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
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
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
Diffusion Leaky Integrate-and-Fire Neuronal Models
Modelling by time-inhomogeneous Diffusion Models
−30
−40
−50
potential [mV]
−20
Membrane Potential recorded by W. Kilb, Mainz
0
1
2
3
4
5
6
time [s]
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
Diffusion Leaky Integrate-and-Fire Neuronal Models
Modelling by time-inhomogeneous Diffusion Models
Diffusion Leaky Integrate-and-Fire Neuronal Model
S
|
{z
} | {z }
T1
...
T2
x0
| {z }
Tn
Assume the process X between spikes is a Diffusion process
given by
dXt =
1
(a − Xt )dt + σ(Xt )dBt
τ
Patrick Jahn - jahn@math.ku.dk
Leaky Integrate-and-Fire Models
Diffusion Leaky Integrate-and-Fire Neuronal Models
Modelling by time-inhomogeneous Diffusion Models
Diffusion Leaky Integrate-and-Fire Neuronal Model
S
|
{z
} | {z }
T1
...
T2
x0
| {z }
Tn
Assume the process X between spikes is a Diffusion process
given by
dXt =
1
(a − Xt )dt + σ(Xt )dBt
τ
Goal: Estimate β(·) := τ1 (a − ·), σ(·), x0 , S from discrete
observations 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
Diffusion Leaky Integrate-and-Fire Neuronal Models
Modelling by time-inhomogeneous Diffusion Models
Diffusion Leaky Integrate-and-Fire Neuronal Model
S
|
{z
} | {z }
T1
...
T2
x0
| {z }
Tn
Assume the process X between spikes is a Diffusion process
given by
dXt =
1
(a − Xt )dt + σ(Xt )dBt
τ
Goal: Estimate β(·) := τ1 (a − ·), σ(·), x0 , S from discrete
observations 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
Diffusion Leaky Integrate-and-Fire Neuronal Models
Modelling by time-inhomogeneous Diffusion Models
−48
−49
−51
−50
S ?
x0 ?
−52
potential [mV]
−47
−46
Problem of Finding Excitation Threshold and Reset Value
1.0
1.5
2.0
2.5
3.0
time [s]
Patrick Jahn - jahn@math.ku.dk
Leaky Integrate-and-Fire Models
3.5
Diffusion Leaky Integrate-and-Fire Neuronal Models
Modelling by time-inhomogeneous Diffusion Models
Problem of Finding Excitation Threshold and Reset Value
−50
−40
−30
[mV]
−20
−10
0
17Sept08_023.asc
alle spikes uebereinanderlegen / spikezeiten auf 0 transformieren
−0.04
−0.02
0.00
0.02
17Sept08_023.asc
Patrick Jahn - jahn@math.ku.dk
Leaky Integrate-and-Fire Models
0.04
Diffusion Leaky Integrate-and-Fire Neuronal Models
Modelling by time-inhomogeneous Diffusion Models
Strategy
θ2
|
{z
T1
} | {z }
...
T2
θ1
| {z }
Tn
1. Fixing drift and diffusion coefficient with nonparametric
methods proposed by R. Höpfner (2006). So assume X is
given by
1
dXt = (a − Xt )dt + σ(Xt )dBt
τ
Patrick Jahn - jahn@math.ku.dk
Leaky Integrate-and-Fire Models
Diffusion Leaky Integrate-and-Fire Neuronal Models
Modelling by time-inhomogeneous Diffusion Models
Strategy
θ2
|
{z
T1
} | {z }
...
T2
θ1
| {z }
Tn
1. Fixing drift and diffusion coefficient with nonparametric
methods proposed by R. Höpfner (2006). So assume X is
given by
1
dXt = (a − Xt )dt + σ(Xt )dBt
τ
2. Estimate θ1 = x0 and θ2 = S from the observation of iid inter
spike times
n
o
(θ )
Ti := inf t ≥ 0 | Xt 1 = θ2 , i = 1, . . . , n.
Patrick Jahn - jahn@math.ku.dk
Leaky Integrate-and-Fire Models
Diffusion Leaky Integrate-and-Fire Neuronal Models
Modelling by time-inhomogeneous Diffusion Models
Strategy
θ2
|
{z
T1
} | {z }
...
T2
θ1
| {z }
Tn
1. Fixing drift and diffusion coefficient with nonparametric
methods proposed by R. Höpfner (2006). So assume X is
given by
1
dXt = (a − Xt )dt + σ(Xt )dBt
τ
2. Estimate θ1 = x0 and θ2 = S from the observation of iid inter
spike times
n
o
(θ )
Ti := inf t ≥ 0 | Xt 1 = θ2 , i = 1, . . . , n.
Patrick Jahn - jahn@math.ku.dk
Leaky Integrate-and-Fire Models
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
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
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
n
L̂n (α) :=
1 X −αTi
e
n
and
Lθ (α) := Eθ [e −αT ],
i=1
then the MDE w.r.t. the LT
θn∗ := arg inf kL̂n − Lθ kH
θ1 <θ2
is strongly consistent and asymptotically normal.
