Abstract

The linking-field neural network model of Eckhorn et al. [Neural Comput. 2, 293–307 (1990)] was introduced to explain the experimentally observed synchronous activity among neural assemblies in the cat cortex induced by feature-dependent visual activity. The model produces synchronous bursts of pulses from neurons with similar activity, effectively grouping them by phase and pulse frequency. It gives a basic new function: grouping by similarity. The synchronous bursts are obtained in the limit of strong linking strengths. The linking-field model in the limit of moderate-to-weak linking characterized by few if any multiple bursts is investigated. In this limit dynamic, locally periodic traveling waves exist whose time signal encodes the geometrical structure of a two-dimensional input image. The signal can be made insensitive to translation, scale, rotation, distortion, and intensity. The waves transmit information beyond the physical interconnect distance. The model is implemented in an optical hybrid demonstration system. Results of the simulations and the optical system are presented.

© 1994 Optical Society of America

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References

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  1. R. Eckhorn, H. J. Reitboeck, M. Arndt, P. Dicke, “Feature linking via synchronization among distributed assemblies: simulations of results from cat cortex,” Neural Comput. 2, 293–307 (1990).
    [CrossRef]
  2. R. Eckhorn, R. Bauer, M. Rosch, W. Jordan, W. Kruse, M. Munk, “Functionally related modules of cat visual cortex shows stimulus-evoked coherent oscillations: a multiple electrode study,” Invest. Ophthalmol. Vis. Sci. 29, 331–343 (1988).
  3. R. Eckhorn, “Stimulus-evoked synchronizations in the visual cortex: linking of local features into global figures?” in Neural Cooperativity, J. Kruger, ed., Vol. 2 of Springer Series in Brain Dynamics (Springer-Verlag, New York, 1989), pp. 267–278.
  4. J. L. Johnson, “Waves in pulse-coupled neural networks,” in Proceedings of the World Congress on Neural NetworksInternational Neural Network Society, Hillsdale, N.J., 1993), Vol. 4, pp. 299–302.
  5. J. L. Johnson, D. Ritter, “Observation of periodic waves in a pulse-coupled neural network,” Opt. Lett. 18, 1253–1255 (1993).
    [CrossRef] [PubMed]
  6. R. Eckhorn, H. J. Reitboeck, M. Arndt, P. Dicke, “A neural network for feature linking via synchronous activity: results from cat visual cortex and from simulations,” in Models of Brain Function, R. M. J. Cotterill, ed. (Cambridge U. Press, Cambridge, 1989), pp. 255–272.
  7. R. Eckhorn, T. Schanze, “Possible neural mechanisms of feature linking in the visual system: stimulus-locked and stimulus-induced synchronizations,” in Self-Organization, Emerging Properties and Learning, A. Babloyantz, ed. (Plenum, New York, to be published).
  8. P. W. Dicke, “Simulation dymanischer Merkmalskopplungen in einem neuronalen Netzwerkmodell,” Ph.D. dissertation (Phillips University, Marburg, Germany, 1992).
  9. A. S. French, R. B. Stein, “A flexible neural analog using integrated circuits,” IEEE Trans. Bio-Med Electron. BME-17, 248–253 (1970).
    [CrossRef]
  10. C. Giles, T. Maxwell, “Learning, invariance, and generalization in high-order neural networks,” Appl. Opt. 26, 4972–4978(1987).
    [CrossRef] [PubMed]
  11. C. Giles, C. Miller, D. Chen, H. Chen, G. Sun, Y. Lee, “Learning and extracting finite state automata with second-order recurrent neural networks,” Neural Comput. 2, 393–405 (1992).
    [CrossRef]
  12. Appendix 1 of Ref. 1 shows that a step function for the pulse generator was used in the simulations there.
  13. S. Grossberg, D. Somers, “Synchronized oscillators during cooperative feature linking in a cortical model of visual perception,” Neural Networks 4, 453–466 (1991).
    [CrossRef]
  14. N. Farhat, M. Eldefrawy, “The bifurcating neuron,” in Annual Meeting, Vol. 17 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), p. 10.
  15. S. Grossberg, Studies of Mind and Brain (Reidel, Dordrecht, The Netherlands, 1982), pp. 49–50.
  16. C. Giles, R. Griffin, T. Maxwell, “Encoding geometrical invariances in higher-order neural networks,” in Proceedings of the IEEE First International Neural Information Processing Systems Conference, D. Anderson, ed. (Institute of Electrical and Electronics Engineers, New York, 1988), pp. 301–306.
  17. A. V. Holden, M. Markus, H. G. Othmer, eds. Nonlinear Wave Processes in Excitable Media (Plenum, New York, to be published).
  18. A. B. Medvinsky, A. V. Panfilov, A. M. Pertsov, “Autowave process characterization in heart tissue,” in Self-Organization, Autowaves and Structures Far From Equilibrium, V. I. Krinsky, ed. (Springer-Verlag, New York, 1984), pp. 195–199.
    [CrossRef]
  19. J. L. Johnson, “Pulse-coupled neural networks,” in Critical Reviews of Optical Science and Technology, S. Chen, H. J. Caulfield, eds., Proc. Soc. Photo-Opt. Instrum. Eng.CR55: 47–76 (1994).
  20. J. L. Johnson, H. Ranganath, H. J. Caulfield, “Pulse-coupled neural networks,” in Temporal Dynamics and Time Variant Pattern Recognition, J. Dayhoff, ed., (Ablex, Norwood, N.J., to be published).

