Statistical Problems Related to Excitation Threshold and Reset Value of Membrane Potentials

Transcription

1 Statistical Problems Related to Excitation Threshold and Reset Value of Membrane Potentials Le Mans - March 18, 2009

2 Contents 1 Biological Background and the Problem Definition 2 3 MDE with respect to the Laplace transform Conditions and Results for MDE 4 Simulated Data Real Data

3 Biological Background Figure: The Neuron

4 Membrane Potential (W. Kilb, Mainz) potential [mv] time [s] Figure: This membrane potential was recorded in vitro from a neuron belonging to a cortical slice preparation from a 6 week old mouse.

5 Usual Model (considered by Ditlevsen, Lansky, Genon-Catalot, Laredo... ) S } {{ } τ 1 }{{}... }{{} τ 2 τ n x 0 Assume the process X between spikes follows an SDE dx t = β θ (X t )dt + σ θ (X t )dw t.

6 Usual Model (considered by Ditlevsen, Lansky, Genon-Catalot, Laredo... ) S } {{ } τ 1 }{{}... }{{} τ 2 τ n x 0 Assume the process X between spikes follows an SDE dx t = β θ (X t )dt + σ θ (X t )dw t. x 0 and S are assumed to be known and θ R d is the parameter to determine by observation of iid inter spike times (level crossing times) τ i, i = 1,...,n.

7 Usual Model (considered by Ditlevsen, Lansky, Genon-Catalot, Laredo... ) S } {{ } τ 1 }{{}... }{{} τ 2 τ n x 0 Assume the process X between spikes follows an SDE dx t = β θ (X t )dt + σ θ (X t )dw t. x 0 and S are assumed to be known and θ R d is the parameter to determine by observation of iid inter spike times (level crossing times) τ i, i = 1,...,n. Nonparametric methods proposed by R. Höpfner (2006).

8 Usual Model (considered by Ditlevsen, Lansky, Genon-Catalot, Laredo... ) S } {{ } τ 1 }{{}... }{{} τ 2 τ n x 0 Assume the process X between spikes follows an SDE dx t = β θ (X t )dt + σ θ (X t )dw t. x 0 and S are assumed to be known and θ R d is the parameter to determine by observation of iid inter spike times (level crossing times) τ i, i = 1,...,n. Nonparametric methods proposed by R. Höpfner (2006).

9 Estimation of Excitation Threshold and Reset Value potential [mv] S? x 0? time [s]

10 The Statistical Problem θ 2 } {{ } τ 1 }{{}... }{{} τ 2 τ n Assume the process X between spikes follows a known SDE dx t = β(x t )dt + σ(x t )dw t. (β( ) and σ( ) are known!) θ 1

11 The Statistical Problem θ 2 } {{ } τ 1 }{{}... }{{} τ 2 τ n Assume the process X between spikes follows a known SDE dx t = β(x t )dt + σ(x t )dw t. (β( ) and σ( ) are known!) θ 1 Observation of iid inter spike times { } τ i := inf t 0 X (θ 1) t = θ 2, i = 1,...,n.

12 The Statistical Problem θ 2 } {{ } τ 1 }{{}... }{{} τ 2 τ n Assume the process X between spikes follows a known SDE dx t = β(x t )dt + σ(x t )dw t. (β( ) and σ( ) are known!) θ 1 Observation of iid inter spike times { } τ i := inf t 0 X (θ 1) t = θ 2, i = 1,...,n. θ 1 = x 0 and θ 2 = S are the parameters to determine.

13 The Statistical Problem θ 2 } {{ } τ 1 }{{}... }{{} τ 2 τ n Assume the process X between spikes follows a known SDE dx t = β(x t )dt + σ(x t )dw t. (β( ) and σ( ) are known!) θ 1 Observation of iid inter spike times { } τ i := inf t 0 X (θ 1) t = θ 2, i = 1,...,n. θ 1 = x 0 and θ 2 = S are the parameters to determine.

14 Assumptions & Strategy We consider the cases where X is a BMD, GBM, OU or CIR.

15 Assumptions & Strategy We consider the cases where X is a BMD, GBM, OU or CIR. We use MLE and LAN theory whenever it is possible.

