An augmented moment method for stochastic ensembles with delayed couplings: II. FitzHugh-Nagumo model

Dynamics of FitzHugh-Nagumo (FN) neuron ensembles with time-delayed couplings subject to white noises, has been studied by using both direct simulations and a semi-analytical augmented moment method (AMM) which has been proposed in a recent paper [H. Hasegawa, E-print: cond-mat/0311021]. For $N$-unit FN neuron ensembles, AMM transforms original $2N$-dimensional {\it stochastic} delay differential equations (SDDEs) to infinite-dimensional {\it deterministic} DEs for means and correlation functions of local and global variables. Infinite-order recursive DEs are terminated at the finite level $m$ in the level-$m$ AMM (AMM$m$), yielding $8(m+1)$-dimensional deterministic DEs. When a single spike is applied, the oscillation may be induced if parameters of coupling strength, delay, noise intensity and/or ensemble size are appropriate. Effects of these parameters on the emergence of the oscillation and on the synchronization in FN neuron ensembles have been studied. The synchronization shows the {\it fluctuation-induced} enhancement at the transition between non-oscillating and oscillating states. Results calculated by AMM5 are in fairly good agreement with those obtained by direct simulations.Comment: 15 pages, 3 figures; changed the title with correcting typos, accepted in Phys. Rev. E with some change

Is the Brain’s Inertia for Motor Movements Different for Acceleration and Deceleration?

The brain’s ability to synchronize movements with external cues is used daily, yet neuroscience is far from a full understanding of the brain mechanisms that facilitate and set behavioral limits on these sequential performances. This functional magnetic resonance imaging (fMRI) study was designed to help understand the neural basis of behavioral performance differences on a synchronizing movement task during increasing (acceleration) and decreasing (deceleration) metronome rates. In the MRI scanner, subjects were instructed to tap their right index finger on a response box in synchrony to visual cues presented on a display screen. The tapping rate varied either continuously or in discrete steps ranging from 0.5 Hz to 3 Hz. Subjects were able to synchronize better during continuously accelerating rhythms than in continuously or discretely decelerating rhythms. The fMRI data revealed that the precuneus was activated more during continuous deceleration than during acceleration with the hysteresis effect significant at rhythm rates above 1 Hz. From the behavioral data, two performance measures, tapping rate and synchrony index, were derived to further analyze the relative brain activity during acceleration and deceleration of rhythms. Tapping rate was associated with a greater brain activity during deceleration in the cerebellum, superior temporal gyrus and parahippocampal gyrus. Synchrony index was associated with a greater activity during the continuous acceleration phase than during the continuous deceleration or discrete acceleration phases in a distributed network of regions including the prefrontal cortex and precuneus. These results indicate that the brain’s inertia for movement is different for acceleration and deceleration, which may have implications in understanding the origin of our perceptual and behavioral limits

Higher Frequency Network Activity Flow Predicts Lower Frequency Node Activity in Intrinsic Low-Frequency BOLD Fluctuations

The brain remains electrically and metabolically active during resting conditions. The low-frequency oscillations (LFO) of the blood oxygen level-dependent (BOLD) signal of functional magnetic resonance imaging (fMRI) coherent across distributed brain regions are known to exhibit features of this activity. However, these intrinsic oscillations may undergo dynamic changes in time scales of seconds to minutes during resting conditions. Here, using wavelet-transform based timefrequency analysis techniques, we investigated the dynamic nature of default-mode networks from intrinsic BOLD signals recorded from participants maintaining visual fixation during resting conditions. We focused on the default-mode network consisting of the posterior cingulate cortex (PCC), the medial prefrontal cortex (mPFC), left middle temporal cortex (LMTC) and left angular gyrus (LAG). The analysis of the spectral power and causal flow patterns revealed that the intrinsic LFO undergo significant dynamic changes over time. Dividing the frequency interval 0 to 0.25 Hz of LFO into four intervals slow- 5 (0.01–0.027 Hz), slow-4 (0.027–0.073 Hz), slow-3 (0.073–0.198 Hz) and slow-2 (0.198–0.25 Hz), we further observed significant positive linear relationships of slow-4 in-out flow of network activity with slow-5 node activity, and slow-3 in-out flow of network activity with slow-4 node activity. The network activity associated with respiratory related frequency (slow- 2) was found to have no relationship with the node activity in any of the frequency intervals. We found that the net causal flow towards a node in slow-3 band was correlated with the number of fibers, obtained from diffusion tensor imaging (DTI) data, from the other nodes connecting to that node. These findings imply that so-called resting state is not ‘entirely’ at rest, the higher frequency network activity flow can predict the lower frequency node activity, and the network activity flow can reflect underlying structural connectivity

Behavior of Dynamical Systems in the Regime of Transient Chaos

The transient chaos regime in a two-dimensional system with discrete time (Eno map) is considered. It is demonstrated that a time series corresponding to this regime differs from a chaotic series constructed for close values of the control parameters by the presence of "nonregular" regions, the number of which increases with the critical parameter. A possible mechanism of this effect is discussed.Comment: 4 pages, 2 figure