1993

1992

C. Giles, C. Miller, D. Chen, H. Chen, G. Sun, Y. Lee, “Learning and extracting finite state automata with second-order recurrent neural networks,” Neural Comput. 2, 393–405 (1992).
[CrossRef]

1991

S. Grossberg, D. Somers, “Synchronized oscillators during cooperative feature linking in a cortical model of visual perception,” Neural Networks 4, 453–466 (1991).
[CrossRef]

1990

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

1988

R. Eckhorn, R. Bauer, M. Rosch, W. Jordan, W. Kruse, M. Munk, “Functionally related modules of cat visual cortex shows stimulus-evoked coherent oscillations: a multiple electrode study,” Invest. Ophthalmol. Vis. Sci. 29, 331–343 (1988).

1987

1970

A. S. French, R. B. Stein, “A flexible neural analog using integrated circuits,” IEEE Trans. Bio-Med Electron. BME-17, 248–253 (1970).
[CrossRef]

Arndt, M.

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

R. Eckhorn, H. J. Reitboeck, M. Arndt, P. Dicke, “A neural network for feature linking via synchronous activity: results from cat visual cortex and from simulations,” in Models of Brain Function, R. M. J. Cotterill, ed. (Cambridge U. Press, Cambridge, 1989), pp. 255–272.

Bauer, R.

R. Eckhorn, R. Bauer, M. Rosch, W. Jordan, W. Kruse, M. Munk, “Functionally related modules of cat visual cortex shows stimulus-evoked coherent oscillations: a multiple electrode study,” Invest. Ophthalmol. Vis. Sci. 29, 331–343 (1988).

Caulfield, H. J.

J. L. Johnson, H. Ranganath, H. J. Caulfield, “Pulse-coupled neural networks,” in Temporal Dynamics and Time Variant Pattern Recognition, J. Dayhoff, ed., (Ablex, Norwood, N.J., to be published).

Chen, D.

C. Giles, C. Miller, D. Chen, H. Chen, G. Sun, Y. Lee, “Learning and extracting finite state automata with second-order recurrent neural networks,” Neural Comput. 2, 393–405 (1992).
[CrossRef]

Chen, H.

C. Giles, C. Miller, D. Chen, H. Chen, G. Sun, Y. Lee, “Learning and extracting finite state automata with second-order recurrent neural networks,” Neural Comput. 2, 393–405 (1992).
[CrossRef]

Dicke, P.

