A Complex-Valued Hopfield Neural Network: Dynamics and Applications

A Complex-Valued Hopfield Neural Network: Dynamics and Applications

V. Srinivasa Chakravarthy (Indian Institute of Technology - Madras, India)
DOI: 10.4018/978-1-60566-214-5.ch004
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This chapter describes Complex Hopfield Neural Network (CHNN), a complex-variable version of the Hopfield neural network, which can exist in both fixed point and oscillatory modes. Memories can be stored by a complex version of Hebbs rule. In the fixed-point mode, CHNN is similar to a continuous-time Hopfield network. In the oscillatory mode, when multiple patterns are stored, the network wanders chaotically among patterns. Presence of chaos in this mode is verified by appropriate time series analysis. It is shown that adaptive connections can be used to control chaos and increase memory capacity. Electronic realization of the network in oscillatory dynamics, with fixed and adaptive connections shows an interesting tradeoff between energy expenditure and retrieval performance. It is shown how the intrinsic chaos in CHNN can be used as a mechanism for annealing when the network is used for solving quadratic optimization problems. The networks applicability to chaotic synchronization is described.
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It has been shown that by extending Hopfield’s real-valued model to complex –variable domain, it is possible to preserve the symmetric Hebbian synapses, while permitting the network to have oscillatory states (Chakravarthy & Ghosh, 1996). Pioneering work on complex-valued versions of Hopfield network was done by Hirose (1992). Other studies in the area of complex neural networks include complex backpropagation algorithm for training complex feedforward networks (Leung & Haykin, 1991; Nitta, 1997) and a similar extension for complex-valued recurrent neural networks (Mandic & Goh, 2004). For a comprehensive review of complex neural models the reader may consult (Hirose, 2003).

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Editorial Advisory Board
Table of Contents
Sven Buchholz
Tohru Nitta
Chapter 1
Masaki Kobayashi
Information geometry is one of the most effective tools to investigate stochastic learning models. In it, stochastic learning models are regarded as... Sample PDF
Complex-Valued Boltzmann Manifold
Chapter 2
Takehiko Ogawa
Network inversion solves inverse problems to estimate cause from result using a multilayer neural network. The original network inversion has been... Sample PDF
Complex-Valued Neural Network and Inverse Problems
Chapter 3
Boris Igelnik
This chapter describes the clustering ensemble method and the Kolmogorovs Spline Complex Network, in the context of adaptive dynamic modeling of... Sample PDF
Kolmogorovs Spline Complex Network and Adaptive Dynamic Modeling of Data
Chapter 4
V. Srinivasa Chakravarthy
This chapter describes Complex Hopfield Neural Network (CHNN), a complex-variable version of the Hopfield neural network, which can exist in both... Sample PDF
A Complex-Valued Hopfield Neural Network: Dynamics and Applications
Chapter 5
Mitsuo Yoshida, Takehiro Mori
Global stability analysis for complex-valued artificial recurrent neural networks seems to be one of yet-unchallenged topics in information science.... Sample PDF
Global Stability Analysis for Complex-Valued Recurrent Neural Networks and Its Application to Convex Optimization Problems
Chapter 6
Yasuaki Kuroe
This chapter presents models of fully connected complex-valued neural networks which are complex-valued extension of Hopfield-type neural networks... Sample PDF
Models of Complex-Valued Hopfield-Type Neural Networks and Their Dynamics
Chapter 7
Sheng Chen
The complex-valued radial basis function (RBF) network proposed by Chen et al. (1994) has found many applications for processing complex-valued... Sample PDF
Complex-Valued Symmetric Radial Basis Function Network for Beamforming
Chapter 8
Rajoo Pandey
The equalization of digital communication channel is an important task in high speed data transmission techniques. The multipath channels cause the... Sample PDF
Complex-Valued Neural Networks for Equalization of Communication Channels
Chapter 9
Cheolwoo You, Daesik Hong
In this chapter, the complex Backpropagation (BP) algorithm for the complex backpropagation neural networks (BPN) consisting of the suitable node... Sample PDF
Learning Algorithms for Complex-Valued Neural Networks in Communication Signal Processing and Adaptive Equalization as its Application
Chapter 10
Donq-Liang Lee
New design methods for the complex-valued multistate Hopfield associative memories (CVHAMs) are presented. The author of this chapter shows that the... Sample PDF
Image Reconstruction by the Complex-Valued Neural Networks: Design by Using Generalized Projection Rule
Chapter 11
Naoyuki Morita
The author proposes an automatic estimation method for nuclear magnetic resonance (NMR) spectra of the metabolites in the living body by magnetic... Sample PDF
A Method of Estimation for Magnetic Resonance Spectroscopy Using Complex-Valued Neural Networks
Chapter 12
Michele Scarpiniti, Daniele Vigliano, Raffaele Parisi, Aurelio Uncini
This chapter aims at introducing an Independent Component Analysis (ICA) approach to the separation of linear and nonlinear mixtures in complex... Sample PDF
Flexible Blind Signal Separation in the Complex Domain
Chapter 13
Nobuyuki Matsui, Haruhiko Nishimura, Teijiro Isokawa
Recently, quantum neural networks have been explored as one of the candidates for improving the computational efficiency of neural networks. In this... Sample PDF
Qubit Neural Network: Its Performance and Applications
Chapter 14
Shigeo Sato, Mitsunaga Kinjo
The advantage of quantum mechanical dynamics in information processing has attracted much interest, and dedicated studies on quantum computation... Sample PDF
Neuromorphic Adiabatic Quantum Computation
Chapter 15
G.G. Rigatos, S.G. Tzafestas
Neural computation based on principles of quantum mechanics can provide improved models of memory processes and brain functioning and is of primary... Sample PDF
Attractors and Energy Spectrum of Neural Structures Based on the Model of the Quantum Harmonic Oscillator
Chapter 16
Teijiro Isokawa, Nobuyuki Matsui, Haruhiko Nishimura
Quaternions are a class of hypercomplex number systems, a four-dimensional extension of imaginary numbers, which are extensively used in various... Sample PDF
Quaternionic Neural Networks: Fundamental Properties and Applications
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