Federico Zertuche

Quantum dynamical entropies in discrete classical chaos

We discuss certain analogies between quantization and discretization of classical systems on manifolds. In particular, we will apply the quantum dynamical entropy of Alicki and Fannes to numerically study the footprints of chaos in discretized versions of hyperbolic maps on the torus.

On the robustness of NK-Kauffman networks against changes in their connections and Boolean functions

NK-Kauffman networks LKN are a subset of the Boolean functions on N Boolean variables to themselves, ΛN={ξ:Z2N→Z2N}. To each NK-Kauffman network it is possible to assign a unique Boolean function on N variables through the function Ψ:LKN→ΛN. The probability PK that Ψ(f)=Ψ(f′), when f′ is obtained through f by a change in one of its K-Boolean functions (bK:Z2K→Z2), and/or connections, is calculated. The leading term of the asymptotic expansion of PK, for N⪢1, turns out to depend on the probability to extract the tautology and contradiction Boolean functions, and in the average value of the distribution of probability of the Boolean functions, the other terms decay as O(1/N). In order to accomplish this, a classification of the Boolean functions in terms of what I have called their irreducible degree of connectivity is established. The mathematical findings are discussed in the biological context, where Ψ is used to model the genotype-phenotype map.

The asymptotic number of attractors in the random map model

The random map model is a deterministic dynamical system in a finite phase space with n points. The map that establishes the dynamics of the system is constructed by randomly choosing, for every point, another one as its image. We derive here explicit formulae for the statistical distribution of the number of attractors in the system. As in related results, the number of operations involved by our formulae increases exponentially with n; therefore, they are not directly applicable to study the behaviour of systems where n is large. However, our formulae can be used to derive useful asymptotic expressions, as we show.

Number of different binary functions generated by NK-Kauffman networks and the emergence of genetic robustness

We determine the average number

Z2-algebras in the Boolean function irreducible decomposition

\vartheta (N, K)

Discrete dynamical systems embedded in Cantor sets

, of \textit{NK}-Kauffman networks that give rise to the same binary function. We show that, for

Recognition of temporal sequences of patterns using state-dependent synapses

N \gg 1

Storage capacity of a neural network with state-dependent synapses

, there exists a connectivity critical value

Categorization and effective perceptron learning in feed-forward neural networks

K_c

On the Pauli-Van Vleck formula for arbitrary quadratic systems with memory in one dimension

such that