Connor Stevens

Main equations

$$c_m\frac{dV}{dt}=-i_m+\frac{I_e}{A}$$

$${\tau }_m(V)\frac{dm}{dt}=m_{\mathrm{\infty }}(V)-m$$

$${\tau }_h(V)\frac{dh}{dt}=h_{\mathrm{\infty }}(V)-h$$

$${\tau }_n(V)\frac{dn}{dt}=n_{\mathrm{\infty }}(V)-n$$

$${\tau }_a(V)\frac{da}{dt}=a_{\mathrm{\infty }}(V)-a$$

$${\tau }_b(V)\frac{db}{dt}=b_{\mathrm{\infty }}(V)-b$$

$$i_m=\overline{g}\mathrm{\_}L(V-E_L)+\overline{g}\mathrm{\_}Na{m}^{3}h(V-E\mathrm{\_}Na)+\overline{g}\mathrm{\_}K{n}^4(V-E\mathrm{\_}K)+\overline{g}\mathrm{\_}A{a}^{3}b(V-E_A)$$

alpha-values

$${\alpha }_n(V)=\frac{0.02m{V}^{-1}(V+45.7mV)}{1-\exp (-0.1m{V}^{-1}(V+45.7mV))}$$

$${\alpha }_m(V)=\frac{0.38m{V}^{-1}(V+29.7mV)}{1-\exp (-0.1m{V}^{-1}(V+29.7mV))}$$

$${\alpha }_h(V)=0.266\exp (-0.05m{V}^{-1}(V+48.0mV))$$

beta-values

$${\beta }_n(V)=0.25\exp (-0.0125m{V}^{-1}(V+55.7mV))$$

$${\beta }_m(V)=15.2\exp (-0.0556m{V}^{-1}(V+54.7mV))$$

$${\beta }_h(V)=\frac{3.8}{1+\exp (-0.1m{V}^{-1}(V+18mV))}$$

time constants

$${\tau }_n(V)=\frac{1.0ms}{{\alpha }_n(V)+{\beta }_n(V)}$$

$${\tau }_m(V)=\frac{1.0ms}{{\alpha }_m(V)+{\beta }_m(V)}$$

$${\tau }_h(V)=\frac{1.0ms}{{\alpha }_h(V)+{\beta }_h(V)}$$

$${\tau }_a(V)=0.3632ms+\frac{1.158ms}{1.0+\exp (0.0497m{V}^{-1}(V+55.96mV))}$$

$${\tau }_b(V)=1.24ms+\frac{2.678ms}{1.0+\exp (0.0624m{V}^{-1}(V+50.0mV))}$$

asymptotic values

$$a_{\mathrm{\infty }}(V)={(\frac{0.0761*\exp (0.0314m{V}^{-1}(V+94.22mV))}{1+\exp (0.0346m{V}^{-1}(V+1.17mV))}))}^{1/3}ms$$

$$b_{\mathrm{\infty }}(V)={\left(\frac{1}{1+\exp (0.0688m{V}^{-1}(V+53.3mV))}\right)}^{4}ms$$

$$n_{\mathrm{\infty }}(V)={\alpha }_n(V){\tau }_n(V)$$

$$m_{\mathrm{\infty }}(V)={\alpha }_m(V){\tau }_m(V)$$

$$h_{\mathrm{\infty }}(V)={\alpha }_h(V){\tau }_h(V)$$

Suitable initial conditions

$$m(t=0)=0.010$$

$$n(t=0)=0.156$$

$$h(t=0)=0.966$$

$$a(t=0)=0.540$$

$$b(t=0)=0.289$$

$$V(t=0)=-68.0mV$$

Parameter

$$c_m=0.1\frac{\mu F}{m{m}^2}$$

$$\frac{I_e}{A}=0.35\frac{\mu A}{m{m}^2}$$

$${\overline{g}}_L=0.003\frac{mS}{m{m}^2}$$

$${\overline{g}}_{Na}=1.2\frac{mS}{m{m}^2}$$

$${\overline{g}}_K=0.2\frac{mS}{m{m}^2}$$

$${\overline{g}}_A=0.477\frac{mS}{m{m}^2}$$

$$E_L=-17.0mV$$

$$E_{Na}=55.0mV$$

$$E_K=-72.0mV$$

$$E_A=-75.0mV$$

Transient Ca2+ Conductance

$$M_{\mathrm{\infty }}=\frac{1}{1+\exp (-(V+57mV)/6.2mV)}$$

$$H_{\mathrm{\infty }}=\frac{1}{1+\exp (-(V+81mV)/4mV)}$$

$${\tau }_M=0.612ms+\frac{1ms}{\exp (-(V+132mV)/16.7mV)+\exp ((V+16.8mV)/18.2mV)}$$

if $V<-80mV$:

$${\tau }_H=1ms\exp ((V+467mV)/66.6mV)$$

else:

$${\tau }_H=28ms+1ms\exp (-(V+22mV)/10.5mV)$$

Ca2+- depentdent K+ Conducatance

$$c_{\mathrm{\infty }}=\left(\frac{[C{a}^{2+}]}{[C{a}^{2+}]+3\mu M}\right)\frac{1}{1+\exp (-(V+28.3mV)/12.6mV)}$$

$${\tau }_C=90.3ms-\frac{75.1ms}{1+\exp ((-V+46mV)/22.7mV)}$$