For over 50 years, sensory neurophysiology has utilized a 'black-box' paradigm, describing the functional relationship between stimuli and neural responses without explicitly considering underlying brain circuitry. In this thesis, we consider two avenues of research which can help to better understand how the functional relationships measured in experiments are ultimately generated by neural circuitry.

In Part I, we demonstrate how the architectural properties of neural networks constrain optimal and invariant stimuli for sensory neurons. In Chapter 2 we demonstrate for neurons whose functional networks are convergent the optimal stimulus must lie on the boundary of a topologically compact set of permissible input unit activities, implying that such a network cannot have a firing rate peak in this space. Applying this to network models of sensory processing predicts monotonic tuning for stimulus contrast, as is observed experimentally. In Chapter 3 we consider quadratic analysis of neural networks and its implications for finding optimal and invariant stimuli. We show how quadratic analysis is constrained by the architecture of the underlying neural network, and how quadratic analysis describes stimulus invariances arising from the first layer but not higher layers of network processing.

In Part II, we consider an approach to sensory neurophysiology based on fitting multiple models in on-line experiments and generating critical stimuli to distinguish competing models. In Chapter 4 we consider the problem of identifying the true parameters of neural network models from finite, noisy data. By solving a differential equation, we show that is only possible for a continuum of different parameters to implement the same input-output function when the hidden unit gains are given by power, exponential or logarithm functions. However, even in networks with standard gains functions continuous parameter confounding can occur as well, making unique recovery of network parameters impossible with standard training sets. Finally, in Chapter 5 we demonstrate how active data collection may overcome the problem of continuum confounding, and illustrate with an experimentally interesting example how on-line stimulus generation may be far more effective for model estimation and comparison than presenting random stimuli.