Polynomials under Ornstein–Uhlenbeck noise and an application to inference in stochastic Hodgkin–Huxley systems

Reinhard Höpfner ()
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Reinhard Höpfner: Universität Mainz

Statistical Inference for Stochastic Processes, 2021, vol. 24, issue 1, No 3, 35-59

Abstract: Abstract We discuss estimation problems where a polynomial $$s\rightarrow \sum _{i=0}^\ell \vartheta _i s^i$$ s → ∑ i = 0 ℓ ϑ i s i with strictly positive leading coefficient is observed under Ornstein–Uhlenbeck noise over a long time interval. We prove local asymptotic normality (LAN) and specify asymptotically efficient estimators. We apply this to the following problem: feeding noise $$dY_t$$ d Y t into the classical (deterministic) Hodgkin–Huxley model in neuroscience, with $$Y_t=\vartheta t + X_t$$ Y t = ϑ t + X t and X some Ornstein–Uhlenbeck process with backdriving force $$\tau $$ τ , we have asymptotically efficient estimators for the pair $$(\vartheta ,\tau )$$ ( ϑ , τ ) ; based on observation of the membrane potential up to time n, the estimate for $$\vartheta $$ ϑ converges at rate $$\sqrt{n^3\,}$$ n 3 .

Keywords: Diffusion models; Local asymptotic normality; Asymptotically efficient estimators; Degenerate diffusions; Stochastic Hodgkin–Huxley model; 62F12; 60J60 (search for similar items in EconPapers)
Date: 2021
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DOI: 10.1007/s11203-020-09226-0

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