Wilson-Cowan model

In computational neuroscience, the Wilson-Cowan model describes the dynamics of interactions between populations of excitatory and inhibitory neuronal populations.

Mathematical description

Cells in refactory period$int\_\{t-r\}^\{t\}E(t\text{'})dt\text{'}$

Sensitive cells$1-int\_\{t-r\}^\{t\}E(t\text{'})dt\text{'}$

Subpopulation response function based on the distribution of neuronal thresholds$S(x)=int\_\{0\}^\{x(t)\}D(\; heta)d\; heta$

Subpopulation response function based on the distribution of afferent synapses per cell$S(x)=int\_\{frac\{\; heta\}\{x(t)^\{infty\}C(w)dw$

Average excitation level$int\_\{-infty\}^\{t\}alpha(t-t\text{'})\; [c\; 1E(t)-c\; 2I(t\text{'})+P(t\text{'})]\; dt\text{'}$

Excitatory subpopulation expression$[1-int\; \{t-r\}^\{t\}E(t\text{'})dt\text{'}]\; S(x)dt$

Complete Wilson-Cowan model$E(t+\; au)=\; [1-int\; \{t-r\}^\{t\}E(t\text{'})dt\text{'}]\; Sleft\; \{int\_\{-infty\}^\{t\}alpha(t-t\text{'})\; [c\; 1E(t)-c\; 2I(t\text{'})+P(t\text{'})]\; dt\text{'}\; ight\; \}$

$I(t+\; au)=\; [1-int\; \{t-r\}^\{t\}I(t\text{'})dt\text{'}]\; Sleft\; \{int\_\{-infty\}^\{t\}alpha(t-t\text{'})\; [c\; 3E(t)-c\; 4I(t\text{'})+Q(t\text{'})]\; dt\text{'}\; ight\; \}$

Time Course Graining

Isocline Equation$c\_2I=c\_1E-S\_e^\{-1\}left\; (frac\{E\}\{k\_e-r\_eE\}\; ight\; )+P$

Sigmoid Function$S(x)=frac\{1\}\{1+exp\; [-a(x-\; heta)]\; \}-frac\{1\}\{1+exp(a\; heta)\}$