This book focuses on the dynamic complexity of neural, genetic networks, and reaction diffusion systems. The author shows that all robust attractors can be realized in dynamics of such systems. In particular, a positive solution of the Ruelle-Takens hypothesis for on chaos existence for large class of reaction-diffusion systems is given. The book considers viability problems for such systems - viability under extreme random perturbations - and discusses an interesting hypothesis of M. Gromov and A. Carbone on biological evolution. There appears a connection with the Kolmogorov complexity theory. As applications, transcription-factors-microRNA networks are considered, patterning in biology, a new approach to estimate the computational power of neural and genetic networks, social and economical networks, and a connection with the hard combinatorial problems.

Auteur

S. Vakulenko, Petersburg State University of Technology and Design, Russian Academy of Sciences, Saint Petersburg.

Contenu

Complexity and evolution of spatially extended systems: analytical approach

Chapter 1: Introduction

Dynamical systems
Attractors
Strange attractors
Neural and genetic networks
Reaction diffusion systems
Systems with random perturbations and Gromov-Carbone problem Chapter 2: Method to control dynamics: Invariant manifolds, realization of vector fields
Invariant manifolds
Method of realization of vector fields
Control of attractor and inertial dynamics for neural networks Chapter 3: Complexity of patterns and attractors in genetic networks Centralized networks and attractor complexity in such network
A connection with computational problems, Turing machines and finite automatons
Graph theory, graph growth and computational power of neural and genetical networks
Mathematical model that shows how positional information can be transformed into body plan of multicellular organism
Applications to TF- microRNA networks. Bifurcation complexity in networks Chapter 4: Viability problem, Robustness under noise and evolution
Here we consider neural and genetic networks under large random perturbations
Viability problem
We show that network should evolve to be viable, and network complexity should increase
A connection with graph growth theory (Erdos-Renyi, Albert-Barabasi)
Relation between robustness, attractor complexity and functioning speed
Why Stalin and Putin's empires fall (as a simple illustration)
The Kolmogorov complexity of multicellular organisms and genetic codes: nontrivial connections
Robustness of multicellular organisms (Drosophila as an example)
A connection with the Hopfield system Chapter 5: Complexity of attractors for reaction diffusion systems and systems with convection
Existence of chemical waves with complex fronts
Existence of complicated attractors for reaction diffusion systems
Applications to Ginzburg Landau systems and natural computing
Existence of complicated attractors for Navier Stokes equations