The need for the emergence of mathematical neuroscience: beyond computation and simulation.

Computational neuroscience, broadly defined, is the mathematical and physical modeling of neural processes at a specific chosen scale, from molecular and cellular to systems, for the purpose of understanding how the brain and related structures represent and process information. The ultimate objective is to provide an understanding of how the organism takes in sensory information, how such information is integrated and used in the brain, and how the output of such processing results in meaningful decisions and behaviors by the organism to allow it to function and thrive in its environment. This endeavor involves the building of computational models that aim to replicate and explain observed or measured data in order to arrive at a deeper understanding of the dynamics of brain function. Beginning with a set of experimental observations or measurements, a model is postulated that aims to provide a set of rules or relationships that if given the initial experimental observations (or at least part of such a set) would be able to describe and explain some desired aspects or properties of the experimental measurements, such as casual, correlative, or mechanistic relationships between the data and underlying molecular, cellular, and systems mechanisms that produced it. In general, this process almost always begins with a qualitative “guess” about how the data fit together and what are the likely rules that govern the relationships between it. This is subject to a number of uncontrollable variables, including the amount and quality (e.g., accuracy and precision) of the data, how general or narrow the acquisition conditions were under which it was collected, which may constrain the generality and applicability of the model, and the degree of understanding and expertise on the part of the investigator constructing the model about the neurobiology which the data describe. This qualitative picture of the model is then “translated” into a quantitative mathematical framework which almost always involves expressing the hypothesized relationships as ordinary or partial differential equations or related objects, such as difference equations, as state variables that evolve in space and/or time. The model, once constructed, is still nothing more than a guess, and so testing it with the goal of building circumstantial support for it (or against it) is then carried out by numerical simulations of the processes being modeled, often where the answers or outputs are known from experiment and can be compared with the outputs computed by the model. At this point several outcomes are possible, assuming the model is at least partially correct. One possibility is that the model is able to describe the data set used to constructed it but cannot make any novel non-trivial predictions or new hypotheses about the system under study. This outcome often provides a modest contribution to the literature that may give some insights into the mechanisms involved if the model or at least parts of it can be experimentally tested and validated. A less desirable outcome is where a model contains terms or is constructed in a way where further experimental testing of the model cannot occur, for example due to limitations in experimental technologies or terms in the mathematics that have no known real world counterparts. A more productive outcome is when the model results in a novel non-trivial or unexpected experimental hypothesis that can be tested and verified. This may lead to the design and carrying out of new experiments and may lead to potentially significant novel experimental findings. In turn, new data sets allow the fine tuning or modification of the model in an iterative way. But in all cases though, the core of the process is the same: one guesses at a model and uses mathematics to justify the guess. The actual validation of the guess is based on numerical simulations, and in an often iterative approach improves the model. Note however, that in the typical way in which computational neuroscience is practiced, the mathematics involved, while in an applied sense is central to the process, is purely descriptive and does not participate in the process of discovery. Given this discussion, we can define computational neuroscience somewhat more provocatively as numerical simulations of postulated models constructed from qualitative hypotheses. In the most limited case this definition extends itself to numerical simulations of postulated models constructed from unverifiable hypotheses. The computational neuroscience literature is full of beautifully mathematically constructed models that have had minimal impact on main stream neuroscience or our understanding of brain function because of this. Here, we propose to define mathematical neuroscience not as the generation of hypotheses based on numerical simulations of postulated models, but as the systematic analytical investigation of data driven theorems, a term we progressively explain in the rest of this commentary. The principle idea being put forward is that mathematical conjectures can be written down and logically proven who’s axioms, i.e., the starting point ground truths, are not unknown or postulated hypotheses about how the system under consideration might work, but, within the limits of experimental verification, are the simplest possible set of experimentally verified knowns which allow the construction of the statement of truth being made by the conjecture. (We apologize outright to mathematicians for the considerable abuse of the term “axiom” here). The initial goal is to set up a conjecture that is mathematically sound and is based on an agreed upon experimentally known set of axioms, and then use any mathematics possible to formally prove it or at least provide a reasonable starting point, such as a