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Cable model

  • experimental data from lobster: [math]\displaystyle{ \lambda_c \approx 1.3 }[/math] mm
  • Thus, infinitesimal electrode means much smaller than 1 mm and remote electrode means much greater than 1 mm
  • The distance a signal can travel without Hodgkin-Huxley regenerative mechanism is quite small
  • Look at dynamics of stimulus pulse
    • Pulse in space and time
    • Solution of form: [math]\displaystyle{ v_m(z,t)=w(z,t) e^{-t/\tau_m} }[/math]
    • where [math]\displaystyle{ \frac{\partial w}{\partial t} = \frac{\lambda_c^2}{\tau_m} \frac{\partial^2 w}{\partial z^2} }[/math]
      • This is the Diffusion Equation with [math]\displaystyle{ D=\frac{\lambda_c^2}{\tau_m} }[/math]
    • The effect of the stimulus diffuses down according to w but also dissipates equally across the membrane
  • In decrement free conduction (HH), space/time relationship follows wave equation whereas in decremental conduction in cable model, it's diffusion-like