Boundary value problems for the nonlinear Schrödinger equation on the half line in laboratory coo... more Boundary value problems for the nonlinear Schrödinger equation on the half line in laboratory coordinates are considered. A class of boundary conditions that lead to linearizable problems is identified by introducing appropriate extensions to initial-value problems on the infinite line, either explicitly or by constructing a suitable Bäcklund transformation. Various soliton solutions are explicitly constructed and studied.
We study the synchronization of a stochastically-driven, current-based, integrate-and-fire neuron... more We study the synchronization of a stochastically-driven, current-based, integrate-and-fire neuronal model on a preferential-attachment network with scale-free characteristics and high clustering. The synchrony is induced by cascading total firing events where every neuron in the network fires at the same instant of time. We show that in the regime where the system remains in this highly synchronous state, the firing rate of the network is completely independent of the synaptic coupling, and depends solely on the external drive. On the other hand, the ability for the network to maintain synchrony depends on a balance between the fluctuations of the external input and the synaptic coupling strength. In order to accurately predict the probability of repeated cascading total firing events we go beyond mean-field and tree-like approximations and conduct a detailed second order calculation taking into account local clustering. Our explicit analytical results are shown to give excellent agreement with direct numerical simulations for the particular preferential-attachment network model investigated.
For a class of integrate-and-fire, pulse-coupled networks with complex topology, we study the dep... more For a class of integrate-and-fire, pulse-coupled networks with complex topology, we study the dependence of the pulse rate on the underlying architectural connectivity statistics. We derive the distribution of the pulse rate from this dependence and determine when the underlying scale-free architectural connectivity gives rise to a scale-free pulse-rate distribution. We identify the scaling of the pairwise coupling between the dynamical units in this network class that keeps their pulse rates bounded in the infinite-network limit. In the process, we determine the connectivity statistics for a specific scale-free network grown by preferential attachment.
We propose a Boussinesq-type model to study the surface/interfacial wave manifestation of an unde... more We propose a Boussinesq-type model to study the surface/interfacial wave manifestation of an underlying, slowly-varying, long-wavelength, baroclinic flow in a two-layer, density-stratified system. The results of our model show numerically that, under strong nonlinearity, surface waves, with their typical wavenumber being the resonant k res , can be generated locally at the leading edge of the underlying slowly-varying, long-wavelength baroclinic flow. Here, the resonant k res satisfies the class 3 triad resonance condition among two short-mode waves and one long-mode wave in which all waves propagate in the same direction. Moreover, when the slope of the baroclinic flow is sufficiently small, only one spatially-localized large-amplitude surface wave packet can be generated at the leading edge. This localized surface wave packet becomes high in amplitude and large in group velocity after the interaction with its surrounding waves. These results are qualitatively consistent with various experimental observations including resonant surface waves at the leading edge of an internal wave. Subsequently, we propose a mechanism, referred to as the modulation-resonance mechanism, underlying these surface phenomena, based on our numerical simulations. The proposed modulation-resonance mechanism combines the linear modulation (ray-based) theory for the spatiotemporal asymmetric behavior of surface waves and the nonlinear class 3 triad resonance theory for the energy focusing of surface waves around the resonant wavenumber k res in Fourier space.
Journal of Dynamics and Differential Equations, Oct 1, 1993
This paper describes a new type of orbits homoclinic to resonance bands in a class of near-integr... more This paper describes a new type of orbits homoclinic to resonance bands in a class of near-integrable Hamiltonian systems. It presents a constructive method for establishing whether small conservative perturbations of a family of heteroclinic orbits that connect pairs of points on a circle of equilibria will yield transverse homoclinic connections between periodic orbits in the resonance band resulting from the perturbation. In any given example, this method may be used to prove the existence of such transverse homoclinic orbits, as well as to determine their precise shape, their asymptotic behavior, and their possible bifurcations. The method is a combination of the Melnikov method and geometric singular perturbation theory for ordinary differential equations.
