Anything that is simulatable with a computer can be simulated on an embedded system. Maybe slower than on a server rack, but, still, possible. So that answers your literal question
I am wondering if it is possible to simulate Hindmarsh-Rose (HR) model on a embedded systems?
with a clear "yes".
I need the output of the embedded system to match the burst/firing rate of a neuron. what protocol could i used to simulate those bursts? like SPI, UART, DMA?
That question makes no sense. "Protocol" only makes sense if you're talking to other systems, but you say you want to simulate a SNN on a embedded system, singular. If you want to link multiple of these, you'd need to start writing down your actual communication requirements, which you haven't done at all. From that you could derive things like the computational hardware you need, the type of link or bus or network between the nodes, finally per-link speeds, and then could finally boil down to a selection of options. You're trying to solve the "flying" part, before you even thought about the "crawling" part.
(And DMA is not a protocol – better research about what these things are once you've finished writing down the requirements would be in order.)