Some (only some!) hints to answer your question can be found here:

1: Tuckwell HC. Biophysical properties and computational modeling of calcium

spikes in serotonergic neurons of the dorsal raphe nucleus. Biosystems. 2013

Jun;112(3):204-13. doi: 10.1016/j.biosystems.2013.01.007. Epub 2013 Feb 4.

Review.

2: Barbuti A, DiFrancesco D. Control of cardiac rate by "funny" channels in

health and disease. Ann N Y Acad Sci. 2008 Mar;1123:213-23. doi:

10.1196/annals.1420.024. Review.

3: Akhavan A. Contribution of pacemaker channels to autonomous electrical

activity of differentiated embryonic stem cells. J Physiol. 2008 May

15;586(10):2425-6. doi: 10.1113/jphysiol.2008.153338. Epub 2008 Mar 27. Review.

4: Ikeda M. Calcium dynamics and circadian rhythms in suprachiasmatic nucleus

neurons. Neuroscientist. 2004 Aug;10(4):315-24. Review.

5: Freeman WJ. Characterization of state transitions in spatially distributed,

chaotic, nonlinear, dynamical systems in cerebral cortex. Integr Physiol Behav

Sci. 1994 Jul-Sep;29(3):294-306. Review.

Many central neurons can generate rhythmic AP firing patterns autonomously. The firing results from one or more types of persistent membrane current that reset themselves either through inactivation or through secondary effects via calcium-sensitive potassium channels that create negative feedback. The cells' output pattern can be either tonic, repetitive firing or bursts of action potentials. Molluscan central neurons are well known for autonomous activity. Neurons in mammalian brainstem that are involved in the respiratory kernel have also been well studied, as have the

cells comprising the vertebrate cardiac pacemaker.

I had questioned the same question before two days, you can it in my account

Some (only some!) hints to answer your question can be found here:

1: Tuckwell HC. Biophysical properties and computational modeling of calcium

spikes in serotonergic neurons of the dorsal raphe nucleus. Biosystems. 2013

Jun;112(3):204-13. doi: 10.1016/j.biosystems.2013.01.007. Epub 2013 Feb 4.

Review.

2: Barbuti A, DiFrancesco D. Control of cardiac rate by "funny" channels in

health and disease. Ann N Y Acad Sci. 2008 Mar;1123:213-23. doi:

10.1196/annals.1420.024. Review.

3: Akhavan A. Contribution of pacemaker channels to autonomous electrical

activity of differentiated embryonic stem cells. J Physiol. 2008 May

15;586(10):2425-6. doi: 10.1113/jphysiol.2008.153338. Epub 2008 Mar 27. Review.

4: Ikeda M. Calcium dynamics and circadian rhythms in suprachiasmatic nucleus

neurons. Neuroscientist. 2004 Aug;10(4):315-24. Review.

5: Freeman WJ. Characterization of state transitions in spatially distributed,

chaotic, nonlinear, dynamical systems in cerebral cortex. Integr Physiol Behav

Sci. 1994 Jul-Sep;29(3):294-306. Review.

Thank you Deforest and Michael for your answers and especially the article titles

If you block voltage gated Na- channel and K-channel with TTX and TEA. You might still see the calcium spikes in depolarizing current pulses. The amplitude of calcium spike never overshoots in case of pyramidal cell of principal cortical neurons. In addition, in zero calcium solution, again the action potential are seen, however they can change to rhythmic oscillation pattern. These both experiments shows that synaptic transmission in not needed to produce either normal/abnormal electrical activity.

Motor neurones in the spinal cord exhibit plateau potentials, and when in the hyper polarized state fire rhythmically. Dave Bennett and Monica Gorassini have done work on this from animal to human models, along with many others.

Dopamine neurons of the substantia nigra and ventral tegmental area also exhibit slow intrinsic pacemaker activity.

Dhruba, thank you and if you could point me towards these studies it would be great.

Thanks also to Kristal and Jonathan.

In hippocampal interneurons spontaneous activation of Na channels produces depolarization that triggers A-type of potassium channels. This cooperation of Na and K channels produces fluctuation of resting membrane potentials that may eventually reach the threshold and trigger "spontaneous" spikes.

These membrane potential fluctuations are very sensitive to mean membrane potential. Thus sonic depolarization of the membrane for a couple of millivolts can significantly change the frequency of intrinsically generated action potentials.

This opens intriguing possibility that neuronal output in form of firing patterns can be set not only by integration of synaptic inputs, but also by extrasynaptic "tonic" signaling.

Song I, Savtchenko L, Semyanov A. Tonic excitation or inhibition is set by GABA(A) conductance in hippocampal interneurons. Nature Commun. 2011 Jul 5;2:376. doi: 0.1038/ncomms1377

Spontaneously firing action potentials need to be driven by pacemaker potentials or similar electrical activities. These pacemaker potentials generate regular small depolarisations. When they reach the threshold for Na or Ca channels, Na or Ca mediated action potentials will be fired with full amplitude. Many ion channels and electrogenic ion exchangers may underlie these pacemaker activities as long as their prevailing equilibrium potentials are more positive than the resting potential of the cell, for example, cation channels, chloride channels and Na/Ca exchange. The activity of these channels is in turn influenced by ion concentrations, intracellular Ca, other second messengers, SR Ca stores. These factors may change in pathological status, for example, Na or Ca overload, which may contribute to abnormal electrical activity of the cell.

Spontaneously active Na channels + A-type of K+ channels cause fluctuations of membrane potential. From time to time these fluctuations reach action potential threshold. 

Interestingly, the magnitude of membrane potential fluctuations is regulated by extrasynaptic receptors which in turn can change the frequency of  spontaneous action potentials.

Here we show such effect for tonic GABA mediated conductance in hippocampal interneurons:

http://www.ncbi.nlm.nih.gov/pubmed/21730957

Thus, neurons integrate not only synaptic inputs but also extrasynaptic. The latter have to be taken into account when one monitors neuronal firing in vitro or in vivo (but rarely done)

You can see some answers similar to you from this link

https://www.researchgate.net/post/What_is_the_source_of_action_potential_in_the_brain