16 Assumptions & Strategy We consider the cases where X is a BMD, GBM, OU or CIR. We use MLE and LAN theory whenever it is possible. In other cases we use MDE.

17 Assumptions & Strategy We consider the cases where X is a BMD, GBM, OU or CIR. We use MLE and LAN theory whenever it is possible. In other cases we use MDE.

18 Brownian Motion with Drift Let σ > 0, a 0 and (B t ) t 0 be a standard Brownian motion. Define X = (X t ) t 0 a BMD starting in X 0 = θ 1 by X t := θ 1 + at + σb t. Lebesgue-density of L(τ): t (0, ) f BMD (t) := (θ 2 θ 1 ) t 3/2 exp { ((θ 2 θ 1 ) at) 2 } 2πσ 2σ 2 t Note that a joint estimation of θ 1 and θ 2 is impossible!

19 Brownian Motion with Drift Let σ > 0, a 0 and (B t ) t 0 be a standard Brownian motion. Define X = (X t ) t 0 a BMD starting in X 0 = θ 1 by X t := θ 1 + at + σb t. Lebesgue-density of L(τ): t (0, ) f BMD (t) := (θ 2 θ 1 ) t 3/2 exp { ((θ 2 θ 1 ) at) 2 } 2πσ 2σ 2 t Note that a joint estimation of θ 1 and θ 2 is impossible!

20 What to do, if we know neither x 0 = θ 1 nor S = θ 2? θ 2 } {{ } τ 1 }{{}... }{{} τ 2 τ n θ 1

21 What to do, if we know neither x 0 = θ 1 nor S = θ 2? θ 2 } {{ } τ 1 }{{}... }{{} τ 2 τ n θ 1 median θ 2 θ 2 = ˆθ 1 τ 1 ˆτ 1 τ 2 ˆτ 2... τ n ˆτ n θ 1

22 Theorem for the Brownian Motion with Drift If X is a BMD, the corresponding sequences of experiments for θ 1 and θ 2 are LAN. Further, the MLEs (a ˆθ 1,n := θ 2 2 τh a ) 2 n 2 τh n + σ2 τ n h, (a ˆθ 2,n := θ τh a ) 2 n + 2 τh n + σ2 τ n h ) 1 where τ n h :=, are strongly consistent and LAM such that ( n i=1 1 τi ) L(θi ) n (ˆθ i,n θ i N n ( 0, (θ 2 θ 1 ) 2 σ 2 ) 2σ 2, i = 1, 2. + a(θ 2 θ 1 )

23 Theorem for the Brownian Motion with Drift If X is a BMD, the corresponding sequences of experiments for θ 1 and θ 2 are LAN. Further, the MLEs (a ˆθ 1,n := θ 2 2 τh a ) 2 n 2 τh n + σ2 τ n h, (a ˆθ 2,n := θ τh a ) 2 n + 2 τh n + σ2 τ n h ) 1 where τ n h :=, are strongly consistent and LAM such that ( n i=1 1 τi ) L(θi ) n (ˆθ i,n θ i N n ( 0, (θ 2 θ 1 ) 2 σ 2 ) 2σ 2, i = 1, 2. + a(θ 2 θ 1 ) With exponential scaling, the case of GMB yields a corresponding result!

24 Theorem for the Brownian Motion with Drift If X is a BMD, the corresponding sequences of experiments for θ 1 and θ 2 are LAN. Further, the MLEs (a ˆθ 1,n := θ 2 2 τh a ) 2 n 2 τh n + σ2 τ n h, (a ˆθ 2,n := θ τh a ) 2 n + 2 τh n + σ2 τ n h ) 1 where τ n h :=, are strongly consistent and LAM such that ( n i=1 1 τi ) L(θi ) n (ˆθ i,n θ i N n ( 0, (θ 2 θ 1 ) 2 σ 2 ) 2σ 2, i = 1, 2. + a(θ 2 θ 1 ) With exponential scaling, the case of GMB yields a corresponding result!

