A first approach to determining the dependency of FPT on the individual parameters comes from
considering mean trajectory plots, obtained by plotting 4 as a function of t and considering the
bands obtained adding 9
VarY
t
Y =
0 or 9 2
VarY
t
Y =
0 to its value. Fig. 1 a – d are examples of these plots obtained by varying the
parameters u, m and V
I
respectively. The values used for these parameters are in a
biologically reasonable range cf. La´nsky´ et al., 1995 and references quoted therein. For the case
considered in Fig. 1a m = 0.5 mV ms
− 1
, s
2
= 0.4
mV ms
− 1
, V
I
= − 10 mV, u = 5 or 7 ms respec-
tively; mean trajectories: u = 7 ms dotted line, u =
5 ms dashed-dotted line; from top to bottom: upper and lower limits for u = 7 ms dashed line,
u = 5 ms thick line, crossing of a boundary in
S = 10 is due to the noise, while for the case of Fig. 1b m = 1.5 mV ms
− 1
, s
2
= 0.4 mV ms
− 1
, V
I
= − 10 mV, u = 5 or 7 ms respectively; same
lines caption as before there are deterministic crossings when u = 7 ms.
In this last instance we can easily deduce from Fig. 1b that FPT corresponding to higher values
of u are faster. However an analogous approach cannot be used for the instances considered in
Fig. 1a, since crossings are due to the noise and the shapes of the mean trajectories look quite
similar.
An analogous problem arises when one at- tempts to understand the dependence of FPTs on
V
I
. Some authors propose a value of − 10 mV cf. La´nsky´ et al., 1995 and references quoted therein
while others suggest V
I
= − 7 mV, but it is not
easy to understand the implications of these two choices on firing times. Indeed, as shown by the
mean trajectory plots in Fig. 1c – d, when s
2
= 0.4
mV ms
− 1
, u = 5 ms, V
I
= − 7 or − 10 mV and
m = 0.5 or 1.5 mV ms
− 1
, respectively; mean tra- jectories: V
I
= − 7 mV coincident with V
I
= − 10
mV dashed-dotted line; from top to bottom: up- per and lower limits for V
I
= − 7 ms dashed line,
V
I
= − 10 ms thick line, for these instances the
boundary crossings are caused by the noise. Fur- thermore use of the analytical expressions 4 and
5 becomes difficult as the parameters values are changed simultaneously. Hence the use of an al-
ternative approach to this analysis seems desirable to improve the understanding of this model. In
the next Section we consider a new method to compare the ISI interval distributions as the
parameters values change.
3. Comparison of FPTs
In many instances it looks difficult to determine closed form expressions for FPTs and hence com-
paring interspike distributions seems a hard task. However, focusing on phenomena driven by the
same noise values, it is obvious that when each sample path of a first process assumes values that
are always larger than the corresponding ones for the second process the interspike intervals will be
smaller for the first model. Furthermore in this instance the FPT distribution for the first model is
Fig. 1. Mean trajectory plots illustrating the dependence of FPT on u a and b and V
I
c and d.
dominated by the FPT distribution of the second one.
One could suppose that the same ordering be- tween FPTs is likely to arise in different in-
stances, corresponding to particular choices for the parameters characterizing the models. A
mathematical approach to these question has been proposed in Sacerdote and Smith 1999
where theorems useful for ordering first passage times of diffusion processes through boundaries
have been proved. Making use of the mathemati- cal
tools provided
in that
paper we
can consider the case of two diffusions Y
1
, Y
2
satisfying the SDE 2 driven by the same noise process W and characterized by parameters
u
1
, m
1
, s
1
, V
I 1
, S, y and
u
2
, m
2
, s
2
, V
I 2
, S, y
respectively. For easy of notation from now on we will set a
k
= 1s
k 2
m
k
− V
I k
u
k
, k = 1, 2 . If the parameters satisfy one of the two
following relationships:
a. u
1
] u
2
a
1
] a
2
6 or
b. u
1
] u
2
a
1
] a
2
S 5 min s
1 2
a
1
− a
2
1u
1
− 1u
2
+ V
I 1
, s
2 2
a
1
− a
2
1u
1
− 1u
2
+ V
I 2
7 then
the first passage times
T
Y
1
S y
and T
Y
2
S y
of the processes Y
1
, Y
2
originating at Y
i
= y
, i = 1, 2, satisfy the inequality: T
Y
1
S y
5 T
Y
2
S y
8 with probability one. This means that if we com-
pare the interspike intervals corresponding to the models characterized by the parameters indexed
1 with the analogous ones for the model indexed 2, always considering processes characterized by
the same driving noise W, we are sure that the last ones are always larger than the others. Fur-
thermore inequality Eq. 8 holds if c.
u
1
] u
2
a
1
B a
2
9 and the lower process is constrained by a reflect-
ing boundary in y = max
s
1 2
2a
2
− a
1
1u
1
− 1u
2
+ V
I 1
, s
2 2
a
2
− a
1
1u
1
− 1u
2
+ V
I 2
10 We can use relationships 6, 7 and 9 to
study the effect of changes in the parameters values on the model features. In particular we
can compare the distributions of interspike inter- vals since, as we have already mentioned, when
the values corresponding to the same driving noise are ordered it follows that the distributions
are ordered cf. Shaked and Shanthikumar, 1988. Hence by means of 6, 7 and 9 we can
determine ranges of parameters values that in- crease the interspike intervals.
Note that condition 10 adds a new hypothe- sis to the model by introducing a boundary that
cannot be justified from a biological point of view. However, as we will see in the next Section
by means of some numerical examples, inequality 8 holds under more general instances and the
interspike intervals comparison can be used to understand the role of the parameters when they
vary in a larger range. Furthermore, when the parameters are in a biologically interesting range,
condition 10 can be removed without introduc- ing serious errors and the validity of the relation-
ship
8 is
kept for
the models
under consideration.
Another approach to the analysis of the parameters dependence of interspike interval dis-
tributions consists in determining possible differ- ent values for the parameters of the same model
that produce equal first passage times. The ratio- nal for this study is strictly related to the prob-
lem of estimating the parameters values from samples of interspike intervals. In order to un-
derstand if it is possible that two different pro- cesses
give rise
to the
same interspike
distribution, we consider two Feller models char- acterized by parameters u
1
,m
1
,s
1
,V
I 1
,S
1
,y
1
and u
2
,m
2
,s
2
,V
I 2
,S
2
,y
2
and we consider the spatial transformations:
g
i
y = 2
s
i
y−V
I i
, i = 1, 2.
11 It is a simple matter of calculus to observe that if
a u
1
= u
2
b S
2
= s
2 2
s
1 2
S
1
− V
I 1
+ V
I 2
c y
2
= s
2 2
s
1 2
y
1
− V
I 1
+ V
I 2
d 2a
1
= 2a
2
12 then
T
Y
1
S
1
y
1
= T
Y
2
S
2
y
2
. 13
Hence we are able to determine sets of parame- ters values for two different processes that lead to
the same interspike distribution.
4. Results and discussion
