## 4 Discussion
To our knowledge, we are the first to introduce this parameter for the study of ON, although it was suggested as a possible parameter by [PERSON] et al. (2015), but then never used in the following pupillometric studies, probably because it was not therein considered as reliable. It is interesting to note that \(A_{tot}\) is instead a very good predictor of ON. Below the first two features, there is a big drop in the importance of around 9% and we can find, with essentially the same ranking, \(max\{v_{contraction}\}\), and \(N_{peaks}\). The first one, as we said previously, is a variable typically collected in pupillometric studies, even on Multiple Sclerosis ([PERSON], 1999; [PERSON], 2000; [PERSON], 2000) and the second one has been introduced in this work as an automatically detected variable. This is again a variable that includes both the sympathetic and parasympathetic nervous system, reflecting the competition between them, because noradrenaline and acetylcholine are both released in the neuromuscular junction of the dilator muscle, when a contraction occurs, and this causes oscillations in absence of a net light stimulus ([PERSON], 2000; [PERSON], 2000). We noticed a prevalence of higher values of \(N_{peaks}\) in the Healthy class, highlighting the importance of this oscillatory trend for healthy subjects. Smaller values of the \(N_{peaks}\) are a direct consequence of the lack of these oscillations or (at least) of the less pronounced condition of this behavior. The comparable role played by the velocity of pupil contraction, which tends to be smaller in ON patients, is strictly related to the role of \(N_{peaks}\), and therefore is not surprising to obtain an equivalent (within the error bars) contribution to the classification. Indeed, the absence of the oscillatory trend and at the same time reduced contraction velocity could be a sign of the reduced release or sensitivity of/to acetylcholine neurotransmitter in the neuromuscular junction of the dilator muscle, causing a decrease of the inhibition of the dilation necessary for the contraction. However, it is interesting how the classification model used for the forecast tends to give priority to both these features' contributions, confirming their complementarity. This means that the characterization of the observed oscillations is not yet perfectly captured by the proposed features and a combination of them is mandatory for the correct classification. Then, there is again a drop of around 4% in importance and we find 4 variables with importance between 3% and 5%. Two of them, \(max\{v_{dilation}\}\), and \(min\{D\}\), are features typically collected in pupillometric studies even on Multiple Sclerosis ([PERSON], 1999; [PERSON], 2000; [PERSON], 2000), and the other two were introduced in this work: \(\mu_{prv}(p)\) and \(\lambda_{t}\). The first one can be seen as the average waiting time between oscillations, and therefore is a signature again of both parasympathetic and sympathetic functions. The second one can be interpreted as the typical time decay of acetylcholine, causing the restoration of the sympathetic function. On the bottom of the obtained ranked list of features, we find the last four significant features for the prediction, even though with an importance of less than 2%. Two of them are features introduced in this work (\(MSD\) and \(\sigma_{prv}\)), while the others are commonly collected features (\(T_{p}\) and \(T_{s}\)). It is worth noticing however that, even if \(T_{p}\) was considered one of the most associated features related to the pathology, in Multiple Sclerosis ([PERSON], 2000), Alzheimer's disease ([PERSON], 2000; [PERSON], 2000), and diabetes ([PERSON], 2000; [PERSON], 2000), it ends up being not very important for the prediction task. The feature obtained from the mean square displacement although not very important for prediction, can be interesting since it is independent of the pupil diameter.