Aim of this project is to study some Partial Differential Equations (PDE) arising as models of biological processes and other systems
(such as e.g., crowds, animal groups, cell colonies)
involving individuals who possess some decisional abilities.
Indeed complex systems have attracted interest in various fields, from Sociology to Economyy and Biology since they pose new and
stimulating scientific challenges with respect to more traditional systems:
"Today, most of science is biology" (Reed, "Mathematical biology
is good for mathematics", Notices of AMS, 62, 2015) and also in Mathematics biological applications are becoming the main driving force of innovation.
Many biological processes are controlled by complex interactions which are not well described by a classical setting even in the case of Euclidean geometry and therefore require differential models with a more flexible structure. In addition often such processes evolve on irregular spatial domains and the Euclidean setting is only a first approximation to the complexity of the problem. Hence the increasing interest in the study of nonlinear differential models on networks, on ramified spaces and involving nonlocal terms.
In this project we concentrate on differential problems as outlined
above with the following objectives:
a) A correct mathematical formulation of interacting models with special attention to
- PDE defined on networks and other irregular geometric structures
- PDE with nonlocal terms
b) The extension to the new setting of classical functional analysis techniques in order to study the well posed-ness of the previous problems.
c) The development of numerical methods and algorithms for the
validation of the theoretical results.
The technological development of the last decades has made it possible to measure biological quantities that were previously out of
reach. Experimentalists, such as molecular biologists, geneticists,
clinicians, have now access to individual cell motion, molecular
content in some proteins, genes controlling these proteins. A very large quantity of experimental data is now available. Mathematical modeling can help to validate predictions and biological insights as well as to reduce the number of experiments needed to confirm them. Hence the theoretical and numerical study of biological processes and the training of young researchers in mathematical biology are important for the future of mathematical research.
Going into detail we expect our research to have a scientific impact
in the following fields:
1) Optimal transport and biological algorithms. The ability of the
slime mold Physarium to solve a maze by finding the shortest path
joining two food sites has attracted attention to a possible
"Physarium computing" (Tero et al., Physarum solver: A biologically
inspired method of road-network navigation, Physica A 363, 2006).
Recently the Physarium algorithm was applied to the network design for a newly established railroad network in Tokyo to reach maximum
efficiency by minimizing cost while maximizing transport and fault
tolerance (Tero et al., Rules for Biologically Inspired Adaptive
Network Design, Science 327, 2010). Hence a deeper insight into the
Physarium model with the possibility of considering multiple and
distributed food sites in the maze, besides helping to understand
better the transport mechanism in the slime mold, can be useful in
network design of infrastructures such as railroads, highways and power grids.
2) Discrete control systems, self-similarity and applications.
Concerning the second point of the project,
on the one hand we expect our contribution to
enrich the family of mathematical models for zoology, on the other
hand this study requires some theoretical developments that can
possibly be applied to wider settings. We also remark that modeling
articulations observed in nature is an important tool for the development of bio-inspired robotic arms. The importance of proposing models for highly articulated manipulators and soft robots relies in their precision in grasping and their ability of moving in constrained environments. These features are particularly relevant for medical applications: snake-like robots are used for endoscopic surgery to access and visualize hard-to-reach
anatomical locations, see the Flex® Robotic System project (http://medrobotics.com/gateway/flex-system-int/)
and the survey (Burgner-Kahrs, Rucker, Choset "Continuum robots
for medical applications: A survey." IEEE Transactions on Robotics, 2015).
Also, the research on dexterous manipulation and precision
grasp, also ensured by scalable highly articulated manipulators, finds
application in rehabilitative medicine as support for the recovery of
hand functions after strokes (Lum et altri "Robotic approaches for rehabilitation of hand function after stroke", American Journal of Physical Medicine & Rehabilitation, 2012).
3) Visco-elasticity and biomechanics
Biomechanics is the study of of biological systems using the methods of mechanics and newer and better measurement techniques continuously improve the available data
about the composition and behavior of bones, cartilages, and ligaments. Indeed biological tissues are viscoelastic materials since their behavior is both viscous, meaning
time- and history-dependent, as well as elastic (Ratner et altri, "Biomaterials science: an introduction
to materials in medicine" Academic, New York, 1996). Hence the study of mathematical models arising in Biomechanics and a better comprehension of viscoelastic properties of biological materials are necessary for understanding, design, fabricate and replace a body part or malfunctioning organ
due to disease processes, trauma, or surgical removal.