Fernando J. Pineda Applied Physics Laboratory, Johns Hopkins University Johns Hopkins Rd., Laurel MD 20707 Abstract A general method for deriving backpropagation algorithms for networks with recurrent and higher order networks is introduced. The propagation of activation in these networks is determined by dissipative differential equations. The error signal is backpropagated by integrating an associated differential equation. The method is introduced by applying it to the recurrent generalization of the feedforward backpropagation network. The method is extended to the case of higher order networks and to a constrained dynamical system for training a content addressable memory. The essential feature of the adaptive algorithms is that adaptive equation has a simple outer product form.

Please address all further correspondence to: John Y. Cheung School of EECS 202 W. Boyd, CEC 219 Norman, OK 73019 (405)325-4721 November,1987 American Institute of Physics 1988 165 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.

The nets are also described as parallel distributed systems!, connectionist models2

An input vector in the feature space is transformed into an internal representation which is a codeword in the code space, and then error correction decoded in this space to classify the input feature vector to its class. Two classes of codes which give high performance are the Hadamard matrix code and the maximal length sequence code. We show that the number of classes stored in an N-neuron system is linear in N and significantly more than that obtainable by using the Hopfield type memory as a classifier. I. INTRODUCTION Associative recall using neural networks has recently received a great deal of attention. Hopfield in his papers [1,2) deSCribes a mechanism which iterates through a feedback loop and stabilizes at the memory element that is nearest the input, provided that not many memory vectors are stored in the machine. He has also shown that the number of memories that can be stored in an N-neuron system is about O.15N for N between 30 and 100. McEliece et al. in their work (3) showed that for synchronous operation of the Hopfield memory about N/(2IogN) data vectors can be stored reliably when N is large. Abu-Mostafa (4) has predicted that the upper bound for the number of data vectors in an N-neuron Hopfield machine is N. We believe that one should be able to devise a machine with M, the number of data vectors, linear in N and larger than the O.15N achieved by the Hopfield method.

Less work has been performed on using neural networks to process floating point numbers and it is sometimes stated that neural networks are somehow inherently inaccurate andtherefore best suited for "fuzzy" qualitative reasoning. Nevertheless, the potential speed of massively parallel operations make neural net "number crunching" an interesting topic to explore. In this paper we discuss some of our work in which we demonstrate that for certain applications neural networks can achieve significantly higher numerical accuracy than more conventional techniques. Inparticular, prediction of future values of a chaotic time series can be performed with exceptionally high accuracy. We analyze how a neural net is able to do this, and in the process show that a large class of functions from Rn. Rffl may be accurately approximated by a backpropagation neural net with just two "hidden" layers. The network uses this functional approximation to perform either interpolation (signal processing applications) or extrapolation (symbol processing applicationsJ. Neural nets therefore use quite familiar methods toperform.

TheSIMD model of parallel computation is chosen, because systems of this type can be built with large numbers of processing elements. However, such systems are not naturally suited to generalized communication. A method is proposed that allows an implementation of neural network connections on massively parallel SIMD architectures. The key to this system is an algorithm that allows the formation of arbitrary connections between the "neurons". A feature is the ability to add new connections quickly.

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 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 theoremsfor 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. Wediscuss an allied problem-the complexity of learning-and close with some open problems and a discussion of the observed limitations of the theoretical approach. Wenext turn to a rigourous classification of how much a network of given structure can do; i.e., the computational capacity of a given construct.

Current knowledge about the activity dependence of the firing threshold, the conditions required for conduction failure, and the similarity of nodes along a single axon will be reviewed. An electronic circuit model for a site of low conduction safety in an axon will be presented. In response to single frequency stimulation the electronic circuit acts as a lowpass filter. I. INTRODUCTION The axon is often modeled as a wire which imposes a fixed delay on a propagating signal. Using this model, neural information processing is performed by synaptically sum m ing weighted contributions of the outputs from other neurons.

Asa means of probing these cerebellar mechanisms, my colleagues and I have been conducting microelectrode studies of the neural messages that flow through the intermediate divisionof the cerebellum and onward to limb muscles via the rubrospinal tract. We regard this cerebellorubrospinal pathwayas a useful model system for studying general problems of sensorimotor integration and adaptive brain function.