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84
From kuramoto to crawford: exploring the onset of synchronization in populations of coupled oscillators
 Phys. D
, 2000
"... The Kuramoto model describes a large population of coupled limitcycle oscillators whose natural frequencies are drawn from some prescribed distribution. If the coupling strength exceeds a certain threshold, the system exhibits a phase transition: some of the oscillators spontaneously synchronize, w ..."
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Cited by 141 (3 self)
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The Kuramoto model describes a large population of coupled limitcycle oscillators whose natural frequencies are drawn from some prescribed distribution. If the coupling strength exceeds a certain threshold, the system exhibits a phase transition: some of the oscillators spontaneously synchronize, while others remain incoherent. The mathematical analysis of this bifurcation has proved both problematic and fascinating. We review 25 years of research on the Kuramoto model, highlighting the false turns as well as the successes, but mainly following the trail leading from Kuramoto’s work to Crawford’s recent contributions. It is a lovely winding road, with excursions through mathematical biology, statistical physics, kinetic theory, bifurcation theory, and plasma physics. © 2000 Elsevier Science B.V. All rights reserved.
The twodimensional spatial structure of simple receptive fields in cat striate cortex
 Journal of Neurophysiology
, 1987
"... method is developed that allows quantitative determination of visual receptivefield structure in two spatial dimensions. This method is applied to simple cells in the cat striate cortex. 2. It is demonstrated that the reverse correlation method yields results with several desirable properties, incl ..."
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Cited by 86 (2 self)
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method is developed that allows quantitative determination of visual receptivefield structure in two spatial dimensions. This method is applied to simple cells in the cat striate cortex. 2. It is demonstrated that the reverse correlation method yields results with several desirable properties, including convergence and reproducibility independent of modest changes in stimulus parameters. 3. In contrast to results obtained with moving stimuli (26), we find that the bright and dark excitatory subregions in simple receptive fields do not overlap to any great extent. This difference in results may be attributed to confounding the independent variables space and time when using moving stimuli. 4. All simple receptive fields have subregions that vary smoothly in all directions in space. There are no sharp transitions either between excitatory subregions or between subregions and the area surrounding the receptive field. 5. Simple receptive fields vary both in the number of subregions observed, in the elongation of each subregion, and in the overall elongation of the field. In contrast with results obtained using moving stimuli (26), we find that subregions within a given receptive field need not be the same length. 6. The hypothesis that simple receptive fields can be modeled as either even symmetric or odd symmetric about a central axis is evaluated. This hypothesis is found to be false in general. Most simple receptive fields are neither even symmetric nor odd symmetric. 7. The hypothesis that simple receptive fields can be modeled as the product of a width response profile and an orthogonal length response profile (Cartesian separability) is evaluated. This hypothesis is found to be true for only50% sample.
Reduction of the HodgkinHuxley Equations to a SingleVariable Threshold Model
 NEURAL COMPUTATION
, 1997
"... It is generally believed that a neuron is a threshold element which fires when some variable u reaches a threshold. Here we pursue the question of whether this picture can be justified and study the fourdimensional neuron model of Hodgkin and Huxley as a concrete example. The model is approximat ..."
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Cited by 67 (22 self)
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It is generally believed that a neuron is a threshold element which fires when some variable u reaches a threshold. Here we pursue the question of whether this picture can be justified and study the fourdimensional neuron model of Hodgkin and Huxley as a concrete example. The model is approximated by a response kernel expansion in terms of a single variable, the membrane voltage. The firstorder term is linear in the input and has the typical form of an elementary postsynaptic potential. Higherorder kernels take care of nonlinear interactions between input spikes. In contrast to the standard Volterra expansion the kernels depend on the firing time of the most recent output spike. In particular, a zeroorder kernel which describes the shape of the spike and the typical afterpotential is included. Our model neuron fires, if the membrane voltage, given by the truncated response kernel expansion crosses a threshold. The threshold model is tested on a spike train generated by t...
On the Stability of the Kuramoto Model of Coupled Nonlinear Oscillators
, 2005
"... We provide an analysis of the classic Kuramoto model of coupled nonlinear oscillators that goes beyond the existing results for alltoall networks of identical oscillators. Our work is applicable to oscillator networks of arbitrary interconnection topology with uncertain natural frequencies. Using ..."
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Cited by 59 (8 self)
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We provide an analysis of the classic Kuramoto model of coupled nonlinear oscillators that goes beyond the existing results for alltoall networks of identical oscillators. Our work is applicable to oscillator networks of arbitrary interconnection topology with uncertain natural frequencies. Using tools from spectral graph theory and control theory, we prove that for couplings above a critical value, the synchronized state is locally asymptotically stable, resulting in convergence of all phase differences to a constant value, both in the case of identical natural frequencies as well as uncertain ones. We further explain the behavior of the system as the number of oscillators grows to infinity.
Generalized IntegrateandFire Models of Neuronal Activity Approximate Spike Trains of a . . .
"... We demonstrate that singlevariable integrateandfire models can quantitatively capture the dynamics of a physiologicallydetailed model for fastspiking cortical neurons. Through a systematic set of approximations, we reduce the conductance based model to two variants of integrateandfire mode ..."
