Relationships between lognormal distributions of neural properties, activity, criticality, and connectivity

Relationships between convergence of inputs onto neurons, divergence of outputs from them, synaptic strengths, nonlinear firing response properties, and randomness of axonal ranges are systematically explored by interrelating means and variances of synaptic strengths, firing rates, and soma voltages. When self-consistency is imposed, it is found that broad distributions of synaptic strength are a necessary concomitant of the known massive convergence of inputs to individual neurons, and observed widths of lognormal distributions of synaptic strength and firing rate are explained provided the brain is in a near-critical state, consistent with independent observations. The strongest individual synapses are shown to have an effect on soma voltage comparable to the effect of all others combined, which supports suggestions that they may have a key role in neural communication. Remarkably, inclusion of moderate randomness in characteristic axonal ranges is shown to account for the observed [Formula: see text]-fold variability in two-point connectivity at a given separation and [Formula: see text]-fold overall when the known mean exponential fall-off is included, consistent with observed near-lognormal distributions. Inferred axonal deviations from straight-line paths are also consistent with independent estimates.