This Membrane Potential article mentions only that Cl- ion is used to model inhibitory GABA synapses. Does it have another role besides hyper-polarizing the cell due to inhibitory neurotransmitter binding?
In normal neurons, Chloride's reversal potential is near the resting potential for the neuron and also happens to be near the leak conductance reversal potential for the neuron. While not exactly the same these three are sometimes confused.
The difference between these three reversal potentials is subtle.
Chloride Reversal Potential: is the potential carried only by the chloride ion. Therefore it given by the Nernst Equilibrium Potential.
Leak Reversal Potential: The conductance that remains (relatively) constant. As useful metaphor the leak conductance is the passive conductance of ions through the membrane. This current can be carried by any Ion.
Resting Potential: The voltage where the neuron is at rest, usually when it is at a steady state. It is defined by all ion's that permeate the membrane at rest. (see GHK)
Because these three reversal potentials are approximately the same that means many computational neuroscience papers will consider only consider the leak current like the two paper's you linked. However, if there is reason to consider Chloride conducting channels specifically (like GABA receptors) then there will be a specific term for the Chloride Ion's Reversal potential.
From a computational perspective, the fact that the Chloride reversal potential is close to the rest potential is actually important. It is capable of shunting inhibition. In short this means that the neurons membrane potential is moved closer to rest, rather than more negative as is the case with regular hyperpolarizing current carried by Potassium Ions. Thus if the membrane potential is below the driving potential for chlorine the shunting inhibition will actually slightly depolarize the cell. Note that this small depolarization will not pass the chlorine reversal potential, thus keeping the cell at rest and preventing firing, despite "depolarizing the cell"
While it is generally true that hyperpolarizing current suppresses firing, there are exceptions to this rule. For example, if a neuron is a resonator (fires in response to certain frequency of input) then hyperpolarizing inputs at the right frequency will cause the neuron to fire. Note, However this is only true for the current carried by Potassium Ions, and shunting inhibition will prevent this type of hyperpolarize-induced firing. The reasoning for this is quite complex, but a good (albeit math-heavy) explanation see Dr. Izhikevich's book Chp 7.
Mostly Cl- is disregarded in calculations of the resting membrane potential and action potential voltage changes, because it is less important for the neural membrane characteristics than Na+ and K+.
In some neurons Cl- is not actively transported. In terms of the resting membrane potential, Cl- hence settles its gradient passively across the membrane according to its Nernst potential, as it has a relatively large permeability in most neurons. The Nernst potential of Cl- is hence governed by the concentration and voltage gradients of Na+ and K+. Na+ and K+ are actively transported out and into the cell, respectively, by the action of the Na+,-K+-ATPase (the sodium potassium pump).
The Nernst potential of Cl- can be calculated with the Nernst equation:
However, many neurons have a K+/Cl- co-transporter that uses the K+ gradient to push Cl- against its concentration gradient out of the cell. In this case there is more Cl- outside the cell than there would be without the active transport. Hence, opening of Cl- channels as in your GABAA example, will hyperpolarize the cell because Cl- will enter upon Cl- channel opening. Secondary active transport of Cl- may also affect the resting membrane potential of these neurons as it will actively participate in the overall membrane potential when actively transported. When taking into account the relative permeabilities of K+, Na+ and Cl-, i.e., 1 : 0.04 : 0.5, the Goldman equation can be used to calculate the membrane potential: