quasicrystal-patterns-in-neural-fields's Introduction

Quasicrystal patterns in neural fields

We show the solutions of the Amari model [1], and the two population neural field model in two spatial dimensions:

$$\frac{\partial u(\textbf{r},t)}{\partial t}=-u(\textbf{r},t) + v(\textbf{r},t)+ \int_{\mathbb{R}^{2}} w(\textbf{r}-\textbf{r}')f(u(\textbf{r}',t)-h){\rm d}\textbf{r}',$$

$$\frac{\partial v(\textbf{r},t)}{\partial t}=-v(\textbf{r},t) + u(\textbf{r},t)- \int_{\mathbb{R}^{2}} w(\textbf{r}-\textbf{r}')f(u(\textbf{r}',t)-h){\rm d}\textbf{r}',$$

where $(\textbf{r} = (r,\phi)) \text{ and } (\textbf{r}' = (r', \phi'))$.

The kernel is given by:

$$w(r) = \alpha_1 p(r,b_1,s_1,1) + \alpha_2 p(r,b_2,s_2,q),$$

where $p$ is an oscillatory function

$$p(r,b,s,q) =e^{-sr}[\cos(qr)+b \sin(qr)].$$

The ratio between two spatial scales of $w$ is determined by the parameter $q$.

We consider the model with initial condition $u(\textbf{r},0) = K(\textbf{r})$ and $v(\textbf{r},0) = 0$, where $K(\textbf{r})$ is given by

$$K(\textbf{r}) = e^{-a|\textbf{r}|^2 /L} \sum^{N}_{j=1}(e^{i\textbf{k}_j \cdot \textbf{r}} + e^{i\textbf{q}_j \cdot \textbf{r}}), \quad \textbf{k}=\begin{bmatrix} \cos \theta_j \\\ \sin \theta_j \end{bmatrix},$$

with $\textbf{q}_j$ = $q\textbf{k}_j$, where $q$ is the ratio between two spatial scales (used also in the formula for the kernel $w(r)$ ). Here, we consider the case where $a=0$, so that $K(\textbf{r})$ results in a pattern with an $N$-fold rotational symmetry. When $a\neq0$, $K(\textbf{r})$ gives a localized spot (bump) perturbed by an $N$-fold rotationally symmetrical pattern.

Figure: Solution of the two-field model with inhomogeneous initial condition $K(\textbf{r})$ with $N=10$ and $h=3$.

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