Are action potentials necessary for experience?
It is generally accepted that all activity having to do with conscious experience is mediated by spiking in the cortex. Sub-threshold activity, such as excitatory postsynaptic potentials (EPSPs) are not carried forward in the nervous system and will 'fade' before having an effect. It all revolves around action potentials.

Quoting from Kandel et al. (2000):


  Action potentials constitute the signals by which the brain receives, analyzes and conveys information. These signals are highly stereotyped throughout the nervous system [...]. The information conveyed by the action potentials is determined not by the form of the signal but by the pathway the signal travels. The brain analyzes patterns of incoming electrical signals and in this way creates our everyday sensations of sight, touch, taste, smell, and sound. 


Reference
Kandel et al. eds. Principles of Neural Science, 4th ed. McGraw-Hill, 2000