It is generally accepted that the number of neurons in a given brain area far exceeds the number of neurons needed to carry any specific function controlled by that area. For example, motor areas of the human brain contain tens of millions of neurons that control the activation of tens or at most hundreds of muscles. This massive redundancy implies the covariation of many neurons, which constrains the population activity to a low-dimensional manifold within the space of all possible patterns of neural activity. To gain a conceptual understanding of the complexity of the neural activity within a manifold, it is useful to estimate its dimensionality, which quantifies the number of degrees of freedom required to describe the observed population activity without significant information loss. While there are many algorithms for dimensionality estimation, we do not know which are well suited for analyzing neural activity. The objective of this study was to evaluate the efficacy of several representative algorithms for estimating the dimensionality of linearly and nonlinearly embedded data. We generated synthetic neural recordings with known intrinsic dimensionality and used them to test the algorithms' accuracy and robustness. We emulated some of the important challenges associated with experimental data by adding noise, altering the nature of the embedding of the low-dimensional manifold within the high-dimensional recordings, varying the dimensionality of the manifold, and limiting the amount of available data. We demonstrated that linear algorithms overestimate the dimensionality of nonlinear, noise-free data. In cases of high noise, most algorithms overestimated the dimensionality. We thus developed a denoising algorithm based on deep learning, the "Joint Autoencoder", which significantly improved subsequent dimensionality estimation. Critically, we found that all algorithms failed when the intrinsic dimensionality was high (above 20) or when the amount of data used for estimation was low. Based on the challenges we observed, we formulated a pipeline for estimating the dimensionality of experimental neural data.

A medical student learning to perform a laparoscopic procedure or a recently paralyzed user of a powered wheelchair must learn to operate machinery via interfaces that translate their actions into commands for an external device. Since the user's actions are selected from a number of alternatives that would result in the same effect in the control space of the external device, learning to use such interfaces involves dealing with redundancy. Subjects need to learn an externally chosen many-to-one map that transforms their actions into device commands. Mathematically, we describe this type of learning as a deterministic dynamical process, whose state is the evolving forward and inverse internal models of the interface. The forward model predicts the outcomes of actions, while the inverse model generates actions designed to attain desired outcomes. Both the mathematical analysis of the proposed model of learning dynamics and the learning performance observed in a group of subjects demonstrate a first-order exponential convergence of the learning process toward a particular state that depends only on the initial state of the inverse and forward models and on the sequence of targets supplied to the users. Noise is not only present but necessary for the convergence of learning through the minimization of the difference between actual and predicted outcomes.

A robust identification algorithm has been developed for linear, time-invariant, multiple-input single-output systems, with an emphasis on how this algorithm can be used to estimate the dynamic relationship between a set of neural recordings and related physiological signals. The identification algorithm provides a decomposition of the system output such that each component is uniquely attributable to a specific input signal, and then reduces the complexity of the estimation problem by discarding those input signals that are deemed to be insignificant. Numerical difficulties due to limited input bandwidth and correlations among the inputs are addressed using a robust estimation technique based on singular value decomposition. The algorithm has been evaluated on both simulated and experimental data. The latter involved estimating the relationship between up to 40 simultaneously recorded motor cortical signals and peripheral electromyograms (EMGs) from four upper limb muscles in a freely moving primate. The algorithm performed well in both cases: it provided reliable estimates of the system output and significantly reduced the number of inputs needed for output prediction. For example, although physiological recordings from up to 40 different neuronal signals were available, the input selection algorithm reduced this to 10 neuronal signals that made significant contributions to the recorded EMGs.

Single unit activity in the striatum of awake monkeys shows a marked dependence on the expected reward that a behavior will elicit. We present a computational model of spiny neurons, the principal neurons of the striatum, to assess the hypothesis that di(cid:173) rect neuromodulatory effects of dopamine through the activation of D 1 receptors mediate the reward dependency of spiny neuron activity. Dopamine release results in the amplification of key ion currents, leading to the emergence of bistability, which not only modulates the peak firing rate but also introduces a temporal and state dependence of the model's response, thus improving the de(cid:173) tectability of temporally correlated inputs.

A better understanding of pruning methods based on a ranking of weights according to their saliency in a trained network requires further information on the statistical properties of such saliencies. We focus on two-layer networks with either a linear or nonlinear output unit, and obtain analytic expressions for the distribution of saliencies and their logarithms. Our results reveal unexpected universal properties of the log-saliency distribution and suggest a novel algorithm for saliency-based weight ranking that avoids the numerical cost of second derivative evaluations.

Achieving good performance in statistical pattern recognition requires matching the capacity of the classifier to the amount of training data. If the classifier has too many adjustable parameters (large capacity), it is likely to learn the training data without difficulty, but will probably not generalize properly to patterns that do not belong to the training set. Conversely, if the capacity of the classifier is not large enough, it might not be able to learn the task at all. In between, there is an optimal classifier capacity which ensures the best expected generalization for a given amount of training data. The method of structural risk minimization (SRM) refers to tuning the capacity of the classifier to the available amount of training data. This paper illustrates the method of SRM with several examples of algorithms. Experiments confirm theoretical predictions of performance improvement in application to handwritten digit recognition.>

Hardware architectures for character recognition are discussed, and choices for possible circuits are outlined. An advanced (and working) reconfigurable neural-net chip that mixes analog and digital processing is described. It is found that different approaches to image recognition often lead to neural-net architectures that have limited connectivity and repeated use of the same set of weights. This architecture is ideal for time-multiplexing (a combined parallel-series processing) on hardware systems that would be too small to evaluate the entire network in parallel. To make this process efficient, a chip needs to have shift registers to format the input data and additional registers to store intermediate results. Within this framework, it is possible to design chips that have broad utility, large connection capacity, and high speed. This was demonstrated by a new chip with 32000 reconfigurable connections

Exhaustive exploration of an ensemble of networks is used to model learning and generalization in layered neural networks. A simple Boolean learning problem involving networks with binary weights is numerically solved to obtain the entropy Sm and the average generalization ability Gm as a function of the size m of the training set. Learning curves Gm vs m are shown to depend solely on the distribution of generalization abilities over the ensemble of networks. Such distribution is determined prior to learning, and provides a novel theoretical tool for the prediction of network performance on a specific task.

A general statistical description of the problem of learning from examples is presented. Learning in layered networks is posed as a search in the network parameter space for a network that minimizes an additive error function of a statistically independent examples. By imposing the equivalence of the minimum error and the maximum likelihood criteria for training the network, the Gibbs distribution on the ensemble of networks with a fixed architecture is derived. The probability of correct prediction of a novel example can be expressed using the ensemble, serving as a measure to the network's generalization ability. The entropy of the prediction distribution is shown to be a consistent measure of the network's performance. The proposed formalism is applied to the problems of selecting an optimal architecture and the prediction of learning curves.>