I am a Senior Lecturer in Mathematics at Loughborough University, United Kingdom.

My research expertise is in nonlinear dynamical systems and their applications to medicine, biology, engineering and physics. I have been studying various self-organisation phenomena that can be described with dynamical systems, with relevant models involving autonomous dynamical systems, systems under deterministic and random forces, systems with time delay, and interacting oscillators. I am a co-author of the book "Synchronization: from simple to complex" (Springer, 2009).

Currently, my research is centered around mathematics of cognition. Recently I proposed a novel mathematics-based approach to explain the brain and artificial neural networks [Janson & Marsden (2017)]. The proposed framework could make the first draft of the unified brain theory and lead to brain-like explainable artificial intelligence.

The theory above suggests that, mathematically, the brain could be viewed as a dynamical system with a plastic self-organising vector field. In a dynamical model of the brain, the velocity vector field embodies both the rules governing the behavior, and the brain's physical structure, hence psychology and physiology can be directly connected via this field. Unlike in all previously known machines and organs, in the brain the vector field self-organises to reflect the self-organising brain plasticity, which required introduction of a previously unknown type of dynamical systems. Thus, it is proposed that to be intelligent, the system should have at least two levels of self-organisation as opposed to only one in all previously known dynamical systems. Namely, there should be conventional self-organisation of the observable behavior, and in addition self-organisation of the (not directly observable) rules governing the behavior and embodied in the vector field. The first theoretical analysis of the new systems has been done in [Janson & Kloeden (2020)], [Janson & Kloeden (2021)].

Among other things, the new theory offers an explicit answer to the question "What are memories and where can they be found in the brain?". It is proposed that memories are traces on the vector field of the brain, and just like this fied they exist in the phase space of the brain. The phase space is a well-known mathematical object, and although it is an abstract non-physicsl space, it can be reconstructed from detailed measurements of the physical parameters of the brain.

My other direction of research is optimisation based on mathematics of complex systems. Recently I proposed a mathematical principle for building an analogue optimising machine, which could potentially be faster than digital computers and much cheaper than quantum ones [Janson & Marsden 2021a, Janson & Marsden 2021b]. The idea is to utilise deterministic chaos instead of traditionally used random forces to make the system explore all available minima of the cost function. To achieve this in a reliable and robust manner not sensitive to the specifics of the problem, we introduced the time delay into the classical gradient descent setting. This is a highly counter-intuitive step given that normally the delay makes an already complex situation much more complex and unpredictable. However, we hypothesized and verified that for a certain broad class of the systems with delay, an increase of the delay can lead to quite predictable robust results, which would be largely independent of the specifics of the equations.

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