Mathematical Description of Human Nervous System Using Fractals, Fractional and Integer Order Calculus

Dear Colleagues,

This Special Issue aims to offer an overview of state-of-the-art developments in the field of mathematical descriptions of the human nervous system using fractals or fractional and integer order derivatives. We invite researchers from the health sector, industry, and academia to contribute to this Special Issue. Original research articles and reviews are welcome. The vision of this Special Issue encompasses the most relevant developments in using novel approaches in neurosciences, including conceptual and theoretical approaches, as well as numerical methods that can be applied in neuroscience. This type of modeling of structure and processes, biomedical signal and image analyses, and experimental research illustrates how to solve specific challenges. This Special Issue also provides a platform for researchers to share their latest findings, exchange ideas, and identify future directions for research in these exciting and rapidly evolving fields.

Topics of interest include, but are not limited to, the following:

• Fractal nature of the nervous system and fractals in the nervous system;
• Fractals in neuroscience, general principles, basic neuroscience, clinical applications;
• Investigating fractal analysis as a diagnostic tool;
• Recurrent fractal neural networks;
• Fractal character of the neural spike train;
• Fractals and chaotic dynamics of the nervous system;
• Fractal analysis of the resting state;
• Fractal analysis and spontaneous activity;
• Fractal design in human brain and nervous tissue;
• Fractal nature of the nervous cell;
• Fractional stochastic models;
• Continuous and discrete time neural networks dynamic models;
• Equilibrium of nervous system and stability of an equilibrium.

We invite you to participate in our open call for papers to share ideas, experiments, and ultimately knowledge in this emerging area of public interest.

Finally, I would like to thank Andreea Valentina Cojocaru and her valuable work for assisting me with this Special Issue.

Prof. Dr. Stefan Balint
Guest Editor

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• human nervous system
• mathematical description using fractals
• mathematical description using integer order and fractional order calculus

Title: Fractals, appearing in the mathematical description of the dynamics of nervous system, using discrete time Hopfield neural network
Authors: Andreea V. Cojocaru1, Stefan Balint2
Affiliation: 1 Independent researcher, [email protected] 2 Department of Computer Science, West University of Timisoara, Blvd. V. Parvan 4, 300223 Timisoara, Roma-nia, [email protected]
Abstract: Abstract: Continuous-time and Discrete-time Hopfield neural networks claims to be mathematical descriptions of electrical phenomena appearing in nervous system. From the point of view of controllability in [1] an extensive analysis of Continuous-time and Discrete-time Hopfield neural networks was undertaken in general. In the present paper, the dynamics of a Discrete-time Hopfield neural network of two neurons with delay and no self-connection, and the dynamics of a Discrete-time Hopfield neural network of five neurons with delay and ring architecture is analyzed from fractals point of view. In the cases presented here, the external inputs are equal to zero and the evolution of the voltage state of neurons is due to the variation of an integrated internal parameter of the network. The internal parameter can be regarded as the consequence of a disease. The novelty is that, in the considered cases, it is shown that, the so called “bifurcation diagrams”, representing the dynamics of the neurons due to the variation of the integrated internal parameter of the network, are fractals. Moreover, for some particular values of the internal integrated parameter, the orbits of neurons or the orbit projection on the coordinate planes are also fractals. For the analysis of these fractals the largest Liapunov exponent is used.