25 MDE with respect to the Laplace transform Conditions and Results for MDE MDE Framework for the cases of OU and CIR Observations (τ i ) i N iid inter spike times of the OU or the CIR neuronal model. An explicit expression for the density of L(τ) is not known! But

26 MDE with respect to the Laplace transform Conditions and Results for MDE MDE Framework for the cases of OU and CIR Observations (τ i ) i N iid inter spike times of the OU or the CIR neuronal model. An explicit expression for the density of L(τ) is not known! But we know the Laplace transform!

27 MDE with respect to the Laplace transform Conditions and Results for MDE MDE Framework for the cases of OU and CIR Observations (τ i ) i N iid inter spike times of the OU or the CIR neuronal model. An explicit expression for the density of L(τ) is not known! But we know the Laplace transform! Θ {θ R 2 : θ 1 < θ 2 } bounded

28 MDE with respect to the Laplace transform Conditions and Results for MDE MDE Framework for the cases of OU and CIR Observations (τ i ) i N iid inter spike times of the OU or the CIR neuronal model. An explicit expression for the density of L(τ) is not known! But we know the Laplace transform! Θ {θ R 2 : θ 1 < θ 2 } bounded H := L 2 (R +, B(R + ), µ), µ has a piecewise continuous Lebesgue density with compact support.

29 MDE with respect to the Laplace transform Conditions and Results for MDE MDE Framework for the cases of OU and CIR Observations (τ i ) i N iid inter spike times of the OU or the CIR neuronal model. An explicit expression for the density of L(τ) is not known! But we know the Laplace transform! Θ {θ R 2 : θ 1 < θ 2 } bounded H := L 2 (R +, B(R + ), µ), µ has a piecewise continuous Lebesgue density with compact support. Reference Values: LTs L θ (α) := E θ [e ατ ], θ Θ continuously parameterized in H,

30 MDE with respect to the Laplace transform Conditions and Results for MDE MDE Framework for the cases of OU and CIR Observations (τ i ) i N iid inter spike times of the OU or the CIR neuronal model. An explicit expression for the density of L(τ) is not known! But we know the Laplace transform! Θ {θ R 2 : θ 1 < θ 2 } bounded H := L 2 (R +, B(R + ), µ), µ has a piecewise continuous Lebesgue density with compact support. Reference Values: LTs L θ (α) := E θ [e ατ ], θ Θ continuously parameterized in H, Empirical Value: Emp. LT ˆL n (α) := 1 n e ατ i, α 0. n i=1

31 Minimum Distance Estimator MDE with respect to the Laplace transform Conditions and Results for MDE The MDE for this experiment is defined by This definition is not unique. θn := arg inf ˆL n L ξ H ξ Θ

32 Conditions MDE with respect to the Laplace transform Conditions and Results for MDE SLLN(θ): L θ ˆL n H P θ a.s. n 0 This is obviously fulfilled for all θ Θ.

33 Conditions MDE with respect to the Laplace transform Conditions and Results for MDE SLLN(θ): L θ ˆL n H P θ a.s. n 0 This is obviously fulfilled for all θ Θ.

34 Conditions MDE with respect to the Laplace transform Conditions and Results for MDE SLLN(θ): L θ ˆL n H P θ a.s. n 0 This is obviously fulfilled for all θ Θ. Identifiability I(θ): inf L θ L ξ H > 0, δ > 0. ξ Θ, θ ξ δ If L θ L ξ as ξ θ, the properties of the LT ensure, that I(θ) holds for all θ Θ.

35 Conditions MDE with respect to the Laplace transform Conditions and Results for MDE SLLN(θ): L θ ˆL n H P θ a.s. n 0 This is obviously fulfilled for all θ Θ. Identifiability I(θ): inf L θ L ξ H > 0, δ > 0. ξ Θ, θ ξ δ If L θ L ξ as ξ θ, the properties of the LT ensure, that I(θ) holds for all θ Θ.

