Where would a cognitively separated person get their brain signals excited from? I suppose there's nothing wrong about the idea that a cognitively separated person could cognitively function as anybody else. But brain as I understand it always needs a potential from some sense. If I suddenly lost all my senses and my brain would stop working for a second would I be able to recover my brain activity? 

Otherwise, the brain keeps the same electrical impulses running in the brain so you can't really think of anything new, right? I am really interested just in the way the brain creates new electrical potentials, "just on his own."
Seems that you are asking:


  I am really interested just in the way the brain creates new
  electrical potentials, "just on his own."


and


  whether [sic] the brain would 'stop working', implying that there no longer is any neural activity due to the lack of external impulses.


If you were to theoretically disconnect the brain from all sensory input, there could be several sources of activity. For example:


Random noise. Neurons are messy objects, there are many small and large molecules floating inside and outside the cells. Furthermore, the membranes themselves are relatively dynamic, and are studded with receptors, ion channels, and other proteins, some of which migrate regularly (see Lipid rafts). The ion channels (e.g. sodium and potassium) that contribute to the electric currents that generate action potentials, can be affected by thermal fluctuations, and thus change their open-closed states in a stochastic manner. Notice the fluxuations in Figure 1 of Horn and Vandenberg (1984) obtained from a single sodium channel.
Some cells fire without any external stimulation. Central pattern generator circuits is one example, and pacemaker neurons is another. Take a look at Figure 1 of Alving (1968) where an isolated neuron exhibits regular spiking without any external input. One can imagine that one such spiking cell could be connected to other neurons and could cause them to spike as well.
Gain control mechanisms might amplify noise. In your scenario, if the brain is not receiving sensory input, the natural gain control mechanisms, attempting to maintain a certain activity range, might increase amplification to such an extent that the thermal noise fluctuations become consciously perceived. For example, sensory deprivation can result in hallucinations, which might be explained by noise amplification by gain control mechanisms. Similar mechanism can also be seen in the Ganzfeld effect


Overall, any one or a combination of the mechanisms described above could cause spontaneous brain activity without external stimulation.