Dynamical origin of independent spiking and bursting activity in neural microcircuits

The relationship between spiking and bursting dynamics is a key question in neuroscience, particularly in understanding the origins of different neural coding strategies and the mechanisms of motor command generation and neural circuit coordination. Experiments indicate that spiking and bursting dynamics can be independent. We hypothesize that different mechanisms for spike and burst generation, intrinsic neuron dynamics for spiking and a modulational network instability for bursting, are the origin of this independence. We tested the hypothesis in a detailed dynamical analysis of a minimal inhibitory neural microcircuit (motif) of three reciprocally connected Hodgkin-Huxley neurons. We reduced this high-dimensional dynamical system to a rate model and showed that both systems have identical bifurcations from tonic spiking to burst generation, which, therefore, does not depend on the details of spiking activity.     

Parameter-dependent synchronization transition of coupled neurons with co-existing spiking and bursting

A firing pattern transition is simulated in the Leech neuron model, firstly from bursting to co-existence of spiking and bursting and then to spiking. The attraction domain of spiking and bursting for three different parameter values are calculated. Synchronization transition processes of two coupled Leech neurons, one is bursting and the other the co-existing spiking, are simulated for the three parameters. The three synchronization processes appear similar as the coupling strength increases, beginning from non-synchronization to complete synchronization through a complex dynamical procedure, but their detailed processes are different depending on the parameter values. The transition procedure is complex and the complete synchronization is in bursting for larger parameter values, while the process is simple with complete synchronization of spiking for smaller values. The potential relationship between complete synchronization and the attraction domain is also discussed. The results are instructive to understanding the synchronization behaviors of the coupled neuronal system with co-existing attractors.     

A model for the transition to baroclinic chaos

The transition to chaos in a 2-layer baroclinic fluid with a slightly different viscosity in each layer is investigated. A low-order model is constructed by truncating the quasi-geostrophic partial differential equations to include one azimuthal wave in each layer while retaining a general zonal-flow correction. When the viscosities are equal the model reduces to a modified form of the Lorenz equations and displays a cusped return map. For slightly different viscosities the transition to chaos follows the Feigenbaum scenario. The return map in the chaotic regime is nearly parabolic and highly contracted. Multiple attractors are found for the same control parameters. The predicted route to chaos is similar to that observed in laboratory experiments, although significant quantitative differences remain.     

Different types of bursting in Chay neuronal model

Based on actual neuronal firing activities, bursting in the Chay neuronal model is considered, in which V _K, reversal potentials for K⁺, V _C, reversal potentials for Ca²⁺, time kinetic constant λ _n and an additional depolarized current I are considered as dynamical parameters. According to the number of the Hopf bifurcation points on the upper branch of the bifurcation curve of fast subsystem, which is associated with the stable limit cycle corresponding to spiking states, different types of bursting and their respective dynamical behavior are surveyed by means of fast-slow dynamical bifurcation analysis. Supported by the National Natural Science Foundation of China (Grant Nos. 10432010, 10526002 and 10702002)     

Transition between tonic spiking and bursting in a neuron model via the blue-sky catastrophe

We study a continuous and reversible transition between periodic tonic spiking and bursting activities in a neuron model. It is described as the blue-sky catastrophe, which is a homoclinic bifurcation of a saddle-node periodic orbit of codimension one. This transition constitutes a biophysically plausible mechanism for the regulation of burst duration that increases with no bound like 1/square root alpha-alpha0 as the transition value alpha0 is approached.     

Metastable chaos: The transition to sustained chaotic behavior in the Lorenz model

The system of equations introduced by Lorenz to model turbulent convective flow is studied here for Rayleigh numbersr somewhat smaller than the critical value required for sustained chaotic behavior. In this regime the system is found to exhibit transient chaotic behavior. Some statistical properties of this transient chaos are examined numerically. A mean decay time from chaos to steady flow is found and its dependence uponr is studied both numerically and (very close to the criticalr) analytically.This work was supported in part by NASA grant NSG 5209; partial support of computer costs was provided by the University of Maryland-Baltimore County Computer Center.     