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

R. Eckhorn, H. J. Reitboeck, M. Arndt, P. Dicke, “A neural network for feature linking via synchronous activity: results from cat visual cortex and from simulations,” in Models of Brain Function, R. M. J. Cotterill, ed. (Cambridge U. Press, Cambridge, 1989), pp. 255–272.

Dicke, P. W.

P. W. Dicke, “Simulation dymanischer Merkmalskopplungen in einem neuronalen Netzwerkmodell,” Ph.D. dissertation (Phillips University, Marburg, Germany, 1992).

Eckhorn, R.

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

R. Eckhorn, R. Bauer, M. Rosch, W. Jordan, W. Kruse, M. Munk, “Functionally related modules of cat visual cortex shows stimulus-evoked coherent oscillations: a multiple electrode study,” Invest. Ophthalmol. Vis. Sci. 29, 331–343 (1988).

R. Eckhorn, T. Schanze, “Possible neural mechanisms of feature linking in the visual system: stimulus-locked and stimulus-induced synchronizations,” in Self-Organization, Emerging Properties and Learning, A. Babloyantz, ed. (Plenum, New York, to be published).

R. Eckhorn, “Stimulus-evoked synchronizations in the visual cortex: linking of local features into global figures?” in Neural Cooperativity, J. Kruger, ed., Vol. 2 of Springer Series in Brain Dynamics (Springer-Verlag, New York, 1989), pp. 267–278.

R. Eckhorn, H. J. Reitboeck, M. Arndt, P. Dicke, “A neural network for feature linking via synchronous activity: results from cat visual cortex and from simulations,” in Models of Brain Function, R. M. J. Cotterill, ed. (Cambridge U. Press, Cambridge, 1989), pp. 255–272.

Eldefrawy, M.

N. Farhat, M. Eldefrawy, “The bifurcating neuron,” in Annual Meeting, Vol. 17 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), p. 10.

Farhat, N.

N. Farhat, M. Eldefrawy, “The bifurcating neuron,” in Annual Meeting, Vol. 17 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), p. 10.

French, A. S.

A. S. French, R. B. Stein, “A flexible neural analog using integrated circuits,” IEEE Trans. Bio-Med Electron. BME-17, 248–253 (1970).
[CrossRef]

Giles, C.

C. Giles, C. Miller, D. Chen, H. Chen, G. Sun, Y. Lee, “Learning and extracting finite state automata with second-order recurrent neural networks,” Neural Comput. 2, 393–405 (1992).
[CrossRef]

C. Giles, T. Maxwell, “Learning, invariance, and generalization in high-order neural networks,” Appl. Opt. 26, 4972–4978(1987).
[CrossRef] [PubMed]

C. Giles, R. Griffin, T. Maxwell, “Encoding geometrical invariances in higher-order neural networks,” in Proceedings of the IEEE First International Neural Information Processing Systems Conference, D. Anderson, ed. (Institute of Electrical and Electronics Engineers, New York, 1988), pp. 301–306.

Griffin, R.

C. Giles, R. Griffin, T. Maxwell, “Encoding geometrical invariances in higher-order neural networks,” in Proceedings of the IEEE First International Neural Information Processing Systems Conference, D. Anderson, ed. (Institute of Electrical and Electronics Engineers, New York, 1988), pp. 301–306.

Grossberg, S.

S. Grossberg, D. Somers, “Synchronized oscillators during cooperative feature linking in a cortical model of visual perception,” Neural Networks 4, 453–466 (1991).
[CrossRef]

S. Grossberg, Studies of Mind and Brain (Reidel, Dordrecht, The Netherlands, 1982), pp. 49–50.

Johnson, J. L.

J. L. Johnson, D. Ritter, “Observation of periodic waves in a pulse-coupled neural network,” Opt. Lett. 18, 1253–1255 (1993).
[CrossRef] [PubMed]

J. L. Johnson, “Waves in pulse-coupled neural networks,” in Proceedings of the World Congress on Neural NetworksInternational Neural Network Society, Hillsdale, N.J., 1993), Vol. 4, pp. 299–302.

