[a
r
Q
1 j
t
j + 1
− a + a
r
Q
2 j
t
j + 1
− a
r
] [ − I
d
Q
1 j
t
j + 1
+ I
d
+ a
r
S] = [a
r
Q
2 j
t
j + 1
− a + a
r
Q
1 j
t
j + 1
+ a + a
r
] [ − I
d
Q
2 j
t
j + 1
− a + a
r
S] This can be written in the form
Saa + 2a
r
[1 − Q
1 j
t
j + 1
] + a
r
I
d
[Q
2 j
t
j + 1
− Q
1 j
t
j + 1
+ 1] [Q
2 j
t
j + 1
+ Q
1 j
t
j + 1
− 1] = 0. This relationship can be obtained from the results
presented by Bressloff and Taylor 1994, and is identical with formula 35 of La´nsky´ and Ro-
driguez 1999a.
2
.
3
.
2
. Example b
As a second example, let us take a model with four compartments: a trigger zone in series with a
dendritic compartment that is connected to a couple
of two
parallel compartments,
X 0 =
X
tz
, X
1
, X
11
, X
12
. The system Eqs. 1 – 3 takes the form
dX
tz
dt = −
a + a
1
X
tz
+ a
1
X
1
+ I
tz
dX
1
dt = −
a + a
1,11
+ a
1,12
+ a
1
X
1
+ a
1,11
X
11
+ a
1,12
X
12
+ a
1
X
tz
+ I
1
dX
11
dt = −
a + a
1,11
X
11
+ a
1,11
X
1
+ I
11
dX
12
dt = −
a + a
1,12
X
12
+ a
1,12
X
1
+ I
12
assuming a identical for all compartments. Fig. 2 shows the frequencies of spike emission as ob-
tained from Eq. 4 points and from numerical simulation of Eqs. 1 – 3 lines using a Euler
method with step dt = 0.01 ms, for inputs applied only on the most distal compartments dendrite 1
and dendrite 2 , namely I
tz
= I
1
= 0, I
11
, I
12
[ − 5, 20] mVs. The threshold value S = 2 mV and the
time constants have been chosen of the order of 10 ms. The values of parameters are a = 0.1
ms
− 1
, a
1,11
= a
1,12
= 116
ms
− 1
, a
1
= 0.9a
1,11
Fig. 2. Transfer function for inputs on two distal compart- ments
ms
− 1
, indicating a small variation of junctional time constants along compartments.
2
.
3
.
3
. Example c
Let us consider a model with N compartments in
series, X
0 =X
tz
, X
1
, X
11
, X
111
, …, X
1…1
, with an input applied to the most distal compart-
ment whose depolarization is X
1...1
. Fig. 3 shows how the transfer functions change
in dependency on the number of compartments the threshold S is kept fixed. The curves were
obtained as solutions of Eq. 4 and verified by numerical integration. It follows from Fig. 3 that
increasing, in this model, the number of compart- ments is accompanied by an increase of minimum
signal intensity necessary to evoke a response.
3. The stochastic system
3
.
1
. White noise perturbations and the filtering process
Let us now consider the model neuron under the action of constant deterministic inputs that
are corrupted by white noise perturbations. For
subthreshold behavior, the deterministic system Eqs. 1 – 3 turns out to be a N
d
+ 1 dimensional
Ornstein – Uhlenbeck process
given by
the stochastic differential equation
dX
t
0 =MX
t
0 +I 0 dt+s˜ dW
t
X 0 =X0
5 where I
0 is constant input and W
t
is a standard Wiener
process with
amplitudes s
˜ = s
tz
, s
i
1
, ..., s
i
1
i
2
…i
B
specific for different com- partments. The mean behavior of the system of
Eq. 5 is the same as the behavior of the deter- ministic system Eqs. 1 – 3. We will derive the
second-order moments of X
t
0 in order to investi- gate the coding characteristics of the model neu-
ron. Denoting Kt the covariance matrix of the process X
t
, 0 Kt=E{[X
t
0 −EX
t
0 ][X
t
0 −EX
t
0 ]
T
} , it satisfies the differential equation dKtdt =
MKt + KtM
T
+ s ˜ s˜
T
with initial value K =
E{[X 0 −EX
0 ][X 0 −EX
0 ]
T
} , Arnold, 1974. The solution of this equation is
Kt = e
Mt
K e
Mt
+
t
e
Mt − s
s ˜ s˜
T
e
Mt − s
ds 6
As an example of application of this formula, we consider the last model of Section 2.3.3 N com-
partments in series. For the elements of Kt K
ij
t = [QtK Qt]
ij
+ s
2 t
Q
iN
t − sQ
Nj
t − s ds 7
where Qt =
k
e
l
k
t j k
[M − l
j
l
k
− l
j
] and l
i
, i = 1, …, N are the negative eigenvalues of the symmetric N × N matrix M. Integration in
Eq. 7 can be performed, and the limiting value of K
mm
t as t , which is the asymptotic value of the variance of the depolarization of the mth
compartment, is obtained as: VarX
m
= − s
2 N
k = 1
L
Nm k
2
2l
k
+
i ji j
L
Nm i
L
Nm j
l
i
+ l
j
8
Fig. 3. Transfer function for a model with N compartments in series. The reciprocals of time constants a
j
between compartments j and j + 1 are such that a
j + 1
= 11.01a
j
with a
1
= 0.2 ms
− 1
; the firing threshold S and constant a are the same as in Fig 2.
