62 research outputs found
Slow Convergence in Generalized Central Limit Theorems
We study the central limit theorem in the non-normal domain of attraction to
symmetric -stable laws for . We show that for i.i.d.
random variables , the convergence rate in of both the
densities and distributions of is at best
logarithmic if is a non-trivial slowly varying function. Asymptotic laws
for several physical processes have been derived using central limit theorems
with scaling and Gaussian limiting distributions. Our result
implies that such asymptotic laws are accurate only for exponentially large
.Comment: To appear in Comptes Rendus de l'Acad\'emie des Sciences,
Math\'ematique
Infinitesimal phase response functions can be misleading
Phase response functions are the central tool in the mathematical analysis of
pulse-coupled oscillators. When an oscillator receives a brief input pulse, the
phase response function specifies how its phase is altered, as a function of
the phase at which the input arrives. When the pulse is weak, it is customary
to linearize the dependence of the response on pulse strength. The result is
called the {\em infinitesimal} phase response function. These ideas have been
used extensively in theoretical biology and also in some areas of engineering.
I give an example in which the infinitesimal phase response function predicts
that two oscillators, as they exchange pulses back and fourth, will converge to
synchrony. Using the true phase response, I prove this prediction to be false
for all positive interaction strengths. For short, the analogue of the
Hartman-Grobman theorem that one might, at first sight, expect to hold is
invalid
Approximate, not perfect synchrony maximizes the downstream effectiveness of excitatory neuronal ensembles
The most basic functional role commonly ascribed to synchrony in the brain is that of amplifying excitatory neuronal signals. The reasoning is straightforward: When positive charge is injected into a leaky target neuron over a time window of positive duration, some of it will have time to leak back out before an action potential is triggered in the target, and it will in that sense be wasted. If the goal is to elicit a firing response in the target using as little charge as possible, it seems best to deliver the charge all at once, i.e., in perfect synchrony. In this article, we show that this reasoning is correct only if one assumes that the input ceases when the target crosses the firing threshold, but before it actually fires. If the input ceases later-for instance, in response to a feedback signal triggered by the firing of the target-the "most economical" way of delivering input (the way that requires the least total amount of input) is no longer precisely synchronous, but merely approximately so. If the target is a heterogeneous network, as it always is in the brain, then ceasing the input "when the target crosses the firing threshold" is not an option, because there is no single moment when the firing threshold is crossed. In this sense, precise synchrony is never optimal in the brain.R01 NS067199 - NINDS NIH HH
Domain imbedding methods for the Stokes equations
We study direct and iterative domain imbedding methods for the Stokes equations on certain non-rectangular domains in two space dimensions. We analyze a continuous analog of numerical domain imbedding for bounded, smooth domains, and give an example of a simple numerical algorithm suggested by the continuous analysis. This algorithms is applicable for simply connected domains which can be covered by rectangular grids, with uniformly spaced grid lines in at least one coordinate direction. We also discuss a related FFT-based fast solver for Stokes problems with physical boundary conditions on rectangles, and present some numerical results.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/46324/1/211_2005_Article_BF01386422.pd
A blue sky bifurcation in the dynamics of political candidates
Political candidates often shift their positions opportunistically in hopes
of capturing more votes. When there are only two candidates, the best strategy
for each of them is often to move towards the other. This eventually results in
two centrists with coalescing views. However, the strategy of moving towards
the other candidate ceases to be optimal when enough voters abstain instead of
voting for a centrist who does not represent their views. These observations,
formalized in various ways, have been made many times. Our own formalization is
based on differential equations. The surprise and main result derived from
these equations is that the final candidate positions can jump discontinuously
as the voters' loyalty towards their candidate wanes. The underlying
mathematical mechanism is a blue sky bifurcation
Neurosystems: brain rhythms and cognitive processing