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Cited by 58 (14 self)
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We demonstrate that singlevariable integrateandfire models can quantitatively capture the dynamics of a physiologicallydetailed model for fastspiking cortical neurons. Through a systematic set of approximations, we reduce the conductance based model to two variants of integrateandfire models. In the first variant (nonlinear integrateandfire model), parameters depend on the instantaneous membrane potential whereas in the second variant, they depend on the time elapsed since the last spike (Spike Response Model). The direct reduction links features of the simple models to biophysical features of the full conductance based model. To quantitatively
Local phase coherence and the perception of blur
 in Adv. Neural Information Processing Systems., 2004
"... Humans are able to detect blurring of visual images, but the mechanism by which they do so is not known. A traditional view is that a blurred image looks “unnatural ” because of the reduction in energy at high frequencies. We argue that the disruption of local phase is a more important factor for de ..."
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Cited by 22 (3 self)
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Humans are able to detect blurring of visual images, but the mechanism by which they do so is not known. A traditional view is that a blurred image looks “unnatural ” because of the reduction in energy at high frequencies. We argue that the disruption of local phase is a more important factor for detecting blur. We first demonstrate that a sharp image with its high frequency energy reduced but local phase preserved appears much sharper than a blurred image with its high frequency energy corrected but local phase uncorrected. We show that precisely localized features such as step edges result in strong local phase coherence structures across scale and space in the complex wavelet transform domain, and blurring causes loss of such phase coherence. We propose a technique for coarsetofine phase prediction of wavelet coefficients, and observe that (1) such predictions are highly effective in natural images, (2) phase coherence increases with the strength of image features, and (3) blurring disrupts the phase coherence relationship in images. We thus lay the groundwork for a new theory of perceptual blur estimation, as well as a variety of algorithms for restoration and manipulation of photographic images. 1
Online prediction of time series data with kernels
 IEEE TRANS. SIGNAL PROCESSING
, 2009
"... Kernelbased algorithms have been a topic of considerable interest in the machine learning community over the last ten years. Their attractiveness resides in their elegant treatment of nonlinear problems. They have been successfully applied to pattern recognition, regression and density estimation. ..."
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Cited by 20 (15 self)
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Kernelbased algorithms have been a topic of considerable interest in the machine learning community over the last ten years. Their attractiveness resides in their elegant treatment of nonlinear problems. They have been successfully applied to pattern recognition, regression and density estimation. A common characteristic of kernelbased methods is that they deal with kernel expansions whose number of terms equals the number of input data, making them unsuitable for online applications. Recently, several solutions have been proposed to circumvent this computational burden in time series prediction problems. Nevertheless, most of them require excessively elaborate and costly operations. In this paper, we investigate a new model reduction criterion that makes computationally demanding sparsification procedures unnecessary. The increase in the number of variables is controlled by the coherence parameter, a fundamental quantity that characterizes the behavior of dictionaries in sparse approximation problems. We incorporate the coherence criterion into a new kernelbased affine projection algorithm for time series prediction. We also derive the kernelbased normalized LMS algorithm as a particular case. Finally, experiments are conducted to compare our approach to existing methods.
Dynamic iv curves are reliable predictors of naturalistic pyramidalneuron voltage traces
 J Neurophysiol 99:656
, 2008
"... MJ. Dynamic IV curves are reliable predictors of naturalistic pyramidalneuron voltage traces. J Neurophysiol 99: 656–666, 2008. First published December 5, 2007; doi:10.1152/jn.01107.2007. Neuronal response properties are typically probed by intracellular measurements of currentvoltage (IV) rela ..."
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Cited by 19 (8 self)
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MJ. Dynamic IV curves are reliable predictors of naturalistic pyramidalneuron voltage traces. J Neurophysiol 99: 656–666, 2008. First published December 5, 2007; doi:10.1152/jn.01107.2007. Neuronal response properties are typically probed by intracellular measurements of currentvoltage (IV) relationships during application of current or voltage steps. Here we demonstrate the measurement of a novel IV curve measured while the neuron exhibits a fluctuating voltage and emits spikes. This dynamic IV curve requires only a few tens of seconds of experimental time and so lends itself readily to the rapid classification of cell type, quantification of heterogeneities in cell populations, and generation of reduced analytical models. We apply this technique to layer5 pyramidal cells and show that their dynamic IV curve comprises linear and exponential components, providing experimental evidence for a recently proposed theoretical model. The approach also allows us to determine the change of neuronal response properties after a spike, millisecond by millisecond, so that postspike refractoriness of pyramidal cells can be quantified. Observations of IV curves during and in absence of refractoriness are cast into a model that is used to predict both the subthreshold response and spiking activity of the neuron to novel stimuli. The predictions of the resulting model are in excellent agreement with experimental data and close to the intrinsic neuronal reproducibility to repeated stimuli.
Asymptotic spectral theory for nonlinear time series
, 2007
"... We consider asymptotic problems in spectral analysis of stationary causal processes. Limiting distributions of periodograms and smoothed periodogram spectral density estimates are obtained and applications to the spectral domain bootstrap are given. Instead of the commonly used strong mixing conditi ..."
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Cited by 15 (5 self)
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We consider asymptotic problems in spectral analysis of stationary causal processes. Limiting distributions of periodograms and smoothed periodogram spectral density estimates are obtained and applications to the spectral domain bootstrap are given. Instead of the commonly used strong mixing conditions, in our asymptotic spectral theory we impose conditions only involving (conditional) moments, which are easily verifiable for a variety of nonlinear time series.