Abstract
Neural field models describe the coarse-grained activity of populations of interacting neurons. Because of the laminar structure of real cortical tissue they are often studied in two spatial dimensions, where they are well known to generate rich patterns of spatiotemporal activity. Such patterns have been interpreted in a variety of contexts ranging from the understanding of visual hallucinations to the generation of electroencephalographic signals. Typical patterns include localized solutions in the form of traveling spots, as well as intricate labyrinthine structures. These patterns are naturally defined by the interface between low and high states of neural activity. Here we derive the equations of motion for such interfaces and show, for a Heaviside firing rate, that the normal velocity of an interface is given in terms of a non-local Biot-Savart type interaction over the boundaries of the high activity regions. This exact, but dimensionally reduced, system of equations is solved numerically and shown to be in excellent agreement with the full nonlinear integral equation defining the neural field. We develop a linear stability analysis for the interface dynamics that allows us to understand the mechanisms of pattern formation that arise from instabilities of spots, rings, stripes and fronts. We further show how to analyze neural field models with linear adaptation currents, and determine the conditions for the dynamic instability of spots that can give rise to breathers and traveling waves.
Highlights
The functional organization of cortex appears to be roughly columnar, with the laminar sub-structure of each column organizing its micro-circuitry
Before we develop the machinery for describing the evolution of interfaces in twodimensional neural field models, it is informative to begin with a discussion in one dimension
In our calculations we have found that the key assumption of a Heaviside firing rate H (u − h) can be relaxed to a degree without fundamentally changing the results
Summary
The functional organization of cortex appears to be roughly columnar, with the laminar sub-structure of each column organizing its micro-circuitry. We show that activity patterns can be described by dynamical equations of reduced dimension, and that these depend only on the shape of the interface (requiring no knowledge of activity away from the interface) Is this description amenable to fast numerical simulation strategies, it allows for the construction of localized states and an analysis of their linear stability. This is useful for introducing the definition of normal velocity from a level-set condition, as well as establishing what it means for an interface to be linearly stable.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