J. L. Johnson, “Pulse-coupled neural networks,” in Critical Reviews of Optical Science and Technology, S. Chen, H. J. Caulfield, eds., Proc. Soc. Photo-Opt. Instrum. Eng.CR55: 47–76 (1994).

J. L. Johnson, H. Ranganath, H. J. Caulfield, “Pulse-coupled neural networks,” in Temporal Dynamics and Time Variant Pattern Recognition, J. Dayhoff, ed., (Ablex, Norwood, N.J., to be published).

Jordan, W.

R. Eckhorn, R. Bauer, M. Rosch, W. Jordan, W. Kruse, M. Munk, “Functionally related modules of cat visual cortex shows stimulus-evoked coherent oscillations: a multiple electrode study,” Invest. Ophthalmol. Vis. Sci. 29, 331–343 (1988).

Kruse, W.

R. Eckhorn, R. Bauer, M. Rosch, W. Jordan, W. Kruse, M. Munk, “Functionally related modules of cat visual cortex shows stimulus-evoked coherent oscillations: a multiple electrode study,” Invest. Ophthalmol. Vis. Sci. 29, 331–343 (1988).

Lee, Y.

C. Giles, C. Miller, D. Chen, H. Chen, G. Sun, Y. Lee, “Learning and extracting finite state automata with second-order recurrent neural networks,” Neural Comput. 2, 393–405 (1992).
[CrossRef]

Maxwell, T.

C. Giles, T. Maxwell, “Learning, invariance, and generalization in high-order neural networks,” Appl. Opt. 26, 4972–4978(1987).
[CrossRef] [PubMed]

C. Giles, R. Griffin, T. Maxwell, “Encoding geometrical invariances in higher-order neural networks,” in Proceedings of the IEEE First International Neural Information Processing Systems Conference, D. Anderson, ed. (Institute of Electrical and Electronics Engineers, New York, 1988), pp. 301–306.

Medvinsky, A. B.

A. B. Medvinsky, A. V. Panfilov, A. M. Pertsov, “Autowave process characterization in heart tissue,” in Self-Organization, Autowaves and Structures Far From Equilibrium, V. I. Krinsky, ed. (Springer-Verlag, New York, 1984), pp. 195–199.
[CrossRef]

Miller, C.

C. Giles, C. Miller, D. Chen, H. Chen, G. Sun, Y. Lee, “Learning and extracting finite state automata with second-order recurrent neural networks,” Neural Comput. 2, 393–405 (1992).
[CrossRef]

Munk, M.

R. Eckhorn, R. Bauer, M. Rosch, W. Jordan, W. Kruse, M. Munk, “Functionally related modules of cat visual cortex shows stimulus-evoked coherent oscillations: a multiple electrode study,” Invest. Ophthalmol. Vis. Sci. 29, 331–343 (1988).

Panfilov, A. V.

A. B. Medvinsky, A. V. Panfilov, A. M. Pertsov, “Autowave process characterization in heart tissue,” in Self-Organization, Autowaves and Structures Far From Equilibrium, V. I. Krinsky, ed. (Springer-Verlag, New York, 1984), pp. 195–199.
[CrossRef]

Pertsov, A. M.

A. B. Medvinsky, A. V. Panfilov, A. M. Pertsov, “Autowave process characterization in heart tissue,” in Self-Organization, Autowaves and Structures Far From Equilibrium, V. I. Krinsky, ed. (Springer-Verlag, New York, 1984), pp. 195–199.
[CrossRef]

Ranganath, H.

J. L. Johnson, H. Ranganath, H. J. Caulfield, “Pulse-coupled neural networks,” in Temporal Dynamics and Time Variant Pattern Recognition, J. Dayhoff, ed., (Ablex, Norwood, N.J., to be published).

Reitboeck, H. J.

