Janson_Kloeden_plastic_accepted.pdf (282.64 kB)
Mathematical consistency and long-term behaviour of a dynamical system with a self-organising vector field
journal contribution
posted on 2020-02-18, 11:25 authored by Natalia JansonNatalia Janson, Peter E KloedenA dynamical system with a plastic self-organising velocity vector field
was introduced in [Janson & Marsden 2017] as a mathematical prototype of new explainable intelligent systems. Although inspired by the brain plasticity, it does not
model or explain any specific brain mechanisms or processes, but instead expresses
a hypothesised principle possibly implemented by the brain. The hypothesis states
that, by means of its plastic architecture, the brain creates a plastic self-organising
velocity vector field, which embodies self-organising rules governing neural activity and through that the behaviour of the whole body. The model is represented
by a two-tier dynamical system, in which the observable behaviour obeys a velocity field, which is itself controlled by another dynamical system. Contrary to
standard brain models, in the new model the sensory input affects the velocity
field directly, rather than indirectly via neural activity. However, this model was
postulated without sufficient explication or theoretical proof of its mathematical
consistency. Here we provide a more rigorous mathematical formulation of this
problem, make several simplifying assumptions about the form of the model and
of the applied stimulus, and perform its mathematical analysis. Namely, we explore
the existence, uniqueness, continuity and smoothness of both the plastic velocity
vector field controlling the observable behaviour of the system, and the of the behaviour itself. We also analyse the existence of pullback attractors and of forward
limit sets in such a non-autonomous system of a special form. Our results verify
the consistency of the problem and pave the way to constructing more models
with specific pre-defined cognitive functions.
History
School
- Science
Department
- Mathematical Sciences
Published in
Journal of Dynamics and Differential EquationsVolume
34Issue
1Pages
63 - 78Publisher
Springer VerlagVersion
- AM (Accepted Manuscript)
Rights holder
© SpringerPublisher statement
This is a post-peer-review, pre-copyedit version of an article published in Journal of Dynamics and Differential Equations. The final authenticated version is available online at: https://doi.org/10.1007/s10884-020-09834-7Acceptance date
2020-02-12Publication date
2020-03-05Copyright date
2020ISSN
1040-7294eISSN
1572-9222Publisher version
Language
- en
Depositor
Dr Natalia Janson Deposit date: 17 February 2020Usage metrics
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