The current research in Neuro is that a memory a synaptic condition, in that some synapse is structured in some way in relation to other neurons. There are W-many GABA receptors and X-many vesicles, Y-many AMPA receptors and Z-many neurotransmitters. Each receptor is in some state of flux in terms of it's age and other chemical modifiers on the protein. The synapse itself is tethered in some way and has structural proteins in it (not all of which are known these days). The astrocytes and glia that surround each synapse are also modifying the region, including some uptake/downtake of receptors and neurotransmitter vesicles (cool, very recent stuff). Additionally, the cells themselves are regulating the synapses in relation to the other synapses in the cell and the extra-cellular environment. Note: not all neurons 'fire' via synapses, some are just directly attached to a cell's soma (main cell body where the nucleus is). It's Bio, everything is crazy.
Still, that synapse is thought to be the main modification area of neuronal firing, in that it acts like a classical memristor. So, though electrical activity is not required and lightning/shocks can be sustained (by some miracle), the 'self'/mind is the network of the synapses (GIANT caveats apply here). Also, even under heavy anesthesia, the neurons are firing and active, jsut not coherently.
If you can preserve the synaptic weights, their dynamics, and their network, you should be able to reconstruct the 'mind' of another person. However, that is one instant in time and space. You'd need to know where pretty much every ion was and it's momentum, and I'm pretty sure that's impossible (Heisenberg and all that jazz).