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

R. Eckhorn, H. J. Reitboeck, M. Arndt, P. Dicke, “A neural network for feature linking via synchronous activity: results from cat visual cortex and from simulations,” in Models of Brain Function, R. M. J. Cotterill, ed. (Cambridge U. Press, Cambridge, 1989), pp. 255–272.

Ritter, D.

Rosch, M.

R. Eckhorn, R. Bauer, M. Rosch, W. Jordan, W. Kruse, M. Munk, “Functionally related modules of cat visual cortex shows stimulus-evoked coherent oscillations: a multiple electrode study,” Invest. Ophthalmol. Vis. Sci. 29, 331–343 (1988).

Schanze, T.

R. Eckhorn, T. Schanze, “Possible neural mechanisms of feature linking in the visual system: stimulus-locked and stimulus-induced synchronizations,” in Self-Organization, Emerging Properties and Learning, A. Babloyantz, ed. (Plenum, New York, to be published).

Somers, D.

S. Grossberg, D. Somers, “Synchronized oscillators during cooperative feature linking in a cortical model of visual perception,” Neural Networks 4, 453–466 (1991).
[CrossRef]

Stein, R. B.

A. S. French, R. B. Stein, “A flexible neural analog using integrated circuits,” IEEE Trans. Bio-Med Electron. BME-17, 248–253 (1970).
[CrossRef]

Sun, G.

C. Giles, C. Miller, D. Chen, H. Chen, G. Sun, Y. Lee, “Learning and extracting finite state automata with second-order recurrent neural networks,” Neural Comput. 2, 393–405 (1992).
[CrossRef]

Appl. Opt.

IEEE Trans. Bio-Med Electron.

A. S. French, R. B. Stein, “A flexible neural analog using integrated circuits,” IEEE Trans. Bio-Med Electron. BME-17, 248–253 (1970).
[CrossRef]

Invest. Ophthalmol. Vis. Sci.

R. Eckhorn, R. Bauer, M. Rosch, W. Jordan, W. Kruse, M. Munk, “Functionally related modules of cat visual cortex shows stimulus-evoked coherent oscillations: a multiple electrode study,” Invest. Ophthalmol. Vis. Sci. 29, 331–343 (1988).

Neural Comput.

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

C. Giles, C. Miller, D. Chen, H. Chen, G. Sun, Y. Lee, “Learning and extracting finite state automata with second-order recurrent neural networks,” Neural Comput. 2, 393–405 (1992).
[CrossRef]

Neural Networks

S. Grossberg, D. Somers, “Synchronized oscillators during cooperative feature linking in a cortical model of visual perception,” Neural Networks 4, 453–466 (1991).
[CrossRef]

Opt. Lett.

Other

R. Eckhorn, H. J. Reitboeck, M. Arndt, P. Dicke, “A neural network for feature linking via synchronous activity: results from cat visual cortex and from simulations,” in Models of Brain Function, R. M. J. Cotterill, ed. (Cambridge U. Press, Cambridge, 1989), pp. 255–272.

R. Eckhorn, T. Schanze, “Possible neural mechanisms of feature linking in the visual system: stimulus-locked and stimulus-induced synchronizations,” in Self-Organization, Emerging Properties and Learning, A. Babloyantz, ed. (Plenum, New York, to be published).

P. W. Dicke, “Simulation dymanischer Merkmalskopplungen in einem neuronalen Netzwerkmodell,” Ph.D. dissertation (Phillips University, Marburg, Germany, 1992).

R. Eckhorn, “Stimulus-evoked synchronizations in the visual cortex: linking of local features into global figures?” in Neural Cooperativity, J. Kruger, ed., Vol. 2 of Springer Series in Brain Dynamics (Springer-Verlag, New York, 1989), pp. 267–278.

J. L. Johnson, “Waves in pulse-coupled neural networks,” in Proceedings of the World Congress on Neural NetworksInternational Neural Network Society, Hillsdale, N.J., 1993), Vol. 4, pp. 299–302.

