Towards a Neuronal Gauge Theory

The free energy formulation enables us (or our brains) to compare a variety of hypotheses about the environment given our sensory samples. So what does a gauge formulation add? After scoring a variety of models according to their variational free energy, one selects the best generative model that explains the data at hand. If symmetry transformations can be evaluated for the best performing model, afforded by the symmetry group, we can possibly go a step further and ask what other models could explain the data with the same model evidence? Such symmetry transformations therefore furnish us with a family of models that explain the data with equal probability [43–45]: see Fig 1 for a schematic illustration of this idea. The challenge of such an approach would be to deduce the symmetry groups of the probability distributions encoded by neuronal activity (S5 Text). If this was possible, one could identify the class of models explaining data using a single inference methodology. From the perspective of biological self-organisation, this means that there are potentially many different models that do the same job of predicting sensory data. It is tempting to associate the equivalence classes afforded by symmetry groups above with the biological species—all doing an equally good job of modelling their (local) environments.

BS and KJF are supported by the Wellcome Trust (088130/Z/09/Z). BS is thankful to the Mathematical Biosciences Institute for hosting him, and facilitating communication with experts working on the symmetry perspective of dynamical systems. We thank John Ashburner for his comments on an initial draft of this paper. GKC is supported by a postdoctoral fellowship from the Swedish Brain Foundation (Hjärnfonden). PD is funded by the Klingenstein Foundation, and the Keck Foundation. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.