We present a topographic product which measures the preservation of neighborhood relations as a criterion to optimize the output space topology of the map with regard to the global dimensionality DA as well as to the dimensions inthe individual directions. We test the topographic product method not only on synthetic mapping examples, but also on speech data.

Network vision systems must make inferences from evidential information acrosslevels of representational abstraction, from low level invariants, through intermediate scene segments, to high level behaviorally relevant object descriptions. This paper shows that such networks can be realized as Markov Random Fields (MRFs). We show first how to construct an MRF functionally equivalent to a Hough transform parameter network, thus establishing a principled probabilistic basis for visual networks. Second, weshow that these MRF parameter networks are more capable and flexible than traditional methods. In particular, they have a well-defined probabilistic interpretation, intrinsically incorporate feedback, and offer richer representations and decision capabilities.

The nature of information processing in complex dendritic trees has remained an open question since the origin of the neuron doctrine 100 years ago. With respect to learning, for example, it is not known whether a neuron is best modeled as 35 36 Mel a pseudo-linear unit, equivalent in power to a simple Perceptron, or as a general nonlinear learning device, equivalent in power to a multi-layered network. In an attempt tocharacterize the input-output behavior of a whole dendritic tree containing voltage-dependent membrane mechanisms, a recent compartmental modeling study in an anatomically reconstructed neocortical pyramidal cell (anatomical data from Douglas et al., 1991; "NEURON" simulation package provided by Michael Hines and John Moore) showed that a dendritic tree rich in NMDA-type synaptic channels isselectively responsive to spatially clustered, as opposed to diffuse, pattens of synaptic activation (Mel, 1992). For example, 100 synapses which were simultaneously activatedat 100 randomly chosen locations about the dendritic arbor were less effective at firing the cell than 100 synapses activated in groups of 5, at each of 20 randomly chosen dendritic locations. The cooperativity among the synapses in each group is due to the voltage dependence of the NMDA channel: Each activated NMDA synapse becomes up to three times more effective at injecting synaptic current whenthe post-synaptic membrane is locally depolarized by 30-40 m V from the resting potential.

Visual object recognition involves the identification of images of 3-D objects seenfrom arbitrary viewpoints. We suggest an approach to object recognition in which a view is represented as a collection of points given by their location in the image. An object is modeled by a set of 2-D views together with the correspondence between the views. We show that any novel view of the object can be expressed as a linear combination of the stored views. Consequently, we build a linear operator that distinguishes between views of a specific object and views of other objects.

Hand-printed digits can be modeled as splines that are governed by about 8 control points.

One way of simplifying neural networks so they generalize better is to add an extra t.erm

Standard pattern recognition systems usually involve a segmentation step prior to the recognition step. For example, it is very common in character recognition to segment characters in a pre-processing step then normalize the individual characters and pass them to a recognition engine such as a neural network, as in the work of LeCun et al. 1988, Martin and Pittman (1988). This separation between segmentation and recognition becomes unreliable if the characters are touching each other, touching bounding boxes, broken, or noisy. Other applications such as scene analysis or continuous speech recognition pose similar and more severe segmentation problems. The difficulties encountered in these applications present an apparent dilemma: one cannot recognize the patterns 496 *keeler@mcc.comReprint

The RBF network consists of an input layer, a hidden layer composed of Gaussian basis functions, and an output layer. Connections from the input layer to the hidden layer are fixed at unity while those from the hidden layer to the output layer are trained by minimizing the overall mean-square error between actual and desired output values. Each RBF output node has a corresponding state in a set of HMM word models which represent the words in the vocabulary. HMM word models are left-to-right with no skip states and have a one-state background noise model at either end. The background noise models are identical for all words.

Recently synchronization phenomena in neural networks have attracted considerable attention. Gray et al. (1989, 1990) as well as Eckhorn et al. (1988) provided electrophysiological evidence that neurons in the visual cortex of cats discharge in a semi-synchronous, oscillatory manner in the 40 Hz range and that the firing activity of neurons up to 10 mm away is phase-locked with a mean phase-shift of less than 3 msec. It has been proposed that this phase synchronization can solve the binding problem for figure-ground segregation (von der Malsburg and Schneider, 1986) and underly visual attention and awareness (Crick and Koch, 1990). A number of theoretical explanations based on coupled (relaxation) oscillator mod-117 118 Schuster and Koch els have been proposed for burst synchronization (Sompolinsky et al., 1990). The crucial issue of phase synchronization has also recently been addressed by Bush and Douglas (1991), who simulated the dynamics of a network consisting of bursty, layer V pyramidal cells coupled to a common pool of basket cells inhibiting all pyramidal cells.

It has been observed in numerical simulations that a weight decay can improve generalizationin a feed-forward neural network.