Robust coding has been proposed as a solution to the problem of minimizing decoding error in the presence of neural noise. Many real-world problems, however, have degradation in the input signal, not just in neural representations. This generalized problem is more relevant to biological sensory coding where internal noise arises from limited neural precision and external noise from distortion of sensory signal such as blurring and phototransduction noise. In this note, we show that the optimal linear encoder for this problem can be decomposed exactly into two serial processes that can be optimized separately. One is Wiener filtering, which optimally compensates for input degradation. The other is robust coding, which best uses the available representational capacity for signal transmission with a noisy population of linear neurons. We also present spectral analysis of the decomposition that characterizes how the reconstruction error is minimized under different input signal spectra, types and amounts of degradation, degrees of neural precision, and neural population sizes.

Capturing statistical regularities in complex, high-dimensional data is an important problem in machine learning and signal processing. Models such as principal component analysis (PCA) and independent component analysis (ICA) make few assumptions about the structure in the data and have good scaling properties, but they are limited to representing linear statistical regularities and assume that the distribution of the data is stationary. For many natural, complex signals, the latent variables often exhibit residual dependencies as well as nonstationary statistics. Here we present a hierarchical Bayesian model that is able to capture higher-order nonlinear structure and represent nonstationary data distributions. The model is a generalization of ICA in which the basis function coefficients are no longer assumed to be independent; instead, the dependencies in their magnitudes are captured by a set of density components. Each density component describes a common pattern of deviation from the marginal density of the pattern ensemble; in different combinations, they can describe nonstationary distributions. Adapting the model to image or audio data yields a nonlinear, distributed code for higher-order statistical regularities that reflect more abstract, invariant properties of the signal.

Nonstationary acoustic features provide essential cues for many auditory tasks, including sound localization, auditory stream analysis, and speech recognition. These features can best be characterized relative to a precise point in time, such as the onset of a sound or the beginning of a harmonic periodicity. Extracting these types of features is a difficult problem. Part of the difficulty is that with standard block-based signal analysis methods, the representation is sensitive to the arbitrary alignment of the blocks with respect to the signal. Convolutional techniques such as shift-invariant transformations can reduce this sensitivity, but these do not yield a code that is efficient, that is, one that forms a nonredundant representation of the underlying structure. Here, we develop a non-block-based method for signal representation that is both time relative and efficient. Signals are represented using a linear superposition of time-shiftable kernel functions, each with an associated magnitude and temporal position. Signal decomposition in this method is a non-linear process that consists of optimizing the kernel function scaling coefficients and temporal positions to form an efficient, shift-invariant representation. We demonstrate the properties of this representation for the purpose of characterizing structure in various types of nonstationary acoustic signals. The computational problem investigated here has direct relevance to the neural coding at the auditory nerve and the more general issue of how to encode complex, time-varying signals with a population of spiking neurons.

In an overcomplete basis, the number of basis vectors is greater than the dimensionality of the input, and the representation of an input is not a unique combination of basis vectors. Overcomplete representations have been advocated because they have greater robustness in the presence of noise, can be sparser, and can have greater flexibility in matching structure in the data. Overcomplete codes have also been proposed as a model of some of the response properties of neurons in primary visual cortex. Previous work has focused on finding the best representation of a signal using a fixed overcomplete basis (or dictionary). We present an algorithm for learning an overcomplete basis by viewing it as probabilistic model of the observed data. We show that overcomplete bases can yield a better approximation of the underlying statistical distribution of the data and can thus lead to greater coding efficiency. This can be viewed as a generalization of the technique of independent component analysis and provides a method for Bayesian reconstruction of signals in the presence of noise and for blind source separation when there are more sources than mixtures.

Identifying and classifying action potential shapes in extracellular neural waveforms have long been the subject of research, and although several algorithms for this purpose have been successfully applied, their use has been limited by some outstanding problems. The first is how to determine shapes of the action potentials in the waveform and, second, how to decide how many shapes are distinct. A harder problem is that action potentials frequently overlap making difficult both the determination of the shapes and the classification of the spikes. In this report, a solution to each of these problems is obtained by applying Bayesian probability theory. By defining a probabilistic model of the waveform, the probability of both the form and number of spike shapes can be quantified. In addition, this framework is used to obtain an efficient algorithm for the decomposition of arbitrarily complex overlap sequences. This algorithm can extract many times more information than previous methods and facilitates the extracellular investigation of neuronal classes and of interactions within neuronal circuits.