- 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 periodint_{t-r}^{t}E(t')dt'
Sensitive cells1-int_{t-r}^{t}E(t')dt'
Subpopulation response function based on the distribution of neuronal thresholdsS(x)=int_{0}^{x(t)}D( heta)d heta
Subpopulation response function based on the distribution of afferent synapses per cellS(x)=int_{frac{ heta}{x(t)^{infty}C(w)dw
Average excitation levelint_{-infty}^{t}alpha(t-t') [c 1E(t)-c 2I(t')+P(t')] dt'
Excitatory subpopulation expression1-int {t-r}^{t}E(t')dt'] S(x)dt
Complete Wilson-Cowan modelE(t+ au)= [1-int {t-r}^{t}E(t')dt'] Sleft {int_{-infty}^{t}alpha(t-t') [c 1E(t)-c 2I(t')+P(t')] dt' ight }
I(t+ au)= [1-int {t-r}^{t}I(t')dt'] Sleft {int_{-infty}^{t}alpha(t-t') [c 3E(t)-c 4I(t')+Q(t')] dt' ight }
Time Course Grainingaufrac{dar{E{dt}=-ar{E}+(1-rar{E})S_e [kc 1ar{E}(t)+kP(t)]
Isocline Equationc_2I=c_1E-S_e^{-1}left (frac{E}{k_e-r_eE} ight )+P
Sigmoid FunctionS(x)=frac{1}{1+exp [-a(x- heta)] }-frac{1}{1+exp(a heta)}
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