The Fall of Rome had a completely negligible impact on the Roman Senate because technically the western empire in Italy didn't "end" it was just resubsumed into a single non-geographically divided empire under the aegis of the emperor of Constantinople. Combine this with the fact that Theoderic was openly seen as a new princeps, and actively engaged with the Senate of Rome, and their role became more relevant than ever as the legitimizer for the (effectively) re-emergent western sovereign.

The real moment of importance for the Senate of Rome was the 30-year-long Gothic Wars 60+ years later, which decimated Italy, senatorial estates, and killed off many (perhaps most) of the Senators themselves (they were taken hostage by Totila and ordered to be killed after his death and defeat). Although I believe they were mentioned in the writings of Gregory the Great around 600 CE, they quickly vaporized from the historical record shortly afterwards, with the pope in full civic leadership role of the city. You won't see a "Senate" again until attempts to revive that classical tradition in the Italian commune era of the 12th century.

But you know, something to think about, senate (senatus in latin) comes from senex, meaning old man. Senate literally means "council of elders." If we're talking about a community (albeit non-formal) of Roman nobility within the city, then they most definitely survived through the "dark ages." These were the elites involved in the political infighting over pope selection that you see in early medieval history, like what led Pope Leo III to seek Frankish support through Charlemagne, because he was losing Roman aristocratic support, or the "pornocracy" of 10th century Rome.

follow up question:

in the middle age most, if not all, of north and central italian cities and later autonomous comunes had a city senate

when those senate originated? were they continuation of roman era senates or just took the name?

Followup question: I seem the recall there was a "roman senate" operating throughout the middle ages and even during the renaissance papal state. Machiavelli mentions it repeatedly in the Prince. Was this a continuation of the institution or simply a new one using the same name to appear as a continuation?