An integral equation describing the time evolution of the population activity in a homogeneous pool of spiking neurons of the integrate-and-fire type is discussed. It is analytically shown that transients from a state of incoherent firing can be immediate. The stability of incoherent firing is analyzed in terms of the noise level and transmission delay and a bifurcation diagram is derived. The response of a population of noisy integrate-and-fire neurons to an input current of small amplitude is calculated and characterized by a linear filter L. The stability of perfectly synchronized `locked' solutions is analyzed.

Introduction In standard neural network theory, neurons are described in terms of mean firing rates. The analog input variable I is mapped via a nonlinear gain function g to an analog output variable = g(I) which may be interpreted as the mean firing rate. If the input consists of output rates j of other neurons weighted by a factor w ij , we arrive at the standard formula i = g( X j w ij j ) (10.1) which is the starting point of most neural network theories. As we have seen in Chapter 1, the firing rate defined by a temporal average over many spikes of a single neuron is a concept which works well if the input is constant or changes on a time scale which is slow with respect to the size of the temporal averaging window. Sensory inpu

How does the brain group together different parts of an object into a coherent visual object representation? Different parts of an object may be processed by the brain at different rates and may thus become desynchronized. Perceptual framing is a process that resynchronizes cortical activities corresponding to the same retinal object. A neural network model is presented that is able to rapidly resynchronize desynchronized neural activities. The model provides a link between perceptual and brain data. Model properties quantitatively simulate perceptual framing data, including psychophysical data about temporal order judgments and the reduction of threshold contrast as a function of stimulus length. Such a model has earlier been used to explain data about illusory contour formation, texture segregation, shape-from-shading, 3-D vision, and cortical receptive fields. The model hereby shows how many data may be understood as manifestations of a cortical grouping process that can rapidly res...

We present a cortical-based model for computing the perceptual salience of contours embedded in noisy images. It has been suggested (Gilbert, 1992; Field, Hayes & Hess, 1993) that horizontal intra-cortical connections in primary visual cortex may modulate contrast detection thresholds and pre-attentive "popout ". In our model, horizontal connections mediate context-dependent facilitatory and inhibitory interactions among oriented cells. Strongly facilitated cells undergo temporal synchronization; and perceptual salience is determined by the level of synchronized activity. The model accounts for a range of reported psychophysical and physiological effects of contour salience (Polat & Sagi, 1993, 1994; Kapadia, Ito, Gilbert & Westheimer, 1995; Field et al., 1993; Kovács, Polat & Norcia, 1996; Pettet, McKee & Grzywacz, 1996). In particular, the model proposes that intrinsic properties of synchronization account for the increased salience of smooth, closed contours (Kovács & Julesz, 1993, ...

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Auditory scene analysis is critical for complex auditory processing. We study auditory segregation from the neural network perspective, and develop a framework for primitive auditory scene analysis. The architecture is a laterally coupled two-dimensional network of relaxation oscillators with a global inhibitor. One dimension represents time and another one represents frequency. We show that this architecture, plus systematic delay lines, can in real time group auditory features into a stream by phase synchrony and segregate different streams by desynchronization. The network demonstrates a set of psychological phenomena regarding primitive auditory scene analysis, including dependency on frequency proximity and the rate of presentation, sequential capturing, and competition among different perceptual organizations. We offer a neurocomputational theory - shifting synchronization theory - for explaining how auditory segregation might be achieved in the brain, and the psychological pheno...

Due to many experimental reports of synchronous neural activity in the brain, there is much interest in understanding synchronization in networks of neural oscillators and its potential for computing perceptual organization. Contrary to Hopfield and Herz (1995), we find that networks of locally coupled integrate-and-fire oscillators can quickly synchronize. Furthermore, we examine the time needed to synchronize such networks. We observe that these networks synchronize at times proportional to the logarithm of their size, and we give the parameters used to control the rate of synchronization. Inspired by locally excitatory globally inhibitory oscillator network (LEGION) dynamics with relaxation oscillators (Terman & Wang, 1995), we find that global inhibition can play a similar role of desynchronization in a network of integrate-and-fire oscillators. We illustrate that a LEGION architecture with integrate-and-fire oscillators can be similarly used to address image analysis.

A relaxation oscillator model of neural spiking dynamics is applied to the task of finding downbeats in rhythmical patterns. The importance of downbeat discovery or beat induction is discussed, and the relaxation oscillator model is compared to other oscillator models. In a set of computer simulations the model is tested on 35 rhythmical patterns from Povel and Essens (1985). The model performs well, making good predictions in 34 of 35 cases. In an analysis we identify some shortcomings of the model and relate model behavior to dynamical properties of relaxation oscillators.

We present a recurrent neural network for feature binding and sensory segmentation, the competitive layer model (CLM). The CLM uses topographically structured competitive and cooperative interactions in a layered network to partition a set of input features into salient groups. The dynamics is formulated within a standard additive recurrent network with linear threshold neurons. Contextual relations among features are coded by pairwise compatibilities which define an energy function to be minimized by the neural dynamics. Due to the usage of dynamical winner-take-all circuits the model gains more flexible response properties than spin models of segmentation by exploiting amplitude information in the grouping process. We prove analytic results on the convergence and stable attractors of the CLM, which generalize earlier results on winner-take-all networks, and incorporate deterministic annealing for robustness against local minima. The piecewise linear dynamics of the CLM allows a linear eigensubspace analysis which we use to analyze the dynamics of binding in conjunction with annealing. For the example of contour detection we show how the CLM can integrate figure-ground segmentation and grouping into a unified model.

Inhibition in oscillatory networks of neurons can have apparently paradoxical e ects, sometimes creating dispersion of phases, sometimes fostering synchrony in the network. We analyze a pair of biophysically modeled neurons and show how the rates of onset and decay of inhibition interact with the time scales of the intrinsic oscillators to determine when stable synchrony is possible. We show that there are two di erent regimes in parameter space in which di erent combinations of the time constants and other parameters regulate whether the synchronous state is stable. We also discuss the construction and stability of non-synchronous solutions, and the implications of the analysis for larger networks. The analysis uses geometric techniques of singular perturbation theory that allow one to combine estimates from slow ows and fast jumps. Key words: oscillations, inhibition, synchronization