Brains are energy-hungry and evolution has driven them to work in an energy-efficient manner. In food restriction experiments, Padamsey et al. [1] have recently observed that individual neurons trade off energy consumption and information transmission. Specifically, neurons appear to reduce energy consumption by changing their integration properties through the adaptation of membrane resistance, resting potential, and synaptic efficacies, which comes at the cost of reduced information transmission. Furthermore, they found that individual neurons assume a broad range of values of these parameters, suggesting many degenerate solutions to this energy-information trade-off. Here we investigate this trade-off theoretically by combining simulations of a generic leaky-integrate-and-fire neuron model with well-established energy calculations [2] and information measures. Concretely, we compute the coefficient of variation of neural activity and mutual information between input and output across the space of neuronal parameter values and relate them to the total energy consumption of a neuron. Our simulations explain the trajectory of neurons in the neuronal parameter space under food restrictions as maintaining a near-optimal energy-information trade-off. Furthermore, our results show that measures of information transmission per expended energy are highest in the very low to medium-low firing rate regime, offering an explanation of the low firing rates typically observed in mammalian neocortex. Our results also predict a high but well-orchestrated variability of these neuronal parameters to maintain a near-optimal energy-information trade-off. Overall, our work elucidates the relationship between information transmission and energy consumption and the resulting degeneracy of neuronal parameters.