Heteroskedasticity of neuronal population responses

Abstract

Typical changes in a neuronal circuit's operating regime - such as shifts in sensorimotor conditions or in brain state - reshuffle firing rates across its neurons. One well-established key organising principle is rate preservation: firing rates correlate strongly across conditions. Here, we demonstrate that the joint distribution of firing log-rates across pairs of conditions exhibits an additional pervasive and previously unrecognised property: strong heteroskedasticity around the rate-preservation axis, whereby slow-firing neurons (< 1 spike/s) display substantially more variable relative responses across conditions than fast-firing neurons (>10 spikes/s). Using large-scale recordings across multiple brain regions (visual cortex, hippocampus, thalamus, and motor cortex) in mice and primates, we demonstrate that this structure is a generic feature of neuronal population responses. A spike subsampling procedure and explainable variance analysis confirm that this heteroskedasticity reflects true biophysical variance rather than statistical estimation bias resulting from low spike counts. Rate and spiking network models reproduce this phenomenon, revealing its key mechanism: while the standard neuronal transfer function (the F-I curve) is convex, its log-rate counterpart (the logF-I curve) is concave. Consequently, for equivalent input current shifts slow-firing neurons exhibit greater relative sensitivity whereas fast-firing neurons exhibit greater absolute sensitivity. Finally, we show that heteroskedasticity strongly influences downstream readout. Under wiring constraints, neurons with intermediate firing rates provide optimal discrimination performance, whereas under metabolic constraints, large assemblies of slow-firing neurons are most advantageous. These findings identify heteroskedasticity as a fundamental organising principle of neuronal population responses, establishing a mechanistic link between non-linear cellular transfer functions, population-level responses and energy-efficient representations.