An SPDE Approach to Self-Excitatory Neuronal Models

This project falls within the EPSRC Statistics and Applied Probability research area.

This project aims to study a system of interacting diffusions on the negative real line of the McKean-Vlasov type with a resetting component and all are influenced by an independent common source of noise. When one of the diffusions of the system hits zero, its value is reset and all other diffusions in the system will receive a continuous impulse to their value over a predetermined timescale. This system is a generalisation of noisy integrate-and-fire models in neuroscience that model the membrane potential of a neuron in biological systems. An important feature of these models is that when the membrane potential reaches a certain threshold it is instantly reset to some predetermined value and all other neurons in the system receive an impulse that affects their voltage. The primary hypothesis of noisy integrate-and-fire models is that the action potential threshold and shape are invariant from spike to spike, and thus a precise description of the spike can be omitted and simply sketched by a spiking threshold. The diffusion model captures solely spike generation and places the biological aspects of the neuron into the noise. It has been observed that neurons react in a predictable and reproducible manner to temporally structured stimuli. The main sources of stochasticity in the model arises from the innate stochasticity in the biological mechanisms which cause voltage changes in neurons and the neurons may receive inputs from other neurons in the system which are not being observed. The latter may be thought of as the common source of noise in the model.

The main objective of this project is to show the existence and uniqueness of the mean-field limit to the system of diffusions described above. Moreover, as the interactions between diffusions occur over predefined time scales, this project will attempt to show that, as the time scales on which interactions occur approach zero, the models where the interaction between diffusions occur instantly are recovered. Aside from these aims, this project will also attempt to prove the convergence of an Euler-Maruyama type method for the numerical solution of the limiting system.

A further novelty of this research project comes from there being a common noise source felt by all diffusions in the finite system which leads to showing well-posedness and solutions to an SPDE that describes the law of the representative particle in the limiting system with infinite diffusions. The SPDE approach which will be employed by this project leads to looking for solutions being elements of the càdlàg functions taking values in the space of tempered distributions endowed with the M1 topology. This approach differs from existing literature as generally solutions to large scale limits of systems interacting diffusions are looked for in the space of probability measures on the space of càdlàg functions endowed with the M1 topology.

Probabilistic modelling permeates the Financial services, healthcare, technology and other Service industries crucial to the UK's continuing social and economic prosperity, which are major users of stochastic algorithms for data analysis, simulation, systems design and optimisation. There is a major and growing skills shortage of experts in this area, and the success of the UK in addressing this shortage in cross-disciplinary research and industry expertise in computing, analytics and finance will directly impact the international competitiveness of UK companies and the quality of services delivered by government institutions.
By training highly skilled experts equipped to build, analyse and deploy probabilistic models, the CDT in Mathematics of Random Systems will contribute to
- sharpening the UK's research lead in this area and
- meeting the needs of industry across the technology, finance, government and healthcare sectors

MATHEMATICS, THEORETICAL PHYSICS and MATHEMATICAL BIOLOGY

The explosion of novel research areas in stochastic analysis requires the training of young researchers capable of facing the new scientific challenges and maintaining the UK's lead in this area. The partners are at the forefront of many recent developments and ideally positioned to successfully train the next generation of UK scientists for tackling these exciting challenges.
The theory of regularity structures, pioneered by Hairer (Imperial), has generated a ground-breaking approach to singular stochastic partial differential equations (SPDEs) and opened the way to solve longstanding problems in physics of random interface growth and quantum field theory, spearheaded by Hairer's group at Imperial. The theory of rough paths, initiated by TJ Lyons (Oxford), is undergoing a renewal spurred by applications in Data Science and systems control, led by the Oxford group in conjunction with Cass (Imperial). Pathwise methods and infinite dimensional methods in stochastic analysis with applications to robust modelling in finance and control have been developed by both groups.
Applications of probabilistic modelling in population genetics, mathematical ecology and precision healthcare, are active areas in which our groups have recognized expertise.

FINANCIAL SERVICES and GOVERNMENT

The large-scale computerisation of financial markets and retail finance and the advent of massive financial data sets are radically changing the landscape of financial services, requiring new profiles of experts with strong analytical and computing skills as well as familiarity with Big Data analysis and data-driven modelling, not matched by current MSc and PhD programs. Financial regulators (Bank of England, FCA, ECB) are investing in analytics and modelling to face this challenge. We will develop a novel training and research agenda adapted to these needs by leveraging the considerable expertise of our teams in quantitative modelling in finance and our extensive experience in partnerships with the financial institutions and regulators.

DATA SCIENCE:

Probabilistic algorithms, such as Stochastic gradient descent and Monte Carlo Tree Search, underlie the impressive achievements of Deep Learning methods. Stochastic control provides the theoretical framework for understanding and designing Reinforcement Learning algorithms. Deeper understanding of these algorithms can pave the way to designing improved algorithms with higher predictability and 'explainable' results, crucial for applications.
We will train experts who can blend a deeper understanding of algorithms with knowledge of the application at hand to go beyond pure data analysis and develop data-driven models and decision aid tools
There is a high demand for such expertise in technology, healthcare and finance sectors and great enthusiasm from our industry partners. Knowledge transfer will be enhanced through internships, co-funded studentships and paths to entrepreneurs