Signal estimation from intracellular recordings in the
Feller neuronal model
Roberta Sirovich
Department of Mathematics G. Peano
University of Torino
Marseille, january 21st, 2010
joint work with E. Bibbona (University of Torino)
and P. Lansky (Academy of Sciences, Prague)
R. Sirovich (Univ. Torino)
Estimation in the Feller model
21.1.2010
1 / 21
What are we talking about?
ESTIMATION
R. Sirovich (Univ. Torino)
Estimation in the Feller model
21.1.2010
2 / 21
What are we talking about?
ESTIMATION
PARAMETERS (signal)
R. Sirovich (Univ. Torino)
Estimation in the Feller model
21.1.2010
2 / 21
What are we talking about?
ESTIMATION
PARAMETERS (signal)
MODEL (Feller or CIR)
R. Sirovich (Univ. Torino)
Estimation in the Feller model
21.1.2010
2 / 21
What are we talking about?
ESTIMATION
PARAMETERS (signal)
MODEL (Feller or CIR)
SAMPLE (intracellular recordings)
R. Sirovich (Univ. Torino)
Estimation in the Feller model
21.1.2010
2 / 21
Neuronal modelling
A neuron and its membran potential
membrane potential [mV]
Dendrites
0
-20
Axon
Cell Body
(Soma)
-40
-60
15.2
Pre-synaptic fiber
(e.g. branch of axon)
15.4
15.6
time [s]
15.8
16
interspike interval
Synapse
Portion of
post-synaptic cell
(e.g. dendrite)
R. Sirovich (Univ. Torino)
Estimation in the Feller model
21.1.2010
3 / 21
The Model
The Feller (CIR) neuronal model.
The membrane potential in an interspike interval is given as a solution of
the SDE:
Feller model SDE
p
Xt
dXt = − + µ dt + σ Xt dWt ,
τ
X0 = x0
with the following transition density
f (x, t|x0 ) = c e
where
c=
−r −s
2
,
τ σ 2 (1 − e−t/τ )
r = c x0 e−t/τ ,
s q
2
r
√
Iq (2 rs)
q=
2µ
−1
σ2
s = c x,
and Iq (·) is the modified Bessel function
R. Sirovich (Univ. Torino)
Estimation in the Feller model
21.1.2010
4 / 21
The Model
The Feller (CIR) process is time-homogeneous, it has state space
(0, ∞) and it is ergodic whenever 2µ ≥ σ 2 (zero is not reached).
Conditional moments
t→∞
E(Xt |X0 , µ) = X0 e−t/τ + µτ (1 − e−t/τ ) −→ µτ
Var(Xt |X0 , µ, σ 2 ) = σ 2 v (Xt |X0 , µ) =
= σ 2 τ2 (1 − e−t/τ )[µτ (1 − e−t/τ ) + 2X0 e−t/τ ]
Cov(Xt , Xs |X0 , µ, σ 2 ) = e−(t−s)/τ Var(Xs |X0 , µ, σ 2 ) for (s < t).
R. Sirovich (Univ. Torino)
Estimation in the Feller model
21.1.2010
5 / 21
The Model
The Feller (CIR) process is time-homogeneous, it has state space
(0, ∞) and it is ergodic whenever 2µ ≥ σ 2 (zero is not reached).
Conditional moments
t→∞
E(Xt |X0 , µ) = X0 e−t/τ + µτ (1 − e−t/τ ) −→ µτ
Var(Xt |X0 , µ, σ 2 ) = σ 2 v (Xt |X0 , µ) =
= σ 2 τ2 (1 − e−t/τ )[µτ (1 − e−t/τ ) + 2X0 e−t/τ ]
Cov(Xt , Xs |X0 , µ, σ 2 ) = e−(t−s)/τ Var(Xs |X0 , µ, σ 2 ) for (s < t).
An action potential is generated when the membrane potential
reaches an absorbing threshold S for the first time.
R. Sirovich (Univ. Torino)
Estimation in the Feller model
21.1.2010
5 / 21
The Model
The Feller (CIR) process is time-homogeneous, it has state space
(0, ∞) and it is ergodic whenever 2µ ≥ σ 2 (zero is not reached).
