Chaotic neural networks have received a great deal of attention these last years. In this paper we establish a precise correspondence between the so-called chaotic iterations and a particular class of artificial neural networks: global recurrent multi-layer perceptrons. We show formally that it is possible to make these iterations behave chaotically, as defined by Devaney, and thus we obtain the first neural networks proven chaotic. Several neural networks with different architectures are trained to exhibit a chaotical behavior.
Since a while neuroscientists discuss the existence of chaos in the brain. In the context of artificial neural networks, this interest has given raise to various works studying the modeling of chaos in neurons. The chaotic neuron model designed by Aihara @cite_11 is particularly used to build chaotic neural networks. For example, in @cite_16 is proposed a feedback ANN architecture which consists of two layers (apart from the input layer) with one of them composed of chaotic neurons. In their experiments, the authors showed that without any input sequence the activation of each chaotic neuron results in a positive average Lyapunov exponent, which means a true chaotic behavior. When an input sequence is given iteratively to the network the chaotic neurons reach stabilized periodic orbits with different periods, and thus potentially provide a recognition state. Similarly, the same authors have recently introduced another model of chaotic neuron: the non-linear dynamic state (NDS) neuron, and used it to build a neural network which is able to recognize learned stabilized periodic orbits identifying patterns @cite_9 .
@cite_9 @cite_16 @cite_11
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This research investigates the potential utility of chaotic dynamics in neural information processing. A novel chaotic spiking neural network model is presented which is composed of non-linear dynamic state (NDS) neurons. The activity of each NDS neuron is driven by a set of non-linear equations coupled with a threshold based spike output mechanism. If time-delayed self-connections are enabled then the network stabilises to a periodic pattern of activation. Previous publications of this work have demonstrated that the chaotic dynamics which drive the network activity ensure that an extremely large number of such periodic patterns can be generated by this network. This paper presents a major extension to this model which enables the network to recall a pattern of activity from a selection of previously stabilised patterns. Chaotic systems are sensitive to initial conditions, system parameters and topological transitivity and these properties are also remarkable for cryptanalysts. Noise like behavior of chaotic systems is the main reason of using these systems in cryptology. However some properties of chaotic systems such as synchronization, fewness of parameters etc. cause serious problems for cryptology. In this paper, to overcome disadvantages of chaotic systems, the dynamics of Chua's circuit namely x , y and z were modeled using Artificial Neural Network (ANN). ANNs have some distinctive capabilities like learning from experiences, generalizing from a few data and nonlinear relationship between inputs and outputs. The proposed ANN was trained in different structures using different learning algorithms. To train the ANN, 24 different sets including the initial conditions of Chua's circuit were used and each set consisted of about 1800 input-output data. The experimental results showed that a feed-forward Multi Layer Perceptron (MLP), trained with Bayesian Regulation backpropagation algorithm, was found as the suitable network structure. As a case study, a message was first encrypted and then decrypted by the chaotic dynamics obtained from the proposed ANN and a comparison was made between the proposed ANN and the numerical solution of Chua's circuit about encrypted and decrypted messages. Abstract This article describes computer simulation of the dynamics of a distributed model of the olfactory system that is aimed at understanding the role of chaos in biological pattern recognition. The model is governed by coupled nonlinear differential equations with many variables and parameters, which allow multiple high-dimensional chaotic states. An appropriate set of the parameters is identified by computer experiments with the guidance of biological measurements, through which this model of the olfactory system maintains a low dimensional global chaotic attractor with multiple “wings.” The central part of the attractor is its basal chaotic activity, which simulates the electroencephalographic (EEG) activity of the olfactory system under zero signal input (exhalation). It provides the system with a ready state so that it is unnecessary for the system to “wake up” from or return to a “dormant” equilibrium state every time that an input is given (by inhalation). Each of the wings may be either a near-limit cycle (a narrow band chaos) or a broad band chaos. The reproducible spatial pattern of each near-limit cycle is determined by a template made in the system. A novel input with no template activates the system to either a nonreproducible near-limit cycle wing or a broad band chaotic wing. Pattern recognition in the system may be considered as the transition from one wing to another, as demonstrated by the computer simulation. The time series of the manifestations of the attractor are EEG-like waveforms with fractal dimensions that reflect which wing the system is placed in by input or lack of input. The computer simulation also shows that the adaptive behavior of the system is scaling invariant, and it is independent of the initial conditions at the transition from one wing to another. These properties enable the system to classify an uninterrupted sequence of stimuli. Abstract A model of a single neuron with chaotic dynamics is proposed by considering the following properties of biological neurons: (1) graded responses, (2) relative refractoriness and (3) spatio-temporal summation of inputs. The model includes some conventional models of a neuron as its special cases; namely, chaotic dynamics is introduced as a natural extension of the former models. Chaotic solutions of both the single chaotic neuron and the chaotic neural network composed of such neurons are numerically demonstrated.