Research
My main research interest is in problems from complex dynamics that have applications
to various fields in the natural sciences. Hence part of my work is purely
theoretical, focused around identifying and understanding new phenomena in
discrete random complex dynamics, and in complex dynamic networks. The other
part centers around using dynamical systems methods and results to derive and
analyze models in a variety of fields, among which mathematical neuroscience
(to model brain networks that govern emotion and reward) medicine
(vision pathologies of the retina), epidemiology (dynamics of Ebola outbreaks),
climate (human impact to green house gas emissions) and the environment
(pharmacokinetics of lead, and effects of contamination on cognitive development
in children).
Work in dynomics. My primary mathematical research interest is in using complex
dynamics to understand "dynomics," that is to study ensemble dynamics in networks
of interacting nodes. The goal is to understand how the global behavior of a
dynamic network emerges from the interplay between the network's connectivity
profile and the node-wise dynamics. This is a crucial question with potential
applications to many fields in which dynamic networks are a central object:
from neuroscience to sociology, from ecology to epidemics. If one can dissociate
which aspects of the temporal evolution of the system are due to specific patterns
in its hard-wired architecture, and which are due to properties of the local
dynamics in each node -- one may increase the ability to predict the long-term
dynamic outcome in such a system, classify it, control it and potentially repair
it if necessary. Progress on dynomics questions has been notoriously difficult,
due to the size and complexity of natural systems. Stepping away from the original
enterprise to address massive biophysical, data-driven models, a trend in recent
research has been regrouping around understanding basic principles in simpler or
smaller networks. While this approach has produced promising results within a
variety of dynamic classes, a lot is still missing in the direction of obtaining
general, universal principles. My work addresses precisely this gap, by
constructing a canonical framework for studying dynomics, based upon one of the
oldest and best understood branches in the field of dynamics: iterated complex
quadratic maps. I am working on (1) development of new and original mathematical
methods for networks, building on specific properties of iterated maps that make
this framework desirable for studying dynomics; (2) comparison with results in
other established network models (some of them currently studied by collaborators),
to investigate whether these can be reproduced and unified within our canonical
framework; (3) application of this consolidated approach to modeling problems in
the life sciences (neuroscience in particular).
Work in template iterations. Random iterations have been studied since the early
1990s, starting with the pioneering work of Fornaess and Sibony, and more recently
by our collaborator Mark Comerford. As a particular problem within the field of
non-autonomous discrete systems, we consider iterations of two quadratic maps,
according to a prescribed binary sequence, which we call template. Our
theoretical setup can be broadly described as iterating in a prescribed order a
"correct" function and an "erroneous" perturbation (or mutation). We consider
problems that a sustainable replication system may have to solve when facing the
potential for errors (e.g., which specific errors are more likely to affect the
system's dynamics, in absence of prior knowledge of their timing). We found, for
example, that within an optimal locus for the correct function, almost no errors
can affect the sustainability of the system. Mathematically, our work complements
broader existing results in non-autonomous dynamics with more specific detail for
the case of two random iterated functions. We have recently started using this
framework to study applications to cancer genetics.
Work in mathematical neuroscience. My main interest in computational
neuroscience is to understand human brain function within the framework of dynomics,
as described above. As an over-arching goal of my research, I aim to deliver model
results and predictions applicable towards crafting novel, neurobiology-based
diagnosis, classification and treatment techniques in psychiatry. At the
physiological scale, a brain circuit can be seen as a network of single neurons,
with edges between nodes representing interneuronal synapses. Understanding spiking
rhythms in populations of cells -- and the factors contributing to triggering and
tuning these rhythms -- is very important, since firing patterns are often the best
indicators of pathological behavior. We are working in a mathematically canonical
context which may lead to new connections and insights; it is not unusual in
mathematics to rephrase a problem in a more general framework to reveal paths to
possible solutions. All our physiological modeling is in collaboration with the
Scimemi lab at SUNY Albany, which is also providing the data. At a macroscopic
scale, the brain may be viewed as a network in which the interconnected nodes
represent entire functional regions (e.g., the amygdala, or the hippocampus),
and the edges correspond to projection bundles between regions, which can be mapped
and quantified (using dyes and various neuroimaging techniques). This may lead to
extremely valuable clinical insights into quantitative, neurobiology-driven
diagnostics of behavior and psychiatric illness. In collaboration with a group at
SUNY Buffalo, I am currently working on using tractography-based human connectomes
available in the public domain to test the efficiency of our complex dynomics
methods to distinguish between connectomes of different individuals,
with different behavioral profiles.
Other models from the natural sciences. A considerable part of my research
resources are allocated to investigating how traditional and novel methods from
dynamical systems can help model, analyze and understand complex natural systems.
In addition to the neurosicence and genetics applications described above, I have
been exploring a variety of applications, in collaboration with both colleagues
from other institutions, and with my own mentored students. One of the most
prominent such efforts is a model of retinal pathologies, on which I began
collaborating with Dr. Camacho (at ASU) in 2017. Our modeling work was recently
supported by the AWM as a Women in Biomathematics team project at their IPAM
workshop in June 2019. The team, co-lead by Dr. Camacho and myself and
composed of six women in mathematics, is currently working on writing results
for publication, and on applying for funds to support further collaborative
meetings. Another major modeling project emerged in 2018 from a local concern
about drinking water quality, and developed throughout the following year into
a pharmacokinetic model of lead contamination and neurotoxicity. Building upon
our (now published) results, we are working on applying for an exploratory NIH
grant that would support our future work on the model. Other recent models, some
representing mentored research with undergraduates, have been centered around
other sustainability problems, such as finding optimal quarantine measures during
an Ebola outbreak, modeling the genetics of yeast proliferation, and identifying
the human behaviors with highest impact on the greenhouse effect.
Current research collaborations