Conditional moments
t→∞
E(Xt |X0 , µ) = X0 e−t/τ + µτ (1 − e−t/τ ) −→ µτ
Var(Xt |X0 , µ, σ 2 ) = σ 2 v (Xt |X0 , µ) =
= σ 2 τ2 (1 − e−t/τ )[µτ (1 − e−t/τ ) + 2X0 e−t/τ ]
Cov(Xt , Xs |X0 , µ, σ 2 ) = e−(t−s)/τ Var(Xs |X0 , µ, σ 2 ) for (s < t).
An action potential is generated when the membrane potential
reaches an absorbing threshold S for the first time.
Parameters (µ, σ 2 ) are related with the input signal that is arriving to
the neuron from the network, while (S, x0 , τ ) are related to properties
of the cell and assumed to be known.
R. Sirovich (Univ. Torino)
Estimation in the Feller model
21.1.2010
5 / 21
Estimation methods
Estimation methods
Let’s introduce and compare the following five methods to estimate
parameters µ and σ 2 from a single trajectory sampled at discrete times
(Xi )ni=0 , where Xi = Xti :
“Naive” Least squares (LS)
Conditional Least Squares (CLS)
Gauss-Markov (GM)
Bibby and Sorensen 1995 (BS)
Maximum Likelihood (ML)
R. Sirovich (Univ. Torino)
Estimation in the Feller model
21.1.2010
6 / 21
Estimation methods
Estimation methods
Let’s introduce and compare the following five methods to estimate
parameters µ and σ 2 from a single trajectory sampled at discrete times
(Xi )ni=0 , where Xi = Xti :
“Naive” Least squares (LS)
Conditional Least Squares (CLS)
Gauss-Markov (GM)
Bibby and Sorensen 1995 (BS)
Maximum Likelihood (ML)
Warning
All the estimation methods proposed above do not account for the
presence of the absorbing threshold.
R. Sirovich (Univ. Torino)
Estimation in the Feller model
21.1.2010
6 / 21
Estimation methods
Goals...
compare the estimation methods
R. Sirovich (Univ. Torino)
Estimation in the Feller model
21.1.2010
7 / 21
Estimation methods
Goals...
compare the estimation methods
show that the absorbing threshold cannot be ignored (at least for µ)
R. Sirovich (Univ. Torino)
Estimation in the Feller model
21.1.2010
7 / 21
Estimation methods
Least Squares
Naive Least Squares (LS)
Let’s define the residuals ei (µ) = Xi − E(Xi |x0 , µ) and minimize the
following functions
LS1 =
n
X
[ei (µ)]2
LS2 =
i=1
and get
µ̂
σ̂
LS
2LS
n
X
2
[ei (µ̂LS )]2 − E [ei (µ̂LS )]2
i=1
Pn
− x0 e−ih/τ )(1 − e−ih/τ )]
Pn
τ i=1 (1 − e−ih/τ )2
i=1 [(Xi
=
(no σ!)
Pn
=
[e (µ̂LS ))]2 v (Xi |x0 , µ̂LS )
i=1
Pni
2
LS
i=1 v (Xi |x0 , µ̂ )
Advantages: µ̂LS is unbiased ∀n, σ̂ 2LS would be if µ → µ̂LS (care!)
Drawbacks: the residuals ei = Xi − E(Xi |x0 , µ) are correlated and not
identically distributed
R. Sirovich (Univ. Torino)
Estimation in the Feller model
21.1.2010
8 / 21
Estimation methods
Conditional Least Squares
Conditional Least Squares (CLS)
Residuals are ηi (µ) = Xi − E(Xi |Xi−1 , µ). We minimize
n
n n
o2
X
X
CLS1 =
[ηi (µ)]2
CLS2 =
[ηi (µ̂CLS )]2 − E[ηi (µ̂CLS )]2
i=1
i=1
and get
µ̂
σ̂
CLS
2CLS
Pn
n
− Xi−1 e−h/τ )
1X
µ̂i
=
n
nτ (1 − e−h/τ )
i=1
i=1 (Xi
=
(no σ!)
Pn
=
[ηi (µ̂CLS )]2 v (Xi |Xi−1 , µ̂CLS )
i=1P
n
2
CLS )
i=1 v (Xi |Xi−1 , µ̂
Advantages: µ̂CLS is unbiased ∀n, σ̂ 2CLS would be if µ → µ̂CLS (!). Uncorrelated
residuals.
