Equivariant Hopf Bifurcation in Symmetric Artificial Neuronal Networks Characterised by Multiple Unbounded Distributed Time Delays
Abstract
Objectives: Symmetries and signal transmission time delays are ubiquitous in neuronal net?works and in other biological, physical, and mechanical systems. Retarded functional differen?tial equations have been widely used to model symmetric neuronal networks with inherent time delays. Broadly speaking, bifurcation theory is primarily concerned with how the properties of solutions to a given multi-parameter system evolve under variation of its parameters. How the properties of solutions evolve is dependent on the inherent symmetries of the system and on the number of parameters. The theory of Hopf bifurcation of symmetric systems or networks with discrete time delays is well-established. For instance, it has been shown that a time-delayed (discrete) Hopfield-Cohen-Grossberg artificial neuronal network, with a symmetric circulant interconnection matrix, admits multiple coexisting large-amplitude periodic solutions. This project will study the phenomenon of Hopf bifurcation in symmetric artificial neuronal net?works that are characterised by unbounded general distributed time delays. The project will gain insight into how unbounded general distributed time delays interact with inherent network symmetries to generate oscillatory spatio-temporal patterns via equivariant Hopf bifurcation in symmetric artificial neuronal networks. Research-Related Education: Throughout the performance period of the proposed project, mathematics undergraduate students will be hired as direct labour to work on various aspects of the research, under the direct supervision of the Pl. This exposure of students to research constitutes an invaluable component of their mathematics education. Methods: This project will employ geometric, probabilistic, and group-theoretic methods to study equivariant Hopf bifurcation, and its interactions with other types of bifurcations, in sym?metric artificial neuronal networks characterised by unbounded general distributed time delays. In particular, we will use mathematical machinery from geometry, probability, groupoids, and compact Lie groups to establish a general mathematical framework in which questions about symmetry, bifurcations, and dynamics in a certain class of artificial neuronal networks can be rigorously investigated. Symmetry groupoids will be employed in the description and analysis of the interaction that exists between network symmetries and unbounded general distributed time delays, and how this interaction gives rise to complex emergent collective behaviour via equivariant Hopf bifurcation. Furthermore, we plan to investigate the spectral properties of the interconnection matrices of symmetric networks, and to use this information to identify a bi?furcation set. By drawing inspiration from the equivariant Hopf bifurcation theory for ordinary differential equations developed by Golubitsky et al. [31, 33], delay differential equations de?veloped by Wu [75], and the normal form theory for functional differential equations developed by Faria et al. [25, 26], we plan to develop an equivariant Hopf bifurcation theory for symmetric networks endowed with unbounded general distributed time delays. This approach will allow us to rigorously describe network bifurcations in terms of inherent statistical properties of the various time delay kernels. Significance: Hopf bifurcation of symmetric artificial neuronal networks endowed with un?bounded general distributed time delays is currently poorly understood. This project will de?velop a concrete mathematical framework in which it will be possible to establish an equivariant Hopf bifurcation theory for these types of networks. The results of this project will provide in?sight into the complex emergent collective dynamics of large symmetric networks of interacting units.
Document Details
- Document Type
- DoD Grant Award
- Publication Date
- Jul 28, 2023
- Source ID
- W911NF2310281
Entities
People
- Israel Ncube
Organizations
- Alabama A & M College
- Army Contracting Command
- Office of the Secretary of Defense