I've found a video that shows how to connect a receiver to the E160's motherboard, but I'm afraid that available DSM and SBUS/PPM ports are not suitable for my Futaba receiver.
The product page of the helicopter states it is compatible with: FUTABA S-FHSS, expandable satellite receiver, and supports DSMX DSM2 PPM S-SUS
Your TX/RX are from Futaba but from the older FASST system so S-FHSS is no option. DSMX and DSM2 are from Spektrum and not compatible with Futaba.
There are two options remaining: PPM and S-BUS. Both do basically the same: send the signal of all channels via one cable. As far as I can tell your TX does not support either and only allows to send one channel via one output.
S-Bus is also from Futaba so you could try to find a FASST receiver that supports S-Bus output. Frsky also has compatible FASST receivers with PPM output (like FrSKY TFR4-B). Then you could solder a connector from the outputs from the RX to the E160 motherboard. The yellow cable in the picture is the S-BUS/PPM wire, red is 5V and black is ground. To find the correct matching pins on the RX I'd look in the manual. Usually you have some labels on the case.
A PPM enconder is also an option. This is a small board that encodes the servo signals to a single ppm stream. The downside is that you still have many cabels between the encoder and the RX and you need space for an additional board.