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Modelling Membrane Potentials by Diffusion Leaky Integrate-and-Fire - - PowerPoint PPT Presentation

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Diffusion Leaky Integrate-and-Fire Neuronal Models Modelling by time-inhomogeneous Diffusion Models Modelling Membrane Potentials by Diffusion Leaky Integrate-and-Fire Models Patrick Jahn DEPARTMENT OF MATHEMATICAL SCIENCES UNIVERSITY OF

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Diffusion Leaky Integrate-and-Fire Neuronal Models Modelling by time-inhomogeneous Diffusion Models

Modelling Membrane Potentials by Diffusion Leaky Integrate-and-Fire Models

Patrick Jahn

DEPARTMENT OF MATHEMATICAL SCIENCES UNIVERSITY OF COPENHAGEN jahn@math.ku.dk

CIRM, Marseille - January 18, 2009

Patrick Jahn - jahn@math.ku.dk Leaky Integrate-and-Fire Models

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Diffusion Leaky Integrate-and-Fire Neuronal Models Modelling by time-inhomogeneous Diffusion Models

Contents

1 Diffusion Leaky Integrate-and-Fire Neuronal Models 2 Modelling Membrane Potentials in Motoneurons by

time-inhomogeneous Diffusion Models

Patrick Jahn - jahn@math.ku.dk Leaky Integrate-and-Fire Models

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Diffusion Leaky Integrate-and-Fire Neuronal Models Modelling by time-inhomogeneous Diffusion Models

Biological Background

Figure: The Neuron

Patrick Jahn - jahn@math.ku.dk Leaky Integrate-and-Fire Models

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Diffusion Leaky Integrate-and-Fire Neuronal Models Modelling by time-inhomogeneous Diffusion Models

Membrane Potential recorded by W. Kilb, Mainz

1 2 3 4 5 6 −50 −40 −30 −20 time [s] potential [mV]

Figure: This membrane potential was recorded in vitro from a pyramidal neuron belonging to cortical slice preparation of a juvenile C57bl/6 mouse.

Patrick Jahn - jahn@math.ku.dk Leaky Integrate-and-Fire Models

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Diffusion Leaky Integrate-and-Fire Neuronal Models Modelling by time-inhomogeneous Diffusion Models

Diffusion Leaky Integrate-and-Fire Neuronal Model

x0 S

  • T1

T2

. . .

Tn

Assume the process X between spikes is a Diffusion process given by dXt = 1 τ (a − Xt)dt + σ(Xt)dBt

Patrick Jahn - jahn@math.ku.dk Leaky Integrate-and-Fire Models

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Diffusion Leaky Integrate-and-Fire Neuronal Models Modelling by time-inhomogeneous Diffusion Models

Diffusion Leaky Integrate-and-Fire Neuronal Model

x0 S

  • T1

T2

. . .

Tn

Assume the process X between spikes is a Diffusion process given by dXt = 1 τ (a − Xt)dt + σ(Xt)dBt Goal: Estimate β(·) := 1

τ (a − ·), σ(·), x0, S from discrete

  • bservations of X and from the observation of iid ISIs (level

crossing times) Ti, i = 1, . . . , n..

Patrick Jahn - jahn@math.ku.dk Leaky Integrate-and-Fire Models

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Diffusion Leaky Integrate-and-Fire Neuronal Models Modelling by time-inhomogeneous Diffusion Models

Diffusion Leaky Integrate-and-Fire Neuronal Model

x0 S

  • T1

T2

. . .

Tn

Assume the process X between spikes is a Diffusion process given by dXt = 1 τ (a − Xt)dt + σ(Xt)dBt Goal: Estimate β(·) := 1

τ (a − ·), σ(·), x0, S from discrete

  • bservations of X and from the observation of iid ISIs (level

crossing times) Ti, i = 1, . . . , n..

Patrick Jahn - jahn@math.ku.dk Leaky Integrate-and-Fire Models

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Diffusion Leaky Integrate-and-Fire Neuronal Models Modelling by time-inhomogeneous Diffusion Models

Problem of Finding Excitation Threshold and Reset Value

1.0 1.5 2.0 2.5 3.0 3.5 −52 −51 −50 −49 −48 −47 −46 time [s] potential [mV]

S ? x0 ?

