Ascribing computational principles to neural feedback circuits is an important problem in theoretical neuroscience. We study symmetric threshold-linearnetworks and derive stability results that go beyond the insights that can be gained from Lyapunov theory or energy functions. By applying linear analysis to subnetworks composed ofcoactive neurons, we determine the stability of potential steady states. We find that stability depends on two types of eigenmodes. Onetype determines global stability and the other type determines whether or not multistability is possible.

We present an improvement of Novikoff's perceptron convergence theorem. Reinterpreting this mistake bound as a margin dependent sparsity guarantee allows us to give a PACstyle generalisation error boundfor the classifier learned by the perceptron learning algorithm. Thebound value crucially depends on the margin a support vector machine would achieve on the same data set using the same kernel. Ironically, the bound yields better guarantees than are currently availablefor the support vector solution itself. 1 Introduction In the last few years there has been a large controversy about the significance of the attained margin, i.e. the smallest real valued output of a classifiers before thresholding, as an indicator of generalisation performance. Results in the YC, PAC and luckiness frameworks seem to indicate that a large margin is a prerequisite for small generalisation error bounds (see [14, 12]). These results caused many researchers to focus on large margin methods such as the well known support vector machine (SYM).

Hebbian and competitive Hebbian algorithms are almost ubiquitous in modeling pattern formation in cortical development. We analyse in theoretical detaila particular model (adapted from Piepenbrock & Obermayer, 1999) for the development of Id stripe-like patterns, which places competitive and interactive cortical influences, and free and restricted initial arborisationonto a common footing. 1 Introduction Cats, many species of monkeys, and humans exibit ocular dominance stripes, which are alternating areas of primary visual cortex devoted to input from (the thalamic relay associated with)just one or the other eye (see Erwin et aI, 1995; Miller, 1996; Swindale, 1996 for reviews of theory and data). These well-known fingerprint patterns have been a seductive targetfor models of cortical pattern formation because of the mix of competition and cooperation they suggest. A wealth of synaptic adaptation algorithms has been suggested to account for them (and also the concomitant refinement of the topography of the map between the eyes and the cortex), many of which are based on forms of Hebbian learning. Critical issues for the models are the degree of correlation between inputs from the eyes, the nature of the initial arborisation of the axonal inputs, the degree and form of cortical competition, and the nature of synaptic saturation (preventing weights from changing sign or getting too large) and normalisation (allowing cortical and/or thalamic cells to support only a certain total synaptic weight).

A stable learner is one for which the learned solution does not change much with small changes in the training set. The bounds we obtain do not depend on any measure of the complexity of the hypothesis space (e.g. VC dimension) but rather depend on how the learning algorithm searches this space, and can thus be applied even when the VC dimension is infinite. We demonstrate that regularization networks possess the required stability property and apply our method to obtain new bounds on their generalization performance.

The resulting combinatorial problem - finding the best agreement half-space over an input sample - is NP hard to approximate to within some constant factor. We suggest a way to circumvent this theoretical bound by introducing a new measure of success for such algorithms. An algorithm is ILmargin successful if the agreement ratio of the half-space it outputs is as good as that of any half-space once training points that are inside the ILmargins of its separating hyper-plane are disregarded. We prove crisp computational complexity resultswith respect to this success measure: On one hand, for every positive IL, there exist efficient (poly-time) ILmargin successful learningalgorithms. On the other hand, we prove that unless P NP, there is no algorithm that runs in time polynomial in the sample size and in 1/IL that is ILmargin successful for all IL O. 1 Introduction We consider the computational complexity of learning linear perceptrons for arbitrary (Le.non -separable) data sets.

