Adaptive Bet-Hedging Revisited: Considerations of Risk and Time Horizon.

Models of adaptive bet-hedging commonly adopt insights from Kelly's famous work on optimal gambling strategies and the financial value of information. In particular, such models seek evolutionary solutions that maximize long-term average growth rate of lineages, even in the face of highly stochastic growth trajectories. Here, we argue for extensive departures from the standard approach to better account for evolutionary contingencies.

The stochastic Fitzhugh-Nagumo neuron model in the excitable regime embeds a leaky integrate-and-fire model.

In this paper, we provide a complete mathematical construction for a stochastic leaky-integrate-and-fire model (LIF) mimicking the interspike interval (ISI) statistics of a stochastic FitzHugh-Nagumo neuron model (FHN) in the excitable regime, where the unique fixed point is stable. Under specific types of noises, we prove that there exists a global random attractor for the stochastic FHN system. The linearization method is then applied to estimate the firing time and to derive the associated radial equation representing a LIF equation.

From typical sequences to typical genotypes.

We demonstrate an application of a core notion of information theory, typical sequences and their related properties, to analysis of population genetic data. Based on the asymptotic equipartition property (AEP) for nonstationary discrete-time sources producing independent symbols, we introduce the concepts of typical genotypes and population entropy and cross entropy rate. We analyze three perspectives on typical genotypes: a set perspective on the interplay of typical sets of genotypes from two populations, a geometric perspective on their structure in high dimensional space, and a statistical learning perspective on the prospects of constructing typical-set based classifiers.

The free energy method and the Wright-Fisher model with 2 alleles.

We systematically investigate the Wright-Fisher model of population genetics with the free energy functional formalism of statistical mechanics and in the light of recent mathematical work on the connection between Fokker-Planck equations and free energy functionals. In statistical physics, entropy increases, or equivalently, free energy decreases, and the asymptotic state is given by a Gibbs-type distribution. This also works for the Wright-Fisher model when rewritten in divergence to identify the correct free energy functional.

The evolution of moment generating functions for the Wright-Fisher model of population genetics.

We derive and apply a partial differential equation for the moment generating function of the Wright-Fisher model of population genetics.

An introduction to the mathematical structure of the Wright-Fisher model of population genetics.

In this paper, we develop the mathematical structure of the Wright-Fisher model for evolution of the relative frequencies of two alleles at a diploid locus under random genetic drift in a population of fixed size in its simplest form, that is, without mutation or selection. We establish a new concept of a global solution for the diffusion approximation (Fokker-Planck equation), prove its existence and uniqueness and then show how one can easily derive all the essential properties of this random genetic drift process from our solution. Thus, our solution turns out to be superior to the local solution constructed by Kimura.