Chaotic systems implemented by artificial neural networks are good candidates for data encryption. In this manner, this paper introduces the cryptographic application of the Hopfield and the Hindmarsh-Rose neurons. The contribution is focused on finding suitable coefficient values of the neurons to generate robust random binary sequences that can be used in image encryption. This task is performed by evaluating the bifurcation diagrams from which one chooses appropriate coefficient values of the mathematical models that produce high positive Lyapunov exponent and Kaplan-Yorke dimension values, which are computed using TISEAN. The randomness of both the Hopfield and the Hindmarsh-Rose neurons is evaluated from chaotic time series data by performing National Institute of Standard and Technology (NIST) tests. The implementation of both neurons is done using field-programmable gate arrays whose architectures are used to develop an encryption system for RGB images. The success of the encryption system is confirmed by performing correlation, histogram, variance, entropy, and Number of Pixel Change Rate (NPCR) tests.In [2] J.J. Hopfield introduced the neuron model that nowadays is known as the Hopfield neural network. Ten years later, a modified model of Hopfield neural network was proposed in [3], and applied in information processing. Immediately, the Hopfield neural network was adapted to generate chaotic behavior in [4] where the authors explored bifurcation diagrams. In [5] the simplified Hopfield neuron model was designed to use a sigmoid as activation function, and three neurons were used to generate chaotic behavior. In addition, the authors performed an optimization process updating the weights of the neurons interconnections. The Hopfield neuron was combined with a chaotic map in [6] to be applied in chaotic masking. More recently, the authors in [7] proposed an image encryption algorithm using the Hopfield neural network. In the same direction, the authors in [8] detailed the behavior of Hindmarsh-Rose neuron to generate chaotic behavior. Its bifurcation diagrams were described in [9], and the results were used to select the values of the model to improve chaotic behavior. Hindmarsh-Rose neurons were synchronized in [10], optimizing the scheme of Lyapunov function with two gain coefficients. In this way, the synchronization region is estimated by evaluating the Lyapunov stability. Two Hindmarsh-Rose neurons were synchronized in [11], and the system was used to mask information in continuous time. To show that the neurons generate chaotic behavior, one must compute Lyapunov exponents, and for the Hindmarsh-Rose neuron they were evaluated by the TISEAN package in [12].The Hopfield neural network has been widely applied in chaotic systems [13][14][15]. This network consists of three neurons, and the authors in [13] proposed a simplified model by removing the synaptic weight connection of the third and second neuron in the original Hopfield network. Numerical simulations were carried out considering values from t...

Artificial neural networks have demonstrated to be very useful in solving problems in artificial intelligence. However, in most cases, ANNs are considered integer-order models, limiting the possible applications in recent engineering problems. In addition, when dealing with fractional-order neural networks, almost any work shows cases when varying the fractional order. In this manner, we introduce the optimization of a fractional-order neural network by applying metaheuristics, namely: differential evolution (DE) and accelerated particle swarm optimization (APSO) algorithms. The case study is a chaotic cellular neural network (CNN), for which the main goal is generating fractional orders of the neurons whose Kaplan-Yorke dimension is being maximized. We propose a method based on Fourier transform to evaluate if the generated time series is chaotic or not. The solutions that do not have chaotic behavior are not passed to the time series analysis (TISEAN) software, thus saving execution time. We show the best solutions provided by DE and APSO of the attractors of the fractional-order chaotic CNNs.

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