Why do electric pickles only glow at one end? Original Question
The electric pickle is often used as an example of a non-ohmic resistor. In the experiment, electric current excites the sodium ions inside pickle, producing very bright and intense light effect. What I am wondering here is that why in many experiments only one end of the pickle glows?
Bounty Starter's Comments
I recently saw this Periodic Video.
The Professor (Martin Polyakoff) notes that, 


*

*The Gherkin normally glows only at one end due to excitation of $Na^+$ ions.

*The end which glows can either be the neutral end or the live end. It is impossible to predict which one will glow.

*During the video, both ends glowed together only for a few seconds.
Even the Professor admitted that probably glowing at one end happens due to pure chance.

Why should the gherkin glow only at one end? Is there any reason? Normally one would expect it to glow symmetrically around the center, that is, at both ends. 

The question has been quite well received. However no one has commented or answered. Does the question need to be re-worded or better explained?
Do let me know in the comments.
 A: Speculation
In the US, the AC mains has a "live wire" and a "neutral wire". One wire is kept near 0V while the other has 120 V RMS (meaning it swings from +170V to -170V). I think that the phenomenon you see takes place at the "live wire" only. That is - there is a local rapidly varying electric field at the tip of the live wire, and that is responsible for initiating the luminescence you are seeing.
You could confirm this if you had the fortitude to do this experiment with the power used for an electric dryer - there, the two wires are both live (that's how you get 240 V... it's necessary to get really high power devices working in the US where the normal mains voltage is 120 V). If my theory is correct, you would see both ends glowing.

Caution! Potentially lethal shock hazard! Because of the use of exposed bare wires in this demonstration, it should be done with great care and capable supervision.

Electric pickle is an extremely dangerous experiment. It will get a lot more dangerous when you double the voltage. It might be better to use an AC/AC transformer with a floating secondary - then the fields ought to distribute evenly without stepping up the voltage. Even at 120 V it's a dangerous experiment.
UPDATE
There was another question (since deleted) on this topic, linking to a video that showed that the pickle could sometimes light at one end, and sometimes at the other end. Looking closely at that video I now speculate as follows:


*

*The light was coming from the place where the metal touched the pickle, and looked like "fire". This could be spontaneous combustion of the hydrogen and oxygen gas that are produced from hydrolysis at the electrode (hydrogen during one half of the AC cycle, and oxygen during the other half). The burning would then create a local hot spot sufficient to cause Na+ ions to luminesce.

*It seemed to me that the light was coming from the side where the fork was inserted less deeply: this would encourage higher current density and thus heat; once one side "catches fire" I imagine the combustion product (steam) could increase the local resistance, so more power would be dissipated at one end - in other words, it might be an unstable equilibrium.

*The one experiment in which first one side, and then the other side, lit up, may have been a result of having the cut (high resistance) in the middle: this made the voltage distribution a bit more even (the moment one side became more resistive, the total current would drop, the voltage drop across the middle cut would be less, and that would "dampen" the effect that normally drives luminescence to one side).

A: In the linked video, there's a brief segment starting around 6:45 where both sides of one pickle are glowing "simultaneously."  In the high-speed video, it's clear that the glow is not simultaneous, but alternates between each end of the pickle. That is consistent with the sodium ions ($\rm Na^+$) being excited at either the anode or the cathode of the pickle, but not at the other end.
(My chemistry is too weak for me to decide confidently which end, sorry.)
I would predict that a hot pickle powered by 60 Hz electricity with a single glowing end would flicker at 30 Hz, 15 Hz, or some other sub-harmonic frequency, when the current is flowing "the right way." You could figure out which way by putting the pickle in parallel with an LED, which would also only glow when the current goes in a particular direction.  If current flow in the pickle went the same direction as in the LED, they'd flicker in phase at 30 Hz; if the two currents were reversed, the pickle would be bright when the LED was dark and vice-versa.
A: It's a sad fact that pickle/nail interface technology is an intellectual backwater, and has had little support from the military/industrial complex, which means that quality control for pickle illumination systems is generally minimal to none. Thus, it will be quite common that one end's resistance will be significantly higher than the other end's.
Call the two pickle ends $1$ and $2$. If the resistances at the nail/pickle interfaces are $R_1$ and $R_2$, the current through the pickle will be $I = {V\over{R_1+R_2}}$, and the power at the two ends will be $P_1 = IR_1 = {VR_1\over{R_1+R_2}}$ and $P_2 = IR_2 = {VR_2\over{R_1+R_2}}$. 
So, what if end $1$ has a 10% higher resistance than end $2$? That means end $1$ will dissipate 10% more power, drying out its electrode/pickle interface faster than end $2$s. That raises end $1$s relative resistance further, increasing the imbalance, until the great majority of the heat is being dissipated from end $1$, leaving end $2$ pyrotechnically challenged.
The solution to this problem is to only use highly uniform pickles, along with precision pickle piercing.
A: My speculation is that since there are Na+ and Cl- ions present after pickling process, when the pickle is connected to a circuit, the Na+ accumulates at one end and Cl- the other. Since the luminescence is created by the excitation of the Na+, only the end where Na+ accumulate will glow.
