Barter economies never had inflationary periods. Inflation as we see it is in it's major component caused by governments debasing the currency.
Barter economies have also never existed in human societies. They have either been gift economies (esp. with smaller tribes) or credit economies (records dating back to the earliest tables from Mesopotamia and Ancient Egypt). See:
* https://en.wikipedia.org/wiki/Debt:_The_First_5000_Years
For a more recent example of a society becoming pure credit, see:
* https://en.wikipedia.org/wiki/Irish_bank_strikes_(1966–1976)