Tayefeh, Vahid

The human brain consists of billions of neurons and these neurons pool together
in groups at different scales. On one hand, these neural entities tend to behave
as single units and on the other hand show collective macroscopic patterns of activity.
The neural units communicate with each other and process information over time.
This communication is through small electrical impulses which at the macroscopic
scale are measurable as brain waves. The electric field that is produced collectively
by macroscopic groups of neurons within the brain can be measured on the surface
of the skull via a brain imaging modality called Electroencephalography (EEG). The
brain as a neural system has variant connection topology, in which an area might not
only be connected to its adjacent neighbors homogeneously but also distant areas can
directly transfer brain activity [16]. Timing of these brain activity communications
between different neural units bring up overall emerging spatiotemporal patterns.
The dynamics of these patterns and formation of neural activities in cortical surface
is influenced by the presence of long-range connections between heterogeneous neural
units. Brain activity at large-scale is thought to be involved in the information processing
and the implementation of cognitive functions of the brain. This research
aims to determine how the spatiotemporal pattern formation phenomena in the brain
depend on its connection topology. This connection topology consists of homogeneous
connections in local cortical areas alongside the couplings between distant functional
units as heterogeneous connections. Homogeneous connectivity or synaptic weight
distribution representing the large-scale anatomy of cortex is assumed to depend on
the Euclidean distance between interacting neural units. Altering characteristics of
inhomogeneous pathways as control parameters guide the brain pattern formation
through phase transitions at critical points. In this research, linear stability analysis
is applied to a macroscopic neural field in a one-dimensional circular and a twodimensional
spherical model of the brain in order to find destabilization mechanism
and subsequently emerging patterns.

The neuronal ensembles in cortical tissue, which tend to behave as single functional units, communicate
with each other and process information over time. Neural activity fields, in form of spatially continuous
networks, can be used to model a variety of neurobiological phenomena. The connection topology of
brain tissue is such that a cortical area is not only connected to its neighbors locally, but also has global
projection to distant areas via a fiber system. Such projections not only serve to organize local dynamics
within cortical areas but timing of these processes at different sites will affect the overall emerging pattern
and contributes to the macroscopic organization and global dynamics of neural activity. The dynamics
of this neural field activity gives rise to pattern formation phenomena and self-organization. Our
macroscopic spatiotemporal pattern formation approach assumes the existence of an order parameter
dynamics and leads to phenomenological models to understand the collective phenomena even though
the microscopic dynamics is not completely known. We are investigating how the emerging patterns
depend on the space-time structure of the coupling between functional units i.e. long-range
heterogeneous pathways coupling strength (space) and the axonal time delay due to propagation with
finite speed between areas (time). We analyze the stability of the rest-state activity of a neural field as
manipulating heterogeneous two-point connections varies network connection topology in two geometries with periodic boundary conditions: a closed one-dimensional loop and a closed spherical 2-
D cortical surface.

Spatially continuous cortical surface is composed of neural ensembles interconnected with a
general connectivity embedded local homogeneous connectivity to nearest neighbors and global
heterogeneous projections to distant areas. Interconnection delay and long-range connectivity in
neural field models shows activity transfer via neural oscillation.
Destabilization mechanism of two-point connected dynamical neural system with spatially
variant connection topology as a control parameter leads to phase transition and macroscopic
coherent spatiotemporal pattern formation of neural activity. Authors are intended to generalize
this neural field dynamic to more realistic geometries such as a sphere.