Integrate-and-fire (I&F) neuronal networks are ubiquitous in diverse image processing application... more Integrate-and-fire (I&F) neuronal networks are ubiquitous in diverse image processing applications, including image segmentation and visual perception. While conventional I&F network image processing requires the number of nodes composing the network to be equal to the number of image pixels driving the network, we determine whether I&F dynamics can accurately transmit image information when there are significantly fewer nodes than network input-signal components. Although compressive sensing (CS) theory facilitates the recovery of images using very few samples through linear signal processing, it does not address whether similar signal recovery techniques facilitate reconstructions through measurement of the nonlinear dynamics of an I&F network. In this paper, we present a new framework for recovering sparse inputs of nonlinear neuronal networks via compressive sensing. By recovering both one-dimensional inputs and two-dimensional images, resembling natural stimuli, we demonstrate that input information can be well-preserved through nonlinear I&F network dynamics even when the number of network-output measurements is significantly smaller than the number of input-signal components. This work suggests an important exten- *
Compressive sensing (CS) theory demonstrates that by using uniformly-random sampling, rather than... more Compressive sensing (CS) theory demonstrates that by using uniformly-random sampling, rather than uniformly-spaced sampling, higher quality image reconstructions are often achievable. Considering that the structure of sampling protocols has such a profound impact on the quality of image reconstructions, we formulate a new sampling scheme motivated by physiological receptive field structure, localized random sampling, which yields significantly improved CS image reconstructions. For each set of localized image measurements, our sampling method first randomly selects an image pixel and then measures its nearby pixels with probability depending on their distance from the initially selected pixel. We compare the uniformly-random and localized random sampling methods over a large space of sampling parameters, and show that, for the optimal parameter choices, higher quality image reconstructions can be consistently obtained by using localized random sampling. In addition, we argue that the localized random CS optimal parameter choice is stable with respect to diverse natural images, and scales with the number of samples used for reconstruction. We expect that the localized random sampling protocol helps to explain the evolutionarily advantageous nature of receptive field structure in visual systems and suggests several future research areas in CS theory and its application to brain imaging. Sampling protocols have drastically changed with the discovery of compressive sensing (CS) data acquisition and signal recovery 1,2 . Prior to the development of CS theory, the Shannon-Nyquist theorem determined the majority of sampling procedures for both audio signals and images, dictating the minimum rate, the Nyquist rate, with which a signal must be uniformly sampled to guarantee successful reconstruction 3 . Since the theorem specifically addresses minimal sampling rates corresponding to uniformly-spaced measurements, signals were typically sampled at equally-spaced intervals in space or time before the discovery of CS. However, using CS-type data acquisition, it is possible to reconstruct a broad class of sparse signals, containing a small number of dominant components in some domain, by employing a sub-Nyquist sampling rate 2 . Instead of applying uniformly-spaced signal measurements, CS theory demonstrates that several types of uniformly-random sampling protocols will yield successful reconstructions with high probability 4-6 . While CS signal recovery is relatively accurate for sufficiently high sampling rates, we demonstrate that, for the recovery of natural scenes, reconstruction quality can be further improved via localized random sampling. In this new protocol, each signal sample consists of a randomly centered local cluster of measurements, in which the probability of measuring a given pixel decreases with its distance from the cluster center. We show that the localized random sampling protocol consistently produces more accurate CS reconstructions of natural scenes than the uniformly-random sampling procedure using the same number of samples. For images containing a relatively large spread of dominant frequency components, the improvement is most pronounced, with localized random sampling yielding a higher fidelity representation of both low and moderate frequency components containing the majority of image information. Moreover, the reconstruction improvements garnered by localized random sampling also extend to images with varying size and spectrum distribution, affording improved reconstruction
We describe a three-dimensional dynamical system, which is obtained as a pseudo-spectral approxim... more We describe a three-dimensional dynamical system, which is obtained as a pseudo-spectral approximation to a free boundary problem modeling solid combustion and rapid solidification, and is capable of generating its major dynamical patterns. These patterns include a Hopf bifurcation followed by a sequence of secondary period doubling and a transition to chaos, reverse sequences, and sequences followed by Shilnikov type trajectories. A computer-assisted bifurcation analysis uncovers some novel mechanisms of stability exchange. The most striking of them is an infinite period bifurcation which resembles the classical Shilnikov bifurcation, but instead of a funnel-shaped spiral along which the period is continually increasing, the continuation produced a series of isolas. Each isola is a closed branch of solutions of roughly the same period, and with the same number of oscillations. The isolas corresponding to consecutive numbers of low amplitude oscillations about the equilibrium are adjacent to each other, and appear to accumulate on a saddle-focus homoclinic connection of Shilnikov type.
