The most prominent feature of this network is the pair of large Mauthner cells whose axons cross the midline and descend down the spinal cord to synapse on primary motoneurons. The Mauthner system also includes inhibitory neurons, the PHP cells, which have a unique and intense field effect inhibition at the spikeinitiating zone of the Mauthner cells (Faber and Korn, 1978). The Mauthner system is part of the full brainstem escape network which also includes two pairs of cells homologous to the Mauthner cell and other populations of reticulospinal neurons. With this network fish initiate escapes only from appropriate stimuli, turn away from the offending stimulus, and do so very rapidly with a latency around 15 msec in goldfish. The Mauthner cells play an important role in these functions.

Biological neurons have a variety of intrinsic properties because of the large number of voltage dependent currents that control their activity. Neuromodulatory substances modify both the balance of conductances that determine intrinsic properties and the strength of synapses. These mechanisms alter circuit dynamics, and suggest that functional circuits exist only in the modulatory environment in which they operate. 1 INTRODUCTION Many studies of artificial neural networks employ model neurons and synapses that are considerably simpler than their biological counterparts.

In many cases the crosscorrelations between the activities of cortical neurons are approximately symmetric about zero time delay. These have been taken as an indication of the presence of "functional connectivity" between the correlated neurons (Fetz, Toyama and Smith 1991, Abeles 1991). However, a quantitative comparison between the observed cross-correlations and those expected to exist between neurons that are part of a large assembly of interacting population has been lacking. Most of the theoretical studies of recurrent neural network models consider only time averaged firing rates, which are usually given as solutions of mean-field equations. They do not account for the fluctuations about these averages, the study of which requires going beyond the mean-field approximations. In this work we perform a theoretical study of the fluctuations in the neuronal activities and their correlations, in a large stochastic network of excitatory and inhibitory neurons. Depending on the model parameters, this system can exhibit coherent undamped oscillations. Here we focus on parameter regimes where the system is in a statistically stationary state, which is more appropriate for modeling non oscillatory neuronal activity in cortex. Our results for the magnitudes and the time-dependence of the correlation functions can provide a basis for comparison with physiological data on neuronal correlation functions.

We prove that except possibly for small exceptional sets, discretetime analog neural nets are globally observable, i.e. all their corrupted pseudo-orbits on computer simulations actually reflect the true dynamical behavior of the network. Locally finite discrete (boolean) neural networks are observable without exception.

We describe the use of smoothing spline analysis of variance (SS ANOVA) in the penalized log likelihood context, for learning (estimating) the probability p of a '1' outcome, given a training set with attribute vectors and outcomes.

We study feed-forward nets with arbitrarily many layers, using the standard sigmoid, tanh x. Aside from technicalities, our theorems are: 1. Complete knowledge of the output of a neural net for arbitrary inputs uniquely specifies the architecture, weights and thresholds; and 2. There are only finitely many critical points on the error surface for a generic training problem. Neural nets were originally introduced as highly simplified models of the nervous system. Today they are widely used in technology and studied theoretically by scientists from several disciplines. However, they remain little understood.

We show how an "Elman" network architecture, constructed from recurrently connected oscillatory associative memory network modules, can employ selective "attentional" control of synchronization to direct the flow of communication and computation within the architecture to solve a grammatical inference problem. Previously we have shown how the discrete time "Elman" network algorithm can be implemented in a network completely described by continuous ordinary differential equations. The time steps (machine cycles) of the system are implemented by rhythmic variation (clocking) of a bifurcation parameter. In this architecture, oscillation amplitude codes the information content or activity of a module (unit), whereas phase and frequency are used to "softwire" the network. Only synchronized modules communicate by exchanging amplitude information; the activity of non-resonating modules contributes incoherent crosstalk noise. Attentional control is modeled as a special subset of the hidden modules with ouputs which affect the resonant frequencies of other hidden modules. They control synchrony among the other modules and direct the flow of computation (attention) to effect transitions between two subgraphs of a thirteen state automaton which the system emulates to generate a Reber grammar. The internal crosstalk noise is used to drive the required random transitions of the automaton.

Four versions of a k-nearest neighbor algorithm with locally adaptive k are introduced and compared to the basic k-nearest neighbor algorithm (kNN). Locally adaptive kNN algorithms choose the value of k that should be used to classify a query by consulting the results of cross-validation computations in the local neighborhood of the query. Local kNN methods are shown to perform similar to kNN in experiments with twelve commonly used data sets. Encouraging results in three constructed tasks show that local methods can significantly outperform kNN in specific applications. Local methods can be recommended for online learning and for applications where different regions of the input space are covered by patterns solving different sub-tasks.

The distance measure and learning problem are formally described as nested objective functions. We derive an efficient algorithm by using optimization techniques that allow us to divide up the objective function into parts which may be minimized in distinct phases. The algorithm has accurately recreated 10 prototypes from a randomly generated sample database of 100 images consisting of 20 points each in 120 experiments. Finally, by incorporating permutation invariance in our distance measure, we have a technique that we may be able to apply to the clustering of graphs. Our goal is to develop measures which will enable the learning of objects with shape or structure. Acknowledgements This work has been supported by AFOSR grant F49620-92-J-0465 and ONR/DARPA grant N00014-92-J-4048.

We show how an "Elman" network architecture, constructed from recurrently connected oscillatory associative memory network modules, can employ selective "attentional" control of synchronization to direct the flow of communication and computation within the architecture to solve a grammatical inference problem. Previously we have shown how the discrete time "Elman" network algorithm can be implemented in a network completely described by continuous ordinary differential equations. The time steps (machine cycles) of the system are implemented by rhythmic variation (clocking) of a bifurcation parameter. In this architecture, oscillation amplitude codes the information content or activity of a module (unit), whereas phase and frequency are used to "softwire" the network. Only synchronized modules communicate by exchanging amplitude information; the activity of non-resonating modules contributes incoherent crosstalk noise. Attentional control is modeled as a special subset of the hidden modules with ouputs which affect the resonant frequencies of other hidden modules. They control synchrony among the other modules and direct the flow of computation (attention) to effect transitions between two subgraphs of a thirteen state automaton which the system emulates to generate a Reber grammar. The internal crosstalk noise is used to drive the required random transitions of the automaton.