| Abstract: |
| This talk deals with the Amari model, an integro-differential equation that is one of the fundamental neural field equations. Neural field equations are mathematical models used to analyze the time evolution of excitation patterns in neural fields and to understand the dynamics of neuronal populations at the tissue level. In recent years, increasing attention has been paid to the influence of the geometry of organs, such as the brain, on pattern formation, and to the constraints on solution behavior imposed by curved surfaces.
To investigate such geometric effects, we consider the sphere and a slightly deformed sphere (spheroid) as model surfaces, and analyze how the structure and stability of spot solutions depend on the geometry of the surface. First, we construct spot solutions on the sphere and characterize their linear stability through spectral analysis of the linearized operator. Next, we perform a perturbation analysis from the sphere to construct spot solutions at the north pole of the spheroid and derive the eigenvalues that determine stability. Finally, numerical simulations based on the theoretical results are carried out to examine the behavior of spot solutions. |
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