Considering that many natural stimuli are sparse, can a sensory system evolve to take advantage o... more Considering that many natural stimuli are sparse, can a sensory system evolve to take advantage of this sparsity? We explore this question and show that significant downstream reductions in the numbers of neurons transmitting stimuli observed in early sensory pathways might be a consequence of this sparsity. First, we model an early sensory pathway using an idealized neuronal network comprised of receptors and downstream sensory neurons. Then, by revealing a linear structure intrinsic to neuronal network dynamics, our work points to a potential mechanism for transmitting sparse stimuli, related to compressed-sensing (CS) type data acquisition. Through simulation, we examine the characteristics of networks that are optimal in sparsity encoding, and the impact of localized receptive fields beyond conventional CS theory. The results of this work suggest a new network framework of signal sparsity, freeing the notion from any dependence on specific componentspace representations. We expect our CS network mechanism to provide guidance for studying sparse stimulus transmission along realistic sensory pathways as well as engineering network designs that utilize sparsity encoding.
Archive for Rational Mechanics and Analysis, Sep 1, 1998
We present an extension of the Melnikov method which can be used for ascertaining the existence o... more We present an extension of the Melnikov method which can be used for ascertaining the existence of homoclinic and heteroclinic orbits with many pulses in a class of near-integrable systems. The Melnikov function in this situation is the sum of the usual Melnikov functions evaluated with some appropriate phase delays. We show that a nonfolding condition which involves the logarithmic derivative of the Melnikov function must be satisfied in addition to the usual transversality conditions in order for homoclinic orbits with more than one pulse to exist.
Journal of Computational Neuroscience, Jan 18, 2014
In order to properly capture spike-frequency adaptation with a simplified point-neuron model, we ... more In order to properly capture spike-frequency adaptation with a simplified point-neuron model, we study approximations of Hodgkin-Huxley (HH) models including slow currents by exponential integrate-and-fire (EIF) models that incorporate the same types of currents. We optimize the parameters of the EIF models under the external drive consisting of AMPA-type conductance pulses using the current-voltage curves and the van Rossum metric to best capture the subthreshold membrane potential, firing rate, and jump size of the slow current at the neuron's spike times. Our numerical simulations demonstrate that, in addition to these quantities, the approximate EIF-type
From among the waves whose dynamics are governed by the nonlinear Schrödinger equation, we find a... more From among the waves whose dynamics are governed by the nonlinear Schrödinger equation, we find a robust, spatiotemporally disordered family, in which waves initialized with increasing amplitudes, on average, over long timescales, effectively evolve as ever more weakly coupled collections of plane waves. In particular, the relative amount of energy contained in their coupling decays to zero with increasing wave amplitude.
We consider stochastically perturbed gradient flows in the limit when the amplitude of random flu... more We consider stochastically perturbed gradient flows in the limit when the amplitude of random fluctuations is small relative to the typical energy scale in the system and the minima of the energy are not isolated but form submanifolds of the phase space. In this case the limiting dynamics may be described in terms of a diffusion process on these manifolds. We derive explicit equations for this limiting dynamics and illustrate them on a few finite-dimensional examples. Finally, we formally extrapolate the reduction technique to several infinite-dimensional examples and derive equations of the stochastic kink motion in Allen-Cahn-type systems.
We present an extension of the Melnikov method which can be used for ascertaining the existence o... more We present an extension of the Melnikov method which can be used for ascertaining the existence of homoclinic and heteroclinic orbits with many pulses in a class of near-integrable systems. The Melnikov function in this situation is the sum of the usual Melnikov functions evaluated with some appropriate phase delays. We show that a nonfolding condition which involves the logarithmic derivative of the Melnikov function must be satisfied in addition to the usual transversality conditions in order for homoclinic orbits with more than one pulse to exist.
