With the Romans (as well as other honor-focused societies), suicide was often seen as a way to either preserve one's honour before it can be sullied with capture or humiliation or to regain some dignity and honour after one has either caused or failed at something which is perceived to have tarnished their honour. In essence, suicide was seen as an honorific way to die, as opposed to being captured or dying as the loser of a particular battle.

If we look at the cases which you have presented, each of them committed suicide either after losing a battle (Brutus after losing against the Triumvirate forces Phillipi, and Cassius at the same battle due to believing that Brutus had lost or failed) or while close to being captured at the end of the war (Antony killed himself as Octavion began to invade Egypt, and after mistakenly hearing that Cleopatra had killed herself prior). Other famous Romans, such as Cato (who killed himself after the battle of Utica, not wanting Caeser to capture him and parade him as a way to legitimize his reign), and Publius Quinctilius Varus killed himself after the ambush at Teutonberg forest. For all of these men, suicide was preferable to being captured by the enemy, even if that enemy was a Roman force led by a leader who was looking to pardon them. Their honour and dignity would not allow them keep on living after the defeats they all oversaw.

Even non-Romans, during the period also committed suicide rather than being captured. Most notable would be Hannibal (of the 2nd Punic War). While he ran from the Romans for most his life after the Carthaginian loss of the war, he would eventually end up very close to capture within Asia Minor/the Caucus region. Rather than being captured by his old enemies, he committed suicide, similar to the many Romans I have noted above.

For the Romans, and those that they faced within their conquests, suicide was often seen as the way to regain honour and ensure that you would not suffer humility at the hands of an enemy. This is why there are many notable Romans whose cause of death comes down to suicide.

Sources Used:

Galasso, Vittorio Nicholas. "Honor and The Performance of Roman State Identity." Foreign Policy Analysis 8, no. 2 (2012): 173-89.

Catharine Edwards (2005) Modelling Roman suicide? The afterlife of Cato, Economy and Society, 34:2, 200-222

Plutarch; Plutarch's Lives

Appian; Roman History

TL;DR: Romans didn't like losing, and many generals/politicians when backed into a corner saw no way out anyways.

In Roman society, social rank was all about honor. Nearly everything you could do in public had some sort of way in which you could gain or lose honor. For example, sexual relationships between members of the same sex were accepted, as long as you affiliate with the dominant partner. As long as you were top, it didn't matter who you had a relationship with. You mention Brutus and Marcus Antonius as politicians who took their own lives; it is necessary to view their circumstances at their time of death.

Regarding Brutus. He was famously a member of the assassins who killed Julius Caesar, hoping to restore democracy and bring an end to the coming one-man tyranny before it began. The assassins saw Caesar as a geniune threat to the democracy of the republic, and really believed they were on the right side of history. After the assassination, Marcus Antonius riled up the population of Rome to such an extent that the assassins had to flee. This resulted in a major civil war between the old officers and family of Caesar, and the assassins. On the side of the Caesarians were Marcus Antonius, Octavian and Lepidus, and on the side of the assassins were Cassius and Brutus (among others). At the battle of Phillipi, it appeared that the battle was losing for the assassins, and Cassius comitted suicide right there. If he lost this major battle, he would be paraded around, probably locked up, and for him it would be a great dishonor. He also saw the destruction of democracy and the rise of tyranny, and maybe lost hope. Brutus managed to escape, as after Cassius' suicide the battle was lost. He faced the same dilemma as Cassius. He had just lost a major battle, would probably be humiliated, and it is not unlikely that he would even be publicly executed. He saw no other way out, and after spending some time debating philosophy and poetry with his friends, he killed himself.

Regarding Marcus Anthony. After the civil war between the Assassins and the Ceasarians, Octavian, Lepidus and Anthony divided the Republic between them - Octavian got Italy, Spain and Gaul, Lepidus got North Africa and Sicily, and Anthony got Anatolia, Greece and Syria. Anthony then got into a relationship with Cleopatra, the queen of Egypt, which was still a self governed vassal state in the Republic. Octavian, who really was quite the power hungry fellow, painted Anthony as a weak simp for an eastern witch queen, who was giving away the lands of the Republic to foreigners. It is fact that Anthony liked to spend time with Cleopatra more than governing, but it is likely that Octavian's propaganda was a bit exaggerated. Anyways, another civil war broke out between Octavian and Anthony, and at the battle of Actium, Anthony was soundly beaten. He and Cleopatra retreated to Alexandria, which was quickly besieged by Octavian. Again, there was no way out. Roman society demanded that there was no choice between capture and death, and Anthony, as well as Brutus, Crassus, Varus, and so many others, followed this strict code of honor to the end. In all of these cases, the man in question is in command, and the battle is fought and over. There is no hope for escape, neither from enemies nor from public shame and humiliation.

In essence, the suicide of prominent Roman politicians is comparable to Hitler's suicide (a modern day example which I hope helps in understanding the motives of suicide by Roman leaders, and not in fact a direct comparison between Hitler and Brutus or Anthony). What choice did Hitler have? He was beaten, it was over, there was nothing left to be done. HOWEVER, it can be argued that Hitler's suicide was mainly motivated by fear (of capture and of facing his failure), whereas for Roman leaders there is a genuine question if it was fear of capture or fear of dishonor that motivated them to commit suicide. There is, for instance, also the case of Cato the Younger, who comitted suicide after Caesar won against Pompey. He was a very prominent politician, who was very much against Caesar's one-man rule. He explicitly stated that he could not bear what was becoming of the Republic before comitting suicide in his villa.

So in conclusion, there were other motives than just depression: political reasons, fear of capture and fear of humiliation/loss of honor. In general, it is difficult to translate current societal rules and mental states to people 2000 years ago (our modern concept of sexuality is a wonderful example of this). But it is of course still possible that some people listed here really were depressed, we will never know.

Edit: replaced Crassus with Cassius, I confused their names