Dissipative quantum chaos: transition from wave packet collapse to explosion

Using the quantum trajectories approach, we study the quantum dynamics of a dissipative chaotic system described by the Zaslavsky map. For strong dissipation the quantum wave function in the phase space collapses onto a compact packet which follows classical chaotic dynamics and whose area is proportional to the Planck constant. At weak dissipation the exponential instability of quantum dynamics on the Ehrenfest time scale dominates and leads to wave packet explosion. The transition from collapse to explosion takes place when the dissipation time scale exceeds the Ehrenfest time. For integrable nonlinear dynamics the explosion practically disappears leaving place to collapse.     

Modulated amplitude waves and the transition from phase to defect chaos

The mechanism for transitions from phase to defect chaos in the one-dimensional complex Ginzburg-Landau equation (CGLE) is presented. We describe periodic coherent structures of the CGLE, called modulated amplitude waves (MAWs). MAWs of various periods P occur in phase chaotic states. A bifurcation study of the MAWs reveals that for sufficiently large period, pairs of MAWs cease to exist via a saddle-node bifurcation. For periods beyond this bifurcation, incoherent near-MAW structures evolve towards defects. This leads to our main result: the transition from phase to defect chaos takes place when the periods of MAWs in phase chaos are driven beyond their saddle-node bifurcation.     

Universal properties of the transition from quasi-periodicity to chaos in dissipative systems

An exact renormalization group transformation is developed for dissipative systems which describes how the transition to chaos may occur in a continuous and universal manner if the frequency ratio in the quasi-periodic regime is held at a fixed irrational value. Our approach is a natural extension of K.A.M. theory to strong coupling. Most of our analysis is for analytic circle maps. We have found a strong coupling fixed point where invertibility is lost, which describes the universal features of the transition to chaos. We find numerically that any two such critical maps with the same winding number are C¹ conjugate. It follows that the low frequency peaks in an experimental spectrum are universal and we determine how their envelope scales with frequency.When the winding number has a periodic continued fraction, our renormalization transform has a fixed point and spectra are self similar in addition. For a set of non-periodic winding numbers with full measure our renormalization transformation yields an ergodic trajectory in a sub-space of all critical maps. Physically one finds singular and universal spectra that do not scale.     

Quantum chaos triggered by precursors of a quantum phase transition: the dicke model

We consider the Dicke Hamiltonian, a simple quantum-optical model which exhibits a zero-temperature quantum phase transition. We present numerical results demonstrating that at this transition the system changes from being quasi-integrable to quantum chaotic. By deriving an exact solution in the thermodynamic limit we relate this phenomenon to a localization-delocalization transition in which a macroscopic superposition is generated. We also describe the classical analogs of this behavior.     

Dynamical studies of highly deuterated betaine phosphate near the antiferroelectric phase transition

The dielectric response and the Raman spectra of single crystals of deuterated betaine phosphate are studied around the antiferroelectric phase transition. The dielectric data between 10 MHz and 11 GHz can be explained on the basis of a simple Debye-relaxation with a critical slowing-down of the relaxation rate on approachingT _C . Using the Cole-Davidson form of the dielectric function we succeeded in fitting the data in the whole frequency range from 10 MHz to 11 GHz and from 64–400 GHz over a temperature range from 145–280 K. Raman spectra clearly indicate that the doubling of the unit cell does not take place at the antiferroelectric transition temperature, but some degrees below.     

Universality in the transition to chaos of dissipative systems

The period-doubling bifurcation process for two-dimensional transforms exhibits a new class of universality when a small dissipation is taken into account. The effective Jacobian is then defined as a function of both the dissipation and the rank n of the cascade (cycle 2ⁿ). Numerical simulations of a simple mechanical system and numerical calculations on the Hénon mapping show that the decrement lies on a continuous curve as function of the effective Jacobian. A method using this result to understand experimental data is explained and a first order approximation of the renormalization process yields an analytic expression of the curve.Among the different transitions to chaos, the period-doubling bifurcation cascade [1, 2] has been extensively studied. This transition is characterized by an experimental convergence rate of the bifurcation threshold sequence to the accumulation point: the threshold of chaos. It is well known that the decrement of this bifurcation cascade can take different values. Each value corresponds to a specific class of systems which can be characterized by some general features of the system undergoing the transition [3, 4, 5]. We are concerned here with the two values; δ(I) = 4.699… the decrement of the well-known one-dimensional transform with a quadratic maximum [2] and δ(II) = 8.721 the decrement of a two-dimensional non-dissipative transforms [3]. These two classes of systems are generic in physics and the two values δ(I) and δ(II) are therefore relevant values of the decrement. However, these two exponents stand for the infinite dissipation case and the conservative one thus leaving out the general physical situation of a finite dissipation. Only hints of the effect of a small dissipation in a two-dimensional mapping have been given [6] before the work of Zisook [7].A thorough study of the effect of dissipation is set forth here. The first two sections deal with the physical model used to perform the numerical investigation and the “experimental” data thus obtained. A study of the renormalization process enables to generalise the relation δ_n(J)=δ_n−(J²), first given by Zisook in [7], to all transforms where the Jacobian does not depend on the linearization point in the phase space. Furthermore a first order approximation gives an excellent analytic expression of the universal function displaying the crossover of the decrement between δ(II) and δ(I).     