Drawbacks: residuals ηi = Xi − E(Xi |Xi−1 , µ) are nor i.d.:
µτ 2
2
−h/τ −ih/τ
−2h/τ
Var(ηi ) = σ (τ x0 − µτ )(1 − e
)e
+
(1 − e
)
2
2
R. Sirovich (Univ. Torino)
Estimation in the Feller model
21.1.2010
9 / 21
Estimation methods
Gauss Markov
Gauss Markov (GM)
We focus on µ. If Var(µ̂i ) = c1 Var(ηi ) = 1/pi was constants, Gauss
Markov
theorem would suggest us to estimate µ minimizing
Pn
2
p
i=1 i [ηi (µ)] getting
Pn
pi µ̂i
GM
µ̂
= Pi=1
(no σ!)
n
i=1 pi
Var(ηi ) in fact depends on µ, however you can use instead
µ̂CLS τ 2
1
2
CLS 2
−h/τ −ih/τ
−2h/τ
= σ (τ x0 − µ̂ τ )(1 − e
)e
+
(1 − e
)
pi
2
Advantages: we tried to avoid the drawbacks of previous methods
Drawbacks: no theoretical result on the properties of µ̂GM
R. Sirovich (Univ. Torino)
Estimation in the Feller model
21.1.2010
10 / 21
Estimation methods
Bibby Sorensen 1995-96
Bibby Sorensen 1995-96 (BS)
In two paper by M. Sorensen and B. M. Bibby in 1995-1996 the following
two estimators were proposed from Martingale Estimating Functions theory
Pn
Pn
−1
−h/τ X
i−1 )/Xi−1 ]
i=1 [Xi−1 µ̂i ]
BS
i=1 [(Xi − e
µ̂ =
=
P
P
n
−1
−1
τ (1 − e−h/τ ) ni=1 Xi−1
i=1 Xi−1
P
2 ni=1 [Xi − e−h/τ Xi−1 − µ̂BS τ (1 − e−h/τ )]2 /Xi−1
2BS
P
σ̂
=
τ (1 − e−h/τ ) ni=1 [µ̂BS τ (1 − e−h/τ ) + 2Xi−1 e−h/τ ]/Xi−1
N.B. µ̂BS differs from µ̂GM just in the choice of the weights
Advantages: BS estimators are consistent and AN for n → ∞
Drawbacks: finite sample properties?
R. Sirovich (Univ. Torino)
Estimation in the Feller model
21.1.2010
11 / 21
Estimation methods
Likelihood
Maximum Likelihood (ML)
The log-likelihood function is the following
L(xi , ..., xn |µ, σ) = log
n
Y
!
f (xi , h|xi−1 )
(1)
i=1
Advantages: MLE estimators are consistent and AN for n → ∞ and they
have the smallest asymptotic variance.
Drawbacks: again, finite sample properties? (Require pretty expensive
numerical maximization).
R. Sirovich (Univ. Torino)
Estimation in the Feller model
21.1.2010
12 / 21
The sample
Simulated samples
Simulated samples
Two samples are simulated:
a trajectory (at discrete times) of the process X ;
a trajectory (at discrete times) of the process X with absorbing
threshold;
Simulations are performed according to the following methods:
very small sampling step h = 0.01 ms;
parameters range coherent with expected values for neuronal data;
both the sample in the presence of the threshold and in its absence
are simulated with the same length;
simulation up to the first passage time are made taking care of
eventual hidden crossings of the threshold
S
1
evaluate the probability p of an
hidden crossing
2
generate a uniform r.v. u
3
if u > p next point, else stop
R. Sirovich (Univ. Torino)
Un
Estimation in the Feller model
U n+1
21.1.2010
13 / 21
Results
Estimation of µ in the absence of the threshold
Estimation of µ in the absence of the threshold
0.04
]
-1
[mVms
]
-1
90
70
τ=
50
τ=
35
[ms]
[mVms -1 ]
D
τ=
25
τ=
0.
0
τ=
[mVms
22
2
99
0
=
0.
07
29
2
=
2
μ=0.7
2
σ =0.0324
σ
=
σ
0.
05
06
0.
03
24
2
σ
0.
