The work presents a novel method for the detection of mutual
phase synchronization in non-stationary time series. We show how the
application of a cluster algorithmthat considers spatio-temporal struc-
tures of data follows from the general condition of phase-synchronized
data. In view of the topology of phasic data, we re-formulate the
K-Means cluster algorithm on a flat torus and apply a segmentation
index derived in an earlier work (Physica D 177,203-232(2003)). This index is extended by means of averaging in order to reflect phase syn-
chronization in ensembles of multivariate time series. The method is
illustrated using simulated multivariate phase dynamics and arrays of
chaotic systems, in which temporal segments of phase-synchronized
states are registered. A comparison with results from an existing bi-
variate synchronization index reveals major advantages of our method.
A stability analysis is presented for neural field equations in the presence
of axonal delays and for a general class of connectivity kernels and synap-
tic properties. Sufficient conditions are given for the stability of equilibrium
solutions. It is shown that the delays play a crucial role in non-stationary
bifurcations of equilibria, whereas the stationary bifurcations depend only on
the kernel. Bounds are determined for the frequencies of bifurcating periodic
solutions. A perturbative scheme is used to calculate the types of bifurca-
tions leading to spatial patterns, oscillatory solutions, and traveling waves.
For high transmission speeds a simple method is derived that allows the de-
termination of the bifurcation type by visual inspection of the Fourier trans-
forms of the connectivity kernel and its first moment. Results are numerically
illustrated on a class of neurologically plausible second order systems with
combinations of Gaussian excitatory and inhibitory connections.
This work studies dynamical properties of spatially extended neu-
ronal ensembles. We first derive an evolution equation from tem-
poral properties and statistical distributions of synapses and somata.
The obtained integro-differential equation considers both synaptic and
axonal propagation delay, while spatial synaptic connectivities ex-
hibit gamma-distributed distributions. This familiy of connectivity
kernels also covers the cases of divergent, finite, and negligible self-
connections. The work derives conditions for both stationary and
nonstationary instabilities for gamma-distributed kernels.It turns out
that the stability conditions can be formulated in terms of the mean spatial interaction ranges and the mean spatial interaction times. In
addition, a numerical study examines the evoked spatiotemporal re-
sponse activity caused by short local stimuli and reveals maximum
response activity after the mean interaction time at a distance from
stimulus offset location equal to the mean interaction range. These
findings propose new insights to neuronal mechanisms of experimen-
tally observed evoked brain activity.
Effects of nonlocal feedback on traveling fronts in neural fields subject to transmission delay
(2004)
The work introduces a model for reciprocal connections in neural fields by a nonlocal feedback
mechanism, while the neural field exhibits nonlocal interactions and intra-areal transmission delays.
We study the speed of traveling fronts with respect to the transmission delay, the spatial feedback
range and the feedback delay for general axonal and feedback connectivity kernels. In addition, we
find a novel shape of traveling fronts due to the applied feedback and criteria for its occurence are
derived.
This work studies the stability of spatially extended neuronal ensembles. We first
derive the model equation from statistical properties of the neuron population. The
obtained integro-differential equation considers synaptic and space-dependent transmission
delay for both general and gamma-distributed synaptic connectivities. The
latter connectivity type reveals infinite, finite and vanishing self-connectivities. The
work derives conditions for stationary and nonstationary instabilities for both kernel
types. In addition, a nonlinear analysis for general kernels yields the order parameter
equation of the Turing instability. To compare the results to findings for partial
differential equations (PDEs), two typical PDE-types are derived from the examined
model equation. In case of the gamma-distributed kernels, the stability conditions
are formulated in terms of the mean excitatory and inhibitory interaction ranges. As
a novel finding, we obtain Turing instabilities in fields with local inhibition-lateral
excitation, while wave instabilities occur in fields with local excitation and lateral
inhibition. Numerical simulations support the analytical results.
This work studies the stability and the stochastic properties of neural activity evoked by external
stimulation. The underlying model describes the spatiotemporal dynamics of neural populations
involving both synaptic delay and axonal transmission delay. We show, that the linear model
recasts to a set of affne delay differential equations in spatial Fourier space. Besides a stability
study for general kernels and general external stimulation, the power spectrum of evoked activity
is derived analytically in case of external Gaussian noise. Further applications to specific kernels
reveal critical
uctuations at Hopf- and Turing bifurcations and allow the numerical detection of
1/f fluctuations near the stability threshold.
The present work introduces an analysis framework for the de-
tection of metastable signal segments in multivariate time series. It
is shown that in case of linear data these segments represent tran-
sient generalized synchronization, while metastable segments in circu-
lar data reflect transient mutual phase synchronization. We propose
a single segmentation approach for both types of data considering the
space-time structure of the data. Applications to both event-related
potentials and single evoked potentials obtained from an auditory odd-
ball experiment reveal the lack of the component P300 in an experi-
mental condition, indicates attention effects in component N100 and
shows dramatic latency jitters in single trials. A comparison of the
proposed method to a conventional index of mutual phase synchro-
nization demonstrates the superiority of considering space-time data
structures.