N. Farhat, M. Eldefrawy, “The bifurcating neuron,” in Annual Meeting, Vol. 17 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), p. 10.

S. Grossberg, Studies of Mind and Brain (Reidel, Dordrecht, The Netherlands, 1982), pp. 49–50.

C. Giles, R. Griffin, T. Maxwell, “Encoding geometrical invariances in higher-order neural networks,” in Proceedings of the IEEE First International Neural Information Processing Systems Conference, D. Anderson, ed. (Institute of Electrical and Electronics Engineers, New York, 1988), pp. 301–306.

A. V. Holden, M. Markus, H. G. Othmer, eds. Nonlinear Wave Processes in Excitable Media (Plenum, New York, to be published).

A. B. Medvinsky, A. V. Panfilov, A. M. Pertsov, “Autowave process characterization in heart tissue,” in Self-Organization, Autowaves and Structures Far From Equilibrium, V. I. Krinsky, ed. (Springer-Verlag, New York, 1984), pp. 195–199.
[CrossRef]

J. L. Johnson, “Pulse-coupled neural networks,” in Critical Reviews of Optical Science and Technology, S. Chen, H. J. Caulfield, eds., Proc. Soc. Photo-Opt. Instrum. Eng.CR55: 47–76 (1994).

J. L. Johnson, H. Ranganath, H. J. Caulfield, “Pulse-coupled neural networks,” in Temporal Dynamics and Time Variant Pattern Recognition, J. Dayhoff, ed., (Ablex, Norwood, N.J., to be published).

Appendix 1 of Ref. 1 shows that a step function for the pulse generator was used in the simulations there.

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Figures (17)

Fig. 1
Fig. 1

Model neuron has three parts: the dendritic tree, the linking, and the pulse generator. The dendritic tree is subdivided into two channels, linking and feeding. All synapses are leaky integrator connections. The inputs are pulses from other neurons, and the output is a pulse. The uniting input modulates the feeding input. When a pulse occurs in the linking input, it briefly raises the total internal activity U j and can cause the model neuron to fire at that time, thus synchronizing it with the neuron transmitting the linking pulse.

Fig. 2
Fig. 2

Pulse generation and linking: The threshold is recharged when it decays to less than the internal activity U j = F j (1 + β j L j ). The output pulse is formed as the threshold turns the step function of Eq. (5) on and then off as the threshold goes below U j , starts recharging, and then rises above U j . If a linking pulse occurs in the capture-zone time, it causes the threshold to recharge sooner than otherwise, and the neuron fires a pulse synchronized with the arrival of the linking pulse.

Fig. 3
Fig. 3

Pulse frequency f j as a function of the internal activity U j : The pulse frequency is a sigmoidal function of the internal activity. Addition of a refractory time period τ r makes the frequency saturate at the refractory frequency. A bias offset θ0 shifts the curve’s origin to that bias point.

Fig. 4
Fig. 4

Formation of a periodic time series: Neurons 1–4 all fire together at t = 0. As time passes, they occasionally link in various combinations. If at time T they link again so as to fire together, the situation will be the same as at t = 0. The system will repeat its behavior, generating a time series. The linear sum of the group’s outputs is the periodic time signature of the input distribution to neurons 1–4.

Fig. 5
Fig. 5

Geometry for scale invariance: A neuron at R receives a linking contribution from a neuron at ρ. When the image is rescaled, the image patch at R goes to k R and the patch at ρ goes to k ρ. Only the latter patch is shown. For the case of an optical image rescaled by a change in the object distance the intensity per image patch is constant. The objective is to design a linking receptive field such that L(k R) = L(R).

Fig. 6
Fig. 6

Geometry used to show that the fixed inner radius ρ0 of the local group L 1 causes a dependency on the rescaling factor k. The external group L 2 is in the annulus from r to R, while L 1 extends from ρ0 to r.

