This is an interactive example of a neural field: a mathematical model of the patterns of electrical activity seen in neural tissue. Drag the cursor over the surface to add electrical impulses, and watch the patterns of synaptic activity grow. Scroll down for a technical description of the model.

Parameter controls

Firing threshold \(h\):

Adaptivity \(g\):

Presets

Choose colour map:

**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 PDEs

\[
\frac{\partial u(\mathbf{x},t)}{\partial t} = -u(\mathbf{x},t) + \int_\Omega w(\mathbf{x},\mathbf{x}^{\prime})f(u(\mathbf{x}^{\prime},t))\,\mathrm{d}\mathbf{x}^{\prime} - g a(\mathbf{x},t),
\]
\[
\frac{\partial a(\mathbf{x},t)}{\partial t} = u(\mathbf{x},t) - a(\mathbf{x},t).
\]
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 \(\mathbf{x}^{\prime}\), and \(f\) 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. The parameter \(g\) controls the strength of adaptivity in the system; increasing \(g\) leads to greater oscillatory behaviour. The particular choice of connectivity kernel for the simulation is
\[
w(\mathbf{x},\mathbf{x}^{\prime}) = \mathrm{e}^{-b||\mathbf{x}-\mathbf{x}^{\prime}||_2}\left(b\,\mathrm{sin}(||\mathbf{x}-\mathbf{x}^{\prime}||_2)+\mathrm{cos(||\mathbf{x}-\mathbf{x}^{\prime}||_2)}\right),
\]
which includes both excitatory and inhibitory behaviour.

For an overview of neural field models, see [1]. For various examples of neural fields in action, see [2-4].

[1] P. C. Bresslof, *Spatiotemporal dynamics of continuum neural fields*, J. Phys. A: Math. Theor. 45 (2012) 033001.

[2] S. Coombes, *Waves, bumps, and patterns in neural field theories*, Biol Cybern 93, 91–108 (2005).

[3] D. Avitabile and H. Schmidt, *Snakes and ladders in an inhomegeneous neural field model*, Physica D, Vol. 294 (2015).

[4] J. Rankin, D. Avitabile, J. Baladroni, G. Faye, D.J.B. Lloyd, *Continuation of localised coherent structures in nonlocal neural field equations*, IAM J. Sci. Comput., 36(1), B70–B93.