Thermodynamic constraints on neural dimensions, firing rates, brain temperature and size

There have been suggestions that heat caused by cerebral metabolic activity may constrain mammalian brain evolution, architecture, and function. This article investigates physical limits on brain wiring and corresponding changes in brain temperature that are imposed by thermodynamics of heat balance determined mainly by Na$^{+}$/K$^{+}$-ATPase, cerebral blood flow, and heat conduction. It is found that even moderate firing rates cause significant intracellular Na$^{+}$ build-up, and the ATP consumption rate associated with pumping out these ions grows nonlinearly with frequency. Surprisingly, the power dissipated by the Na$^{+}$/K$^{+}$ pump depends biphasically on frequency, which can lead to the biphasic dependence of brain temperature on frequency as well. Both the total power of sodium pumps and brain temperature diverge for very small fiber diameters, indicating that too thin fibers are not beneficial for thermal balance. For very small brains blood flow is not a sufficient cooling mechanism deep in the brain. The theoretical lower bound on fiber diameter above which brain temperature is in the operational regime is strongly frequency dependent but finite due to synaptic depression. For normal neurophysiological conditions this bound is at least an order of magnitude smaller than average values of empirical fiber diameters, suggesting that neuroanatomy of the mammalian brains operates in the thermodynamically safe regime. Analytical formulas presented can be used to estimate average firing rates in mammals, and relate their changes to changes in brain temperature, which can have important practical applications. In general, activity in larger brains is found to be slower than in smaller brains.

💡 Research Summary

The paper investigates how the thermodynamics of heat production and dissipation constrain neural architecture, firing activity, brain temperature, and ultimately brain size across mammals. The authors build a quantitative framework that links three principal processes: (1) the influx of Na⁺ during action potentials, (2) the energetic cost of restoring ionic gradients via the Na⁺/K⁺‑ATPase, and (3) the removal of the resulting heat by cerebral blood flow and conductive cooling through brain tissue.

First, they derive an expression for the intracellular Na⁺ concentration increase per unit time as a function of axonal diameter (d) and average firing rate (f). Because thinner fibers have a larger surface‑to‑volume ratio, a given spike introduces more Na⁺ per unit volume, leading to a faster rise in intracellular Na⁺. The ATP consumption required to pump this Na⁺ out is modeled as a nonlinear function of the Na⁺ load; the pump’s rate saturates at high loads, producing a biphasic dependence of power dissipation P(f) on firing frequency. At low f, P rises roughly linearly with f, reaches a maximum at an intermediate frequency (typically 20–30 Hz for realistic parameters), and then declines as the pump approaches its maximal turnover rate.

Second, the authors formulate heat balance equations. Convective cooling by blood flow (Q_b) is proportional to cerebral blood flow, blood specific heat, and the temperature difference between blood and brain tissue. Conductive cooling (Q_c) follows Fourier’s law, depending on tissue thermal conductivity, brain surface area, and thickness. The steady‑state brain temperature T_brain is then given by

 T_brain = T_body +