Spot solutions to a neural field equation on oblate spheroids

TL;DR

This work analyzes how surface geometry affects pattern formation in an Amari-type neural field on curved surfaces. By perturbing from the unit sphere to an oblate spheroid with small flattening $\epsilon$, the authors establish the existence and stability of stationary spot solutions, deriving a stability criterion for poles via a spectral analysis that leverages spherical harmonics. They derive explicit perturbation formulas for geodesic distances and surface Jacobians and show that the leading eigenvalue $\mu_1$ dictates whether a pole spot drifts toward the pole or toward the equator, corroborated by numerical simulations. The results demonstrate that localization dynamics in neural fields depend on both the curved geometry and the spatial interaction kernel, highlighting geometry-kernel interplay and suggesting directions for modeling cortical patterns on non-flat geometries.

Abstract

Understanding the dynamics of excitation patterns in neural fields is an important topic in neuroscience. Neural field equations are mathematical models that describe the excitation dynamics of interacting neurons to perform the theoretical analysis. Although many analyses of neural field equations focus on the effect of neuronal interactions on the flat surface, the geometric constraint of the dynamics is also an attractive topic when modeling organs such as the brain. This paper reports pattern dynamics in a neural field equation defined on spheroids as model curved surfaces. We treat spot solutions as localized patterns and discuss how the geometric properties of the curved surface change their properties. To analyze spot patterns on spheroids with small flattening, we first construct exact stationary spot solutions on the spherical surface and reveal their stability. We then extend the analysis to show the existence and stability of stationary spot solutions in the spheroidal case. One of our theoretical results is the derivation of a stability criterion for stationary spot solutions localized at poles on oblate spheroids. The criterion determines whether a spot solution remains at a pole or moves away. Finally, we conduct numerical simulations to discuss the dynamics of spot solutions with the insight of our theoretical predictions. Our results show that the dynamics of spot solutions depend on the curved surface and the coordination of neural interactions.

Figures (7)

• Figure 1: The graph of \ref{['kernel']} with Mexican hat-type shapes. Left: $c_0=0.6$, $c_1=2.0$, $c_2=c_3=0.0$. Right: $c_0=0.14$, $c_1=0.9$, $c_2=1.2$, $c_3=0.45$.
• Figure 2: Parameter areas for the existence of stationary spot solutions \ref{['eq:spot']} when the kernel is \ref{['kernel']} with $c_1>0$ and $c_2=c_3=0$. Left: $u_T=0.5$. Right: $u_T=-0.5$.
• Figure 3: Parameter areas where there exist stationary spot solutions satisfying \ref{['condi:sta']} when the kernel is \ref{['kernel']} with $c_1>0$ and $c_2=c_3=0$. In Area I (light gray), there exist a spot solution that is linearized stable. In Area II (dark gray), there are spot solutions, but they are all unstable. Left: $u_T=0.5$. Right: $u_T=-0.5$.
• Figure 4: The graph of $\mu_1$. The white and black points correspond to the cases when the numerically obtained stationary spot solution $U(\theta;\theta_T)$ on ${\mathbb{S}}^2$ satisfies and does not satisfy Condition \ref{['condi:sta']}, respectively. That is, the white points correspond to the case of a stable spot solution on the sphere. Left: $c_0=0.6$, $c_1=2.0$, $c_2=c_3=0.0$. Right: $c_0=0.14$, $c_1=0.9$, $c_2=1.2$, $c_3=0.45$. The graphs of connection kernels with these parameters are shown in Fig. \ref{['fig:kernel']}.
• Figure 5: Numerical simulations of Equation \ref{['eq:main']} with $\epsilon=0.01$. The red dot and the gray line represent the north pole and the equator, respectively. The black dot corresponds to the center of the spot solution obtained numerically. Parameters of the kernel function \ref{['kernel']} are given as $c_0=0.14$, $c_1=0.9$, $c_2=1.2$, $c_3=0.45$ in both panels. Left: $\theta_T=1.0$, which corresponds to $\mu_1>0$ and $u_T\simeq -0.1898$. Right: $\theta_T=2.0$, which corresponds to $\mu_1<0$ and $u_T\simeq -2.6068$.

Theorems & Definitions (17)

• Example 3.1
• Proposition 3.3: AT
• Proposition 3.4
• proof
• Remark 3.5
• Proposition 3.6
• proof
• Remark 3.7
• Corollary 3.8
• Example 3.9