The present work considers systems whose dynamics are governed by the nonlinear interactions amon... more The present work considers systems whose dynamics are governed by the nonlinear interactions among groups of 6 nonlinear waves, such as those described by the unforced quintic nonlinear Schrödinger equation. Specific parameter regimes in which ensemble-averaged dynamics of such systems with finite size are accurately described by a wave kinetic equation, as used in wave turbulence theory, are theoretically predicted. In addition, the underlying reasons that the wave kinetic equation may be a poor predictor of wave dynamics outside these regimes are also discussed. These theoretical predictions are directly verified by comparing ensemble averages of solutions to the dynamical equation with corresponding solutions of the wave kinetic equation.
From among the waves whose dynamics are governed by the nonlinear Schrödinger (NLS) equation, we ... more From among the waves whose dynamics are governed by the nonlinear Schrödinger (NLS) equation, we find a robust, spatiotemporally disordered family, in which waves initialized with increasing amplitudes, on average, over long time scales, effectively evolve as ever more weakly coupled collections of plane waves. In particular, the relative amount of energy contained in their coupling decays to zero with increasing wave amplitude.
Frontiers in Computational Neuroscience, Jun 28, 2018
Some previous studies have shown that chaotic dynamics in the balanced state, i.e., one with bala... more Some previous studies have shown that chaotic dynamics in the balanced state, i.e., one with balanced excitatory and inhibitory inputs into cortical neurons, is the underlying mechanism for the irregularity of neural activity. In this work, we focus on networks of current-based integrate-and-fire neurons with delta-pulse coupling. While we show that the balanced state robustly persists in this system within a broad range of parameters, we mathematically prove that the largest Lyapunov exponent of this type of neuronal networks is negative. Therefore, the irregular firing activity can exist in the system without the chaotic dynamics. That is the irregularity of balanced neuronal networks need not arise from chaos.
The quantum mechanical harmonic oscillator Hamiltonian H = (t 2 − ∂ 2 t)/2 generates a one-parame... more The quantum mechanical harmonic oscillator Hamiltonian H = (t 2 − ∂ 2 t)/2 generates a one-parameter unitary group W (θ) = e iθH in L 2 (R) which rotates the time-frequency plane. In particular, W (π/2) is the Fourier transform. When W (θ) is applied to any frame of Gabor wavelets, the result is another such frame with identical frame bounds. Thus each Gabor frame gives rise to a one-parameter family of frames, which we call a deformation of the original. For example, beginning with the usual tight frame F of Gabor wavelets generated by a compactly supported window g(t) and parameterized by a regular lattice in the time-frequency plane, one obtains a family {F θ : 0 ≤ θ < 2π} of frames generated by the non-compactly supported windows g θ = W (θ)g, parameterized by rotated versions of the original lattice. This gives a method for constructing tight frames of Gabor wavelets for which neither the window nor its Fourier transform have compact support. When θ = π/2, F θ is the well-known Gabor frame generated by a window with compactly supported Fourier transform. The family {F θ } therefore interpolates these two familiar examples.
From among the waves whose dynamics are governed by the nonlinear Schrödinger (NLS) equation, we ... more From among the waves whose dynamics are governed by the nonlinear Schrödinger (NLS) equation, we find a robust, spatiotemporally disordered family, in which waves initialized with increasing amplitudes, on average, over long time scales, effectively evolve as ever more weakly coupled collections of plane waves. In particular, the relative amount of energy contained in their coupling decays to zero with increasing wave amplitude.
Boundary value problems for the nonlinear Schrödinger equation on the half line in laboratory coo... more Boundary value problems for the nonlinear Schrödinger equation on the half line in laboratory coordinates are considered. A class of boundary conditions that lead to linearizable problems is identified by introducing appropriate extensions to initial-value problems on the infinite line, either explicitly or by constructing a suitable Bäcklund transformation. Various soliton solutions are explicitly constructed and studied.
