  The current-voltage (I-V) conversion characterizes the physiology of cellular
microdomains and reflects cellular communication, excitability, and electrical
transduction. Yet deriving such I-V laws remains a major challenge in most
cellular microdomains due to their small sizes and the difficulty of accessing
voltage with a high nanometer precision. We present here novel analytical
relations derived for different numbers of ionic species inside a neuronal
micro/nano-domains, such as dendritic spines. When a steady-state current is
injected, we find a large deviation from the classical Ohm's law, showing that
the spine neck resistance is insuficent to characterize electrical properties.
For a constricted spine neck, modeled by a hyperboloid, we obtain a new I-V law
that illustrates the consequences of narrow passages on electrical conduction.
Finally, during a fast current transient, the local voltage is modulated by the
distance between activated voltage-gated channels. To conclude,
electro-diffusion laws can now be used to interpret voltage distribution in
neuronal microdomains.
