Energetically Optimal Action Potentials

Most action potentials in the nervous system take on the form of strong, rapid, and brief voltage deflections known as spikes, in stark contrast to other action potentials, such as in the heart, that are characterized by broad voltage plateaus. We derive the shape of the neuronal action potential from first principles, by postulating that action potential generation is strongly constrained by the brain’s need to minimize energy expenditure. For a given height of an action potential, the least energy is consumed when the underlying currents obey the bang-bang principle: the currents giving rise to the spike should be intense, yet short-lived, yielding spikes with sharp onsets and offsets. Energy optimality predicts features in the biophysics that are not per se required for producing the characteristic neuronal action potential: sodium currents should be extraordinarily powerful and inactivate with voltage; both potassium and sodium currents should have kinetics that have a bell-shaped voltage-dependence; and the cooperative action of multiple ‘gates’ should start the flow of current. 1 The paradox Nerve cells communicate with each other over long distances using spike-like action potentials, which are brief electrical events traveling rapidly down axons and dendrites. Each action potential is caused by an accelerating influx of sodium or calcium ions, depolarizing the cell membrane by forty millivolts or more, followed by repolarization of the cell membrane caused by an efflux of potassium ions. As different species of ions are swapped across the membrane during the action potential, ion pumps shuttle the excess ions back and restore the ionic concentration gradients. If we label each ionic species by α, the work ∆E done to restore the ionic concentration gradients is ∆E = RTV ∑ α ∆[α]in ln [α]out [α]in , (1) where R is the gas constant, T is the temperature in Kelvin, V is the cell volume, [α]in|out is the concentration of ion α inside or outside the cell, and ∆[α]in is the concentration change inside the cell, which is assumed to be small relative to the total concentration. The sum ∑ α zα∆[α] = 0, where zα is the charge on ion α, as no net charge accumulates during the action potential and no net work is done by or on the electric field. Often, sodium (Na) and potassium (K) play the dominant role in generating action potentials, in which case ∆E = ∆[Na]inFV(ENa − EK), where F is Faraday’s constant, ENa = RT/F ln ( [Na]out/[Na]in ) is the reversal potential for Na, at which no net sodium current flows, and EK = RT/F ln ( [K]out/[K]in ) . This estimate of the work done does not include heat (due to loss through the membrane resistance) or the work done by the ion channel proteins in changing their conformational state during the action potential. Hence, the action potential’s energetic cost to the cell is directly proportional to ∆[Na]in; taking into account that each Na ion carries one elementary charge, the cost is also proportional to the