Inhibitory
Excitatory
Local coupling strengths ($\sigma_l$)
$w_{ii}$
$w_{ie}$
$w_{ee}$
$w_{ei}$
Spatial spread
$\sigma_i$
$\sigma_e$
$\sigma_l$
Lateral coupling strengths
$a_{ii}$
$a_{ie}$
$a_{ee}$
$a_{ei}$
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
$A$
$f$
Time stepping
$\Delta t$
skip
size
location
Boundary
radius
hardness
Presets
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
0
1
0
1
Eigenvalues at selected point: Real; Imaginary
Angular spatial frequency (1/pixels)
Temporal frequency; Lines: multiples of $1/\tau_e$
Click to start
\[\textrm{E-cells: }
\tau_e \dot{U_e} = -U_e + f\left[
(w_{ee} K_l + a_{ee} K_e) \star U_e
- (w_{ie} K_l + a_{ie} K_i) \star U_i
- \theta_e + g_e S(t) - \beta_e V_e\right]\]
\[\textrm{I-cells: }
\tau_i \dot{U_i} = -U_i + f\left[
(w_{ei}K_l +a_{ei} K_e) \star U_e
- (w_{ii}K_l + a_{ii} K_i) \star U_i
- \theta_i + g_i S(t) - \beta_i V_i \right]\]
\[\textrm{E-cell adaptation: } \tau_{ve} \dot{V_e} = U_e - V_e,\,\,
\tau_{ve} = \begin{cases} \alpha_e & \text{if }V_e< U_e \\
\gamma_e & \text{if }V_e\ge U_e\end{cases} \]
\[\textrm{I-cell adaptation: } \alpha_i \dot{V_i} = U_i - V_i \]
\[\textrm{Interaction kernels (normalized to 1): } K_{e,i,l}(x,y)
\propto \exp\left(\frac{1}{2} \frac{x^2}{\sigma_{e,i,l}^2}
+\frac{1}{2} \frac{y^2}{\sigma_{e,i,l}^2}\right)\]
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.Note: Beware forward-Euler integration, especially with limit cycles (where the error is always excitatory). Do not set Δt larger than 1/10th the shortest time constant. Values larger than 1/5th the shortest time consant are unstable.
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
Local linearization report