I have created a world where the main character is a cyborg, a petri dish with neurons inside a machine, the problem is most neuronal cultures only live for weeks, 60 days or up to 5 years, that's what all the articles I've read say. How could I prolong the life of my cyborg/neurons?
By building a structure that sustains the neural structures and keeps it active.
I dont know what articles you read since you didnt post them so I dont know what the problems were in those instances. But since those neural structures can function for years in humans (centuries in some other animals!) It should be possible to give them a framework where they last longer. The most likely culprits would be improper feeding (nutrients, vitamins, hormones etc) and being purposefully used. In biology "if you dont use it you lose it" is an important aspect to keep efficiency up and not keep feeding cells that dont serve a purpose. Neurons have a high plasticity, this is important in streamlining the brain's processes and switching tasks to things the brain is performing more often. So of the neurons arent used for specific neuron stuff they'll degrade naturally.
If you make sure the neurons can replicate (say stemcells are included that keep replenishing the neurons to facilitate their plasticity and natural death) and are properly fed and stimulated in their performance you can have a long lifetime for these cyborgs.