173 research outputs found
Multi-class oscillating systems of interacting neurons
We consider multi-class systems of interacting nonlinear Hawkes processes
modeling several large families of neurons and study their mean field limits.
As the total number of neurons goes to infinity we prove that the evolution
within each class can be described by a nonlinear limit differential equation
driven by a Poisson random measure, and state associated central limit
theorems. We study situations in which the limit system exhibits oscillatory
behavior, and relate the results to certain piecewise deterministic Markov
processes and their diffusion approximations.Comment: 6 figure
Estimation in the partially observed stochastic Morris-Lecar neuronal model with particle filter and stochastic approximation methods
Parameter estimation in multidimensional diffusion models with only one
coordinate observed is highly relevant in many biological applications, but a
statistically difficult problem. In neuroscience, the membrane potential
evolution in single neurons can be measured at high frequency, but biophysical
realistic models have to include the unobserved dynamics of ion channels. One
such model is the stochastic Morris-Lecar model, defined by a nonlinear
two-dimensional stochastic differential equation. The coordinates are coupled,
that is, the unobserved coordinate is nonautonomous, the model exhibits
oscillations to mimic the spiking behavior, which means it is not of
gradient-type, and the measurement noise from intracellular recordings is
typically negligible. Therefore, the hidden Markov model framework is
degenerate, and available methods break down. The main contributions of this
paper are an approach to estimate in this ill-posed situation and nonasymptotic
convergence results for the method. Specifically, we propose a sequential Monte
Carlo particle filter algorithm to impute the unobserved coordinate, and then
estimate parameters maximizing a pseudo-likelihood through a stochastic version
of the Expectation-Maximization algorithm. It turns out that even the rate
scaling parameter governing the opening and closing of ion channels of the
unobserved coordinate can be reasonably estimated. An experimental data set of
intracellular recordings of the membrane potential of a spinal motoneuron of a
red-eared turtle is analyzed, and the performance is further evaluated in a
simulation study.Comment: Published in at http://dx.doi.org/10.1214/14-AOAS729 the Annals of
Applied Statistics (http://www.imstat.org/aoas/) by the Institute of
Mathematical Statistics (http://www.imstat.org
The space-clamped Hodgkin-Huxley system with random synaptic input: inhibition of spiking by weak noise and analysis with moment equations
We consider a classical space-clamped Hodgkin-Huxley model neuron stimulated
by synaptic excitation and inhibition with conductances represented by
Ornstein-Uhlenbeck processes. Using numerical solutions of the stochastic model
system obtained by an Euler method, it is found that with excitation only there
is a critical value of the steady state excitatory conductance for repetitive
spiking without noise and for values of the conductance near the critical value
small noise has a powerfully inhibitory effect. For a given level of inhibition
there is also a critical value of the steady state excitatory conductance for
repetitive firing and it is demonstrated that noise either in the excitatory or
inhibitory processes or both can powerfully inhibit spiking. Furthermore, near
the critical value, inverse stochastic resonance was observed when noise was
present only in the inhibitory input process.
The system of 27 coupled deterministic differential equations for the
approximate first and second order moments of the 6-dimensional model is
derived. The moment differential equations are solved using Runge-Kutta methods
and the solutions are compared with the results obtained by simulation for
various sets of parameters including some with conductances obtained by
experiment on pyramidal cells of rat prefrontal cortex. The mean and variance
obtained from simulation are in good agreement when there is spiking induced by
strong stimulation and relatively small noise or when the voltage is
fluctuating at subthreshold levels. In the occasional spike mode sometimes
exhibited by spinal motoneurons and cortical pyramidal cells the assunptions
underlying the moment equation approach are not satisfied
Warning of a forthcoming collapse of the Atlantic meridional overturning circulation
Tipping to an undesired state in the climate when a control parameter slowly
approaches a critical value is a growing concern with increasing greenhouse gas
concentrations. Predictions rely on detecting early warning signals (EWSs) in
observations of the system. The primary EWSs are increase in variance, (loss of
resilience), and increased autocorrelation (critical slow down). These measures
are statistical in nature, which implies that the reliability and statistical
significance of the detection depends on the sample size in observations and
the magnitude of the change away from the base value prior to the approach to
the tipping point. Thus, the possibility of providing useful early warning
depends on the relative magnitude of several interdependent time scales in the
system. These are (a) the time before the critical value is reached, (b) the
(inverse) rate of approach to the tipping point, (c) the size of the time
window required to detect a significant change in the EWS and finally, (d) the
escape time for noise-induced transition (prior to the tipping). Conditions for
early warning of tipping of the Atlantic meridional overturning circulation
(AMOC) are marginally fulfilled for the existing past 150 years of proxy
observations where indicators of tipping have recently been reported. Here we
