A graph of neural output as a function of the logarithm of stimulus intensity often produces an S-shaped function, which is frequently modeled by the hyperbolic ratio equation. The response of neurons in early vision to stimuli of varying contrast is an important example of this. Here, the hyperbolic ratio equation with a response exponent of two is derived exactly by considering the balance between information rate and the neural costs of making that information available, where neural costs are a function of synaptic strength and spike rate. The maximal response and semisaturation constant of the neuron can be related to the stimulus ensemble and therefore shift accordingly to exhibit contrast gain control and normalization.

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