01
82
2
=
σ
0.
00
81
=
2
2
=
LS
CLS
BS
GM
*
0
σ
C
0.04
τ=
Average bias avg( μ ) − μ
μ=0.5
τ=35
μ=
1.
4
μ=
1
μ=
0.
7
μ=
0.
5
μ=
0.
4
-1
]
*
[mVms
[mVms -1 ]
LS
CLS
BS
GM
both µ̂BS ∼ µ̂ML
have a small bias
but the smallest
variance
0
B
σ
τ=35
σ 2=0.0324
μ=0.5
τ=35
µ̂GM is both
unbiased and
with a small
variance
µ̂CLS and µ̂LS are
unbiased.
Variance is
2-15% larger
1.20
μ=0.7
2
σ =0.0324
^
^
var(μ
) / var(μ
)
LS
BS
^
^
var(μ
) / var(μ
)
CLS
BS
^
^
var(μ
) / var(μ
)
GM
BS
^
^
var(μ
) / var(μ
)
WCLS
BS
R. Sirovich (Univ. Torino)
0
τ=
9
70
τ=
50
τ=
25
τ=
92
τ=
22
.0
9
72
9
.0
σ2
=0
[mVms -1 ]
σ2
=0
6
.0
50
σ2
=0
18
2
.0
σ2
=0
81
.0
0
σ2
=0
1.
4
μ=
1
0.
7
[mVms -1 ]
μ=
μ=
0.
5
μ=
5
0.
4
μ=
0.
0.95
4
1
μ=
Average bias avg( μ ) − μ
0.04
τ=35
σ 2=0.0324
LS
CLS
BS
GM
*
μ=
0.
45
Average bias avg( μ ) − μ
A
[ms]
Estimation in the Feller model
21.1.2010
14 / 21
Results
Estimation of µ in the presence of the threshold
Estimation of µ in the presence of the threshold
µ̂BS ∼ µ̂ML
A
0
4
τ=
15
μ=
1.
0.
7
0
−0.02
C
[mVms -1 ]
[ms]
0.05
LS
CLS
BS
GM
-1
corrections?
[mVms
R. Sirovich (Univ. Torino)
0
τ=
9
0
0
τ=
7
τ=
5
5
5
τ=
3
−0.02
τ=
2
2
0.
2
=
σ
[mVms ]
09
9
07
29
05
06
0.
0.
2
=
2
=
σ
-1
σ
03
24
0.
2
=
σ
00
81
0.
0.
2
=
=
σ
2
σ
01
82
0
−0.02
LS
CLS
BS
GM
τ=
2
0
μ=0.7
2
σ =0.0324
2
[mVms
μ=0.5
τ=35
for large τ , µ̂CLS
settles to a
known constant
value
]
*
-1
]
*
Average bias avg( μ ) − μ
0.14
Average bias avg( μ ) − μ
μ=
1
μ=
0.
5
0.
μ=
B
μ=
μ=
0.
4
−0.02
45
LS
CLS
BS
GM
E
0
0
0.05
*
[mVms -1 ]
τ=35
σ 2=0.0324
Average bias avg( μ ) − μ
*
µ̂BS , µ̂GM and
µ̂CLS are biased:
bias is positive,
linear in σ 2 ,
weekly dep. on µ
τ=
30
Average bias avg( μ ) − μ
[mVms -1 ]
0.05
µ̂LS is much less
biased than the
others!!
[ms]
Estimation in the Feller model
21.1.2010
15 / 21
Results
Estimation of µ in the presence of the threshold
Estimation of µ in the presence of the threshold
1.20
1
^
^
var(μ
) / var(μ
)
LS
BS
^
^
var(μ
) / var(μ
)
CLS
BS
90
τ=
70
50
τ=
25
τ=
τ=
22
2
.0
2
=0
σ
[mVms -1 ]
τ=
9
99
6
72
.0
2
σ
=0
2
σ
=0
.0
50
2
1
.0
2
σ
=0
.0
18
08
4
1.
1
]
2
-1
μ=0.7
2
σ =0.0324
μ=0.5
τ=35
σ
[mVms
μ=
.7
μ=
5
0.
μ=
0
45
0.
μ=
4
0.
