Choose colour map:

Technical description

The colour of the surface encodes the neural activity $u(\mathbf{x},t)$ at location $\mathbf{x}\in \mathrm{\Omega }$ in the neural tissue at time $t$. The evolution of $u$ is given by the PDEs
$$\frac{\partial u(\mathbf{x},t)}{\partial t}=-u(\mathbf{x},t)+{\int }_{\mathrm{\Omega }}w(\mathbf{x},{\mathbf{x}}^{\prime })f(u({\mathbf{x}}^{\prime },t))\mathrm{d}{\mathbf{x}}^{\prime }-ga(\mathbf{x},t),$$ $$\frac{\partial a(\mathbf{x},t)}{\partial t}=u(\mathbf{x},t)-a(\mathbf{x},t).$$ The domain $\mathrm{\Omega }$ 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 $$w(\mathbf{x},{\mathbf{x}}^{\prime })={\mathrm{e}}^{-b||\mathbf{x}-{\mathbf{x}}^{\prime }|{|}_2}(b\sin (||\mathbf{x}-{\mathbf{x}}^{\prime }|{|}_2)+\cos (||\mathbf{x}-{\mathbf{x}}^{\prime }|{|}_2)),$$ 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.