Despite the central importance of answering this question to analyzing the emergence of neural functionality, it's not clear that this question has been mathematically well posed to date. Given the current state of affairs, it's not clear it can be. What's missing in my opinion is a basic understanding of how nonlinear processes manuipulate information. This includes understanding not only how much information is produced and transmitted by neural systems, but also how much memory they use to maintain a given level of production and, perhaps of more interest, how that memory is structured to support (and preclude) particular classes of computation.
Imagine you could measure any number of neurophysiological observables to any accuracy. Does it necessarily follow that you understand how the underlying network operates? More specifically, even without experimental limitations, how would you detect that some computation was being performed? Over the last decade or so a synthesis of methods from dynamical systems theory, computation theory, and inductive inference has been developed that suggests a constructive approach to addressing this problem. The approach --- computational mechanics --- is nicely complementary to information theoretic methods that, for example, focus on information production rates and various forms of mutual information. It proposes, however, a rather different answer to the search for neural codes, since it highlights their temporal structure and use of memory and gives explicit expression of "codes" in terms of stochastic computational models.
The approach arose in trying to answer a basic question in nonlinear physics, How do dynamical systems process information? My research at the Sloan Center is premised on the hypotheses that (i) neural systems can be modeled as stochastic dynamical systems and (ii) the past theoretical developments now allow us to consistently frame this basic question. The goal is to apply computational mechanics to detect high-level information processing structures embedded in neurobiological systems. In this way we hope to understand the mechanims by which raw neural behavior supports useful neural computation.