pytutorial/advanced_programming/connor_stevens

Main equations

c_m \frac{d V}{dt} = -i_m + \frac{I_e}{A}

t_m(V) \frac{d m}{dt} = m_{\infty}(V) - m

t_h(V) \frac{d h}{dt} = h_{\infty}(V) - h

t_n(V) \frac{d n}{dt} = n_{\infty}(V) - n

t_a(V) \frac{d a}{dt} = a_{\infty}(V) - a

t_b(V) \frac{d b}{dt} = b_{\infty}(V) - b

i_m =\bar{g}_L (V - E_L) + \bar{g}_{Na} m^3 h (V - E_{Na}) + \bar{g}_{K} n^4 (V - E_{K}) + \bar{g}_{A} a^3 b (V-E_A)

alpha-values

\alpha_n(V) = \frac{0.02mV^{-1} (V + 45.7mV)}{1 - \exp(-0.1mV^{-1} (V + 45.7mV))}

\alpha_m(V) = \frac{0.38mV^{-1} (V + 29.7mV)}{1 - \exp(-0.1mV^{-1} (V + 29.7mV))}

\alpha_h(V) = 0.266 \exp(-0.05mV^{-1} (V + 48.0mV))

beta-values

\beta_n(V) = 0.25 \exp(-0.0125mV^{-1} (V + 55.7mV))

\beta_m(V) = 15.2 \exp(-0.0556mV^{-1} (V + 54.7mV))

\beta_h(V) = \frac{3.8}{1 + \exp(-0.1mV^{-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(0.0497mV^{-1} (V + 55.96mV))}

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

asymptotic values

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

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

n_\infty(V) = \alpha_n(V) \tau_n(V)

m_\infty(V) = \alpha_m(V) \tau_m(V)

h_\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) = -68.0 mV

Parameter

c_m = 0.1 \frac{\mu F}{mm^2}

\frac{I_e}{A} = 0.35 \frac{\mu A}{mm^2}

\bar{g}_L = 0.003  \frac{mS}{mm^2}

\bar{g}_{Na} = 1.2 \frac{mS}{mm^2}

\bar{g}_{K} =  0.2 \frac{mS}{mm^2}

\bar{g}_{A} =  0.477 \frac{mS}{mm^2}

E_L = -17.0 mV

E_{Na} = 55.0 mV

E_K = -72.0 mV

E_A = -75.0 mV

Transient Ca2+ Conductance

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

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

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

if V < -80 mV:

\tau_H = 1ms \exp((V+467mV)/66.6mV)

else:

\tau_H = 28ms + 1ms \exp(-(V+22mV)/10.5mV)

Ca2+- depentdent K+ Conducatance

c_\infty = \left( \frac{[Ca^{2+}]}{[Ca^{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)}