provide statistical significance and data driven estimators for the time of
tipping. We estimate a collapse of the AMOC to occur around the year 2057 under
the assumption of a "business as usual" scenario of future emissions.Comment: 18 pages, 7 figure
The Morris-Lecar neuron model embeds a leaky integrate-and-fire model
We show that the stochastic Morris-Lecar neuron, in a neighborhood of its
stable point, can be approximated by a two-dimensional Ornstein-Uhlenbeck (OU)
modulation of a constant circular motion. The associated radial OU process is
an example of a leaky integrate-and-fire (LIF) model prior to firing. A new
model constructed from a radial OU process together with a simple firing
mechanism based on detailed Morris-Lecar firing statistics reproduces the
Morris-Lecar Interspike Interval (ISI) distribution, and has the computational
advantages of a LIF. The result justifies the large amount of attention paid to
the LIF models.Comment: 19 pages, 6 figure
Neural decoding with visual attention using sequential Monte Carlo for leaky integrate-and-fire neurons
How the brain makes sense of a complicated environment is an important question, and a first step is to be able to reconstruct the stimulus that give rise to an observed brain response. Neural coding relates neurobiological observations to external stimuli using computational methods. Encoding refers to how a stimulus affects the neuronal output, and entails constructing a neural model and parameter estimation. Decoding refers to reconstruction of the stimulus that led to a given neuronal output. Existing decoding methods rarely explain neuronal responses to complicated stimuli in a principled way. Here we perform neural decoding for a mixture of multiple stimuli using the leaky integrate-and-fire model describing neural spike trains, under the visual attention hypothesis of probability mixing in which the neuron only attends to a single stimulus at any given time. We assume either a parallel or serial processing visual search mechanism when decoding multiple simultaneous neurons. We consider one or multiple stochastic stimuli following Ornstein-Uhlenbeck processes, and dynamic neuronal attention that switches following discrete Markov processes. To decode stimuli in such situations, we develop various sequential Monte Carlo particle methods in different settings. The likelihood of the observed spike trains is obtained through the first-passage time probabilities obtained by solving the Fokker-Planck equations. We show that the stochastic stimuli can be successfully decoded by sequential Monte Carlo, and different particle methods perform differently considering the number of observed spike trains, the number of stimuli, model complexity, etc. The proposed novel decoding methods, which analyze the neural data through psychological visual attention theories, provide new perspectives to understand the brain
Weighted Reduced Rank Estimators Under Cointegration Rank Uncertainty
Cointegration analysis was developed for non-stationary linear processes that
exhibit stationary relationships between coordinates. Estimation of the
cointegration relationships in a multi-dimensional cointegrated process
typically proceeds in two steps. First the rank is estimated, then the
cointegration matrix is estimated, conditionally on the estimated rank (reduced
rank regression). The asymptotics of the estimator is usually derived under the
assumption of knowing the true rank. In this paper, we quantify the asymptotic
bias and find the asymptotic distributions of the cointegration estimator in
case of misspecified rank. Furthermore, we suggest a new class of weighted
reduced rank estimators that allow for more flexibility in settings where rank
selection is hard. We show empirically that a proper choice of weights can lead
to increased predictive performance when there is rank uncertainty. Finally, we
illustrate the estimators on empirical EEG data from a psychological experiment
on visual processing
Uniform Inference for Cointegrated Vector Autoregressive Processes
Uniformly valid inference for cointegrated vector autoregressive processes
has so far proven difficult due to certain discontinuities arising in the
asymptotic distribution of the least squares estimator. We show how asymptotic
results from the univariate case can be extended to multiple dimensions and how
inference can be based on these results. Furthermore, we show that the novel
instrumental variable procedure proposed by [20] (IVX) yields uniformly valid
confidence regions for the entire autoregressive matrix. The results are
applied to two specific examples for which we verify the theoretical findings
and investigate finite sample properties in simulation experiments
Fokker–Planck and Fortet Equation-Based Parameter Estimation for a Leaky Integrate-and-Fire Model with Sinusoidal and Stochastic Forcing
Abstract
Analysis of sinusoidal noisy leaky integrate-and-fire models and comparison with experimental data are important to understand the neural code and neural synchronization and rhythms. In this paper, we propose two methods to estimate input parameters using interspike interval data only. One is based on numerical solutions of the Fokker–Planck equation, and the other is based on an integral equation, which is fulfilled by the interspike interval probability density. This generalizes previous methods tailored to stationary data to the case of time-dependent input. The main contribution is a binning method to circumvent the problems of nonstationarity, and an easy-to-implement initializer for the numerical procedures. The methods are compared on simulated data.
List of Abbreviations
LIF: Leaky integrate-and-fire
ISI: Interspike interval
SDE: Stochastic differential equation
PDE: Partial differential equatio
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