  A neuron transforms its input into output spikes, and this transformation is
the basic unit of computation in the nervous system. The spiking response of
the neuron to a complex, time-varying input can be predicted from the detailed
biophysical properties of the neuron, modeled as a deterministic nonlinear
dynamical system. In the tradition of neural coding, however, a neuron or
neural system is treated as a black box and statistical techniques are used to
identify functional models of its encoding properties. The goal of this work is
to connect the mechanistic, biophysical approach to neuronal function to a
description in terms of a coding model. Building from preceding work at the
single neuron level, we develop from first principles a mathematical theory
mapping the relationships between two simple but powerful classes of models:
deterministic integrate-and-fire dynamical models and linear-nonlinear coding
models. To do so, we develop an approach for studying a nonlinear dynamical
system by conditioning on an observed linear estimator. We derive asymptotic
closed-form expressions for the linear filter and estimates for the nonlinear
decision function of the linear/nonlinear model. We analytically derive the
dependence of the linear filter on the input statistics and we show how
deterministic nonlinear dynamics can be used to modulate the properties of a
probabilistic code. We demonstrate that integrate-and-fire models without any
additional currents can perform perfect contrast gain control, a sophisticated
adaptive computation, and we identify the general dynamical principles
responsible. We then design from first principles a nonlinear dynamical model
that implements gain control. While we focus on the integrate-and-fire models
for tractability, the framework we propose to relate LN and dynamical models
generalizes naturally to more complex biophysical models.