Patrick Jahn - jahn@math.ku.dk Leaky Integrate-and-Fire Models

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Diffusion Leaky Integrate-and-Fire Neuronal Models Modelling by time-inhomogeneous Diffusion Models

Problem of Finding Excitation Threshold and Reset Value

−0.04 −0.02 0.00 0.02 0.04 −50 −40 −30 −20 −10

17Sept08_023.asc alle spikes uebereinanderlegen / spikezeiten auf 0 transformieren

17Sept08_023.asc [mV]

Patrick Jahn - jahn@math.ku.dk Leaky Integrate-and-Fire Models

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Diffusion Leaky Integrate-and-Fire Neuronal Models Modelling by time-inhomogeneous Diffusion Models

Strategy

θ1 θ2

  • T1

T2

. . .

Tn

  • 1. Fixing drift and diffusion coefficient with nonparametric

methods proposed by R. H¨

  • pfner (2006). So assume X is

given by dXt = 1 τ (a − Xt)dt + σ(Xt)dBt

Patrick Jahn - jahn@math.ku.dk Leaky Integrate-and-Fire Models

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Diffusion Leaky Integrate-and-Fire Neuronal Models Modelling by time-inhomogeneous Diffusion Models

Strategy

θ1 θ2

  • T1

T2

. . .

Tn

  • 1. Fixing drift and diffusion coefficient with nonparametric

methods proposed by R. H¨

  • pfner (2006). So assume X is

given by dXt = 1 τ (a − Xt)dt + σ(Xt)dBt

  • 2. Estimate θ1 = x0 and θ2 = S from the observation of iid inter

spike times Ti := inf

  • t ≥ 0 | X (θ1)

t

= θ2

  • ,

i = 1, . . . , n.

Patrick Jahn - jahn@math.ku.dk Leaky Integrate-and-Fire Models

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Diffusion Leaky Integrate-and-Fire Neuronal Models Modelling by time-inhomogeneous Diffusion Models

Strategy

θ1 θ2

  • T1

T2

. . .

Tn

  • 1. Fixing drift and diffusion coefficient with nonparametric

methods proposed by R. H¨

  • pfner (2006). So assume X is

given by dXt = 1 τ (a − Xt)dt + σ(Xt)dBt

  • 2. Estimate θ1 = x0 and θ2 = S from the observation of iid inter

spike times Ti := inf

  • t ≥ 0 | X (θ1)

t

= θ2

  • ,

i = 1, . . . , n.

Patrick Jahn - jahn@math.ku.dk Leaky Integrate-and-Fire Models

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Diffusion Leaky Integrate-and-Fire Neuronal Models Modelling by time-inhomogeneous Diffusion Models

Some Facts. . .

In the most famous cases, where X is an Ornstein-Uhlenbeck (OU) or a Cox-Ingersoll-Ross (CIR) process, no explicit expression for the density of the level crossing time T is

  • known. However, we know its Laplace transfrom (LT) as a

ratio of special functions. (Jahn, PhD-Thesis 2009): For the OU and CIR cases the estimation problem of θ1 = x0 and θ2 = S was solved by using Millar’s framework (1984) for minimum distance estimators via the comparison of empirical and theoretical LT with respect to a suitable Hilbert space norm . . . .

Patrick Jahn - jahn@math.ku.dk Leaky Integrate-and-Fire Models

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Diffusion Leaky Integrate-and-Fire Neuronal Models Modelling by time-inhomogeneous Diffusion Models

Some Facts. . .

In the most famous cases, where X is an Ornstein-Uhlenbeck (OU) or a Cox-Ingersoll-Ross (CIR) process, no explicit expression for the density of the level crossing time T is

  • known. However, we know its Laplace transfrom (LT) as a

ratio of special functions. (Jahn, PhD-Thesis 2009): For the OU and CIR cases the estimation problem of θ1 = x0 and θ2 = S was solved by using Millar’s framework (1984) for minimum distance estimators via the comparison of empirical and theoretical LT with respect to a suitable Hilbert space norm . . . .

Patrick Jahn - jahn@math.ku.dk Leaky Integrate-and-Fire Models

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Diffusion Leaky Integrate-and-Fire Neuronal Models Modelling by time-inhomogeneous Diffusion Models

The Minimum Distance Estimator w.r.t. the LT

Theorem (J. 2009) Let X be an OU or a CIR process. Define the empirical LT of the level crossing time T of X from θ1 to θ2 and the corresponding family of possibly true LTs by ˆ Ln(α) := 1 n

n

  • i=1

e−αTi and Lθ(α) := Eθ[e−αT], then the MDE w.r.t. the LT θ∗

n := arg inf θ1<θ2 ˆ

Ln − LθH is strongly consistent and asymptotically normal.