Fig. 7
Fig. 7

Linking field model neuron as a multirule logical system: A dendrite receives inputs from many receptive fields along its length. Each input modulates the dendritic signal by the factor (1 + β j L j n ) for the nth input. The receptive fields can give the same signal for more than one input distribution and thus correspond to a logical or. The product term in the modulation factors corresponds to a logical and. These logic-gate correspondences are not exact but can be used effectively, as shown by the example discussed in Section 4.

Fig. 8
Fig. 8

Schematic of the model used in the simulations: An image pixel feeds the neuron. Two receptive-field inputs, a nearest-neighbor 3 × 3 field L 1 and an inverse-square field input L 2, modulate the feeding input on both an excitatory and an inhibitory dendrite. This system is a third-order neural network. The total internal activity is the sum of the two dendritic signals. The pulse generator sends the output pulse to the neuron located in the pulsed array at the same relative location as the feeding input pixel in the input image.

Fig. 9
Fig. 9

Periodic time signatures and invariances for the cross image: The signatures are the periodic part of the total-output time signal of the pulsed array. SC is the scale factor, and AC is the rotation angle in degrees. Good scale invariance was found for scales over 1:0.46 and for large rotations of 30° and 45°. The five blocks arranged to form the image were scaled from 11 × 11, to 9 × 9, to 7 × 7, to 5 × 5 block sizes. The 33 × 33 slab had a background intensity level of zero. Grid coarseness effects were expected for 7 × 7 and smaller block sizes in scale and for 14 × 14 block sizes in rotation. Grid effects were not severe in this image.

Fig. 10
Fig. 10

Periodic time signature and invariances for the T image: The same setup as for Fig. 11 is used, but with the five blocks rearranged to form a T. The signature was very distinct as compared with the first case, showing that the net makes unique time signatures for different images even when they are rearrangements of the same components. The scale invariance was good down to the 7 × 7 block size. The rotated images’ signatures still followed the overall T signature shape in contrast to the cross signature. Their variation from ideal is strictly caused by grid effects.

Fig. 11
Fig. 11

Intensity invariance: The 9 × 9 block-size images were multiplied by an intensity factor I 0 corresponding to a change in scene illumination. From I 0 = 2 to I 0 = 0.01 the signature was invariant in its shape, though the period of the signature varied from 13 to 40 time units.

Fig. 12
Fig. 12

Image distortion: A coordinate transform approximating a 30° out-of-plane rotation was used for both test images. Their signatures were still distinct and recognizable as belonging to the correct image classification.

Fig. 13
Fig. 13

Signature of the T image with two blocks interchanged: The two lower blocks of the full-scale unrotated T image were interchanged, simulating the effect of a shadow moving down the image. The new signature is similar to that of the 7 × 7 block-size T image and still has an initial peak followed by a valley and then a higher peak. In contrast the cross image’s second peak was lower than its first peak, so this signature would still be classified as a T and not as a cross.

Fig. 14
Fig. 14

Effect of combined image changes: The original images were located at coordinates (16, 16) with scale factors sc = 1, they were unrotated, and they had no distortion (RD is the approximate out-of-plane rotation). The signatures were sufficiently insensitive to the combined changes for the images to still be correctly classified.

Fig. 15
Fig. 15

Optical pulse images showing segmentation and linking: This object was chosen at random in the laboratory. The pulse-coupled optical network was operated at high illumination levels so as to produce multiple pulses in the bright areas of the image. (a) The input image, (b) the pulse output image after one cycle, and (c) the output after several cycles. Here the tendency of the network to outline object rogions is apparent. (d) The sum of 30 pulse images, all similar to (b) or (c). The original image is recovered. Its intensity is remapped according to the sigmoid function. There was some noise reduction owing to the averaging inherent in the summation.

Fig. 16
Fig. 16

Optical pulse images of a model truck: A low-intensity low-contrast image was both edge and contrast enhanced by the optical network. Even small intensity differences caused adjacent regions to pulse at different times, giving a momentary contrast enhancement. The network also enhanced edges because the linking waves tended to accumulate in regions of larger gradients.