36 Condition AN(θ) MDE with respect to the Laplace transform Conditions and Results for MDE Asymptotic Normality AN(θ): Define W n, n N: Wα n := ) n (ˆL n (α) L θ (α), then there exists a gaussian process W = W(θ) with covariance function K(, ) s.t. If we define W n L(θ) W in H. n K(α 1, α 2 ) := L θ (α 1 +α 2 ) L θ (α 1 )L θ (α 2 ) = Cov θ [ e α 1τ, e α 2 τ ] and use results from Cremers & Kadelka (1986) then AN(θ) holds for all θ Θ.

37 Condition AN(θ) MDE with respect to the Laplace transform Conditions and Results for MDE Asymptotic Normality AN(θ): Define W n, n N: Wα n := ) n (ˆL n (α) L θ (α), then there exists a gaussian process W = W(θ) with covariance function K(, ) s.t. If we define W n L(θ) W in H. n K(α 1, α 2 ) := L θ (α 1 +α 2 ) L θ (α 1 )L θ (α 2 ) = Cov θ [ e α 1τ, e α 2 τ ] and use results from Cremers & Kadelka (1986) then AN(θ) holds for all θ Θ.

38 The Ornstein-Uhlenbeck Case MDE with respect to the Laplace transform Conditions and Results for MDE Ornstein-Uhlenbeck Process dx t = (a bx t )dt + σdb t, X 0 = θ 1 The level-crossing time τ := inf Laplace transform { t 0 X (θ 1) t = θ 2 } has the (Roy / Smith 1969) L θ (α) := E θ [e ατ ] = H α/b ( ( θ 1 b) a ) b σ H α/b ( ( θ 2 a b) b σ ), α 0, where H is the Hermite function.

39 The Cox-Ingersoll-Ross Case MDE with respect to the Laplace transform Conditions and Results for MDE Cox-Ingersoll-Ross Process dx t = (a bx t )dt + σ X + t db t, X 0 = θ 0 0 The level-crossing time τ := inf Laplace transform { t 0 X (θ 1) t = θ 2 } has the (Göing-Jaeschke / Yor 1999,2003) ( α φ L θ (α) := E θ [e ατ b, 2a ) 2b σ2; σ ] = 2θ 1 ( α φ b, 2a ), α 0, 2b σ2; σ 2θ 2 where φ is the confluent hypergeometric function.

40 Condition D(θ) MDE with respect to the Laplace transform Conditions and Results for MDE Differentiability D(θ): The function Θ ξ L ξ H is Fréchet-differentiable at θ with derivation DL θ = (D 1 L θ,...,d d L θ ) (linearly independent components in H). In the OU and CIR case D(θ) holds for all θ Θ where the derivative consists of ratios of the corresponding special functions.

41 Condition D(θ) MDE with respect to the Laplace transform Conditions and Results for MDE Differentiability D(θ): The function Θ ξ L ξ H is Fréchet-differentiable at θ with derivation DL θ = (D 1 L θ,...,d d L θ ) (linearly independent components in H). In the OU and CIR case D(θ) holds for all θ Θ where the derivative consists of ratios of the corresponding special functions.

42 Results for MDE (Millar 1984) MDE with respect to the Laplace transform Conditions and Results for MDE Let the Conditions SLLN(θ), I(θ), D(θ), AN(θ) hold for every θ Θ. Then every sequence of MDE θ n, n N for θ is strongly consistent and is asymptotically normal where Σ θ := Λ 1 θ V θλ 1 θ with n (θ n θ) L(θ) N (0, Σ θ), n Λ θ := ( D i L θ, D j L θ H )1 i,j d the Gramian matrix of the derivatives and V R d d defined by (V θ ) i,j := 0 0 D i L θ (α 1 )K(α 1, α 2 )D j L θ (α 2 )µ(dα 1 )µ(dα 2 ).

43 Simulation of CIR ISI s Simulated Data Real Data Simulation of ISI s of the CIR neuronal model, dx t = (a b(x t c))dt + σ X t c db t, x 0 = 52.7, S = 49.3, σ = 1.8, a = 93.1, b = 28.7, c = 53.7

44 Simulation of CIR ISI s Simulated Data Real Data Simulation of ISI s of the CIR neuronal model, dx t = (a b(x t c))dt + σ X t c db t, x 0 = 52.7, S = 49.3, σ = 1.8, a = 93.1, b = 28.7, c = 53.7 H := L 2 (R +, B(R + ), µ), where µ := 100 j=1 δ α j and α j are suitable points.

