5,841 research outputs found
Weak convergence of marked point processes generated by crossings of multivariate jump processes. Applications to neural network modeling
We consider the multivariate point process determined by the crossing times
of the components of a multivariate jump process through a multivariate
boundary, assuming to reset each component to an initial value after its
boundary crossing. We prove that this point process converges weakly to the
point process determined by the crossing times of the limit process. This holds
for both diffusion and deterministic limit processes. The almost sure
convergence of the first passage times under the almost sure convergence of the
processes is also proved. The particular case of a multivariate Stein process
converging to a multivariate Ornstein-Uhlenbeck process is discussed as a
guideline for applying diffusion limits for jump processes. We apply our
theoretical findings to neural network modeling. The proposed model gives a
mathematical foundation to the generalization of the class of Leaky
Integrate-and-Fire models for single neural dynamics to the case of a firing
network of neurons. This will help future study of dependent spike trains.Comment: 20 pages, 1 figur
Derivation of mean-field equations for stochastic particle systems
We study stochastic particle systems on a complete graph and derive effective
mean-field rate equations in the limit of diverging system size, which are also
known from cluster aggregation models. We establish the propagation of chaos
under generic growth conditions on particle jump rates, and the limit provides
a master equation for the single site dynamics of the particle system, which is
a non-linear birth death chain. Conservation of mass in the particle system
leads to conservation of the first moment for the limit dynamics, and to
non-uniqueness of stationary distributions. Our findings are consistent with
recent results on exchange driven growth, and provide a connection between the
well studied phenomena of gelation and condensation.Comment: 26 page
Some simple but challenging Markov processes
In this note, we present few examples of Piecewise Deterministic Markov
Processes and their long time behavior. They share two important features: they
are related to concrete models (in biology, networks, chemistry,. . .) and they
are mathematically rich. Their math-ematical study relies on coupling method,
spectral decomposition, PDE technics, functional inequalities. We also relate
these simple examples to recent and open problems
Quantitative ergodicity for some switched dynamical systems
We provide quantitative bounds for the long time behavior of a class of
Piecewise Deterministic Markov Processes with state space Rd \times E where E
is a finite set. The continuous component evolves according to a smooth vector
field that switches at the jump times of the discrete coordinate. The jump
rates may depend on the whole position of the process. Under regularity
assumptions on the jump rates and stability conditions for the vector fields we
provide explicit exponential upper bounds for the convergence to equilibrium in
terms of Wasserstein distances. As an example, we obtain convergence results
for a stochastic version of the Morris-Lecar model of neurobiology
A deep learning integrated Lee-Carter model
In the field of mortality, the Lee–Carter based approach can be considered the milestone
to forecast mortality rates among stochastic models. We could define a “Lee–Carter model family”
that embraces all developments of this model, including its first formulation (1992) that remains the
benchmark for comparing the performance of future models. In the Lee–Carter model, the kt parameter,
describing the mortality trend over time, plays an important role about the future mortality behavior.
The traditional ARIMA process usually used to model kt shows evident limitations to describe the future
mortality shape. Concerning forecasting phase, academics should approach a more plausible way in
order to think a nonlinear shape of the projected mortality rates. Therefore, we propose an alternative
approach the ARIMA processes based on a deep learning technique. More precisely, in order to catch
the pattern of kt series over time more accurately, we apply a Recurrent Neural Network with a Long
Short-Term Memory architecture and integrate the Lee–Carter model to improve its predictive capacity.
The proposed approach provides significant performance in terms of predictive accuracy and also allow
for avoiding the time-chunks’ a priori selection. Indeed, it is a common practice among academics to
delete the time in which the noise is overflowing or the data quality is insufficient. The strength of
the Long Short-Term Memory network lies in its ability to treat this noise and adequately reproduce it
into the forecasted trend, due to its own architecture enabling to take into account significant long-term
patterns
A new stochastic STDP Rule in a neural Network Model
Thought to be responsible for memory, synaptic plasticity has been widely
studied in the past few decades. One example of plasticity models is the
popular Spike Timing Dependent Plasticity (STDP). The huge litterature of STDP
models are mainly based deterministic rules whereas the biological mechanisms
involved are mainly stochastic ones. Moreover, there exist only few
mathematical studies on plasticity taking into account the precise spikes
timings. In this article, we aim at proposing a new stochastic STDP rule with
discrete synaptic weights which allows a mathematical analysis of the full
network dynamics under the hypothesis of separation of timescales. This model
attempts to answer the need for understanding the interplay between the weights
dynamics and the neurons ones
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