1,007 research outputs found
Generalized Turing Patterns and Their Selective Realization in Spatiotemporal Systems
We consider the pattern formation problem in coupled identical systems after
the global synchronized state becomes unstable. Based on analytical results
relating the coupling strengths and the instability of each spatial mode
(pattern) we show that these spatial patterns can be selectively realized by
varying the coupling strengths along different paths in the parameter space.
Furthermore, we discuss the important role of the synchronized state (fixed
point versus chaotic attractor) in modulating the temporal dynamics of the
spatial patterns.Comment: 9 pages, 3 figure
Frequency decomposition of conditional Granger causality and application to multivariate neural field potential data
It is often useful in multivariate time series analysis to determine
statistical causal relations between different time series. Granger causality
is a fundamental measure for this purpose. Yet the traditional pairwise
approach to Granger causality analysis may not clearly distinguish between
direct causal influences from one time series to another and indirect ones
acting through a third time series. In order to differentiate direct from
indirect Granger causality, a conditional Granger causality measure in the
frequency domain is derived based on a partition matrix technique. Simulations
and an application to neural field potential time series are demonstrated to
validate the method.Comment: 18 pages, 6 figures, Journal publishe
Analyzing Stability of Equilibrium Points in Neural Networks: A General Approach
Networks of coupled neural systems represent an important class of models in
computational neuroscience. In some applications it is required that
equilibrium points in these networks remain stable under parameter variations.
Here we present a general methodology to yield explicit constraints on the
coupling strengths to ensure the stability of the equilibrium point. Two models
of coupled excitatory-inhibitory oscillators are used to illustrate the
approach.Comment: 20 pages, 4 figure
Strong convergence and control condition of modified Halpern iterations in Banach spaces
Let C
be a nonempty closed convex subset of a real Banach space
X
which has a uniformly Gâteaux differentiable norm. Let
T∈ΓC
and f∈ΠC. Assume that {xt}
converges
strongly to a fixed point z
of T
as t→0, where
xt
is the unique element of C
which satisfies
xt=tf(xt)+(1−t)Txt. Let {αn}
and {βn} be two real sequences in (0,1) which satisfy the following conditions: (C1)limn→∞αn=0;(C2)∑n=0∞αn=∞;(C6)0<liminfn→∞βn≤limsupn→∞βn<1. For arbitrary x0∈C, let the sequence
{xn}
be defined iteratively by
yn=αnf(xn)+(1−αn)Txn, n≥0,
xn+1=βnxn+(1−βn)yn, n≥0. Then {xn}
converges strongly to a fixed point of T
- …