We show how to estimate (1) the number of functions that can be implemented by a particular network architecture, (2) how much analog precision is needed in the connections in the network, and (3) the number of training examples the network must see before it can be expected to form reliable generalizations.

The network model considered consists of interconnected groups of neurons, where each group could be fully interconnected (it could have feedback connections, with possibly asymmetric weights), but no loops between the groups are allowed. A stochastic descent algorithm is applied, under a certain inequality constraint on each intragroup weight matrix which ensures for the network to possess a unique equilibrium state for every input. Introduction It has been shown in the last few years that large networks of interconnected "neuron" -like elemp.nts

Abstract: We propose that the back propagation algorithm for supervised learning can be generalized, put on a satisfactory conceptual footing, and very likely made more efficient by defining the values of the output and input neurons as probabilities and varying the synaptic weights in the gradient direction of the log likelihood, rather than the'error'. In the past thirty years many researchers have studied the question of supervised learning in'neural'-like networks. Recently a learning algorithm called'back propagation In these applications, it would often be natural to consider imperfect, or probabilistic information. The problem of supervised learning is to model some mapping between input vectors and output vectors presented to us by some real world phenomena. To be specific, coqsider the question of medical diagnosis.

This viewpoint allows the class of probability distributions, P, the neural network can acquire to be explicitly specified. Learning algorithms for the neural network which search for the "most probable" member of P can then be designed. Statistical tests which decide if the "true" or environmental probability distribution is in P can also be developed. Example applications of the theory to the highly nonlinear back-propagation learning algorithm, and the networks of Hopfield and Anderson are discussed. INTRODUCTION A connectionist system is a network of simple neuron-like computing elements which can store and retrieve information, and most importantly make generalizations. Using terminology suggested by Rumelhart & McClelland 1, the computing elements of a connectionist system are called units, and each unit is associated with a real number indicating its activity level. The activity level of a given unit in the system can also influence the activity level of another unit. The degree of influence between two such units is often characterized by a parameter of the system known as a connection strength. During the information retrieval process some subset of the units in the system are activated, and these units in turn activate neighboring units via the inter-unit connection strengths.

Please address all further correspondence to: John Y. Cheung School of EECS 202 W. Boyd, CEC 219 Norman, OK 73019 (405)325-4721 MATHEMATICAL ANALYSIS OF LEARNING BEHAVIOR OF NEURONAL MODELS John Y. Cheung and Massoud Omidvar School of Electrical Engineering and Computer Science ABSTRACT In this paper, we wish to analyze the convergence behavior of a number of neuronal plasticity models. Recent neurophysiological research suggests that the neuronal behavior is adaptive. In particular, memory stored within a neuron is associated with the synaptic weights which are varied or adjusted to achieve learning. A number of adaptive neuronal models have been proposed in the literature. Three specific models will be analyzed in this paper, specifically the Hebb model, the Sutton-Barto model, and the most recent trace model.

There is a widespread misconception that the delta-rule is in some sense guaranteed to work on networks without hidden units. As previous authors have mentioned, there is no such guarantee for classification tasks. We will begin by presenting explicit counterexamples illustrating two different interesting ways in which the delta rule can fail. We go on to provide conditions which do guarantee that gradient descent will successfully train networks without hidden units to perform two-category classification tasks. We discuss the generalization of our ideas to networks with hidden units and to multicategory classification tasks.

In this article we consider two aspects of computation with neural networks. Firstly we consider the problem of the complexity of the network required to compute classes of specified (structured) functions. We give a brief overview of basic known complexity theorems for readers familiar with neural network models but less familiar with circuit complexity theories. We argue that there is considerable computational and physiological justification for the thesis that shallow circuits (Le., networks with relatively few layers) are computationally more efficient. We hence concentrate on structured (as opposed to random) problems that can be computed in shallow (constant depth) circuits with a relatively few number (polynomial) of elements, and demonstrate classes of structured problems that are amenable to such low cost solutions. We discuss an allied problem-the complexity of learning-and close with some open problems and a discussion of the observed limitations of the theoretical approach. We next turn to a rigourous classification of how much a network of given structure can do; i.e., the computational capacity of a given construct.

SYNCHRONIZATION IN NEURAL NETS Jacques J. Vidal University of California Los Angeles, Los Angeles, Ca. 90024 John Haggerty· ABSTRACT The paper presents an artificial neural network concept (the Synchronizable Oscillator Networks) where the instants of individual firings in the form of point processes constitute the only form of information transmitted between joining neurons. This type of communication contrasts with that which is assumed in most other models which typically are continuous or discrete value-passing networks. Limiting the messages received by each processing unit to time markers that signal the firing of other units presents significant implemen tation advantages. When interaction is present, the scheduled firings are advanced or delayed by the firing of neighboring neurons. Networks of such neurons become global oscillators which exhibit multiple synchronizing attractors.

Centric Models of the Orientation Map in Primary Visual Cortex William Baxter Department of Computer Science, S.U.N.Y. at Buffalo, NY 14620 Bruce Dow Department of Physiology, S.U.N.Y. at Buffalo, NY 14620 Abstract In the visual cortex of the monkey the horizontal organization of the preferred orientations of orientation-selective cells follows two opposing rules: 1) neighbors tend to have similar orientation preferences, and 2) many different orientations are observed in a local region. Several orientation models which satisfy these constraints are found to differ in the spacing and the topological index of their singularities. Using the rate of orientation change as a measure, the models are compared to published experimental results. Introduction It has been known for some years that there exist orientation-sensitive neurons in the visual cortex of cats and mOnkeysl,2. These cells react to highly specific patterns of light occurring in narrowly circumscribed regiOns of the visual field, i.e., the cell's receptive field. The best patterns for such cells are typically not diffuse levels of illumination, but elongated bars or edges oriented at specific angles.

Being able to record the electrical activities of a number of neurons simultaneously is likely to be important in the study of the functional organization of networks of real neurons. Using one extracellular microelectrode to record from several neurons is one approach to studying the response properties of sets of adjacent and therefore likely related neurons. However, to do this, it is necessary to correctly classify the signals generated by these different neurons. This paper considers this problem of classifying the signals in such an extracellular recording, based upon their shapes, and specifically considers the classification of signals in the case when spikes overlap temporally. Introduction How single neurons in a network of neurons interact when processing information is likely to be a fundamental question central to understanding how real neural networks compute.