Boundary value problems for the nonlinear Schrödinger equation on the half line in laboratory coo... more Boundary value problems for the nonlinear Schrödinger equation on the half line in laboratory coordinates are considered. A class of boundary conditions that lead to linearizable problems is identified by introducing appropriate extensions to initial-value problems on the infinite line, either explicitly or by constructing a suitable Bäcklund transformation. Various soliton solutions are explicitly constructed and studied.
We study the synchronization of a stochastically-driven, current-based, integrate-and-fire neuron... more We study the synchronization of a stochastically-driven, current-based, integrate-and-fire neuronal model on a preferential-attachment network with scale-free characteristics and high clustering. The synchrony is induced by cascading total firing events where every neuron in the network fires at the same instant of time. We show that in the regime where the system remains in this highly synchronous state, the firing rate of the network is completely independent of the synaptic coupling, and depends solely on the external drive. On the other hand, the ability for the network to maintain synchrony depends on a balance between the fluctuations of the external input and the synaptic coupling strength. In order to accurately predict the probability of repeated cascading total firing events we go beyond mean-field and tree-like approximations and conduct a detailed second order calculation taking into account local clustering. Our explicit analytical results are shown to give excellent agreement with direct numerical simulations for the particular preferential-attachment network model investigated.
For a class of integrate-and-fire, pulse-coupled networks with complex topology, we study the dep... more For a class of integrate-and-fire, pulse-coupled networks with complex topology, we study the dependence of the pulse rate on the underlying architectural connectivity statistics. We derive the distribution of the pulse rate from this dependence and determine when the underlying scale-free architectural connectivity gives rise to a scale-free pulse-rate distribution. We identify the scaling of the pairwise coupling between the dynamical units in this network class that keeps their pulse rates bounded in the infinite-network limit. In the process, we determine the connectivity statistics for a specific scale-free network grown by preferential attachment.
We propose a Boussinesq-type model to study the surface/interfacial wave manifestation of an unde... more We propose a Boussinesq-type model to study the surface/interfacial wave manifestation of an underlying, slowly-varying, long-wavelength, baroclinic flow in a two-layer, density-stratified system. The results of our model show numerically that, under strong nonlinearity, surface waves, with their typical wavenumber being the resonant k res , can be generated locally at the leading edge of the underlying slowly-varying, long-wavelength baroclinic flow. Here, the resonant k res satisfies the class 3 triad resonance condition among two short-mode waves and one long-mode wave in which all waves propagate in the same direction. Moreover, when the slope of the baroclinic flow is sufficiently small, only one spatially-localized large-amplitude surface wave packet can be generated at the leading edge. This localized surface wave packet becomes high in amplitude and large in group velocity after the interaction with its surrounding waves. These results are qualitatively consistent with various experimental observations including resonant surface waves at the leading edge of an internal wave. Subsequently, we propose a mechanism, referred to as the modulation-resonance mechanism, underlying these surface phenomena, based on our numerical simulations. The proposed modulation-resonance mechanism combines the linear modulation (ray-based) theory for the spatiotemporal asymmetric behavior of surface waves and the nonlinear class 3 triad resonance theory for the energy focusing of surface waves around the resonant wavenumber k res in Fourier space.
Journal of Dynamics and Differential Equations, Oct 1, 1993
This paper describes a new type of orbits homoclinic to resonance bands in a class of near-integr... more This paper describes a new type of orbits homoclinic to resonance bands in a class of near-integrable Hamiltonian systems. It presents a constructive method for establishing whether small conservative perturbations of a family of heteroclinic orbits that connect pairs of points on a circle of equilibria will yield transverse homoclinic connections between periodic orbits in the resonance band resulting from the perturbation. In any given example, this method may be used to prove the existence of such transverse homoclinic orbits, as well as to determine their precise shape, their asymptotic behavior, and their possible bifurcations. The method is a combination of the Melnikov method and geometric singular perturbation theory for ordinary differential equations.