Dynamical stability of local gauge symmetry Creation of light from chaos

We show that the large distance behavior of gauge theories is stable, within certain limits, with respect to addition of gauge noninvariant interactions at small distances.     

Hyperlabyrinth chaos: from chaotic walks to spatiotemporal chaos

In this paper we examine a very simple and elegant example of high-dimensional chaos in a coupled array of flows in ring architecture that is cyclically symmetric and can also be viewed as an N-dimensional spatially infinite labyrinth (a "hyperlabyrinth"). The scaling laws of the largest Lyapunov exponent, the Kaplan-Yorke dimension, and the metric entropy are investigated in the high-dimensional limit (3相似文献   

Dynamical transition from a quasi-one-dimensional Bose-Einstein condensate to a Tonks-Girardeau gas

We analyze in detail the expansion of a 1D Bose gas after removing the axial confinement. We show that during its one-dimensional expansion the density of the Bose gas does not follow a self-similar solution. Our analysis is based on a nonlinear Schr?dinger equation with variable nonlinearity whose validity is discussed for the expansion problem, by comparing with an exact Bose-Fermi mapping for the case of an initial Tonks-Girardeau gas. For this case, the gas is shown to expand self-similarly, with a different scaling law compared to the one-dimensional Thomas-Fermi condensate.     

Piecewise linear models for the quasiperiodic transition to chaos

We formulate and study analytically and computationally two families of piecewise linear degree one circle maps. These families offer the rare advantage of being non-trivial but essentially solvable models for the phenomenon of mode locking and the quasiperiodic transition to chaos. For instance, for these families, we obtain complete solutions to several questions still largely unanswered for families of smooth circle maps. Our main results describe (1) the sets of maps in these families having some prescribed rotation interval; (2) the boundaries between zero and positive topological entropy and between zero length and non-zero length rotation interval; and (3) the structure and bifurcations of the attractors in one of these families. We discuss the interpretation of these maps as low-order spline approximations to the classic "sine-circle" map and examine more generally the implications of our results for the case of smooth circle maps. We also mention a possible connection to recent experiments on models of a driven Josephson junction. (c) 1996 American Institute of Physics.     

Distinguishing the transition to chaos in a spherical pendulum

Complex responses are studied for a spherical pendulum whose support is excited with a translational periodic motion. Governing equations are studied analytically to allow prediction of responses under various excitation conditions. Stability for certain cases of damping is predicted by means of existing analysis and compared with experimental data. Numerical time-step integration of the governing equations is developed to predict responses for various types of excitation and damping conditions. Predicted results are compared with corresponding motions measured in an experimental spherical pendulum system. A data acquisition system is included whereby detailed digitized time histories of the pendulum motion can be established and various parameters can be computed to characterize the type of motion present. Two new vector spaces are defined for describing complex responses which occur for certain specified excitation conditions. It is shown in these parameter spaces that the transition from quasiperiodic to chaotic motions can be carefully quantified in systems with very light damping. This discovery provides a convenient means for comparison of complex motions in the numerical and experimental air pendulum systems. The implications of the results are important for dynamic response in various applications, including fluid motions in satellite tanks and other nonlinear time-dependent physical processes which include very light damping. (c) 1995 American Institute of Physics.     

A simple derivation of the stochastic eigenvalue equation in the transition from quasiperiodicity to chaos

A simple derivation of the stochastic eigenvalue equation, previously obtained by irrational decimation of functional integrals, is given to show the universal scaling behavior of external noise in the transition from quasiperiodicity to chaos in dissipative systems.