Patrick Jahn - jahn@math.ku.dk Leaky Integrate-and-Fire Models

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Diffusion Leaky Integrate-and-Fire Neuronal Models Modelling by time-inhomogeneous Diffusion Models

The Pearson Diffusion Case (work with Jesper L. Pedersen. . . )

Pearson Diffusion dXt = 1

τ (a − Xt)dt + σ

  • (Xt − c)2 + ddBt,

X0 = θ1 ≥ 0 Jesper computed the Laplace transform of T: Eθ[e−αT] = gRe(θ1, α) − K(α) · hRe(θ1, α) gRe(θ2, α) − K(α) · hRe(θ2, α) where g, h and K are expressions of hypergemetric functions where the parameters of the diffusion X enter.

Patrick Jahn - jahn@math.ku.dk Leaky Integrate-and-Fire Models

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Diffusion Leaky Integrate-and-Fire Neuronal Models Modelling by time-inhomogeneous Diffusion Models

The Pearson Diffusion Case (work with Jesper L. Pedersen. . . )

Pearson Diffusion dXt = 1

τ (a − Xt)dt + σ

  • (Xt − c)2 + ddBt,

X0 = θ1 ≥ 0 Jesper computed the Laplace transform of T: Eθ[e−αT] = gRe(θ1, α) − K(α) · hRe(θ1, α) gRe(θ2, α) − K(α) · hRe(θ2, α) where g, h and K are expressions of hypergemetric functions where the parameters of the diffusion X enter. Goal: Solve the identifiability problem and apply the developed Framework to this case...

Patrick Jahn - jahn@math.ku.dk Leaky Integrate-and-Fire Models

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Diffusion Leaky Integrate-and-Fire Neuronal Models Modelling by time-inhomogeneous Diffusion Models

The Pearson Diffusion Case (work with Jesper L. Pedersen. . . )

Pearson Diffusion dXt = 1

τ (a − Xt)dt + σ

  • (Xt − c)2 + ddBt,

X0 = θ1 ≥ 0 Jesper computed the Laplace transform of T: Eθ[e−αT] = gRe(θ1, α) − K(α) · hRe(θ1, α) gRe(θ2, α) − K(α) · hRe(θ2, α) where g, h and K are expressions of hypergemetric functions where the parameters of the diffusion X enter. Goal: Solve the identifiability problem and apply the developed Framework to this case...

Patrick Jahn - jahn@math.ku.dk Leaky Integrate-and-Fire Models

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Diffusion Leaky Integrate-and-Fire Neuronal Models Modelling by time-inhomogeneous Diffusion Models

Contents

1 Diffusion Leaky Integrate-and-Fire Neuronal Models 2 Modelling Membrane Potentials in Motoneurons by

time-inhomogeneous Diffusion Models

Patrick Jahn - jahn@math.ku.dk Leaky Integrate-and-Fire Models

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Diffusion Leaky Integrate-and-Fire Neuronal Models Modelling by time-inhomogeneous Diffusion Models

Joint work with R. Berg, J. Hounsgaard and S. Ditlevsen

5000 10000 15000 20000 25000 30000 −110 −90 −80 −70 −60 −50

trace = 32

time (ms) V(t) (mV)

Figure: Data collected by Rune Berg form a motoneuron of an active network in the spinal cord of a turtle, during mechanical stimulation.

Patrick Jahn - jahn@math.ku.dk Leaky Integrate-and-Fire Models

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Diffusion Leaky Integrate-and-Fire Neuronal Models Modelling by time-inhomogeneous Diffusion Models

A First Approach

Assumption: X is a Diffusion process with time inhomogeneous drift, dXt = β(Xt, t)dt + σ(Xt)dBt. Analysis: Nonparametric estimation of drift and diffusion

  • coefficient. For the drift analysis: Cutting the trajectory into

”homogeneous” parts and analyse them separately.

Patrick Jahn - jahn@math.ku.dk Leaky Integrate-and-Fire Models

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Diffusion Leaky Integrate-and-Fire Neuronal Models Modelling by time-inhomogeneous Diffusion Models

A First Approach

Assumption: X is a Diffusion process with time inhomogeneous drift, dXt = β(Xt, t)dt + σ(Xt)dBt. Analysis: Nonparametric estimation of drift and diffusion

  • coefficient. For the drift analysis: Cutting the trajectory into

”homogeneous” parts and analyse them separately.

