Relationships Between Lognormal Distributions of Neural Properties and Connectivities*

Relationships between convergence of inputs onto neurons, divergence of outputs from them, synaptic strengths, nonlinear firing properties, and randomness of axonal ranges are systematically explored by interrelating means and variances of synaptic strengths, firing rates, and soma voltages. Imposition of self-consistency yields broad distributions of synaptic strength as a necessary concomitant of the massive convergence of inputs to individual neurons, and widths of lognormal distributions of synaptic strength and firing rate are explained. The strongest individual synapses are shown to have an effect on soma voltage comparable to the standard deviation of of all others combined. Remarkably, inclusion of moderate randomness in axonal ranges accounts for the observed ~10³-fold variability in two-point connectivity at a given separation, and ~10⁵-fold overall when the known mean exponential fall-off is included, consistent with observed near-lognormal distributions.