45 Simulation of CIR ISI s Simulated Data Real Data Simulation of ISI s of the CIR neuronal model, dx t = (a b(x t c))dt + σ X t c db t, x 0 = 52.7, S = 49.3, σ = 1.8, a = 93.1, b = 28.7, c = 53.7 H := L 2 (R +, B(R + ), µ), where µ := 100 j=1 δ α j and α j are suitable points. Θ 1 := { 53.7, 53.69, 53.68,..., 50} Θ 2 := { 51, 50.99,..., 47} Θ := Θ 1 Θ 2 {θ 1 < θ 2 }.

46 Results of the MDE for θ 2 = S Simulated Data Real Data θ 2,n n Figure: Trajectory of the MDE for θ 2. The dashed line is the true simulation parameter S = 49.3.

47 Simulated Data Real Data Results of the MDE for θ 1 = x 0 θ 1,n e+00 2e+04 4e+04 6e+04 8e+04 1e+05 n Figure: Trajectory of the MDE for θ 1 = x 0. Only every hundredth step is plotted. The dashed line is the true simulation parameter x 0 = 52.7.

48 Simulated Data Real Data Asymptotic Variance of the MDE for θ 2 (Σ θ ) 2, θ θ 1 Figure: The asymptotic variance of θ 2,n depending on the true value θ Θ. The mean value of the stationary distribution of X is a/b + c

49 Simulated Data Real Data Asymptotic Variance of the MDE for θ 1 log 10 [(Σ θ ) 1,1 ] θ θ 1 Figure: The asymptotic variance of θ 1,n in logarithmic scale depending on the true value θ Θ. a/b + c

50 Simulated Data Real Data Asymptotic Correlation of the MDE Components Cor θ (θ ) < 0 Cor θ (θ ) > 0 Cor θ (θ ) θ Figure: The asymptotic correlation coefficient between θ 1,n and θ 2,n. a/b + c θ 1 0.0

51 Simulated Data Real Data A Real Data Set (W. Kilb, Mainz) Figure: This membrane potential includes 46 ISIs. After fixing the CIR model by a nonparametric method, proposed by R. Höpfner (2006), the MDE finds θ = ( 53.74, 49.28) for x 0 and S given by the dashed lines.

52 Fit of the Laplace Transform Simulated Data Real Data Figure: The black rings are the values of the empirical LT ˆL n for n = 46 at the points α j, j = 1,..., 100 which are the mass points of µ. The red line is the estimated LT L θ for θ = ( 53.74, 49.28) that minimizes the distance of these two functions in H, where ˆL n L θ H

53 Résumé Simulated Data Real Data In the cases BMD and GBM the MLE of θ 1 and θ 2 is explicit and its asymptotics are optimal (LAN LAM). In the cases OU and CIR a MLE is not explicitly given, so we take the MDE which is easier to handle because the LT is explicit. A joint estimation of θ 1 and θ 2 is possible, but we need lots of observations (τ 1,...,τ n ) to estimate θ 1. However, in applications θ 1 is a parameter of secondary importance. Further, the distance between emp. and estimated LT gives an indication if the model reproduces the true spiking behavior.

54 Simulated Data Real Data Thank you for your attention! Questions?

55 Simulated Data Real Data Cremers, Kadelka: On weak convergence of integral functionals of stoch. processes with appl. to processes taking paths in L E p. Stoch. Proc. Appl. Vol 21 (1986). A.Göing-Jaeschke, M.Yor: A Survey and Some Generalizations of Bessel Processes. Bernoulli 9 (2003). R.Höpfner: Mathematische Statistik 07/08: hoepfner/material-mathstat.html R.Höpfner: On a set of data for the membrane potential in a neuron. Math. Biosci. 207 (2007). P.Millar: A General Approach to the Optimality of Minimum Distance Estimators. Trans. Amer. Math. Soc., Vol. 286, No. 1. (1984). Roy, Smith: Analysis of the Exponential Decay Model.... Bull. Math. Biophys., Vol. 31 (1969).