Integrate-and-fire (I&F) neuronal networks are ubiquitous in diverse image processing application... more Integrate-and-fire (I&F) neuronal networks are ubiquitous in diverse image processing applications, including image segmentation and visual perception. While conventional I&F network image processing requires the number of nodes composing the network to be equal to the number of image pixels driving the network, we determine whether I&F dynamics can accurately transmit image information when there are significantly fewer nodes than network input-signal components. Although compressive sensing (CS) theory facilitates the recovery of images using very few samples through linear signal processing, it does not address whether similar signal recovery techniques facilitate reconstructions through measurement of the nonlinear dynamics of an I&F network. In this paper, we present a new framework for recovering sparse inputs of nonlinear neuronal networks via compressive sensing. By recovering both one-dimensional inputs and two-dimensional images, resembling natural stimuli, we demonstrate that input information can be well-preserved through nonlinear I&F network dynamics even when the number of network-output measurements is significantly smaller than the number of input-signal components. This work suggests an important exten- *
Compressive sensing (CS) theory demonstrates that by using uniformly-random sampling, rather than... more Compressive sensing (CS) theory demonstrates that by using uniformly-random sampling, rather than uniformly-spaced sampling, higher quality image reconstructions are often achievable. Considering that the structure of sampling protocols has such a profound impact on the quality of image reconstructions, we formulate a new sampling scheme motivated by physiological receptive field structure, localized random sampling, which yields significantly improved CS image reconstructions. For each set of localized image measurements, our sampling method first randomly selects an image pixel and then measures its nearby pixels with probability depending on their distance from the initially selected pixel. We compare the uniformly-random and localized random sampling methods over a large space of sampling parameters, and show that, for the optimal parameter choices, higher quality image reconstructions can be consistently obtained by using localized random sampling. In addition, we argue that the localized random CS optimal parameter choice is stable with respect to diverse natural images, and scales with the number of samples used for reconstruction. We expect that the localized random sampling protocol helps to explain the evolutionarily advantageous nature of receptive field structure in visual systems and suggests several future research areas in CS theory and its application to brain imaging. Sampling protocols have drastically changed with the discovery of compressive sensing (CS) data acquisition and signal recovery 1,2 . Prior to the development of CS theory, the Shannon-Nyquist theorem determined the majority of sampling procedures for both audio signals and images, dictating the minimum rate, the Nyquist rate, with which a signal must be uniformly sampled to guarantee successful reconstruction 3 . Since the theorem specifically addresses minimal sampling rates corresponding to uniformly-spaced measurements, signals were typically sampled at equally-spaced intervals in space or time before the discovery of CS. However, using CS-type data acquisition, it is possible to reconstruct a broad class of sparse signals, containing a small number of dominant components in some domain, by employing a sub-Nyquist sampling rate 2 . Instead of applying uniformly-spaced signal measurements, CS theory demonstrates that several types of uniformly-random sampling protocols will yield successful reconstructions with high probability 4-6 . While CS signal recovery is relatively accurate for sufficiently high sampling rates, we demonstrate that, for the recovery of natural scenes, reconstruction quality can be further improved via localized random sampling. In this new protocol, each signal sample consists of a randomly centered local cluster of measurements, in which the probability of measuring a given pixel decreases with its distance from the cluster center. We show that the localized random sampling protocol consistently produces more accurate CS reconstructions of natural scenes than the uniformly-random sampling procedure using the same number of samples. For images containing a relatively large spread of dominant frequency components, the improvement is most pronounced, with localized random sampling yielding a higher fidelity representation of both low and moderate frequency components containing the majority of image information. Moreover, the reconstruction improvements garnered by localized random sampling also extend to images with varying size and spectrum distribution, affording improved reconstruction
We describe a three-dimensional dynamical system, which is obtained as a pseudo-spectral approxim... more We describe a three-dimensional dynamical system, which is obtained as a pseudo-spectral approximation to a free boundary problem modeling solid combustion and rapid solidification, and is capable of generating its major dynamical patterns. These patterns include a Hopf bifurcation followed by a sequence of secondary period doubling and a transition to chaos, reverse sequences, and sequences followed by Shilnikov type trajectories. A computer-assisted bifurcation analysis uncovers some novel mechanisms of stability exchange. The most striking of them is an infinite period bifurcation which resembles the classical Shilnikov bifurcation, but instead of a funnel-shaped spiral along which the period is continually increasing, the continuation produced a series of isolas. Each isola is a closed branch of solutions of roughly the same period, and with the same number of oscillations. The isolas corresponding to consecutive numbers of low amplitude oscillations about the equilibrium are adjacent to each other, and appear to accumulate on a saddle-focus homoclinic connection of Shilnikov type.
