It is estimated that maintenance of the resting potential of neurons consumes between 15% (in gray matter) and 44% (in fully myelinated white matter) of the brain's total energy budget [1]. This poses the intriguing question why evolution has not strived to lower the permeability of passive ion channels to cut the high resting-state energy budget of the brain. Based on a conceptual mathematical model of neuronal ion currents and action potential (AP) firing we demonstrate that a neuron endowed with small leak currents and correspondingly low energy consumption by the Na(+)/K(+)-ATPase in the resting state may indeed recapitulate all features of normal AP firing. However, the activation and inactivation of such a "low-energy-cost neuron" turns out to be extremely sensitive to small fluctuation of Na(+) currents associated with Na(+)-dependent secondary-active transport that is indispensable for the metabolic integrity of the cell and neurotransmitter recycling. We provide evidence that sufficiently large leak currents function as important stabilizers of the membrane potential and thus are required to allow robust AP firing. Our simulations suggest that the energy demand of the Na(+)/K(+)-ATPase needed to counterbalance passive leak currents cannot be significantly dropped below observed values.