= Passive dendritic cable
{title2=$\lambda=\sqrt{r_m/r_a}$}
A <passive dendritic cable> models a uniform <dendrite> with axial <electrical resistance>, membrane leak and <membrane capacitance>, excluding active <ion channels>. Let $r_a$ be axial <electrical resistance> per length, $r_m$ membrane <electrical resistance> times length, and $c_m$ <membrane capacitance> per length. For voltage deviation $v=V-E_L$ from the leak <reversal potential>, <electric charge conservation> gives the <partial differential equation>
$$
c_m\frac{\partial v}{\partial t}=\frac{1}{r_a}\frac{\partial^2v}{\partial x^2}-\frac{v}{r_m}+i(x,t),\qquad \tau_m=r_mc_m,\qquad \lambda=\sqrt{\frac{r_m}{r_a}}.
$$
$i$ is imposed inward <electric current> per length. Units are $r_a:\Omega/\mathrm m$, $r_m:\Omega\,\mathrm m$ and $c_m:\mathrm F/\mathrm m$, so $\tau_m$ is a time and $\lambda$ a length. On a source-free semi-infinite cable with imposed $v(0)=v_0$ and $v(x)\to0$ as $x\to\infty$, the <steady state> is
$$
v(x)=v_0e^{-x/\lambda}.
$$
Differentiate twice to verify the <ordinary differential equation> $\lambda^2v''-v=0$. At $x=\lambda$, the amplitude is $e^{-1}\approx0.368$ of the boundary amplitude. This is a stationary attenuation law, not the complete waveform of a brief <EPSP>. Transient passive spread also broadens and delays responses. Finite ends, branch points and the <soma>'s load alter the solution; distance alone cannot specify every real <synaptic potential>.<Neuronal Dynamics: From Single Neurons to Networks and Models of Cognition by Wulfram Gerstner and others/3.2 Spatial Structure: The Dendritic Tree>[§3.2.1, equations (3.17)–(3.20): passive cable and length scale; §3.2.2, Figure 3.7: transient attenuation and delay]{ref}
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