Fig. 4. Trajectories of depolarizations in the stochastic model. The parameters are s
11
= 5.0 mV
ms, a=0.1 ms
− 1
, a
r
= 116
ms
− 1,
I
d
= 10 mV ms
− 1
.
where L
Nm k
are elements of L
k
=
j k
[M − l
j
l
k
− l
j
]. The asymptotic covariances can be ob- tained in the same way. It follows from Fig. 4, in
which the behavior of the depolarization in this model is illustrated N = 3, X
0 =X
tz
, X
1
, X
11 T
and s˜ = 0, 0, s
11 T
, that a decrease of stochastic fluctuations occurs from the dendritic part to the
trigger zone. This decrease of variability along three compartments can be analyzed by using Eq.
8. Of course, this effect occurs also for the two-compartment model for which the formulas
are simpler and thus presented here. In that case,
M = −
a + a
r
a
r
a
r
− a + a
r
with eigenvalues l
1
= − a , l
2
= − a − 2a
r
, where a
and a
r
correspond to the reciprocals of time constants for transmembrane current and for cur-
rent between the trigger zone and the other den- dritic compartment. We have
VarX
tz
= − s
2
4 1
2l
1
+ 1
2l
2
− 2
l
1
+ l
2
and VarX
1
= − s
2
4 1
2l
1
+ 1
2l
2
+ 2
l
1
+ l
2
The decreasing variability from the dendrite to the trigger zone follows clearly from these formulas;
these results were obtained by another method by La´nsky´ and Rodriguez 1999a.
The filtering of white noise from distal to prox- imal compartment is a rather natural mechanism
due to the passive transfer of current between neighboring compartments. These analytical re-
sults concern two-compartment systems. How- ever, Eq. 8 may be a useful tool to analyze this
filtering for models with N compartments in series. The most general case, with arbitrary
branching structure, can be handled directly using Eq. 6.
3
.
2
. Poissonian inputs and mean transfer functions The continuous trajectories of the membrane
depolarization are the main characteristic of the already described multicompartmental model. The
reason is that the model does not consider the soma of the neuron as its specific part, where the
contributions to the membrane depolarization are not so frequent not so many synaptic endings are
located there as on the dendrite but of substan- tial size. While the input at the dendrite causes
small changes of the membrane potential, and thus the system is well characterized by the white
noise, it would be natural to expect that the incoming signal located at the soma and near the
soma has discontinuous effect on the depolariza- tion. This leads to the investigation of a model
neuron where the stochastic part also includes Poissonian inputs. Let us consider the following
system:
dX
t
0 =MX
t
0 +I 0 dt+dz0
9 where I
0 is constant input, z0=z
tz
, z
i
1
, z
i
1
i
2
, z
i
1
i
2
i
3
, ....., z
i
1
i
2
…i
B
T
is a
multidimensional stochastic process with components of the form
z
j
= s
j
W
j
+ a
j
N
j +
+ b
j
N
j −
. Here, s
j
is the ampli- tude of the white noise part with W
j
as Brownian motion acting on the jth compartment, and N
j +
resp. N
j −
are independent Poisson processes with intensities l
j +
resp. l
j −
that drive synapses on this compartment for which excitatory and
inhibitory postsynaptic potentials have their am- plitudes a
j
resp. b
j
. For this neuronal model, one can derive a relation between estimate of the
mean interspike interval and the constant input I
0 . This can be done by using Eq. 4, with I0 identified as I
0 =I 0 +K0, where K
j
= a
j
l
j +
+ b
j
l
j −
. The transfer functions deduced in this way were
compared with those obtained by simulation of system of Eq. 9. Here, only a simple example is
presented, in which the neuron is composed of three compartments in series, X
0 =X
tz
, X
1
, X
11 T
, with constant plus white noise input acting on X
11
and Poissonian impulses impinging on X
1
. Using the same notation for the parameters as before,
the stochastic system of Eq. 9 is dX
11
= [ − a + a
1,11
X
11
+ a
1,11
X
1
+ I
11
] dt + s
11
dW dX
1
= [ − a + a
1,11
+ a
1
X
1
+ a
1,11
X
11
+ a
1
X
tz
] dt +
a
1
dN
1 +
+ b
1
dN
1 −
dX
tz
= [ − a + a
1
X
tz
+ a
1
X
1
] dt The input – output functions for this system are
plotted in Fig. 5, where a comparison with the single-compartment model with the same time
constant a
− 1
is also shown curve on the left. The dashed curve on the right corresponds to the
deterministic case s
11
= a
1
= b
1
= 0. The dashed
curve in the middle corresponds to the mean frequencies of the stochastic system as obtained
from Eq. 4. Finally, the crosses were obtained from simulated mean interspike intervals.
A numerical procedure of Euler type was used in order to solve this set of stochastic equations.
For the Brownian part, in the first equation, at each time step of length Dt, a random variable
z = s
11
Dtu was generated where u is a Gaussian variable of zero mean and variance equal to one.
In the second equation, a set of exponentially distributed random time intervals between jump
instants, with parameters l
1 +
resp. l
1 −
, was gen- erated. The process N
1 +
t resp. N
1 −
t was built with jumps of amplitude unity at the jump in-
stants. At each time step of length Dt, at time t, the new value of X
1
, for the Poissonian part, was achieved by adding a
1
[N
1 +
tt + Dt−N
1 +
tt] and b
1
[N
1 −
tt + Dt−N
1 −
tt] to the preceding value.
4. Conclusions
