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Non-parametric estimation of the spiking rate in systems of interacting neurons
- Source :
- Statistical Inference for Stochastic Processes, Statistical Inference for Stochastic Processes, Springer Verlag, 2018, 21 (1), pp.81-111. ⟨10.1007/s11203-016-9150-4⟩, Statistical Inference for Stochastic Processes, Springer Verlag, 2018, 21 (1), pp.81-111. 〈10.1007/s11203-016-9150-4〉, Statistical Inference for Stochastic Processes, 2018, 21 (1), pp.81-111. ⟨10.1007/s11203-016-9150-4⟩
- Publication Year :
- 2018
- Publisher :
- HAL CCSD, 2018.
-
Abstract
- We consider a model of interacting neurons where the membrane potentials of the neurons are described by a multidimensional piecewise deterministic Markov process with values in $${\mathbb {R}}^N, $$ where N is the number of neurons in the network. A deterministic drift attracts each neuron’s membrane potential to an equilibrium potential m. When a neuron jumps, its membrane potential is reset to a resting potential, here 0, while the other neurons receive an additional amount of potential $$\frac{1}{N}.$$ We are interested in the estimation of the jump (or spiking) rate of a single neuron based on an observation of the membrane potentials of the N neurons up to time t. We study a Nadaraya–Watson type kernel estimator for the jump rate and establish its rate of convergence in $$L^2 .$$ This rate of convergence is shown to be optimal for a given Holder class of jump rate functions. We also obtain a central limit theorem for the error of estimation. The main probabilistic tools are the uniform ergodicity of the process and a fine study of the invariant measure of a single neuron.
- Subjects :
- Statistics and Probability
Kernel density estimation
biological neural nets
Mathematics - Statistics Theory
Statistics Theory (math.ST)
Type (model theory)
01 natural sciences
010104 statistics & probability
62G05, 60J75, 62M0
[MATH.MATH-ST]Mathematics [math]/Statistics [math.ST]
FOS: Mathematics
62G05, 60J75, 62M05
Piecewise-deterministic Markov process
[ MATH.MATH-ST ] Mathematics [math]/Statistics [math.ST]
Piecewise deterministic Markov processes
0101 mathematics
ComputingMilieux_MISCELLANEOUS
Mathematics
Central limit theorem
Quantitative Biology::Neurons and Cognition
010102 general mathematics
Mathematical analysis
Ergodicity
Estimator
nonparametric estimation
Kernel estimation
Rate of convergence
nervous system
Invariant measure
Subjects
Details
- Language :
- English
- ISSN :
- 13870874 and 15729311
- Database :
- OpenAIRE
- Journal :
- Statistical Inference for Stochastic Processes, Statistical Inference for Stochastic Processes, Springer Verlag, 2018, 21 (1), pp.81-111. ⟨10.1007/s11203-016-9150-4⟩, Statistical Inference for Stochastic Processes, Springer Verlag, 2018, 21 (1), pp.81-111. 〈10.1007/s11203-016-9150-4〉, Statistical Inference for Stochastic Processes, 2018, 21 (1), pp.81-111. ⟨10.1007/s11203-016-9150-4⟩
- Accession number :
- edsair.doi.dedup.....d3cd1e4707bd2529be280de9032f03a0
- Full Text :
- https://doi.org/10.1007/s11203-016-9150-4⟩