An Entropic Gradient Structure in the Network Dynamics of a Slime Mold

The approach to equilibrium in certain dynamical systems can be usefully described in terms of information-theoretic functionals. Well-studied models of this kind are Markov processes, chemical reaction networks, and replicator dynamics, for all of which it can be proven, under suitable assumptions, that the relative entropy (informational divergence) of the state of the system with respect to an equilibrium is nonincreasing over time. This work reviews another recent result of this type, which emerged in the study of the network optimization dynamics of an acellular slime mold, Physarum polycephalum. In this setting, not only the relative entropy of the state is nonincreasing, but its evolution over time is crucial to the stability of the entire system, and the equilibrium towards which the dynamics is attracted proves to be a global minimizer of the cost of the network.

STOCHASTICS OF ORDER n IN BIOLOGICAL SYSTEMS: APPLICATIONS TO POPULATION DYNAMICS, THERMODYNAMICS, NONEQUILIBRIUM PHASE AND COMPLEXITY

In the present paper, a modeling in the complex space is combined with complex-valued fractional Brownian motion to get some new results in biological systems. The rational of this approach is as follows. Biological dynamics which evolve continuously in time but are not time differentiable, necessarily exhibit random properties. These random features appear also as a result of the randomness of the proper time of biological systems. Usually, this is taken into account by using white noises that is to say fractals of order two. Fractals of order n larger than two are more suitable for increments with large amplitudes, and they may be introduced by using either real-valued fractal noises with long range memory or Brownian motions with independent increments, which are necessarily complex-valued. In the later case, we are then led to describe biological systems in the complex plane. After some background on the complex-valued fractional Brownian motion, we shall deal successively with population growth, information thermodynamics of order n, nonequilibrium phase transition via fractal noises and complexity of Markovian processes via the concept of informational divergence.

Mean anisotropy of homogeneous Gaussian random fields and anisotropic norms of linear translation-invariant operators on multidimensional integer lattices

Sensitivity of output of a linear operator to its input can be quantified in various ways. In Control Theory, the input is usually interpreted as disturbance and the output is to be minimized in some sense. In stochastic worst-case design settings, the disturbance is considered random with imprecisely known probability distribution. The prior set of probability measures can be chosen so as to quantify how far the disturbance deviates from the white-noise hypothesis of Linear Quadratic Gaussian control. Such deviation can be measured by the minimal Kullback-Leibler informational divergence from the Gaussian distributions with zero mean and scalar covariance matrices. The resulting anisotropy functional is defined for finite power random vectors. Originally, anisotropy was introduced for directionally generic random vectors as the relative entropy of the normalized vector with respect to the uniform distribution on the unit sphere. The associated a-anisotropic norm of a matrix is then its maximum root mean square or average energy gain with respect to finite power or directionally generic inputs whose anisotropy is bounded above by a≥0. We give a systematic comparison of the anisotropy functionals and the associated norms. These are considered for unboundedly growing fragments of homogeneous Gaussian random fields on multidimensional integer lattice to yield mean anisotropy. Correspondingly, the anisotropic norms of finite matrices are extended to bounded linear translation invariant operators over such fields.