The singularity does not need a model of a human brain, just a model of our senses. Touch, sight, smell, taste and sound. Sight and sound are in just about every video game you play now. Touch is modeled is many VR games as well.

With today's tech, and a few brave volunteers the singularity can happen now. Right now we have a robot that is controlled completely by brain tissue from a rat. 'Frankenrobot'
What we need is the brain from a patient of the Alcor Life Extension Foundation. Using similar tech as the frankenrobot, the human brain can be used in a multi-electrode array. Once that happens it would be a matter of tuning the brain with the computer. The best way would be a 3d engine where signals from the brain control a human model. If the brain tried speaking, the signals the brain makes would control an algorithm that converts them into sound on the computer side of the array. Same with visuals, everything in line of sight of the 3d model of the patient would send signals to the brain letting them see. Tuning the rest of the senses would be a matter of getting feedback from the patient.

Patient "The color is wrong here, the picture of the mona lisa your showing me has a red hue."

Scientist says into a microphone sending signals to the patients brain "Ok we are adjusting the colors for you now. Just keep giving us feedback."

Once the tuning is complete we would have a human being living inside a computer. Immortal, linked with the internet, learning everything they can. A hunger for knowledge. The Singularity begins.



edit:
Imagine playing an MMORPG with characters controlled by these multi-electrode array brains.

edit: I am naming these new brain/computers The Singularity Genus or Syg for short.