Wilson-Cowan model

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

[edit] Mathematical description

Cells in refactory period \int_{t-r}^{t}E(t')dt'

Sensitive cells 1-\int_{t-r}^{t}E(t')dt'

Subpopulation response function based on the distribution of neuronal thresholds S(x)=\int_{0}^{x(t)}D(\theta)d\theta

Subpopulation response function based on the distribution of afferent synapses per cell S(x)=\int_{\frac{\theta}{x(t)}}^{\infty}C(w)dw

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

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

Complete Wilson-Cowan model E(t+\tau)=[1-\int {t-r}^{t}E(t')dt']S\left \{\int_{-\infty}^{t}\alpha(t-t')[c 1E(t)-c 2I(t')+P(t')]dt'\right \}

I(t+\tau)=[1-\int {t-r}^{t}I(t')dt']S\left \{\int_{-\infty}^{t}\alpha(t-t')[c 3E(t)-c 4I(t')+Q(t')]dt'\right \}

Time Course Graining \tau\frac{d\bar{E}}{dt}=-\bar{E}+(1-r\bar{E})S_e[kc 1\bar{E}(t)+kP(t)]

Isocline Equation c_2I=c_1E-S_e^{-1}\left (\frac{E}{k_e-r_eE} \right )+P

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

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