Synchronized dynamics of cortical neurons with time-delay feedback

The dynamics of three mutually coupled cortical neurons with time delays in the coupling are explored numerically and analytically. The neurons are coupled in a line, with the middle neuron sending a somewhat stronger projection to the outer neurons than the feedback it receives, to model for instance the relay of a signal from primary to higher cortical areas. For a given coupling architecture, the delays introduce correlations in the time series at the time-scale of the delay. It was found that the middle neuron leads the outer ones by the delay time, while the outer neurons are synchronized with zero lag times. Synchronization is found to be highly dependent on the synaptic time constant, with faster synapses increasing both the degree of synchronization and the firing rate. Analysis shows that presynaptic input during the interspike interval stabilizes the synchronous state, even for arbitrarily weak coupling, and independent of the initial phase. The finding may be of significance to synchronization of large groups of cells in the cortex that are spatially distanced from each other.Comment: 21 pages, 11 figure

Grouping time series by pairwise measures of redundancy

A novel approach is proposed to group redundant time series in the frame of causality. It assumes that (i) the dynamics of the system can be described using just a small number of characteristic modes, and that (ii) a pairwise measure of redundancy is sufficient to elicit the presence of correlated degrees of freedom. We show the application of the proposed approach on fMRI data from a resting human brain and gene expression profiles from HeLa cell culture.Comment: 4 pages, 8 figure

Non-parametric directionality analysis: extension for removal of a single common predictor and application to time series

BACKGROUND: The ability to infer network structure from multivariate neuronal signals is central to computational neuroscience. Directed network analyses typically use parametric approaches based on auto-regressive (AR) models, where networks are constructed from estimates of AR model parameters. However, the validity of using low order AR models for neurophysiological signals has been questioned. A recent article introduced a non-parametric approach to estimate directionality in bivariate data, non-parametric approaches are free from concerns over model validity. NEW METHOD: We extend the non-parametric framework to include measures of directed conditional independence, using scalar measures that decompose the overall partial correlation coefficient summatively by direction, and a set of functions that decompose the partial coherence summatively by direction. A time domain partial correlation function allows both time and frequency views of the data to be constructed. The conditional independence estimates are conditioned on a single predictor. RESULTS: The framework is applied to simulated cortical neuron networks and mixtures of Gaussian time series data with known interactions. It is applied to experimental data consisting of local field potential recordings from bilateral hippocampus in anaesthetised rats. COMPARISON WITH EXISTING METHOD(S): The framework offers a non-parametric approach to estimation of directed interactions in multivariate neuronal recordings, and increased flexibility in dealing with both spike train and time series data. CONCLUSIONS: The framework offers a novel alternative non-parametric approach to estimate directed interactions in multivariate neuronal recordings, and is applicable to spike train and time series data

Estimating Granger causality from Fourier and wavelet transforms of time series data

Experiments in many fields of science and engineering yield data in the form of time series. The Fourier and wavelet transform-based nonparametric methods are used widely to study the spectral characteristics of these time series data. Here, we extend the framework of nonparametric spectral methods to include the estimation of Granger causality spectra for assessing directional influences. We illustrate the utility of the proposed methods using synthetic data from network models consisting of interacting dynamical systems.Comment: 6 pages, 2 figure

Controlling chaotic transients: Yorke's game of survival

5 pages, 4 figures.-- PACS nr.: 05.45.Gg, 05.45.Pq.-- PMID: 14995689 [PubMed].We consider the tent map as the prototype of a chaotic system with escapes. We show analytically that a small, bounded, but carefully chosen perturbation added to the system can trap forever an orbit close to the chaotic saddle, even in presence of noise of larger, although bounded, amplitude. This problem is focused as a two-person, mathematical game between two players called "the protagonist" and "the adversary." The protagonist's goal is to survive. He can lose but cannot win; the best he can do is survive to play another round, struggling ad infinitum. In the absence of actions by either player, the dynamics diverge, leaving a relatively safe region, and we say the protagonist loses. What makes survival difficult is that the adversary is allowed stronger "actions" than the protagonist. What makes survival possible is (i) the background dynamics (the tent map here) are chaotic and (ii) the protagonist knows the action of the adversary in choosing his response and is permitted to choose the initial point x(0) of the game. We use the "slope 3" tent map in an example of this problem. We show that it is possible for the protagonist to survive.J.A. and M.S.J. acknowledge financial support from the Spanish Ministry of Science and Technology under project BFM2000-0967, and from the Universidad Rey Juan Carlos under projects URJC-PGRAL-2001/02 and URJC-PIGE-02-04. F.d'O. acknowledges financial support from MCyT (Spain) and FEDER, project REN2001-0802-C02-01/MAR (IMAGEN).Peer reviewe