Patrick Jahn - jahn@math.ku.dk
Leaky Integrate-and-Fire Models
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 = τ1 (a − Xt )dt + σ
p
(Xt − c)2 + ddBt ,
X0 = θ1 ≥ 0
Jesper computed the Laplace transform of T :
Eθ [e −αT ] =
g Re (θ1 , α) − K (α) · hRe (θ1 , α)
g Re (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
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 = τ1 (a − Xt )dt + σ
p
(Xt − c)2 + ddBt ,
X0 = θ1 ≥ 0
Jesper computed the Laplace transform of T :
Eθ [e −αT ] =
g Re (θ1 , α) − K (α) · hRe (θ1 , α)
g Re (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
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 = τ1 (a − Xt )dt + σ
p
(Xt − c)2 + ddBt ,
X0 = θ1 ≥ 0
Jesper computed the Laplace transform of T :
Eθ [e −αT ] =
g Re (θ1 , α) − K (α) · hRe (θ1 , α)
g Re (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
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
Diffusion Leaky Integrate-and-Fire Neuronal Models
Modelling by time-inhomogeneous Diffusion Models
Joint work with R. Berg, J. Hounsgaard and S. Ditlevsen
−70
−80
−90
−110
V(t) (mV)
−60
−50
trace = 32
0
5000
10000
15000
20000
25000
30000
time (ms)
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
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
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
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 [Höpfner (2006),
Florens-Zmirou (1993)] are given by
Pi1 −M Xi∆ −x X(i+M)∆ −Xi∆ i=i0 K
h
∆M
b
β(x)
:= βb(∆,M,h) (x) =
Pi1 −M Xi∆ −x i=i0 K
h
Pi1 −M Xi∆ −x X(i+M)∆ −Xi∆ 2
√
i=i0 K
h
∆M
c2 (x) := σ
c2
σ
(x)
=
(∆,M,h)
Pi1 −M
Xi∆ −x
i=i0 K
h
We use for K a rectangular and a triangular kernel.
Patrick Jahn - jahn@math.ku.dk
Leaky Integrate-and-Fire Models
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 [Höpfner (2006),
Florens-Zmirou (1993)] are given by
Pi1 −M Xi∆ −x X(i+M)∆ −Xi∆ i=i0 K
h
∆M
b
β(x)
:= βb(∆,M,h) (x) =
Pi1 −M Xi∆ −x i=i0 K
h
Pi1 −M Xi∆ −x X(i+M)∆ −Xi∆ 2
√
i=i0 K
h
∆M
c2 (x) := σ
c2
σ
(x)
=
(∆,M,h)
Pi1 −M
Xi∆ −x
i=i0 K
h
We use for K a rectangular and a triangular kernel.
Patrick Jahn - jahn@math.ku.dk
Leaky Integrate-and-Fire Models
Diffusion Leaky Integrate-and-Fire Neuronal Models
Modelling by time-inhomogeneous Diffusion Models
Estimation of the Diffusion coefficient
sigma^2: 05414032.acm_V.txt , LT: 500 , h: 0.1 , M: 20
500
500
1000
1500
sigma^2
1500
1000
sigma^2
2000
2000
2500
2500
sigma^2: 05414013.acm_V.txt , LT: 500 , h: 0.1 , M: 20
−85
−80
−75
−70
−65
−60
−55
−50
potential
−90
−80
−70
−60
potential
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
Diffusion Leaky Integrate-and-Fire Neuronal Models
Modelling by time-inhomogeneous Diffusion Models
Analysis of the Drift:
Cutting the trajectory into ”homogeneous” parts. . .
−90 −80 −70 −60 −50
−110
potential
05414032.acm_V.txt
10
15
20
time
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
Diffusion Leaky Integrate-and-Fire Neuronal Models
Modelling by time-inhomogeneous Diffusion Models
Analysis of the Drift:
Cutting the trajectory into ”homogeneous” parts. . .
