0000000000816203

AUTHOR

Michèle Thieullen

showing 3 related works from this author

Transition densities for stochastic Hodgkin-Huxley models

2012

We consider a stochastic Hodgkin-Huxley model driven by a periodic signal as model for the membrane potential of a pyramidal neuron. The associated five dimensional diffusion process is a time inhomogeneous highly degenerate diffusion for which the weak Hoermander condition holds only locally. Using a technique which is based on estimates of the Fourier transform, inspired by Fournier 2008, Bally 2007 and De Marco 2011, we show that the process admits locally a strictly positive continuous transition density. Moreover, we show that the presence of noise enables the stochastic system to imitate any possible deterministic spiking behavior, i.e. mixtures of regularly spiking and non-spiking ti…

Quantitative Biology::Neurons and CognitionProbability (math.PR)FOS: Mathematics60 J 60Mathematics - Probability
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Ergodicity for a stochastic Hodgkin–Huxley model driven by Ornstein–Uhlenbeck type input

2013

We consider a model describing a neuron and the input it receives from its dendritic tree when this input is a random perturbation of a periodic deterministic signal, driven by an Ornstein-Uhlenbeck process. The neuron itself is modeled by a variant of the classical Hodgkin-Huxley model. Using the existence of an accessible point where the weak Hoermander condition holds and the fact that the coefficients of the system are analytic, we show that the system is non-degenerate. The existence of a Lyapunov function allows to deduce the existence of (at most a finite number of) extremal invariant measures for the process. As a consequence, the complexity of the system is drastically reduced in c…

Statistics and ProbabilityDegenerate diffusion processesWeak Hörmander conditionType (model theory)01 natural sciencesPeriodic ergodicity010104 statistics & probability60H0760J25FOS: Mathematics0101 mathematicsComputingMilieux_MISCELLANEOUSMathematical physicsMathematics60J60Quantitative Biology::Neurons and CognitionProbability (math.PR)010102 general mathematicsErgodicityOrnstein–Uhlenbeck processHodgkin–Huxley model[MATH.MATH-PR]Mathematics [math]/Probability [math.PR]Hodgkin–Huxley model60J60 60J25 60H07Statistics Probability and UncertaintyTime inhomogeneous diffusion processesMathematics - Probability
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Ergodicity and limit theorems for degenerate diffusions with time periodic drift. Application to a stochastic Hodgkin−Huxley model

2016

We formulate simple criteria for positive Harris recurrence of strongly degenerate stochastic differential equations with smooth coefficients on a state space with certain boundary conditions. The drift depends on time and space and is periodic in the time argument. There is no time dependence in the diffusion coefficient. Control systems play a key role, and we prove a new localized version of the support theorem. Beyond existence of some Lyapunov function, we only need one attainable inner point of full weak Hoermander dimension. Our motivation comes from a stochastic Hodgkin−Huxley model for a spiking neuron including its dendritic input. This input carries some deterministic periodic si…

Statistics and ProbabilityLyapunov function010102 general mathematicsErgodicityDegenerate energy levels01 natural sciencesPeriodic function010104 statistics & probabilitysymbols.namesakeStochastic differential equationsymbolsState spaceApplied mathematicsLimit (mathematics)0101 mathematicsBrownian motionMathematicsESAIM: Probability and Statistics
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