Considering that many natural stimuli are sparse, can a sensory system evolve to take advantage o... more Considering that many natural stimuli are sparse, can a sensory system evolve to take advantage of this sparsity? We explore this question and show that significant downstream reductions in the numbers of neurons transmitting stimuli observed in early sensory pathways might be a consequence of this sparsity. First, we model an early sensory pathway using an idealized neuronal network comprised of receptors and downstream sensory neurons. Then, by revealing a linear structure intrinsic to neuronal network dynamics, our work points to a potential mechanism for transmitting sparse stimuli, related to compressed-sensing (CS) type data acquisition. Through simulation, we examine the characteristics of networks that are optimal in sparsity encoding, and the impact of localized receptive fields beyond conventional CS theory. The results of this work suggest a new network framework of signal sparsity, freeing the notion from any dependence on specific componentspace representations. We expect our CS network mechanism to provide guidance for studying sparse stimulus transmission along realistic sensory pathways as well as engineering network designs that utilize sparsity encoding.
Archive for Rational Mechanics and Analysis, Sep 1, 1998
We present an extension of the Melnikov method which can be used for ascertaining the existence o... more We present an extension of the Melnikov method which can be used for ascertaining the existence of homoclinic and heteroclinic orbits with many pulses in a class of near-integrable systems. The Melnikov function in this situation is the sum of the usual Melnikov functions evaluated with some appropriate phase delays. We show that a nonfolding condition which involves the logarithmic derivative of the Melnikov function must be satisfied in addition to the usual transversality conditions in order for homoclinic orbits with more than one pulse to exist.
Journal of Computational Neuroscience, Jan 18, 2014
In order to properly capture spike-frequency adaptation with a simplified point-neuron model, we ... more In order to properly capture spike-frequency adaptation with a simplified point-neuron model, we study approximations of Hodgkin-Huxley (HH) models including slow currents by exponential integrate-and-fire (EIF) models that incorporate the same types of currents. We optimize the parameters of the EIF models under the external drive consisting of AMPA-type conductance pulses using the current-voltage curves and the van Rossum metric to best capture the subthreshold membrane potential, firing rate, and jump size of the slow current at the neuron's spike times. Our numerical simulations demonstrate that, in addition to these quantities, the approximate EIF-type
From among the waves whose dynamics are governed by the nonlinear Schrödinger equation, we find a... more From among the waves whose dynamics are governed by the nonlinear Schrödinger equation, we find a robust, spatiotemporally disordered family, in which waves initialized with increasing amplitudes, on average, over long timescales, effectively evolve as ever more weakly coupled collections of plane waves. In particular, the relative amount of energy contained in their coupling decays to zero with increasing wave amplitude.
We consider stochastically perturbed gradient flows in the limit when the amplitude of random flu... more We consider stochastically perturbed gradient flows in the limit when the amplitude of random fluctuations is small relative to the typical energy scale in the system and the minima of the energy are not isolated but form submanifolds of the phase space. In this case the limiting dynamics may be described in terms of a diffusion process on these manifolds. We derive explicit equations for this limiting dynamics and illustrate them on a few finite-dimensional examples. Finally, we formally extrapolate the reduction technique to several infinite-dimensional examples and derive equations of the stochastic kink motion in Allen-Cahn-type systems.
We present an extension of the Melnikov method which can be used for ascertaining the existence o... more We present an extension of the Melnikov method which can be used for ascertaining the existence of homoclinic and heteroclinic orbits with many pulses in a class of near-integrable systems. The Melnikov function in this situation is the sum of the usual Melnikov functions evaluated with some appropriate phase delays. We show that a nonfolding condition which involves the logarithmic derivative of the Melnikov function must be satisfied in addition to the usual transversality conditions in order for homoclinic orbits with more than one pulse to exist.
