Estimating the energy of dissipative neural systems

There is, at present, a lack of consensus regarding precisely what is meant by the term 'energy' across the sub-disciplines of neuroscience. Definitions range from deficits in the rate of glucose metabolism in consciousness research to regional changes in neuronal activity in cognitive neuroscience. In computational neuroscience virtually all models define the energy of neuronal regions as a quantity that is in a continual process of dissipation to its surroundings. This, however, is at odds with the definition of energy used across all sub-disciplines of physics: a quantity that does not change as a dynamical system evolves in time. Here, we bridge this gap between the dissipative models used in computational neuroscience and the energy-conserving models of physics using a mathematical technique first proposed in the context of fluid dynamics. We go on to derive an expression for the energy of the linear time-invariant (LTI) state space equation. We then use resting-state fMRI data obtained from the human connectome project to show that LTI energy is associated with glucose uptake metabolism. Our hope is that this work paves the way for an increased understanding of energy in the brain, from both a theoretical as well as an experimental perspective.