Patrick Jahn - jahn@math.ku.dk Leaky Integrate-and-Fire Models

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Diffusion Leaky Integrate-and-Fire Neuronal Models Modelling by time-inhomogeneous Diffusion Models

Nonparametric estimators

Consider the SDE dXt = β(Xt)dt + σ(Xt)dBt then the nonparametric estimators [H¨

  • pfner (2006),

Florens-Zmirou (1993)] are given by

  • β(x) :=

β(∆,M,h)(x) = i1−M

i=i0 K

  • Xi∆−x

h

X(i+M)∆−Xi∆

∆M

  • i1−M

i=i0 K

  • Xi∆−x

h

  • σ2(x) :=

σ2(∆,M,h)(x) = i1−M

i=i0 K

  • Xi∆−x

h

X(i+M)∆−Xi∆

√ ∆M

2 i1−M

i=i0 K

  • Xi∆−x

h

  • We use for K a rectangular and a triangular kernel.

Patrick Jahn - jahn@math.ku.dk Leaky Integrate-and-Fire Models

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Diffusion Leaky Integrate-and-Fire Neuronal Models Modelling by time-inhomogeneous Diffusion Models

Nonparametric estimators

Consider the SDE dXt = β(Xt)dt + σ(Xt)dBt then the nonparametric estimators [H¨

  • pfner (2006),

Florens-Zmirou (1993)] are given by

  • β(x) :=

β(∆,M,h)(x) = i1−M

i=i0 K

  • Xi∆−x

h

X(i+M)∆−Xi∆

∆M

  • i1−M

i=i0 K

  • Xi∆−x

h

  • σ2(x) :=

σ2(∆,M,h)(x) = i1−M

i=i0 K

  • Xi∆−x

h

X(i+M)∆−Xi∆

√ ∆M

2 i1−M

i=i0 K

  • Xi∆−x

h

  • We use for K a rectangular and a triangular kernel.

Patrick Jahn - jahn@math.ku.dk Leaky Integrate-and-Fire Models

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Diffusion Leaky Integrate-and-Fire Neuronal Models Modelling by time-inhomogeneous Diffusion Models

Estimation of the Diffusion coefficient

−85 −80 −75 −70 −65 −60 −55 −50 500 1000 1500 2000 2500

sigma^2: 05414013.acm_V.txt , LT: 500 , h: 0.1 , M: 20

potential sigma^2 −90 −80 −70 −60 500 1000 1500 2000 2500

sigma^2: 05414032.acm_V.txt , LT: 500 , h: 0.1 , M: 20

potential sigma^2

Figure: Nonparametric estimation of the Diffusion coefficient σ(·) works, since the estimator only considers the squared increments. Hence the effect of the inhomogeneous drift (finite variation) is negligible.

Patrick Jahn - jahn@math.ku.dk Leaky Integrate-and-Fire Models

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Diffusion Leaky Integrate-and-Fire Neuronal Models Modelling by time-inhomogeneous Diffusion Models

Analysis of the Drift: Cutting the trajectory into ”homogeneous” parts. . .

10 15 20 −110 −90 −80 −70 −60 −50

05414032.acm_V.txt

time potential

Figure: Separate analysis of the drift in the quiescent period, On- and Off-cycles.

Patrick Jahn - jahn@math.ku.dk Leaky Integrate-and-Fire Models

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Diffusion Leaky Integrate-and-Fire Neuronal Models Modelling by time-inhomogeneous Diffusion Models

Analysis of the Drift: Cutting the trajectory into ”homogeneous” parts. . .

12.5 13.0 13.5 14.0 14.5 15.0 −90 −80 −70 −60 −50

05414032.acm_V.txt

time potential Patrick Jahn - jahn@math.ku.dk Leaky Integrate-and-Fire Models

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Diffusion Leaky Integrate-and-Fire Neuronal Models Modelling by time-inhomogeneous Diffusion Models

