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.
To help evaluate the hypothesis that the central respiratory rhythm is generated by a network of interacting neurons, a network model of respiratory rhythmogenesis is formulated and examined computationally. The neural elements of the network are driven by tonic inputs and generate a continuous variable representing firing rate. Each neural element in the model can be described by an activation time constant, an adaptation time constant, and a step nonlinearity. Initial network connectivity was based on an earlier proposed model of the central respiratory pattern generator. These connections were adjusted interactively until the model trajectories resembled those observed electrophysiologically. The properties of the resulting network were examined computationally by simulation, determination of the phase resetting behavior of the network oscillator, and examination of the localized eigenstructure of the network. These results demonstrate that the network model can account for a number of diverse physiological observations, and, thus, support the network hypothesis of respiratory rhymogenesis.