−70
−80
−90
potential
−60
−50
05414032.acm_V.txt
12.5
13.0
13.5
14.0
14.5
time
Patrick Jahn - jahn@math.ku.dk
Leaky Integrate-and-Fire Models
15.0
Diffusion Leaky Integrate-and-Fire Neuronal Models
Modelling by time-inhomogeneous Diffusion Models
Results:
data set
τ ON-cycle
τ OFF-cycle
Q-τ
σ 2 (x)
02.acmV
03.acmV
11.acmV
13.acmV
14.acmV
17.acmV
20.acmV
26.acmV
27.acmV
28.acmV
31.acmV
32.acmV
35.acmV
36.acmV
38.acmV
41.acmV
7.498
10.461
8.074
7.025
5.648
8.767
8.391
9.318
8.751
8.731
9.869
8.077
9.977
7.866
8.928
10.088
13.465
11.813
13.59
16.496
10.443
14.513
14.339
9.356
11.282
11.242
9.042
9.679
9.179
13.528
9.123
14.102
19.966
31.783
27.388
33.156
22.203
25.954
38.494
17.27
33.192
23.89
31.393
26.122
28.975
16.238
17.581
22.892
45.59 · ( x - -88.662 )
39.542 · ( x - -93.113 )
25.426 · ( x - -116.197 )
54.204 · ( x - -87.967 )
57.916 · ( x - -81.793 )
34.146 · ( x - -114.114 )
40.984 · ( x - -101.679 )
48.239 · ( x - -89.122 )
47.846 · ( x - -99.089 )
65.637 · ( x - -83.247 )
38.66 · ( x - -101.546 )
44.244 · ( x - -103.347 )
23.315 · ( x - -135.819 )
73.516 · ( x - -75.379 )
69.325 · ( x - -75.265 )
43.987 · ( x - -93.613 )
Patrick Jahn - jahn@math.ku.dk
Leaky Integrate-and-Fire Models
Diffusion Leaky Integrate-and-Fire Neuronal Models
Modelling by time-inhomogeneous Diffusion Models
An Extended LIF-model
dXt =
p
1
(a + g (t) − Xt )dt + σ Xt − c dBt
τ (Xt )
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
Diffusion Leaky Integrate-and-Fire Neuronal Models
Modelling by time-inhomogeneous Diffusion Models
An Extended LIF-model
dXt =
p
1
(a + g (t) − Xt )dt + σ Xt − c dBt
τ (Xt )
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
Diffusion Leaky Integrate-and-Fire Neuronal Models
Modelling by time-inhomogeneous Diffusion Models
An Extended LIF-model
dXt =
p
1
(a + g (t) − Xt )dt + σ Xt − c dBt
τ (Xt )
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
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−1 e −∆/τ (Xi−1 ) +(g (ti−1 )+a)(1−e −∆/τ (Xi−1 ) )
Hence we derive an estimator
ĝ (ti−1 ) =
Xi − Xi−1 e −∆/τi−1
−a
(1 − e −∆/τi−1 )
Patrick Jahn - jahn@math.ku.dk
Leaky Integrate-and-Fire Models
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−1 e −∆/τ (Xi−1 ) +(g (ti−1 )+a)(1−e −∆/τ (Xi−1 ) )
Hence we derive an estimator
ĝ (ti−1 ) =
Xi − Xi−1 e −∆/τi−1
−a
(1 − e −∆/τi−1 )
Finally we smooth ĝ (·) by a smooth spline g s (·).
Patrick Jahn - jahn@math.ku.dk
Leaky Integrate-and-Fire Models
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−1 e −∆/τ (Xi−1 ) +(g (ti−1 )+a)(1−e −∆/τ (Xi−1 ) )
Hence we derive an estimator
ĝ (ti−1 ) =
Xi − Xi−1 e −∆/τi−1
−a
(1 − e −∆/τi−1 )
Finally we smooth ĝ (·) by a smooth spline g s (·).
Patrick Jahn - jahn@math.ku.dk
Leaky Integrate-and-Fire Models
Diffusion Leaky Integrate-and-Fire Neuronal Models
Modelling by time-inhomogeneous Diffusion Models
Comparing Network Activity and Estimated Input
1.0
Network Activity
0
5000
10000
15000
20000
time
10
20
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.
0
input (mV)
30
−1.0
40
−0.5
0.0
50
0.5
trace = 32
0
5000
10000 15000 20000 25000 30000 35000
time (ms)
Patrick Jahn - jahn@math.ku.dk
We can think of the network activity
to be a threshold version of the
Input!
Leaky Integrate-and-Fire Models
Diffusion Leaky Integrate-and-Fire Neuronal Models
Modelling by time-inhomogeneous Diffusion Models
Comparing Network Activity and Estimated Input
1.0
Network Activity
0
5000
10000
15000
20000
time
10
20
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.
0
input (mV)
30
−1.0
40
−0.5
0.0
50
0.5
trace = 32
0
5000
10000 15000 20000 25000 30000 35000
time (ms)
Patrick Jahn - jahn@math.ku.dk
We can think of the network activity
to be a threshold version of the
Input!
Leaky Integrate-and-Fire Models
Diffusion Leaky Integrate-and-Fire Neuronal Models
Modelling by time-inhomogeneous Diffusion Models
Simulated Data using the estimated quantities compared to real data:
−80
V(t) (mV)
−70
−60
a,γ,τ=
−99.9 , 0.029 , 21.1
−110
−100
−90
−80
−90
−100
−110
V(t) (mV)
−70
−60
−50
trace = 32
−50
trace = 32
0
5000
10000
15000
20000
25000
30000
time (ms)
Patrick Jahn - jahn@math.ku.dk
0
5000
10000
15000
time (ms)
Leaky Integrate-and-Fire Models
20000
25000
30000
Diffusion Leaky Integrate-and-Fire Neuronal Models
Modelling by time-inhomogeneous Diffusion Models
A Spiking Dataset
V(t) (mV)
−60
−50
−40
a,γ,τ=
−89 , 0.04 , 31.6
−100
−90
−80
−70
−60
−70
−80
−90
−100
V(t) (mV)
−50
−40
−30
trace = 13
−30
trace = 13
0
5000
10000
15000
20000
25000
30000
time (ms)
Patrick Jahn - jahn@math.ku.dk
0
5000
10000
15000
time (ms)
Leaky Integrate-and-Fire Models
20000
25000
30000
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.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).
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
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