Results:

data set τ ON-cycle τ OFF-cycle Q-τ σ2(x) 02.acmV 7.498 13.465 19.966 45.59 · ( x - -88.662 ) 03.acmV 10.461 11.813 31.783 39.542 · ( x - -93.113 ) 11.acmV 8.074 13.59 27.388 25.426 · ( x - -116.197 ) 13.acmV 7.025 16.496 33.156 54.204 · ( x - -87.967 ) 14.acmV 5.648 10.443 22.203 57.916 · ( x - -81.793 ) 17.acmV 8.767 14.513 25.954 34.146 · ( x - -114.114 ) 20.acmV 8.391 14.339 38.494 40.984 · ( x - -101.679 ) 26.acmV 9.318 9.356 17.27 48.239 · ( x - -89.122 ) 27.acmV 8.751 11.282 33.192 47.846 · ( x - -99.089 ) 28.acmV 8.731 11.242 23.89 65.637 · ( x - -83.247 ) 31.acmV 9.869 9.042 31.393 38.66 · ( x - -101.546 ) 32.acmV 8.077 9.679 26.122 44.244 · ( x - -103.347 ) 35.acmV 9.977 9.179 28.975 23.315 · ( x - -135.819 ) 36.acmV 7.866 13.528 16.238 73.516 · ( x - -75.379 ) 38.acmV 8.928 9.123 17.581 69.325 · ( x - -75.265 ) 41.acmV 10.088 14.102 22.892 43.987 · ( x - -93.613 )

Patrick Jahn - jahn@math.ku.dk Leaky Integrate-and-Fire Models

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Diffusion Leaky Integrate-and-Fire Neuronal Models Modelling by time-inhomogeneous Diffusion Models

An Extended LIF-model

dXt = 1 τ(Xt)(a + g(t) − Xt)dt + σ

  • Xt − c dBt

τ(Xt) = τ ∗e−γ(Xt−a) σ(·) = σ√· − c was already estimated. a is the mean of the quiescent period. τ ∗ and γ are found by regressing log(τ) on (Xt − a).

Patrick Jahn - jahn@math.ku.dk Leaky Integrate-and-Fire Models

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Diffusion Leaky Integrate-and-Fire Neuronal Models Modelling by time-inhomogeneous Diffusion Models

An Extended LIF-model

dXt = 1 τ(Xt)(a + g(t) − Xt)dt + σ

  • Xt − c dBt

τ(Xt) = τ ∗e−γ(Xt−a) σ(·) = σ√· − c was already estimated. a is the mean of the quiescent period. τ ∗ and γ are found by regressing log(τ) on (Xt − a). Goal: Estimate g(t) at the observation time points t1, . . . , tn.

Patrick Jahn - jahn@math.ku.dk Leaky Integrate-and-Fire Models

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Diffusion Leaky Integrate-and-Fire Neuronal Models Modelling by time-inhomogeneous Diffusion Models

An Extended LIF-model

dXt = 1 τ(Xt)(a + g(t) − Xt)dt + σ

  • Xt − c dBt

τ(Xt) = τ ∗e−γ(Xt−a) σ(·) = σ√· − c was already estimated. a is the mean of the quiescent period. τ ∗ and γ are found by regressing log(τ) on (Xt − a). Goal: Estimate g(t) at the observation time points t1, . . . , tn.

Patrick Jahn - jahn@math.ku.dk Leaky Integrate-and-Fire Models

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Diffusion Leaky Integrate-and-Fire Neuronal Models Modelling by time-inhomogeneous Diffusion Models

Fitting the extended Model

If we assume that between observations g(t) and τ(Xt) are ”constant”, where ∆ = ti − ti−1 is small then we can write Xi ≈ E[Xi|Xi−1] ≈ Xi−1e−∆/τ(Xi−1)+(g(ti−1)+a)(1−e−∆/τ(Xi−1)) Hence we derive an estimator ˆ g(ti−1) = Xi − Xi−1e−∆/τi−1 (1 − e−∆/τi−1) − a

Patrick Jahn - jahn@math.ku.dk Leaky Integrate-and-Fire Models

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Diffusion Leaky Integrate-and-Fire Neuronal Models Modelling by time-inhomogeneous Diffusion Models

Fitting the extended Model

If we assume that between observations g(t) and τ(Xt) are ”constant”, where ∆ = ti − ti−1 is small then we can write Xi ≈ E[Xi|Xi−1] ≈ Xi−1e−∆/τ(Xi−1)+(g(ti−1)+a)(1−e−∆/τ(Xi−1)) Hence we derive an estimator ˆ g(ti−1) = Xi − Xi−1e−∆/τi−1 (1 − e−∆/τi−1) − a Finally we smooth ˆ g(·) by a smooth spline gs(·).

