L3

What can maths tell us about this topic? Do mathematicians even have a seat at the table, and should we? What do we know about directed networks and dynamical systems that can contribute to this?

We consider the implications of the mathematical modelling and analysis of neurone-to-neurone dynamical complex networks. We explain how the dynamical behaviour of relatively small scale strongly connected networks lead naturally to non-binary information processing and thus to multiple hypothesis decision making, even at the very lowest level of the brain’s architecture. This all looks a like a a loose  coupled array of  k-dimensional clocks. There are lots of challenges for maths here. We build on these ideas to address the "hard problem" of consciousness - which other disciplines say is beyond any mathematical explanation for ever! 

We discuss how a proposed “dual hierarchy model”, made up from both externally perceived, physical, elements of increasing complexity, and internally experienced, mental elements (which we argue are equivalent to feelings), may support a leaning and evolving consciousness. We introduce the idea that a human brain ought to be able to re-conjure subjective mental feelings at will. An immediate consequence of this model  is that finite human brains must always be learning and forgetting and that any possible subjective internal feeling that might be fully idealised only with a countable infinity of facets, could never be learned completely a priori by zombies or automata: it may be experienced more and more fully by an evolving human brain (yet never in totality, not even in a lifetime). 

Considerable attention has been recently focused on the study of muscle tissues viewed as prototypes of new materials that can actively generate stresses. The intriguing mechanical properties of such systems can be linked to hierarchical internal architecture. To complicate matters further, they are driven internally by endogenous mechanisms supplying energy and maintaining non-equilibrium.  In this talk we review the principal mechanisms of force generation in muscles and discuss the adequacy of the available mathematical models.

Networks of interacting oscillators give rise to collective dynamics such as localized frequency synchrony. In networks of neuronal oscillators, for example, the location of frequency synchrony could encode information. We discuss some recent persistence results for certain dynamically invariant sets called weak chimeras, which show localized frequency synchrony of oscillators. We then explore how the network structure and interaction allows for dynamic switching of the spatial location of frequency synchrony: these dynamics are induced by stable heteroclinic connections between weak chimeras. Part of this work is joined with Peter Ashwin (Exeter).

Building on advancements in computer vision we now have an array of visual tracking methods that allow the reliable estimation of cellular motion in high-throughput settings as well as more complex biological specimens. In many cases the underlying assumptions of these methods are still not well defined and result in failures when analysing large scale experiments.

Using organotypic co-culture systems we can now mimic more physiologically relevant microenvironments in vitro.  The robust analysis of cellular dynamics in such complex biological systems remains an open challenge. I will attempt to outline some of these challenges and provide some very preliminary results on analysing more complex cellular behaviours.

In the first part of my presentation, I will briefly summarize a dynamic view of the cell cycle created in collaboration with Prof John Tyson over the past 25 years. 
In our view, the decisions a cell must make during DNA synthesis and mitosis are controlled by bistable switches, which provide abrupt and irreversible transition 
between successive cell cycle phases. In addition, bistability provides the foundation for 'checkpoints' that can stop cell proliferation if problems arise 
(e.g., DNA damage by UV irradiation). In the second part of my talk, I will highlight a few representative examples from our ongoing BBSRC Strategic LoLa grant 
(http://cellcycle.org.uk/) in which we are testing the predictions of our theoretical ideas in human cells in collaboration with four experimental groups.

All data intelligence processes are designed for processing a finite amount of data within a time period. In practice, they all encounter
some difficulties, such as the lack of adequate techniques for extracting meaningful information from raw data; incomplete, incorrect 
or noisy data; biases encoded in computer algorithms or biases of human analysts; lack of computational resources or human resources; urgency in 
making a decision; and so on. While there is a great enthusiasm to develop automated data intelligence processes, it is also known that
many of such processes may suffer from the phenomenon of data processing inequality, which places a fundamental doubt on the credibility of these 
processes. In this talk, the speaker will discuss the recent development of an information-theoretic measure (by Chen and Golan) for optimizing 
the cost-benefit ratio of a data intelligence process, and will illustrate its applicability using examples of data analysis and 
visualization processes including some in bioinformatics.

The past few years have seen a surge of interest in nonconvex reformulations of convex optimizations using nonlinear reparameterizations of the optimization variables. Compared with the convex formulations, the nonconvex ones typically involve many fewer variables, allowing them to scale to scenarios with millions of variables. However, one pays the price of solving nonconvex optimizations to global optimality, which is generally believed to be impossible. In this talk, I will characterize the nonconvex geometries of several low-rank matrix optimizations. In particular, I will argue that under reasonable assumptions, each critical point of the nonconvex problems either corresponds to the global optimum of the original convex optimizations, or is a strict saddle point where the Hessian matrix has a negative eigenvalue. Such a geometric structure ensures that many local search algorithms can converge to the global optimum with random initializations. Our analysis is based on studying how the convex geometries are transformed under nonlinear parameterizations.