The Four Per Cent Budget

01The evidence

• Sparse population responses are not confined to one region. Surveys across striate and extrastriate visual cortex find that only a small fraction of neurons respond strongly at any time, and the fraction is similar across areas with very different stimulus statistics.Journal of Neurophysiology
• Highly responsive neurons are sparse across cortical regions, and the sparsity has a characteristic distribution that is conserved despite differences in connectivity and function.Cerebral Cortex

02The tension

Efficient coding predicts that sparsity should match the statistics of the input, so areas seeing different statistics should differ. They do not, much. Something is holding the fraction near a common value that the statistics do not explain.

03The theory

Every spike costs ATP, mostly at the sodium-potassium pump, and the supply of glucose and oxygen is delivered through a capillary bed whose density varies by a factor of two or three across cortex. If the brain is choosing an operating point, the constraint is energy per unit time, not information per spike.

An energy-constrained network maximising information per unit ATP will settle at a small active fraction, because the marginal cost of adding active neurons rises while the marginal information falls. The optimum is not universal in principle, but if the cost per spike and the supply per capillary are similar across cortex, the optimum comes out similar, which is what the data show.

The distinctive prediction is spatial: sparsity should follow the vasculature, not the function. Areas with denser capillary beds should sustain larger active fractions. That is measurable, and it is not predicted by any coding account.

The theory also makes metabolism causal rather than correlational. If you reduce glucose delivery, sparsity should rise; if you increase perfusion, sparsity should fall. Those are experiments, not correlations, and they can be done with existing tools.

04What this is not

05Predictions

• P1Across cortical areas and species, the active fraction should scale with capillary density, with a slope close to one.
• P2Pharmacological hyperaemia should reduce sparsity measurably within minutes.
• P3Impaired glucose transport should raise sparsity while leaving stimulus selectivity largely intact.

06What would refute it

Sparsity that is invariant to metabolic supply, or that varies with stimulus statistics but not with vasculature, would leave the budget account with nothing to explain.

07What it would change

It links the blood-oxygen signal to a sparsity parameter, and predicts which regions degenerate first when perfusion fails.