κ m m ) A j + m = 1 n κ j m A m - 2 g A j m = 1 n A m 2 + g A j 2 A j - κ j j A j             for j = 1 , , n ,
d d t V = ( λ + T r κ ) V + κ V - 2 g V 2 V + f ,
A j ( 0 ) = ϕ j , η = Ω ϕ j ( x ) η ( x ) d x ,
V s ( 0 ) = j = 1 n A j s ( 0 ) ϕ j .
κ l m ( T ) ~ exp ( - β T ) 0 δ exp ( β T ) A l s ( T ) A ¯ m s ( T ) d T .
κ l m = C A l s A ¯ m s or κ = C V s V s H ,
κ = C I = 1 M V I V I H
V ˜ = J = 1 K α J V J
V I , V J a = j = 1 n A j I A j J .
V ˜ = α V I ,             α 2 = C / g .
κ = C I = 1 M ( U I + W I ) ( U I + W I ) H .
2 ψ - 1 c 2 ( μ ) 2 t 2 ψ = 0
2 v + k 2 ( μ , i ω ) v = 0
k 2 ( μ , i ω ) = ω 2 c 2 ( μ )
t + 2 T i ω c + 2 T : t { v ( x , t ) exp ( i ω c t ) } = ( i ω c + 2 T ) v ( x , T ) exp ( i ω c t ) .
k 2 ( μ , i ω c + 2 T ) v [ k 2 ( μ , i ω c ) - i 2 ( k 2 ω ) c T ] v .
h v exp ( i ω c t ) v ¯ exp ( - i ω c t ) v exp ( i ω c t ) = h v 2 v exp ( i ω c t )
i 2 ( k 2 ω ) c v T = [ 2 + k 2 ( μ , i ω c ) ] v - h v 2 v ,

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