Experimental data have shown that synapses are heterogeneous: different synapses respond with different sequences of amplitudes of postsynaptic responses to the same spike train. Neither the role of synaptic dynamics itself nor the role of the heterogeneity of synaptic dynamics for computations inneural circuits is well understood. We present in this article methods that make it feasible to compute for a given synapse with known synaptic parameters the spike train that is optimally fitted to the synapse, for example in the sense that it produces the largest sum of postsynaptic responses.To our surprise we find that most of these optimally fitted spike trains match common firing patterns of specific types of neurons that are discussed in the literature. 1 Introduction A large number of experimental studies have shown that biological synapses have an inherent dynamics,which controls how the pattern of amplitudes of postsynaptic responses depends on the temporal pattern of the incoming spike train. Various quantitative models have been proposed involving a small number of characteristic parameters, that allow us to predict the response of a given synapse to a given spike train once proper values for these characteristic synaptic parameters have been found. The analysis of this article is based on the model of [1], where three parameters U, F, D control the dynamics of a synapse and a fourth parameter A - which corresponds to the synaptic "weight" in static synapse models - scales the absolute sizes of the postsynaptic responses. The resulting model predicts theamplitude Ak for the kth spike in a spike train with interspike intervals (lSI's) .60

We present a computational model of the neural mechanisms in the parietal andtemporal lobes that support spatial navigation, recall of scenes and imagery of the products of recall. Long term representations are stored in the hippocampus, and are associated with local spatial and object-related features in the parahippocampal region. Viewer-centered representations are dynamically generated from long term memory in the parietal part of the model. The model thereby simulates recall and imagery oflocations and objects in complex environments. After parietal damage, the model exhibits hemispatial neglect in mental imagery that rotates with the imagined perspective of the observer, as in the famous Milan Square experiment [1]. Our model makes novel predictions for the neural representations in the parahippocampal and parietal regions and for behavior in healthy volunteers and neuropsychological patients.

Neurons in area V4 have relatively large receptive fields (RFs), so multiple visualfeatures are simultaneously "seen" by these cells. Recordings from single V4 neurons suggest that simultaneously presented stimuli compete to set the output firing rate, and that attention acts to isolate individual features by biasing the competition in favor of the attended object. We propose that both stimulus competition and attentional biasing arisefrom the spatial segregation of afferent synapses onto different regions of the excitable dendritic tree of V4 neurons. The pattern of feedforward, stimulus-driveninputs follows from a Hebbian rule: excitatory afferents with similar RFs tend to group together on the dendritic tree, avoiding randomly located inhibitory inputs with similar RFs. The same principle guides the formation of inputs that mediate attentional modulation.

We implement this description for the Hodgkin Huxley model, identify the most relevant dimensions and find the nonlinearity. A two dimensional description already captures a significant fraction of the information that spikes carry about dynamic inputs.This description also shows that computation in the Hodgkin-Huxley model is more complex than a simple integrateand-fire orperceptron model. 1 Introduction Classical neural network models approximate neurons as devices that sum their inputs and generate a nonzero output if the sum exceeds a threshold. From our current state of knowledge in neurobiology it is easy to criticize these models as oversimplified: whereis the complex geometry of neurons, or the many different kinds of ion channel, each with its own intricate multistate kinetics? Indeed, progress at this more microscopic level of description has led us to the point where we can write (almost) exact models for the electrical dynamics of neurons, at least on short time scales. These nearly exact models are complicated by any measure, including tens if not hundreds of differential equations to describe the states of different channels in different spatial compartments of the cell. Faced with this detailed microscopic description, we need to answer a question which goes well beyond the biological context: given a continuous dynamical system, what does it compute? Our goal in this paper is to make this question about what a neuron computes somewhat moreprecise, and then to explore what we take to be the simplest example, namely the Hodgkin-Huxley model [1],[2] (and refs therein).

Theories of object recognition often assume that only one representation schemeis used within one visual-processing pathway. Versatility of the visual system comes from having multiple visual-processing pathways, each specialized in a different category of objects. We propose a theoretically simpler alternative, capable of explaining the same set of data and more. A single primary visual-processing pathway, loosely modular, is assumed. Memory modules are attached to sites along this pathway.