Reproducing natural cognitive systems - OK, I can see that discrete time suffices (speech recognition works as you wrote), thanks for clarifying it. However, AFAIK, the whole idea of AI is to go beyond that and make such a "brain" learn new things to itself through interacting with other "brains" and some environment with some receptors connected to its central unit or something... Now the discrete time doesn't seem to be a natural choice. I'm just wondering (after Cuchulainn) why AI/ML techniques use it if it isn't so complicated to use continuous time, I believe (can't you just integrate a continuous input impulse? it could excite consecutive modes in a vector of neurons, the summed value of which would be an input to some activation functions - I'm thinking it up as I'm writing
I'm sure if it was so simple, someone would have already done it).
I am not in this space really, but it seems to me that most mainstream AI centres around linear algebra (nothing wrong with LA, BTW, it does have its moments). I think professional neuroscientists use more advanced models as in Handbook of Brain Theory and Neural Networks
where ODEs and Control Theory play centre stage. Maybe people are happy with the current methods and maybe they have (greedily) converged to a local minimum and are stuck in a valley.
It's not necessarily so that ODE solvers are slow. This urban myth is not a reason for not using them. In true engineering spirit, it does no harm to test them against other candidate solutions, like how they test cars on Top Gear.