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Higher-Order Energy Expansions and Spike Locations
We consider the following singularly perturbed semilinear elliptic problem:
(I)\left\{
\begin{array}{l}
\epsilon^{2} \Delta u - u + f(u)=0 \ \ \mbox{in} \ \Omega, \\
u>0 \ \ \mbox{in} \ \ \Omega \ \ \mbox{and} \
\frac{\partial u}{\partial \nu} =0 \ \mbox{on} \ \partial \Omega,
\end{array}
\right.
where \Om is a bounded domain in R^N with smooth boundary \partial \Om, \ep>0 is a small constant and f is some superlinear but subcritical nonlinearity.
Associated with (I) is the energy functional J_\ep defined by
J_\ep [u]:= \int_\Om \left(\frac{\ep^2}{2} |\nabla u|^2 + \frac{1}{2} u^2- F(u)\right) dx
\ \ \ \ \ \mbox{for} \ u \in H^1 (\Om),
where F(u)=\int_0^u f(s)ds. Ni and Takagi proved that for a single boundary spike solution u_\ep, the following asymptotic expansion holds:
J_\ep [u_\ep] =\ep^{N} \Bigg[ \frac{1}{2} I[w] -c_1 \ep H(P_\ep) + o(\ep)\Bigg],
where c_1>0 is a generic constant, P_\ep is the unique local maximum point of u_\ep and H(P_\ep) is the boundary mean curvature function at P_\ep \in \partial \Om.
In this paper, we obtain a higher-order expansion of J_\ep [u_\ep]:
J_\ep [u_\ep] =\ep^{N} \Bigg[ \frac{1}{2} I[w] -c_1 \ep H(P_\ep) + \ep^2 [c_2 (H(P_\ep))^2 + c_3 R (P_\ep)]+ o(\ep^2)\Bigg]
where c_2, c_3 are generic constants
and R(P_\ep) is the Ricci scalar curvature at P_\ep.
In particular c_3 >0. Some applications of this expansion are given
Spikes in the Mixmaster regime of G_2 cosmologies
We produce numerical evidence that spikes in the Mixmaster regime of G_2
cosmologies are transient and recurring, supporting the conjecture that the
generalized Mixmaster behavior is asymptotically non-local where spikes occur.
Higher order spike transitions are observed to split into separate first order
spike transitions.Comment: Minor corrections. Matches the published versio
Spikes for the Gierer-Meinhardt system with discontinuous diffusion coefficients
The original publication is available at http://www.springerlink.com/content/vw2m382276u4g814/We rigorously prove results on spiky patterns for the Gierer-Meinhardt system with a jump discontinuity in the diffusion coefficient of the inhibitor. Firstly, we show the existence of an interior spike located away from the jump discontinuity, deriving a necessary condition for the
position of the spike. In particular we show
that the spike is located in one-and-only-one
of the two subintervals created by the jump
discontinuity of the inhibitor diffusivity.
This localisation principle for a spike
is a new effect which does not occur for
homogeneous diffusion coefficients. Further, we show that this interior spike is stable.
Secondly, we establish the existence of a spike whose distance from the jump discontinuity is of the same order as its spatial extent. The existence of such a spike near the jump discontinuity is the second new effect presented in this paper.
To derive these new effects in a mathematically rigorous way, we use analytical tools like Liapunov-Schmidt reduction and nonlocal eigenvalue problems which have been developed in our previous work.
Finally, we confirm our results by numerical
computations for the dynamical behavior of the system. We observe a moving spike which
converges to a stationary spike located in the interior of one of the subintervals or near the jump discontinuity
Emergence of spike correlations in periodically forced excitable systems
In sensory neurons the presence of noise can facilitate the detection of weak
information-carrying signals, which are encoded and transmitted via correlated
sequences of spikes. Here we investigate relative temporal order in spike
sequences induced by a subthreshold periodic input, in the presence of white
Gaussian noise. To simulate the spikes, we use the FitzHugh-Nagumo model, and
to investigate the output sequence of inter-spike intervals (ISIs), we use the
symbolic method of ordinal analysis. We find different types of relative
temporal order, in the form of preferred ordinal patterns which depend on both,
the strength of the noise and the period of the input signal. We also
demonstrate a resonance-like behavior, as certain periods and noise levels
enhance temporal ordering in the ISI sequence, maximizing the probability of
the preferred patterns. Our findings could be relevant for understanding the
mechanisms underlying temporal coding, by which single sensory neurons
represent in spike sequences the information about weak periodic stimuli
An AER Spike-Processing Filter Simulator and Automatic VHDL Generator Based on Cellular Automata
Spike-based systems are neuro-inspired circuits implementations
traditionally used for sensory systems or sensor signal processing. Address-Event-
Representation (AER) is a neuromorphic communication protocol for transferring
asynchronous events between VLSI spike-based chips. These neuro-inspired
implementations allow developing complex, multilayer, multichip neuromorphic
systems and have been used to design sensor chips, such as retinas and cochlea,
processing chips, e.g. filters, and learning chips. Furthermore, Cellular Automata
(CA) is a bio-inspired processing model for problem solving. This approach
divides the processing synchronous cells which change their states at the same time
in order to get the solution. This paper presents a software simulator able to gather
several spike-based elements into the same workspace in order to test a CA
architecture based on AER before a hardware implementation. Furthermore this
simulator produces VHDL for testing the AER-CA into the FPGA of the USBAER
AER-tool.Ministerio de Ciencia e Innovación TEC2009-10639-C04-0
Coherent response of the Hodgkin-Huxley neuron in the high-input regime
We analyze the response of the Hodgkin-Huxley neuron to a large number of
uncorrelated stochastic inhibitory and excitatory post-synaptic spike trains.
In order to clarify the various mechanisms responsible for noise-induced spike
triggering we examine the model in its silent regime. We report the coexistence
of two distinct coherence resonances: the first one at low noise is due to the
stimulation of "correlated" subthreshold oscillations; the second one at
intermediate noise variances is instead related to the regularization of the
emitted spike trains.Comment: 5 pages - 5 eps figures, contribution presented to the conference CNS
2006 held in Edinburgh (UK), to appear on Neurocomputin
Neural Correlates of Social Behavior in Mushroom Body Extrinsic Neurons of the Honeybee Apis mellifera
The social behavior of honeybees (Apis mellifera) has been extensively investigated, but little is known about its neuronal correlates. We developed a method that allowed us to record extracellularly from mushroom body extrinsic neurons (MB ENs) in a freely moving bee within a small but functioning mini colony of approximately 1,000 bees. This study aimed to correlate the neuronal activity of multimodal high-order MB ENs with social behavior in a close to natural setting. The behavior of all bees in the colony was video recorded. The behavior of the recorded animal was compared with other hive mates and no significant differences were found. Changes in the spike rate appeared before, during or after social interactions. The time window of the strongest effect on spike rate changes ranged from 1 s to 2 s before and after the interaction, depending on the individual animal and recorded neuron. The highest spike rates occurred when the experimental animal was situated close to a hive mate. The variance of the spike rates was analyzed as a proxy for high order multi-unit processing. Comparing randomly selected time windows with those in which the recorded animal performed social interactions showed a significantly increased spike rate variance during social interactions. The experimental set-up employed for this study offers a powerful opportunity to correlate neuronal activity with intrinsically motivated behavior of socially interacting animals. We conclude that the recorded MB ENs are potentially involved in initiating and controlling social interactions in honeybees
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