Neuronal rhythms are ubiquitous features of brain dynamics, and are highly correlated with cognitive processing. However, the relationship between the physiological mechanisms producing these rhythms and the functions associated with the rhythms remains mysterious. This article investigates the contributions of rhythms to basic cognitive computations (such as filtering signals by coherence and/or frequency) and to major cognitive functions (such as attention and multi-modal coordination). We offer support to the premise that the physiology underlying brain rhythms plays an essential role in how these rhythms facilitate some cognitive operations.098352 - Wellcome Trust; 5R01NS067199 - NINDS NIH HH
ODEs and Mandatory Voting
This paper presents mathematics relevant to the question whether voting
should be mandatory. Assuming a static distribution of voters' political
beliefs, we model how politicians might adjust their positions to raise their
share of the vote. Various scenarios can be explored using our web-based app
(see text for the link). Abstentions are found to have great impact on the
dynamics of candidates, and in particular to introduce the possibility of
discontinuous jumps in optimal candidate positions. This is a paper intended
for undergraduate students. It is an unusual application of ODEs. We hope that
it might help engage some students who may find it harder to connect with the
more customary applications from the natural sciences.Comment: 16 page
Minimal Size of Cell Assemblies Coordinated by Gamma Oscillations
In networks of excitatory and inhibitory neurons with mutual synaptic coupling, specific drive to sub-ensembles of cells often leads to gamma-frequency (25–100 Hz) oscillations. When the number of driven cells is too small, however, the synaptic interactions may not be strong or homogeneous enough to support the mechanism underlying the rhythm. Using a combination of computational simulation and mathematical analysis, we study the breakdown of gamma rhythms as the driven ensembles become too small, or the synaptic interactions become too weak and heterogeneous. Heterogeneities in drives or synaptic strengths play an important role in the breakdown of the rhythms; nonetheless, we find that the analysis of homogeneous networks yields insight into the breakdown of rhythms in heterogeneous networks. In particular, if parameter values are such that in a homogeneous network, it takes several gamma cycles to converge to synchrony, then in a similar, but realistically heterogeneous network, synchrony breaks down altogether. This leads to the surprising conclusion that in a network with realistic heterogeneity, gamma rhythms based on the interaction of excitatory and inhibitory cell populations must arise either rapidly, or not at all. For given synaptic strengths and heterogeneities, there is a (soft) lower bound on the possible number of cells in an ensemble oscillating at gamma frequency, based simply on the requirement that synaptic interactions between the two cell populations be strong enough. This observation suggests explanations for recent experimental results concerning the modulation of gamma oscillations in macaque primary visual cortex by varying spatial stimulus size or attention level, and for our own experimental results, reported here, concerning the optogenetic modulation of gamma oscillations in kainate-activated hippocampal slices. We make specific predictions about the behavior of pyramidal cells and fast-spiking interneurons in these experiments.Collaborative Research in Computational NeuroscienceNational Institutes of Health (U.S.) (grant 1R01 NS067199)National Institutes of Health (U.S.) (grant DMS 0717670)National Institutes of Health (U.S.) (grant 1R01 DA029639)National Institutes of Health (U.S.) (grant 1RC1 MH088182)National Institutes of Health (U.S.) (grant DP2OD002002)Paul G. Allen Family FoundationnGoogle (Firm
The response of a classical Hodgkin–Huxley neuron to an inhibitory input pulse
A population of uncoupled neurons can often be brought close to synchrony by a single strong inhibitory input pulse affecting all neurons equally. This mechanism is thought to underlie some brain rhythms, in particular gamma frequency (30–80 Hz) oscillations in the hippocampus and neocortex. Here we show that synchronization by an inhibitory input pulse often fails for populations of classical Hodgkin–Huxley neurons. Our reasoning suggests that in general, synchronization by inhibitory input pulses can fail when the transition of the target neurons from rest to spiking involves a Hopf bifurcation, especially when inhibition is shunting, not hyperpolarizing. Surprisingly, synchronization is more likely to fail when the inhibitory pulse is stronger or longer-lasting. These findings have potential implications for the question which neurons participate in brain rhythms, in particular in gamma oscillations
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