In this talk we review recent work on integrable models for Microtubule (MT) networks, subneural paracrystalline cytosceletal structures, which seem to play a fundamental role in the neurons. We cast here the complicated MT dynamics in the form of a 1+1-dimensional non-critical string theory, which can be formulated in terms of (dual) Dirichlet branes, according to modern perspectives. We suggest that the MTs are the microsites in the brain, for the emergence of stable, macroscopic quantum coherent states, identifiable with the ``preconscious states''. Quantum space-time effects, as described by non-critical string theory, trigger then an ``organized collapse'' of the coherent states down to a specific or ``conscious state''. The whole process we estimate to take O(1 sec), in excellent agreement with a plethora of experimental/observational findings. The complete integrability of the stringy model for MT proves sufficient in providing a satisfactory solution to memory coding and capacity. Such features might turn out to be important for a model of the brain as a quantum computer.