I am wondering whether increasing the volume would result in (a) a neuron that was already firing to now increase its spike rate, (b) a different group of neurons to add their activity to the population total, (c) a different group of neurons, coding for the new volume, to become active while the first group of neurons silences, or (d) some combination of the above. In particular, I am curious about the subcortical portion of the auditory pathway.
There are quite a few stations between cochlea and the brain and I will focus on the auditory nerve. That said, your theories (a) and (b) are both correct, and therefore (d) applies as well.
(a) Neurons in the auditory nerve increase their firing rate when sound level is increased (Heil et al, 2011). This can be regarded as the primary mechanism for encoding sound level.
(b) Increased sound levels are accompanied by a larger area in the cochlea being activated due to the low-frequency tail (Kiang & Moxon, 1974). Hence, more neurons start firing at higher sound levels when the tone frequency is the same. Although this may add in the perceived sound level, it is likely of less importance than (a), as it is basically a reflection of the mechanical properties of the basilar membrane in the cochlea and primarily reduces frequency resolution.
(c) There are no neurons ever identified that are dedicated to sound level encoding. Instead, auditory nerve fibers encode sound frequency (according to the place-frequency map of the cochlea) and they encode sound intensity via their firing rate.
(d) Hence, since both (a) and (b) are correct, your hypothesis (d) holds, as it is a combination of the above.
One can safely assume these intensity-coding mechanisms hold up in the responses of neurons in the next station - the cochlear nucleus. However, the higher up you go in the auditory system, the less likely a 1:1 relationship as found in the far periphery applies.
Siebert (1968) modelled level discrimination based on the information in the firing rate of auditory nerve fibers. The model does a reasonable job over a narrow range of conditions, but misses a large number of effects. Since Siebert's original effort, a number of more advance models have been developed. A more recent model by Colburn et. al (2003) highlights just how complicated sound level encoding is at the level of the auditory nerve:
It is shown that the rate information provided by individual AN fibers is more constrained by increases in variance with increases in rate than by saturation. As noted in previous studies, there is sufficient average-rate information within a narrow-CF region to account for robust behavioral performance over a wide dynamic range; however, there is no model based on a simple limitation or use of AN information consistent with parametric variations in performance. This issue is explored in the current study through analysis of performance based on different aspects of AN patterns. For example, we show that performance predicted from use of all rate information degrades significantly as level increases above low–medium levels, inconsistent with Weber’s Law. At low frequencies, synchrony information extends the range over which behavioral performance can be explained by 10–15 dB, but only at low levels. In contrast to rate and synchrony, nonlinear-phase cues are shown to provide robust information at medium and high levels in near-CF fibers for low-frequency stimuli. The level dependence of the discharge rate and phase properties of AN fibers are influenced by the compressive nonlinearity of the inner ear.
Siebert, W. M. (1968). Stimulus transformations in the peripheral auditory system. Recognizing patterns, 104-133.