Fig. 17
Fig. 17

Horizontal gradient-wave dynamics: These display the internal activity rather than the pulse images, showing what the neurons actually receive at the pulse generator. (a) The horizontal gradient strip against a dark background. (b) After 20 cycles, the gradient waves are beginning to push the initial pulse pattern to the left as they establish themselves. (c) The established pattern. (d), (e), (f), and (g) The propagation of a disturbance. It originated from a boundary interaction and was carried by the gradient waves from its point of origin to the left. It moved more slowly than the carrier waves but was taken over distances beyond the linking receptive field’s interconnection distance. This shows that the waves can carry information with them. (h) The gradient waves after the disturbance was pushed off the left end of the strip. The system was run for 300 time cycles.

Equations (34)

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

L j = k L k j = k [ W k j exp ( α k j L t ) ] * Y k ( t ) ,
F j = k F k j = k [ M k j exp ( α k j F t ) ] * Y k ( t ) + I j ,
U j = F j ( 1 + β j L j ) ,
θ j = [ V T exp ( α j T t ) ] * Y j ( t ret ) + θ 0 ,
Y j ( t ) = step [ U j ( t ) θ j ( t ) ] ,
τ j = 1 α T ln ( 1 + V T F j θ 0 ) .
f j = ( τ j + τ r ) 1 .
τ c = 1 α T ln ( 1 + β L ) ,
| τ 2 τ 1 | < τ c ,
| ϕ | < α T τ c ,
| m τ 2 n τ 1 | < τ c ,
S ( t ) = n k = 1 K a k δ ( t n T φ k ) ,
FT ( S ) = [ k = 1 K a k exp ( i ωϕ k ) ] [ n exp ( i ω n T ) ] .
Y ( k R ) = Y ( R ) .
L ( k R ) = 0 2 π ρ 0 1 ( k ρ ) 2 Y [ k ( R + ρ ) ] k ρ k d ρ d θ = 0 2 π ρ 0 1 ρ 2 Y ( R + ρ ) ρ d ρ d θ + L ( R ) .
F ( k R ) = f ( R ) .
U ( k R ) = F ( k R ) [ 1 + β L ( k R ) ] = F ( R ) [ 1 + β L ( R ) ] = U ( R ) .
L = 2 π ρ 0 r Y 1 ρ 2 ρ d ρ + 2 π r R Y 1 ρ 2 ρ d ρ = 2 π Y 1 ln r ρ 0 + 2 π Y 2 ln R r .
U tot = F ( 1 + β 1 L 1 ) ( 1 + β 2 L 2 ) ( 1 + β n L n ) .
U exc = + a 1 F ( 1 + β 1 L 1 ) ( 1 + β 2 L 2 ) ,
U inh = a 2 F ( 1 + β 3 L 1 ) ( 1 + β 4 L 2 ) ,
U tot = U exc + U inh .
U tot = F ( 1 + β L 1 + β { 1 L 1 / [ L 1 ( max ) ] } L 2 ) .
F = image ( j , k ) / 255 ,
L loc ( t + 1 ) = A 1 L loc ( t ) + V L L 1 ( t ) ,
L ext ( t + 1 ) = A 1 L ext ( t ) + V L L 2 ( t ) ,
θ ( t + 1 ) = A 2 θ ( t ) + V T Y ( t ) ,
Y ( t ) = step [ U tot ( t ) θ ( t ) ] .
Y j ( t ) = ( 1 1 + exp { A [ U j ( t ) θ j ( t ) ] } ) P ( t ) ,
P ( t ) = n P 0 ( t n τ r ) ,
P 0 ( t ) = 1 τ w t [ δ ( t ) δ ( t τ w ) d t ] .
1 / α T τ r τ FWHM τ w ,
τ FWHM = FWHM / [ U ( t 0 ) α T ] .
FWHM = 2 A ln ( 1 3 2 2 ) .

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