Scaling of cortical neuron density and white matter volume in mammals

A prior scaling model, based on repeating cortical units, whose number and size increase with brain size, gave discrete exponents for cortical thickness (1/9), outer (visible) surface area (2/3), folded cortical surface area (8/9) and cortical volume (1), each as a function of brain volume. These exponents are in reasonable agreement with a diversity of empirical data (Prothero, 1997). Rockel et al. (1980) reported that neuron number, assayed in a narrow column across cortex (pia to white matter) is invariant over several differing brain regions and species. Since cortical thickness scales, empirically, as about the 1/9 power of brain volume, their data imply that neuron line density (across cortex) scales with an exponent of about -1/9. Rockel et al. (1980) also urged that cortical neuron surface density is invariant. This extrapolation implies that neuron volume density scales, like line density, as the -1/9 power of brain volume, in marked disparity with the data of Haug (1987) and Tower (1954). The present model assumes an invariant number of neurons per repeating unit. Thus neuron number, assayed across cortical thickness, is independent of brain size, in accord with Rockel et al. (1980). The model predicts that neuron line density (in any direction) scales as the -1/9 power of brain volume. Now neuron volume density scales as the -1/3 power of brain volume, in reasonable agreement with the results of Haug (1987) and Tower (1954). For white matter, I assume that mean axon length scales with brain diameter (exponent of 1/3). The number of white matter axons scales in proportion to the number of repeating units (exponent of 2/3). Given an invariant size distribution of white matter axons, white matter volume thus scales with an exponent of one, in reasonable accord with Haug (1970).