In the traditional view, the action potential is considered a "choke-point" for information: it's fired (usually at the axon hillock), travels down the axon, and by the time it reaches the synapse there's no information from the pre-synaptic cell left to be passed through the synapse besides the timing of the spike.
This isn't exactly true, however. Recent research has shown that within a few hundred microns of the soma the shape of the action potential and the "the average amplitude of the postsynaptic potential" evoked can vary depending on the "somatic membrane potential of the presynaptic neuron". Indeed, even without an action potential, voltage changes created by synaptic inputs can propagate a similar distance, modifying the synaptic activity within that range.
We now find out, in a recent paper, that there's another analog signal that can be carried by the axon: small delays in propagation can be introduced as part of a "learning process" on a time-scale of seconds to minutes. In Long-Term Activity-Dependent Plasticity of Action Potential Propagation Delay and Amplitude in Cortical Networks (by Douglas J. Bakkum, Zenas C. Chao, and Steve M. Potter) we discover that various types of activity experienced by "cortical neurons cultured on extracellular multi-electrode arrays (MEAs)" can modify the timing of signals received at their synapses. ...
The actual experiments done by Bakkum et al. involved "varying a simple low frequency stimulation pattern every 40 minutes", which "induced changes in the time elapsed for dAPs [direct electrically-evoked action potentials] to propagate from a probe electrode to a recording electrode and also in their extracellularly recorded amplitude (Fig. 3)."
Delays ranged up to 4 milliseconds (ms), and changes in amplitude reached 80% (20µV). This is enough to produce major differences in how the signal from one axon is treated by the receiving dendrites, as dendrites perform very tight calculations of the relative timing of signals from different axons. The changes in delay, introduced in response to activity, took place over a period of minutes, consistent with the activity of short-term memory.
There were also much more transient delays, from "the impedance mismatch due to the change in volume from the axon into the soma causes a delay in propagation proportional to the somatic membrane potential, which varies with synaptic input [ref]." This actually may be more interesting from a calculative point of view, as it occurs with a time-frame measured in milliseconds, the same as that of action potentials.
I've discussed the ways in which the brain's functioning depends on much more than action potentials, the intelligence of individual synapses, and the need for a "membrane centric" approach to the brain's functioning. This discovery integrates into these discussions.
It's not a totally new discovery, of course, that some nerve cells use more analog types of communication than just action potentials.  Sensory nerves especially, often make use of analog calculations. Nevertheless, except for the sensory margins of cognition, the central nervous system is usually pictured as communicating through action potentials that carry only their relative timing as information. Over long distances, this is probably true.
However, a great many of the connections in the brain, including the neocortex, arguably the most important part of the relative expansion of the human brain, are within a few hundred microns, the distance discussed here.
This means that not only are the axons (and their membranes) integrated into the system of membrane-level analog computation I discussed in Nerve Cells and Glial Cells: Redefining the Foundation of Intelligence and Beyond the Synapse, but even their action potentials carry much more analog information than the notion of an invariant action potential suggests.
They can alter their propagation time to synchroninze (or not) with other neurons, and their output at synapses within a few hundred microns is partially analog depending on the current activity at the soma (or axon hillock).
This information constitutes another nail in the coffin of the more traditional picture of neurons communicating by means of only the timing of their action potentials. Quite to the contrary, the vast majority of calculations in the brain are linked into an analog signaling network where even the axons carry much more analog information than just the timing of action potentials.
Bakkum, D., Chao, Z., & Potter, S. (2008). Long-Term Activity-Dependent Plasticity of Action Potential Propagation Delay and Amplitude in Cortical Networks PLoS ONE, 3 (5) DOI: 10.1371/journal.pone.0002088
Links: (Not all of these have been called out in the text. Many are derived from the main article. Use the back key if you came via clicking a footnote.) (I've included only the links referenced in this leader.)
1. Long-Term Activity-Dependent Plasticity of Action Potential Propagation Delay and Amplitude in Cortical Networks
2. Modulation of intracortical synaptic potentials by presynaptic somatic membrane potential
3. Combined Analog and Action Potential Coding in Hippocampal Mossy Fibers Requires Free Registration
4. Impulse-Coded and Analog Signaling in Single Mechanoreceptor Neurons Requires subscription. Otherwise, click on Ref 3, go to foonote #2 in the article, and click on the /Free Full Text] link.
5. Information processing by graded-potential transmission through tonically active synapses
6. Enhancement of presynaptic neuronal excitability by correlated presynaptic and postsynaptic spiking
7. Recurrent Excitation in Neocortical Circuits Requires subscription. Otherwise, click on Ref 3, go to foonote #6 in the article, and click on the /Free Full Text] link.