Monday 28 October 2024, 13:00-14:00
Watson B16
The brain interdependencies can be studied from either a structural or functional perspective. The former focuses typically on structural connectivity (SC), while the second considers statistical interactions (usually functional connectivity, FC). While SC is inherently pairwise because it describes white-matter fibers projecting from one region to another, FC is not limited to pairwise interdependencies. Despite this, FC analyses predominantly concentrate on pairwise statistics, usually neglecting the possibility of higher-order interactions. Moreover, the precise relationship between high-order and SC is largely unknown, partly due to the absence of mechanistic models that can efficiently map brain connectomics to functional connectivity.
To investigate these interlinked issues, we have built whole-brain computational models using anatomical and functional MRI data in two applications: healthy aging and transcranial ultrasound stimulation (TUS). We show that non-linear variations in the structural connectome can largely explain the differences in high-order functional interactions between age groups. Moreover, we showed the extent of perturbations in dynamical models to describe the high-order effects of TUS in two different brain targets.
Monday 4 November 2024, 13:00-14:00
Arts 201
Stochastic differential equations (SDE) driven by white noise are important models for stochastic dynamical systems in natural science and engineering. The statistical inference of the parameters of such models based on noisy observations has also attracted considerable interest in the machine learning community. Using Girsanov's change of measure approach one can apply powerful variational techniques to solve the inference problem. A limitation of standard SDE models is the fact that they show typically a fast decay of correlation functions. If one is interested in stochastic processes with a long-time memory, a well-known possibility is to replace the Brownian motion in the SDE by the so called fractional Brownian motion (fBM) which is no longer a Markov process. Unfortunately, variational inference for this case is much less straightforward. Our approach to this problem utilises a somewhat overlooked idea by Carmona and Coutin (1998) who showed that fBM can be exactly represented as an infinite-dimensional linear combination of Ornstein-Uhlenbeck processes with different time constants. Using an appropriate discretisation, we arrive at a finite dimensional approximation which is an 'ordinary' SDE model in an augmented space. For this new model we can apply (more or less) off-the shelve variational inference approaches.
Monday 4 November 2024, 14:00-15:00
Arts 201
In this talk, I will give an overview of the field of graph-based learning, a field that has matured over the last 15 years and is rich in both practical applications and theoretical underpinnings. The key idea of graph-based learning is to understand interrelated data as a graph, to solve variational problems and PDEs on that graph to analyse that data, and to study the limits of such models as the number of nodes goes to infinity. I will begin by motivating the approach and then will discuss the mathematical framework, three classic methods in the field, the nuances of implementing these methods, and finally the theoretical underpinnings of this field.
Monday 18 November 2024, 13:00-14:00
Watson B16
In this talk, I will develop a comprehensive geometrically-exact theory for an end-loaded elastic rod constrained to deform on a cylindrical surface. By viewing the rod-cylinder system as a special case of an elastic braid, it will be shown that all forces and moments imparted by the deforming rod to the cylinder as well as all contact reactions can be obtained. This framework allows us to give a complete treatment of static friction consistent with force and moment balance. In addition to the commonly considered model of hard frictionless contact, I analyse two friction models in which the rod, possibly with intrinsic curvature, experiences either lateral or tangential friction. Applications of the theory include studying buckling of the constrained rod under compressive and torsional loads, finding critical loads to depend on Coulomb-like friction parameters, as well as the tendency of the rod to lift off the cylinder under further loading. The cylinder can also have arbitrary orientation relative to the direction of gravity. The cases of a horizontal and vertical cylinder, with gravity having only a lateral or axial component, are amenable to exact analysis, while numerical results map out the transition in buckling mechanism between the two extremes. Weight has a stabilising effect for near-horizontal cylinders, while for near-vertical cylinders it introduces the possibility of buckling purely due to self-weight. The results are relevant for many engineering and medical applications in which a slender structure winds inside or outside a cylindrical boundary.