I am mainly interested in problems that involve complex systems, especially those that explore the interdisciplinary area between physics and biology. Dynamical systems, formation of spatio-temporal patterns, self-organization and emergence of order in chemical and biological systems, chaos, adaptive systems, are among my research interests. In the following I summarize my contribution to some of these fields.
I have also studied dynamical systems based on small-world networks, which seem to be specially suited to model social interactions. I have focused on two models: a socially-inspired imitation game, and the spread of an infectious disease. In both cases promising results have been found, especially in the latter, where a transition in the epidemic behavior exists at a finite value of the small-world parameter. In a neural network with small-world topology, we discovered that the disorder parameter induces a phase transition in the ability of the system to perform as an associative memory device. There critical value of the parameter below which the network cannot recover any of the stored patterns, a fact that we have dubbed: Order kills memory!
I have worked in the mathematical modeling of the Hantavirus epidemics, a common disease that chronically affects the deer mouse in North America, and which is potentially lethal when transmitted to humans. We focused on a few significant field observations and were able to develop a simple mathematical model that accounts for them. The observations are the fact that the infection sporadically disappears and reappears at most regions in the wild, but that some spots act as "refugia," with persistent infection. In our model, the role of the environment functions as a control parameter that determines a phase transition between an infected and a non-infected phase. The wild populations seem to live near criticality in the arid desert of the Southwest, and either seasonal or extraordinary variations drive the extent of the disease.
I have worked in models of biological evolution. Some of these have been observed to evolve to a self-organized critical state, a very interesting possibility for a real ecosystem. Realizing that the concept of "fitness" being used by a growing number of physicists in this field was not the biological one, I developed a more realistic model, one in which there is not even the need of computing a fitness function. The model is a dynamical system of predator-prey interactions between the species. The interactions are allowed to mutate, and the system to evolve. Interesting results show that avalanches of extinction without a characteristic size occur in the system. The absence of a typical scale is often a signature of critical states. More importantly, it shows that an ecological system can suffer extinctions at all scales, for reasons that arise exclusively from the interactions and the dynamics of the populations. Subsequent work in the field of coupled chaotic elements also has been inspired by the modeling of ecological systems.
Interesting collective behavior arises in networks of coupled chaotic elements. I have mainly studied systems of coupled logistic maps, which are both simple in form, and rich and universal in behavior. I have performed some work in this field about the clusterization mechanism of globally coupled maps. I have also studied systems in which the local dynamics is supplemented with an evolutionary rule, inspired by the biological mechanisms of learning or evolution. This additional dynamics affects the local one by either changing the parameters of an element's map or the coupling intensity of it with the rest of the system. In the first case, self-organized states of localized activity develop. In the second case, the system is able to "learn" to self-synchronize.
During the realization of my PhD thesis I worked on chemical kinetics, studying model systems of interacting particles described by reaction-diffusion equations, that behave anomalously in low dimensions. I developed an analytical mesoscopic framework to describe two-component systems. I applied it to models of the A+B -> B+C reaction, in different situations of mutual diffusion of the substances A and B, and systems related to this, like systems of "fixed traps", "quasi-dynamical traps" and the annihilation A+B -> C. Analytic and numeric solutions of these equations were compared with numerical simulations of the systems, of which I also made a detailed study.
I performed analysis of DNA sequences within the framework of statistical mechanics. When the first complete genome of a higher organism (S. cerevisiae) was sequenced, I developed a mapping of the DNA onto a two-dimensional pseudo random walk. I analyzed several properties of the DNA walks thus obtained, like its mean square deviation and the fractality of the sites visited by the walker. I found that long range correlations effectively exist in the sequences, and that the portions of DNA that codes for proteins have different statistical properties than those which do not, although the biological implications of this are far from known. There exist controversial results in this field and a lot of work remains to be done. A subsequent investigation based on a multifractal analysis of the sequences confirmed the previous results.
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