Rigatos G.G. Advanced Models of Neural Networks. Nonlinear Dynamics and Stochasticity in Biological Neurons

Springer, 2015, -296 p.This book provides a complete study on neural structures exhibiting nonlinear and stochastic dynamics. The book elaborates on neural dynamics by introducing advanced models of neural networks. It overviews the main findings in the modeling of neural dynamics in terms of electrical circuits and examines their stability properties with the use of dynamical systems theory. Such electric circuit models are characterized by attractors and fixed points while in certain cases they exhibit bifurcations and chaotic dynamics. Moreover, solutions of the neural dynamics in the form of travelling waves equations are derived. The dynamics of interconnected neurons is analyzed with the use of forced oscillator and coupled oscillator models. It is shown that by introducing stochastic uncertainty and variations in the previous deterministic coupled oscillator model, stochastic coupled oscillator models can be derived. Next, going into a more refined analysis level it is shown how neural dynamics can be interpreted with the use of quantum and stochastic mechanics. It is proven that the model of interacting coupled neurons becomes equivalent to the model of interacting Brownian particles that is associated with the equations and dynamics of quantum mechanics. It is shown that such neural networks with dynamics compatible to quantum mechanics principles exhibit stochastic attractors. Furthermore, the spectral characteristics of such neural networks are analyzed. Additionally, a stochastic mechanics approach to stabilization of particle systems with quantum dynamics is presented. It also shown that the eigenstates of the quantum harmonic oscillator (QHO) can be used as activation functions in neural networks. Moreover, a gradient-based approach to stabilization of particle systems with quantum dynamics is provided. There are remarkable new results in the book concerned with: (1) nonlinear synchronizing control of coupled neural oscillators, (2) neural structures based on stochastic mechanics or quantum mechanics principles, (3) nonlinear estimation of the wave-type dynamics of neurons, (4) neural and wavelet networks with basis functions localized both in space and frequency, (5) stochastic attractors in neural structures, and (6) engineering applications of advanced neural network models.
This book aims at analyzing advanced models of neural networks, starting with methods from dynamical systems theory and advancing progressively to stochasticity-based models and models compatible with principles of quantum mechanics. Advanced models of neural networks enable on the one side to understand patterns of neuronal activity seen in experiments in the area of neuroscience and on the other side to develop neurocomputing methods in the area of information sciences that remain consistent with physics principles governing biological neural structures.Modeling Biological Neurons in Terms of Electrical Circuits
Systems Theory for the Analysis of Biological Neuron Dynamics
Bifurcations and Limit Cycles in Models of Biological Systems
Oscillatory Dynamics in Biological Neurons
Synchronization of Circadian Neurons and Protein Synthesis Control
Wave Dynamics in the Transmission of Neural Signals
Stochastic Models of Biological Neuron Dynamics
Synchronization of Stochastic Neural Oscillators Using Lyapunov Methods
Synchronization of Chaotic and Stochastic Neurons Using Differential Flatness Theory
Attractors in Associative Memories with Stochastic Weights
Spectral Analysis of Neural Models with Stochastic Weights
Neural Networks Based on the Eigenstates of the Quantum Harmonic Oscillator
Quantum Control and Manipulation of Systems and Processes at Molecular Scale