1. Automata Networks -- 1.1. Introduction -- 1.2. Definitions Regarding Automata Networks -- 1.3. Cellular Automata -- 1.4. Complexity Results for Automata Networks -- 1.5. Neural Networks -- 1.6. Examples of Automata Networks -- 2. Algebraic Invariants on Neural Networks -- 2.1. Introduction -- 2.2. K-Chains in 0-1 Periodic Sequences -- 2.3. Covariance in Time -- 2.4. Algebraic Invariants of Synchronous Iteration on Neural Networks -- 2.5. Algebraic Invariants of Sequential Iteration on Neural Networks -- 2.6. Block Sequential Iteration on Neural Networks -- 2.7. Iteration with Memory -- 2.8. Synchronous Iteration on Majority Networks -- 3. Lyapunov Functionals Associated to Neural Networks -- 3.1. Introduction -- 3.2. Synchronous Iteration -- 3.3. Sequential Iteration -- 3.4. Tie Rules for Neural Networks -- 3.5. Antisymmetrical Neural Networks -- 3.6. A Class of Symmetric Networks with Exponential Transient Length for Synchronous Iteration -- 3.7. Exponential Transient Classes for Sequential Iteration -- 4. Uniform One and Two Dimensional Neural Networks -- 4.1. Introduction -- 4.2. One-Dimensional Majority Automata -- 4.3. Two-Dimensional Majority Cellular Automata -- 4.4. Non-Symmetric One-Dimensional Bounded Neural Networks -- 4.5. Two-Dimensional Bounded Neural Networks -- 5. Continuous and Cyclically Monotone Networks -- 5.1. Introduction -- 5.2. Positive Networks -- 5.3. Multithreshold Networks -- 5.4. Approximation of Continuous Networks by Multithreshold Networks -- 5.5. Cyclically Monotone Networks -- 5.6. Positive Definite Interactions. The Maximization Problem -- 5.7. Sequential Iteration for Decreasing Real Functions and Optimization Problems -- 5.8. A Generalized Dynamics -- 5.9. Chain-Symmetric Matrices -- 6. Applications on Thermodynamic Limits on the Bethe Lattice -- 6.1. Introduction -- 6.2. The Bethe Lattice -- 6.3. The Hamiltonian -- 6.4. Thermodynamic Limits of Gibbs Ensembles -- 6.5. Evolution Equations -- 6.6. The One-Site Distribution of the Thermodynamic Limits -- 6.7. Distribution of the Thermodynamic Limits -- 6.8.Period ? 2 Limit Orbits of Some Non Linear Dynamics on $$ \mathbb{R}_{ + }̂{s} $$ -- 7. Potts Automata -- 7.1. The Potts Model -- 7.2. Generalized Potts Hamiltonians and Compatible Rules -- 7.3. The Complexity of Synchronous Iteration on Compatible Rules -- 7.4. Solvable Classes for the Synchronous Update -- References -- Author and Subject Index