Low-probability states, data statistics, and entropy estimation.

A fundamental problem in the analysis of complex systems is getting a reliable estimate of the entropy of their probability distributions over the state space. This is difficult because unsampled states can contribute substantially to the entropy, while they do not contribute to the maximum likelihood estimator of entropy, which replaces probabilities by the observed frequencies. Bayesian estimators overcome this obstacle by introducing a model of the low-probability tail of the probability distribution. Which statistical features of the observed data determine the model of the tail, and hence the output of such estimators, remains unclear. Here we show that well-known entropy estimators for probability distributions on discrete state spaces model the structure of the low-probability tail based largely on a few statistics of the data: the sample size, the maximum likelihood estimate, the number of coincidences among the samples, and the dispersion of the coincidences. We derive approximate analytical entropy estimators for undersampled distributions based on these statistics, and we use the results to propose an intuitive understanding of how the Bayesian entropy estimators work.

A dynamical model of C. elegans thermal preference reveals independent excitatory and inhibitory learning pathways.

Caenorhabditis elegans is capable of learning and remembering behaviorally relevant cues such as smells, tastes, and temperature. This is an example of associative learning, a process in which behavior is modified by making associations between various stimuli. Since the mathematical theory of conditioning does not account for some of its salient aspects, such as spontaneous recovery of extinguished associations, accurate modeling of behavior of real animals during conditioning has turned out difficult. Here, we do this in the context of the dynamics of the thermal preference of C. elegans. We quantify C. elegans thermotaxis in response to various conditioning temperatures, starvation durations, and genetic perturbations using a high-resolution microfluidic droplet assay. We model these data comprehensively, within a biologically interpretable, multi-modal framework. We find that the strength of the thermal preference is composed of two independent, genetically separable contributions and requires a model with at least four dynamical variables. One pathway positively associates the experienced temperature independently of food and the other negatively associates with the temperature when food is absent. The multidimensional structure of the association strength provides an explanation for the apparent classical temperature-food association of C. elegans thermal preference and a number of longstanding questions in animal learning, including spontaneous recovery, asymmetric response to appetitive vs. aversive cues, latent inhibition, and generalization among similar cues.

A theoretical approach to understand spatial organization in complex ecologies.

Predicting the fate of ecologies is a daunting, albeit extremely important, task. As part of this task one needs to develop an understanding of the organization, hierarchies, and correlations among the species forming the ecology. Focusing on complex food networks we present a theoretical method that allows to achieve this understanding. Starting from the adjacency matrix the method derives specific matrices that encode the various inter-species relationships. The full potential of the method is achieved in a spatial setting where one obtains detailed predictions for the emerging space-time patterns. For a variety of cases these theoretical predictions are verified through numerical simulations.