Oscillatory electrical system using a chain of neurons
Short answer
A single pacemaker neuron can generate oscillatory behavior. 

Background
Given our exchange in the comments, I will focus on single neurons with intrinsic oscillatory behavior. For example, thalamocortical relay neurons and inferior olive neurons have intrinsic oscillatory properties, mainly through the interaction of a hyperpolarization-activated cation current (Ih) and a low-threshold activated Ca2+ current (IT) (McCormick &amp; Huguenard, 1992, and Fig. 1).


Fig. 1 Model for an oscillatory inferior olive neuron. Source: Bal &amp; McCormick (1997)

Rhythmic burst firing is created through generation of a low-threshold Ca2+ current (IT), which activates fast action potentials mediated by Na+, K+, and Ca2+ currents (INa, IK, and ICa, respectively). Entry of Ca2+ in the cell during fast action-potential firing and low-threshold Ca2+ spiking result in after-hyperpolarization through activation of a Ca2+-activated K+ current. This afterhyperpolarization activates Ih. Combined activation of Ih and decrease of a Ca2+ activated K+ current (IK,Ca) results in depolarization of neuron, which activates another low-threshold Ca2+ spike. In this manner, Ih modulates frequency of oscillation. In addition, tonic activation of Ih also contributes to average membrane potential, and therefore to presence, strength, and frequency of oscillation
in these neurons.   

Regarding the series of sub-questions - these are quite varied. Perhaps it is better to generate another one or two focused, well-formulated questions on this. For now, I will answer them curtly below. Further, the papers I linked contain a wealth of information, and the McCormick &amp; Huguenard (1992) paper is a detailed modeling study. Both papers are awesome and the authors are three of my all-time favorites so far.

I want to know if these kind of oscillations are unique to some special kinds of neuron?
Answer: Yes quite unique, because Ih channels are quite unique (they have been called "funny currents") and they are not present in every cell.

Is the control of oscillation parameter possible in every neurons and isolated neurons?
Answer: No, as the molecular machinery such as Ih channels may be lacking in most other cells.

Can a neuron or any other cell exhibit change in action potential independently?
Answer: That depends whether the cells are coupled through synapses or other means.

Do other cells except neuron and pacemaker cells exhibit oscillation?
Answer: Yes, some cells start to oscillate, for example in the degenerated retina (modeling work only) and there are non-neural cells that when cultured spontaneously start to oscillate. 

Can a single neuron exhibit memory?
Answer: Yes, through long-term potentiation or depression.

References
- Bal &amp; McCormick, J Neurophysiol (1997); 77:3145-56
- McCormick &amp; Huguenard, J Neurophysiol (1992), 68(4): 1384-1400