Sure; we have artificial neural network algorithms. Check out this letter-recognition (backpropagation) network using 80 neurons that I wrote in JavaScript during a boring Christmas vacation with my parents. (And it sucks- not because it's JavaScript, but it makes embarrassing mistakes, which are the fault of the huge string literal of neuron weights at the end of the code).
Biologically, the process with a real neuronal cell body reaching a certain (unpredictable) voltage and firing is extremely complex. The firing mechanism is an analogue process, unstable and unreliable (which is how it works). It produces a digital signal which has an unpredictable time lag (the axon length and density of boundaries between glial cells affect this) before it reaches synapses (cesspools of quantum indeterminism) and tickles dendrites of other cells. The development and positioning of cell processes (axons, dendrites, synaptic junctions) is a necessary consequence of learning, but these are affected by gene expression and are extremely hard to predict.
Still, given this entire messy system, people's thoughts, free will, and reactions to stimuli are much more deterministic than they realize. But I suspect that if you wanted to make a robot that acts like a person does, you would at the very least need a prolific stream of very high quality random numbers. Maybe you can simulate the brain of Stephen Hawking with a PRNG; I haven't tried. (I sure as hell wouldn't use JavaScript!)