Baum and Haussler [4] used these results to give sample size bounds for multi-layer threshold networks Generalization and the Size of the Weights in Neural Networks 135 that grow at least as quickly as the number of weights (see also [7]). However, for pattern classification applications the VC-bounds seem loose; neural networks often perform successfully with training sets that are considerably smaller than the number of weights. This paper shows that for classification problems on which neural networks perform well, if the weights are not too big, the size of the weights determines the generalization performance. In contrast with the function classes and algorithms considered in the VC-theory, neural networks used for binary classification problems have real-valued outputs, and learning algorithms typically attempt to minimize the squared error of the network output over a training set. As well as encouraging the correct classification, this tends to push the output away from zero and towards the target values of { -1, I}.

The parameter space of neural networks has a Riemannian metric structure. The natural Riemannian gradient should be used instead of the conventional gradient, since the former denotes the true steepest descent direction of a loss function in the Riemannian space. The behavior of the stochastic gradient learning algorithm is much more effective if the natural gradient is used. The present paper studies the information-geometrical structure of perceptrons and other networks, and prove that the online learning method based on the natural gradient is asymptotically as efficient as the optimal batch algorithm. Adaptive modification of the learning constant is proposed and analyzed in terms of the Riemannian measure and is shown to be efficient.

A general feature of the cerebral cortex is its massive interconnectivity - it has been estimated anatomically [19] that cortical neurons receive upwards of 5,000 synapses, the majority of which originate from other nearby cortical neurons. Numerous experiments in primary visual cortex (VI) have revealed strongly nonlinear interactions between stimulus elements which activate classical and nonclassical receptive field regions. Recurrent cortical connections likely contribute substantially to these effects. However, most theories of visual processing have either assumed a feedforward processing scheme [7], or have used recurrent interactions to account for isolated effects only [1, 16, 18]. Since nonlinear systems cannot in general be taken apart and analyzed in pieces, it is not clear what one learns by building a recurrent model that only accounts for one, or very few phenomena. Here we develop a relatively simple model of recurrent interactions in VI, that reflects major anatomical and physiological features of intracortical connectivity, and simultaneously accounts for a wide range of phenomena observed physiologically. All phenomena we address are strongly nonlinear, and cannot be explained by linear feedforward models.

Recently, there has been a vigorous debate concerning the nature of neural coding (Rieke et al. 1996; Stevens and Zador 1995; Shadlen and Newsome 1994). The prevailing view has been that the mean firing rate conveys all information about the sensory stimulus in a spike train and the precise timing of the individual spikes is noise. This belief is, in part, based on a lack of correlation between the precise timing of the spikes and the sensory qualities of the stimulus under study, particularly, on a lack of spike timing repeatability when identical stimulation is delivered. This view has been challenged by a number of recent studies, in which highly repeatable temporal patterns of spikes can be observed both in vivo (Bair and Koch 1996; Abeles et al. 1993) and in vitro (Mainen and Sejnowski 1994). Furthermore, application of information theory to the coding problem in the frog and house fly (Bialek et al. 1991; Bialek and Rieke 1992) suggested that additional information could be extracted from spike timing. In the absence of direct evidence for a timing code in the cerebral cortex, the role of spike timing in neural coding remains controversial.

A linear architectural model of cortical simple cells is presented. The model evidences how mutual inhibition, occurring through synaptic coupling functions asymmetrically distributed in space, can be a possible basis for a wide variety of spatiotemporal simple cell response properties, including direction selectivity and velocity tuning. While spatial asymmetries are included explicitly in the structure of the inhibitory interconnections, temporal asymmetries originate from the specific mutual inhibition scheme considered. Extensive simulations supporting the model are reported.

Coarse codes are widely used throughout the brain to encode sensory and motor variables. Methods designed to interpret these codes, such as population vector analysis, are either inefficient, i.e., the variance of the estimate is much larger than the smallest possible variance, or biologically implausible, like maximum likelihood. Moreover, these methods attempt to compute a scalar or vector estimate of the encoded variable. Neurons are faced with a similar estimation problem. They must read out the responses of the presynaptic neurons, but, by contrast, they typically encode the variable with a further population code rather than as a scalar. We show how a nonlinear recurrent network can be used to perform these estimation in an optimal way while keeping the estimate in a coarse code format. This work suggests that lateral connections in the cortex may be involved in cleaning up uncorrelated noise among neurons representing similar variables.

Biophysical modeling studies have previously shown that cortical pyramidal cells driven by strong NMDA-type synaptic currents and/or containing dendritic voltage-dependent Ca or Na channels, respond more strongly when synapses are activated in several spatially clustered groups of optimal size-in comparison to the same number of synapses activated diffusely about the dendritic arbor [8]- The nonlinear intradendritic interactions giving rise to this "cluster sensitivity" property are akin to a layer of virtual nonlinear "hidden units" in the dendrites, with implications for the cellular basis of learning and memory [7, 6], and for certain classes of nonlinear sensory processing [8]- In the present study, we show that a single neuron, with access only to excitatory inputs from unoriented ONand OFFcenter cells in the LGN, exhibits the principal nonlinear response properties of a "complex" cell in primary visual cortex, namely orientation tuning coupled with translation invariance and contrast insensitivity_ We conjecture that this type of intradendritic processing could explain how complex cell responses can persist in the absence of oriented simple cell input [13]- 84 B. W. Mel, D. L. Ruderman and K. A. Archie

Various experiments have suggested that simple input competition cannot be the whole story.

The weakly electric fish, Eigenmannia, generates a quasi sinusoidal, dipole-like electric field at individually fixed frequencies (250 - 600 Hz) by discharging an electric organ located in its tail (see Bullock and Heilgenberg, 1986 for reviews). The fish sense local changes in the electric field by means of two types of tuberous electroreceptors located on the body surface.

This paper develops arguments for a family of temporal log-linear models to represent spatiotemporal correlations among the spiking events in a group of neurons. The models can represent not just pairwise correlations but also correlations of higher order. Methods are discussed for inferring the existence or absence of correlations and estimating their strength. A frequentist and a Bayesian approach to correlation detection are compared.