This is an interactive example of a neural field (NF) model: a mathematical model of the patterns of electrical activity seen in neural tissue. This particular NF model was investigated by Gökçe et al.  and is capable of generating so-called quasi-patterns.

Quasi-patterns, or quasi-periodic crystals, are lattice structures that are aperiodic. The patterns do not repeat in space but do have rotational symmetry.

Drag the cursor over the surface beneath to add electrical impulses, and watch the quasi-periodic patterns of synaptic activity grow. Scroll down for a technical description of the model.

Firing threshold $$h$$:

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Technical description

The colour of the surface encodes the neural activity $$u(\mathbf{x},t)$$ at location $$\mathbf{x}\in\Omega$$ in the neural tissue at time $$t$$. The evolution of $$u$$ is given by the PDE
$\frac{\partial u(\mathbf{x},t)}{\partial t} = -u(\mathbf{x},t) + \int_\Omega w(\mathbf{x},\mathbf{x}^{\prime})f(u(\mathbf{x}^{\prime},t))\,\mathrm{d}\mathbf{x}^{\prime},$ The domain $$\Omega$$ is the 2D plane with periodic boundary conditions. Fast Fourier transforms are used to numerically evaluate the integral. The connectivity kernel $$w(\mathbf{x},\mathbf{x}^{\prime})$$ describes the strength of interactions between points $$\mathbf{x}$$ and $$\mathbf{x}^{\prime}$$. The kernel typically encodes both excitatory and inhibitory interactions. $$f$$ is a sigmoidal firing rate function: $f(u)=\frac{1}{1+\mathrm{e}^{-\mu(u-h)}},$ where $$h$$ is the firing threshold. Increasing $$h$$ means higher membrane potentials are required to make neurons fire, leading to less activity.

It is known from solid state physics that quasi-patterns can emerge when two different periodic modes with different spatial length scales are excited together. Gökçe et al.  discovered that by carefully choosing a connectivity kernel $$w$$ whose Fourier transform is doubly degenerate, quasi-patterns can therefore be generated within a NF model. One such kernel is $w(\mathbf{r})=\alpha_1p(\mathbf{r}, b_1, s_1, 1) + \alpha_2p(\mathbf{r}, b_2, s_2, q),$ where $$\mathbf{r}=|\mathbf{x}-\mathbf{x}^{\prime}|$$ and $p(\mathbf{r},b,s,q) = \mathrm{e}^{-s\mathbf{r}}\left[\mathrm{cos}(q\mathbf{r}) + b\mathrm{sin}(q\mathbf{r}) \right].$ The parameter choices in the above simulation are $$(\alpha_1,\alpha_2,b_1,b_2,s_1,s_2,q)=(2.14,0.51,0.69,0.61,0.57,0.23,\mathrm{cos}(2\pi/z))$$, where $$z=10$$ or $$z=12$$ for patterns with 10-fold or 12-fold rotational symmetry, respectively. The above simulation follows the work of Gökçe, Coombes and Avitabile , who discovered both the existence of quasi-patterns in NF models and the above procedure to generate them. Please take a look at their paper for more details:

 A. Gökçe, S. Coombes and D. Avitabile. Quasicrystal patterns in a neural field model, J. Phys. Rev. Res. 2, 013234 (2020).

This simulation was implemented using a combination of JavaScript for the interactive elements, and Rust for the numerical computations.