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.

We have presented a locally interconnected network which minimizes a function that is analogous to the log likelihood function near the global minimum. The results of simulations demonstrate that the network can successfully decode input sequences containing no noise at least as well as the globally connected Hopfield-Tank [6] decomposition network.Simulations also strongly support the conjecture that in the noiseless case, the network can be guaranteed to converge to the global minimum. In addition, for low error rates, the network can also decode noisy received sequences. We have been able to apply the Cohen-Grossberg proof of the stability of "oncenter off-surround"networks to show that each stage will maximize the desired local "likelihood" for noisy received sequences. We have also shown that, in the large gain limit, the network as a whole is stable and that the equilibrium points correspond to the MLSE decoder output.

One criticism of back-propagation is that it requires a teacher to specify the desired output vectors. It is possible to dispense with the teacher in the case of "encoder" networks2 in which the desired output vector is identical with the input vector (see Figure 1). The purpose of an encoder network is to learn good "codes" in the intermediate, hidden units. If for, example, there are less hidden units than input units, an encoder network will perform data-compression3.

W. Scott Stornetta Stanford University, Physics Department, Stanford, Ca., 94305 Tad Hogg and B. A. Huberman Xerox Palo Alto Research Center, Palo Alto, Ca. 94304 ABSTRACT Recognizing patterns with temporal context is important for such tasks as speech recognition, motion detection and signature verification. We propose an architecture in which time serves as its own representation, and temporal context is encoded in the state of the nodes. We contrast this with the approach of replicating portions of the architecture to represent time. As one example of these ideas, we demonstrate an architecture with capacitive inputs serving as temporal feature detectors in an otherwise standard back propagation model. Experiments involving motion detection and word discrimination serve to illustrate novel features of the system.

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 retrievalprocess some subset of the units in the system are activated, and these units in turn activate neighboring units via the inter-unit connection strengths.

Instead, a signal space approach is used to gain new insights via ease of analysis and geometrical interpretation.