Neurons are rather impressive and it's hard to see how they can learn. Processor companies have been studying neurons by putting them on chips for years, but if you had a web of neurons with input and output terminals what would you feed into it and train it to do? I found a somewhat old source which is supposedly the first time they have trained neurons outside of the body:

>Learning in Networks of Cortical Neurons [https://doi.org/10.1523/JNEUROSCI.21-22-08782.2001]
>We focally stimulate the network at a low frequency (0.3–1 Hz) until a desired predefined response is observed 50 ± 10 msec after a stimulus, at which point the stimulus is stopped for 5 min. Repeated cycles of this procedure ultimately lead to the desired response being directly elicited by the stimulus.

In another paper they used rat neurons on a grid of electrodes to control a simulated animal in a virtual world. The real neurons are able to interact with the simulation:
>The Neurally Controlled Animat: Biological Brains Acting with Simulated Bodies [https://doi.org/10.1023/A:1012407611130]
>Training of the neural pattern recognizer (clustering algorithm) began by interfacing a neural culture to the system for approximately 8 minutes without feedback, allowing the clustering algorithm to discover some of the recurring activity patterns. The feedback system was then activated and the Animat ran for approximately 50 minutes.
>Movements of the Animat often occurred in rapid succession, in most cases as a result of periods where the culture exhibited bursts of neural activity.
They noticed a bunch of patterns forming but they weren't training it for a particular goal in this experiment.

Of course these are from 2001, I've seen other weird stuff in the news but I'm still trying to find those papers.

I've also found some free programs that can simulate various neuron models:
>NEURON
>NEST (NEural Simulation Tool)