μ=
μ=
0.7
τ=35
σ 2=0.0324
=0
^
^
var(μ
) / var(μ
)
GM
BS
[ms]
In subthreshold regimen µ̂LS has not only the smallest bias but also by far
the smallest variability.
R. Sirovich (Univ. Torino)
Estimation in the Feller model
21.1.2010
16 / 21
Estimation of σ 2
Results
Estimation of σ 2
Average bias avg( σ 2 ) − σ 2
0.00008
CLSc
BSc
*
[mVms
-1
]
Estimation of σ 2 is not
affected by the presence
of the threshold
0
92
29
σ
]
1
μ=0.5
τ=35
τ=35
σ 2=0.0324
2
2
var(σ^BSc
) / var(σ^ CLSc
)
09
2
=
2
=
σ
-1
0.
07
06
[mVms
0.
24
05
0.
2
σ
=
2
=
σ
0.
03
81
01
0.
0.
2
=
σ
σ
=
2
00
82
μ=0.5
τ=35
−0.00004
μ=0.7
2
σ =0.0324
σ̂ 2LS is such heavily
biased that it cannot be
plotted on the same
scale
σ̂ 2BS and σ̂ 2CLS are both
unbiased
[mVms
]
R. Sirovich (Univ. Torino)
[mVms
70
τ=
90
50
τ=
25
τ=
τ=
22
99
2
σ
]
σ̂ 2BS is much better
=0
.0
.0
2
σ
=0
-1
τ=
9
2
6
72
2
σ
2
=0
.0
50
1
18
2
σ
=0
.0
08
.0
2
=0
μ=
1.
1
4
σ
-1
μ=
5
.7
0.
μ=
0
μ=
4
0.
0.
μ=
μ=
45
0.86
[ms]
Estimation in the Feller model
21.1.2010
17 / 21
Back to the sample
Presence/Absence of the threshold
The sample: presence/absence of the threshold
The presence of the threshold S constraints the trajectories inside the
interval (0, S) and their length becomes random. In this sense neuronal
data are “unusual sample”.
28
28
26
26
24
24
22
22
20
20
μ=0.73
τ=25
σ 2 =0.0324
18
18
16
16
14
14
12
12
10
10
absence of threshold
8
0
20
40
60
80
100
presence of threshold
8
120
140
6
0
20
40
60
80
100
120
140
e.g. In red E(Xt ) is plotted according to the formula valid in the absence
of the threshold and it clearly does’t fit in its presence!.
⇒ Statistical inference for the conditioned process is different w.r.t. the
ordinary case.
R. Sirovich (Univ. Torino)
Estimation in the Feller model
21.1.2010
18 / 21
Back to the sample
Why biases?
Why the threshold brings biases?
The threshold introduces two problems that should be solved:
Model misspecification: the threshold is not accounted for in the
inference methods;
Short trajectories: asymptotic properties are not yet reached. Even
with high frequency data the drift is hard to be estimated in a finite
time interval.
Why σ 2 is not affected? σ 2 is a ’local’ parameter.
R. Sirovich (Univ. Torino)
Estimation in the Feller model
21.1.2010
19 / 21
Conclusions
Conclusions
Dedicated methods for statistical inference for processes conditioned
to non absorption are needed.
R. Sirovich (Univ. Torino)
Estimation in the Feller model
21.1.2010
20 / 21
References
E. Bibbona, P. Lansky, L. Sacerdote and R. Sirovich. Errors in
estimation of the input signal for integrate-and-fire neuronal models,
Physical Review E, 78(1):1–10, 2008.
E. Bibbona, P. Lansky and R. Sirovich. Signal estimation from
intracellular recordings in the Feller neuronal model. To appear in
Physical Review E.
R. Sirovich (Univ. Torino)
Estimation in the Feller model
21.1.2010
21 / 21
References
E. Bibbona, P. Lansky, L. Sacerdote and R. Sirovich. Errors in
estimation of the input signal for integrate-and-fire neuronal models,
Physical Review E, 78(1):1–10, 2008.
E. Bibbona, P. Lansky and R. Sirovich. Signal estimation from
intracellular recordings in the Feller neuronal model. To appear in
Physical Review E.
THANK YOU!
R. Sirovich (Univ. Torino)
Estimation in the Feller model
21.1.2010
21 / 21