Patrick Jahn - jahn@math.ku.dk Leaky Integrate-and-Fire Models

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Diffusion Leaky Integrate-and-Fire Neuronal Models Modelling by time-inhomogeneous Diffusion Models

Fitting the extended Model

If we assume that between observations g(t) and τ(Xt) are ”constant”, where ∆ = ti − ti−1 is small then we can write Xi ≈ E[Xi|Xi−1] ≈ Xi−1e−∆/τ(Xi−1)+(g(ti−1)+a)(1−e−∆/τ(Xi−1)) Hence we derive an estimator ˆ g(ti−1) = Xi − Xi−1e−∆/τi−1 (1 − e−∆/τi−1) − a Finally we smooth ˆ g(·) by a smooth spline gs(·).

Patrick Jahn - jahn@math.ku.dk Leaky Integrate-and-Fire Models

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Diffusion Leaky Integrate-and-Fire Neuronal Models Modelling by time-inhomogeneous Diffusion Models

Comparing Network Activity and Estimated Input

5000 10000 15000 20000 25000 30000 35000 10 20 30 40 50

trace = 32

time (ms) input (mV)

5000 10000 15000 20000 −1.0 −0.5 0.0 0.5 1.0

Network Activity

time

The cyan line is the squared and smoothed network activity from

  • above. It was rescaled to be

compared to our estimate g s(·) given by the black line. We can think of the network activity to be a threshold version of the Input!

Patrick Jahn - jahn@math.ku.dk Leaky Integrate-and-Fire Models

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Diffusion Leaky Integrate-and-Fire Neuronal Models Modelling by time-inhomogeneous Diffusion Models

Comparing Network Activity and Estimated Input

5000 10000 15000 20000 25000 30000 35000 10 20 30 40 50

trace = 32

time (ms) input (mV)

5000 10000 15000 20000 −1.0 −0.5 0.0 0.5 1.0

Network Activity

time

The cyan line is the squared and smoothed network activity from

  • above. It was rescaled to be

compared to our estimate g s(·) given by the black line. We can think of the network activity to be a threshold version of the Input!

Patrick Jahn - jahn@math.ku.dk Leaky Integrate-and-Fire Models

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Diffusion Leaky Integrate-and-Fire Neuronal Models Modelling by time-inhomogeneous Diffusion Models

Simulated Data using the estimated quantities compared to real data:

5000 10000 15000 20000 25000 30000 −110 −100 −90 −80 −70 −60 −50

trace = 32

time (ms) 5000 10000 15000 20000 25000 30000 −110 −100 −90 −80 −70 −60 −50

trace = 32

time (ms) V(t) (mV) a,γ,τ= −99.9 , 0.029 , 21.1

Patrick Jahn - jahn@math.ku.dk Leaky Integrate-and-Fire Models

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Diffusion Leaky Integrate-and-Fire Neuronal Models Modelling by time-inhomogeneous Diffusion Models

A Spiking Dataset

5000 10000 15000 20000 25000 30000 −100 −90 −80 −70 −60 −50 −40 −30

trace = 13

time (ms) 5000 10000 15000 20000 25000 30000 −100 −90 −80 −70 −60 −50 −40 −30

trace = 13

time (ms) V(t) (mV) a,γ,τ= −89 , 0.04 , 31.6

Patrick Jahn - jahn@math.ku.dk Leaky Integrate-and-Fire Models

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Diffusion Leaky Integrate-and-Fire Neuronal Models Modelling by time-inhomogeneous Diffusion Models

R.Berg, S.Ditlevsen, J.Hounsgaard: “Intense Synaptic Activity Enhances Temporal Resolution in Spinal Motoneurons”. PLoS ONE (2008). R.H¨

  • pfner: “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).

P.Jahn, PhD. Thesis: “Statistical Problems Related to Excitation Threshold and Reset Value of Membrane Potentials”. http://archimed.uni-mainz.de (2009). P.Jahn: “Estimation of Excitation Threshold and Reset Value in Diffusion Leaky Integrate-and-Fire Neuronal Models”. submitted in J. Math. Biol.

Patrick Jahn - jahn@math.ku.dk Leaky Integrate-and-Fire Models

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Diffusion Leaky Integrate-and-Fire Neuronal Models Modelling by time-inhomogeneous Diffusion Models

Thank you for your attention! Questions?

Patrick Jahn - jahn@math.ku.dk Leaky Integrate-and-Fire Models