The present work considers systems whose dynamics are governed by the nonlinear interactions amon... more The present work considers systems whose dynamics are governed by the nonlinear interactions among groups of 6 nonlinear waves, such as those described by the unforced quintic nonlinear Schrödinger equation. Specific parameter regimes in which ensemble-averaged dynamics of such systems with finite size are accurately described by a wave kinetic equation, as used in wave turbulence theory, are theoretically predicted. In addition, the underlying reasons that the wave kinetic equation may be a poor predictor of wave dynamics outside these regimes are also discussed. These theoretical predictions are directly verified by comparing ensemble averages of solutions to the dynamical equation with corresponding solutions of the wave kinetic equation.
From among the waves whose dynamics are governed by the nonlinear Schrödinger (NLS) equation, we ... more From among the waves whose dynamics are governed by the nonlinear Schrödinger (NLS) equation, we find a robust, spatiotemporally disordered family, in which waves initialized with increasing amplitudes, on average, over long time scales, effectively evolve as ever more weakly coupled collections of plane waves. In particular, the relative amount of energy contained in their coupling decays to zero with increasing wave amplitude.
Frontiers in Computational Neuroscience, Jun 28, 2018
Some previous studies have shown that chaotic dynamics in the balanced state, i.e., one with bala... more Some previous studies have shown that chaotic dynamics in the balanced state, i.e., one with balanced excitatory and inhibitory inputs into cortical neurons, is the underlying mechanism for the irregularity of neural activity. In this work, we focus on networks of current-based integrate-and-fire neurons with delta-pulse coupling. While we show that the balanced state robustly persists in this system within a broad range of parameters, we mathematically prove that the largest Lyapunov exponent of this type of neuronal networks is negative. Therefore, the irregular firing activity can exist in the system without the chaotic dynamics. That is the irregularity of balanced neuronal networks need not arise from chaos.
The quantum mechanical harmonic oscillator Hamiltonian H = (t 2 − ∂ 2 t)/2 generates a one-parame... more The quantum mechanical harmonic oscillator Hamiltonian H = (t 2 − ∂ 2 t)/2 generates a one-parameter unitary group W (θ) = e iθH in L 2 (R) which rotates the time-frequency plane. In particular, W (π/2) is the Fourier transform. When W (θ) is applied to any frame of Gabor wavelets, the result is another such frame with identical frame bounds. Thus each Gabor frame gives rise to a one-parameter family of frames, which we call a deformation of the original. For example, beginning with the usual tight frame F of Gabor wavelets generated by a compactly supported window g(t) and parameterized by a regular lattice in the time-frequency plane, one obtains a family {F θ : 0 ≤ θ < 2π} of frames generated by the non-compactly supported windows g θ = W (θ)g, parameterized by rotated versions of the original lattice. This gives a method for constructing tight frames of Gabor wavelets for which neither the window nor its Fourier transform have compact support. When θ = π/2, F θ is the well-known Gabor frame generated by a window with compactly supported Fourier transform. The family {F θ } therefore interpolates these two familiar examples.
From among the waves whose dynamics are governed by the nonlinear Schrödinger (NLS) equation, we ... more From among the waves whose dynamics are governed by the nonlinear Schrödinger (NLS) equation, we find a robust, spatiotemporally disordered family, in which waves initialized with increasing amplitudes, on average, over long time scales, effectively evolve as ever more weakly coupled collections of plane waves. In particular, the relative amount of energy contained in their coupling decays to zero with increasing wave amplitude.
Boundary value problems for the nonlinear Schrödinger equation on the half line in laboratory coo... more Boundary value problems for the nonlinear Schrödinger equation on the half line in laboratory coordinates are considered. A class of boundary conditions that lead to linearizable problems is identified by introducing appropriate extensions to initial-value problems on the infinite line, either explicitly or by constructing a suitable Bäcklund transformation. Various soliton solutions are explicitly constructed and studied.
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Papers by Gregor Kovacic