Complex neural fields with a Hopfield-like energy function and an analogy to optical fields generated in phase-conjugate resonators

Takeda, Mitsuo; Kishigami, Takaaki

doi:10.1364/josaa.9.002182

Cited by 20 publications

(9 citation statements)

References 20 publications

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“…Notice, however, that we do not discuss the storage of complex-valued patterns. The changes may be included without difficulty by going to complexvalued Hermitean weights (Noest, 1988a(Noest, , 1988bTakeda & Kishigami, 1992;Chakravarathy & Gosh, 1994;Aoyagi, 1995). We present the models in a form that will allow a convenient comparison with equations 1.1 in section 3.…”

Section: The Generalized Ginzburg-landau Modelmentioning

confidence: 99%

“…allows us to express the dynamics of equations 2.2 as a gradient system (Noest, 1988a(Noest, , 1988bTakeda & Kishigami, 1992;Chakravarathy & Gosh, 1994;Aoyagi, 1995;Hoppensteadt & Izhikevitch, 1996). Assuming that the g kl are symmetric, one obtains GGL:…”

Section: The Ggl Model As Gradient System the Discrete Complexmentioning

confidence: 99%

“…Using pure phase models as an adiabatic approximation, associative memory has been implemented by identifying the coupling strengths with Hebbian weights (Abbott, 1990;Sompolinsky, Golomb, & Kleinfeld, 1990, 1991Baldi & Meir, 1990;Kuramoto, Aoyagi, Nishikawa, Chawanya, & Okuda, 1992;Sompolinsky & Tsodyks, 1992;Kuzmina & Surina, 1994;Kuzmina, Manykin, & Surina, 1995). The corresponding step in the context of phase and amplitude dynamics was taken by using a complexified analog of the Cohen-Grossberg-Hopfield function (Takeda & Kishigami, 1992;Chakravaraty & Gosh, 1994;Aoyagi, 1995;Hoppensteadt & Izhikevitch, 1996). This approach imitated the method that was used for discrete dynamics (Noest, 1988a(Noest, , 1988b.…”

Section: Introductionmentioning

confidence: 99%

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Oscillatory Networks: Pattern Recognition Without a Superposition Catastrophe

Burwick

2006

Neural Computation

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Using an oscillatory network model that combines classical network models with phase dynamics, we demonstrate how the superposition catastrophe of pattern recognition may be avoided in the context of phase models. The model is designed to meet two requirements: on and off states should correspond, respectively, to high and low phase velocities, and patterns should be retrieved in coherent mode. Nonoverlapping patterns can be simultaneously active with mutually different phases. For overlapping patterns, competition can be used to reduce coherence to a subset of patterns. The model thereby solves the superposition problem.

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Section: The Generalized Ginzburg-landau Modelmentioning

confidence: 99%

Section: The Ggl Model As Gradient System the Discrete Complexmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Oscillatory Networks: Pattern Recognition Without a Superposition Catastrophe

Burwick

2006

Neural Computation

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“…Actually, in the very early stage of the CVNN research, a pioneering idea and a basic experiment was reported concerning this important feature. That is, in 1992, M. Takeda and T. Kishigami [34] pointed out the fact that the electromagnetic field in a phase-conjugate resonator is formulated in the same manner as that of an associative memory, and that the resonant system realizes a quite fast recall. In this case, the limitation in the energy supply causes amplitude saturation, which realizes the neural nonliearity in the signal amplitude in a natural way.…”

mentioning

confidence: 99%

Nature of complex number and complex-valued neural networks

Hirose

2011

Front. Electr. Electron. Eng. China

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We discuss the nature of complex number and its effect on complex-valued neural networks (CVNNs). After we review some examples of CVNN applications, we look back at the mathematical history to elucidate the features of complex number, in particular to confirm the importance of the phaseand-amplitude viewpoint for designing and constructing CVNNs to enhance the features. This viewpoint is essential in general to deal with waves such as electromagnetic wave and lightwave. Then, we point out that, although we represent a complex number as an ordered pair of real numbers for example, we can reduce ineffective degree of freedom in learning or self-organization in CVNNs to achieve better generalization characteristics. This merit is significantly useful not only for waverelated signal processing but also for general processing with frequency-domain treatment through Fourier transform.Keywords electromagnetic wave, lightwave, coherence, adaptive processing in sensing and imaging, learning logic, neural hardware IntroductionComplex-valued neural networks (CVNNs) [1-6] extend the application fields steadily. We have various application systems employing CVNNs in the field of, for example, ultrasonic fault detection to find defects in metals and other materials [7], blind separation based on principal component analysis (PCA) in sonar [8] and voice processing [9], radars including ground penetrating

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“…What would be worth mentioning is the relation between the covariance phasor model and the oscillatory models such as in [4], [5], [7], [9], [12]. When the firing probability of eq.…”

Section: Property 2 Let X Be a Random Process And Letmentioning

confidence: 99%

Covariance phasor neural network

Takahashi

Proceedings of the 2002 International Joint Conference on Neural Networks. IJCNN'02 (Cat. No.02CH37290)

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We present a phase covariance model that can well represent stimulus intensity as well as feature binding (i.e., covariance). The model is represented by complex neural equations, which is a mean field model of stochastic neural model such as Boltzman machine and sigmoid belief networks. The covariance model can represrnt covariance between two units of stochastic machines as cosine of the phase difference. This enables us to calculate the covariance between two units in a deterministic manner as well as avarage activation. The covariance model could give an elaborate mean field approximation in that to calculate higer moments we have to invoke a higher order mean field model. A covariance Hebbian selforganizing rule and Boltzman learning rule are then investigated on this model.

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Complex neural fields with a Hopfield-like energy function and an analogy to optical fields generated in phase-conjugate resonators

Cited by 20 publications

References 20 publications

Oscillatory Networks: Pattern Recognition Without a Superposition Catastrophe

Oscillatory Networks: Pattern Recognition Without a Superposition Catastrophe

Nature of complex number and complex-valued neural networks

Covariance phasor neural network

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