Wilson-Cowan on the GPU and in your browser

Inhibitory

Excitatory

Lateral E–I coupling strengths ($\sigma_{e,i}$)

$a_{ii}$

$a_{ie}$

$a_{ee}$

$a_{ei}$

Lateral spatial spread

$\sigma_i$

$\sigma_e$

Local E–I coupling strengths ($\sigma_l$)

$w_{ii}$

$w_{ie}$

$w_{ee}$

$w_{ei}$

Local spatial scale (pixels)

$\sigma_l$

Time constants (ms)

$\tau_i$

$\tau_e$

Thresholds

$\theta_i$

$\theta_e$

Stimulus gain

$g_i$

$g_e$

Noise

$N_i$

$N_e$

Adaptation

$\alpha_i$

$\beta_i$

$\alpha_e$

$\gamma_e$

$\beta_e$

Stimulation

$A$

$f$

size

where

wave

square

Time stepping

skip

$\Delta t$

Boundary

radius

hardness

Firing rate variables $U_e$, $U_i$, $3|U_e-U_i|$

Adaptation variables $U_e$, $U_i$, $3|U_e-U_i|$

Phase space (steady-state $V_{e,i}$)

E-cell firing rate

I-cell firing rate

Eigenvalues at selected point: Real; Imaginary

Spatial frequency

Temporal frequency

Click to start

\[ \begin{aligned} \text{Firing-rates:} && \tau_e \dot{U_e} &= -U_e + f\left[ D _e - \theta_e + g_e S(t) - \beta_e V_e\right] \\&& \tau_i \dot{U_i} &= -U_i + f\left[ D_i - \theta_i + g_i S(t) - \beta_i V_i \right] \\ \text{Synaptic drives:}&& D_e &= (w_{ee} K_l + a_{ee} K_e) \star U_e - (w_{ie} K_l + a_{ie} K_i) \star U_i \\&& D_i &= (w_{ei}K_l +a_{ei} K_e) \star U_e - (w_{ii}K_l + a_{ii} K_i) \star U_i \\ \text{Synaptic adaptation:}&& \tau_{ve} \dot{V_e} &= U_e - V_e,\,\, \tau_{ve} = \begin{cases} \alpha_e & \text{if }V_e< U_e \\ \gamma_e & \text{if }V_e\ge U_e \end{cases} \\&& \alpha_i \dot{V_i} &= U_i - V_i \\ \text{Interaction kernels:}&& K_{e,i,l}(x,y) &\propto \exp\big(\tfrac{1}{2} \tfrac{x^2}{\sigma_{e,i,l}^2} +\tfrac{1}{2} \tfrac{y^2}{\sigma_{e,i,l}^2}\big) \end{aligned} \]

Wave State Demonstration

This page is part of the github.com/michaelerule/neuralfield repository, which collects various implementations of the Wilson-Cowan neural field and similar equations over the years. The project can be browsed as a website here.

We used these and similar equations to model flicker-induced geometric hallucinations, optogenetic stimulation in motor cortex, and retinal waves.

This code uses webGL shaders and encodes floats as 16-bit fixed point in an ad-hoc way. Results are for demonstration and exploration only and not guaranteed to be especially accurate.

Simulation

Presets