Some addtional troubleshooting hints with the 4051:
Important! Not all versions of the 4051 will work stable in a 3,3 envirionment If Vee == GND.
e.g. the 4051 HCT has specification:
VCC-GND min: 4.5, typical: 5.0 max:5.5 V
e.g. the 4051 HC has specification:
VCC-GND min:2.0, typical: 5.0, max:10.0 V
So when you use the HCT version with the AXO, you'll run into a lot of strange problems . It works, sometimes - but outsite the specification. This is good if you plan to build an Lunetta :-), but for a multiplexer this can be worse.
BTW: 100nf between Vcc and GND could save a lot of problems in such breadboard environments. Put the cap close as possible to the pins! Or better solder the cap on top over the chip.
FYI: More about the voltages on page 6:http://www.nxp.com/documents/data_sheet/74HC_HCT4051.pdf
This is if you like to use DIPs.
But better would be a version like this:HD74LV4051A
the shortcut LV / LVC gives you a hint of LowVoltage. Those Chips are made for the use with 3,3V. They are hard to get sometimes and hard to solder. Only SMD available. If you have luck, SOIC14 is available. This is one of the disadvantages of the 3,3 voltage for tinkerer (and breadboard testing)
Or use a "level shifter" between the Axo board and the "normal" CMOS 5V world. Since I work with 3,3 I allway have some 8 Line Levershifter breakouts in stock......
The last: Be aware, if you use a 4051, you lose aprox. 0,4 of you incomming signal inside the chip. This is normal!