As many as needed. Since there is no set number for this as it depends on timing, where and what types of inputs the neuron is subjected to, the neurons own level of excitability and where in the brain the neuron is located.

Dear colleague,

Not many, however, you'll have to consider the presence of a wide variety of excitatory and inhibitory synapses working against each other, causing this number to increase. I don't know of anyone investigating this question in depth, which is an very interesting one nonetheless.

Sincerely,

Hércules Freitas

In addition to the presence of excitatory or inhibitory synapses, "time-dependent" and "space-dependent" components for generation depolarizing potentials used to elicit action potential are also important.

This question can be partially answered by the following references.

As many as needed. Since there is no set number for this as it depends on timing, where and what types of inputs the neuron is subjected to, the neurons own level of excitability and where in the brain the neuron is located.

Thanks guys. The thing baffles me is that, if all of these issues are crucial, like timing, excitatory/inhibitory, basal excitability, other neurotransimitters et al.. Considering so many variables, how does a neuron manage to fire a action potential at a specific time when animals undergoing specific tasks or cognitive process?

This is a tough question. There are many variables to affect the axon hillock's elicitation. We still have a long way to go for it.

There is at least one example that a single input neuron can elicit an action potential and even a group of action potentials in the target neuron. In the cerebellum each of the Purkinje cells has an input from a single climbing fiber (a projection from the inferior olivary nucleus). A single climbing fiber makes multiple (>300) synaptic contacts with a single Purkinje cell. The result of the climbing fiber activation is a complex spike ( a group of several action potentials) in the Purkinje cell and Ca2+ concentration increase. This signal is thought to be an error signal, necessary for the mechanism of motor learning  through the mechanism of LTD

And, if just speaking about synaptic efficacy, the safety factor of the neuro-muscular junction is about five, so a single action potential in the motor nerve always elicits an action potential in the innervated muscle

Bear in mind that inhibitory synapses are likely to be much more powerful than excitatory synapses. One inhibitory synapse can silence a neuron and removing inhibition may be all it takes to generate an action potential. I personally suspect that disinhibition is more important to memory than strengthening excitatory synapses.

What does it mean SYNAPSE in your question?

Please, look at the Table 1 (Grantyn, Shapovalov, Shiriaev, 1984); Table 1 ( Babalian, Chmykhova, 1987), Table 4 (Dityatev, Chmykhova et al.,2001), Table 2 (Dityatev, Birinyi et al.,2001. You’ll see the numbers of contacting boutons at different synapses in the connexions:

One fiber – one motoneuron:

Dorsal root fiber – lumbar motoneuron; propriospinal fiber- lumbar motoneuron; reticulospinal fiber – lumbar motoneuron.

The fiber action potential evokes the EPSPs only in a frog lumbar motoneuron.