Unclear what it is, but maybe this schematic is of help.
Suggest probing the R1..R15 pins, with other scope or via other channel, and look for something that pulses in sync with what you see on the scope.
Does the problem go & come with the 'Gain' relay being on or off?
The relays are latching type. They only need a short (~40 ms ish) pulse, and they're powered from the + and - 2.5 rails. The ADG361 has digital outputs, and they're set on/off through SPI commands. If after the 'on' time they're not turned off, then quite large current is drained by the relay, and that can kill the digital outputs. Do all relays still click ok? Is any relay getting warm?
'GND' does not have a relay, but is some solid state switch, controlled by one of those R1..R15 pins.