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
We consider here some particular choices of parameters values for which 6 and 7 or 9
hold. If we do not explicitly state different values for the boundary and the initial value, we always
assume S = 10 mV, y =
0 mV. As a first scenario we analyze the dependence of
interspike intervals on u which is function of the membrane time constant t and of the IPSPs via
3. Consider two models characterized by equal values for all the parameters except that u
1
\ u
2
. In this case condition 9 is verified. Hence first
passage times are inversely proportional to the value of u for a model with a reflecting boundary
at y. However, if the absorbing boundary S is far enough from V
I
the presence of the reflecting boundary 10 does not effect the order relation-
ships between FPTs and can be disregarded. Hence 8 holds for the original model. In Fig. 2a
we give an example of some instances where 8 holds. We plot the first passage time distribution
for u = 5,7 ms, m = 0,1 mV ms
− 1
, s
2
= 0.4 mV
ms
− 1
and V
I
= − 10 mV. Fig. 2b exemplifies a
case where the effect of 10 cannot be disre- garded. Indeed high depolarization values of the
Fig. 2. FPT distributions a from top to bottom: u = 7 ms, m =
1 mV ms
− 1
; u = 5 ms, m = 1 mV ms
− 1
; u = 7 ms, m = 0 mV ms
− 1
; u = 5 ms, m = 0 mV ms
− 1
; b u = 3.5 ms, continu- ous line, and u = 8 ms, dashed line.
membrane potential become attainable because of the smaller value of V
I
= − 7 mV and of the
larger noise coefficient s
2
= 1.6 mV ms
− 1
. In this instance the spikes times are not ordered, for
example if u = 3.5 and 8 ms as shown in Fig. 2b. The second scenario considers the effect of
decreasing the value of V
I
. Making use of 6 with all the parameters equal in the two models except
V
I 1
B V
I 2
, the FPT for the first model is smaller than for the second. Note that this result cannot
be easily explained by resorting to intuition. In- deed we have smaller firing times when we con-
sider a larger diffusion interval for the membrane potential process. However we emphasizes that
the role of V
I
is more complex since it changes the effect of IPSPs on the membrane potential behav-
ior. In Fig. 3 we illustrate by means of two different examples the theoretical results concern-
ing the role of V
I
obtained from 6. The two curves on the top compare the FPT distributions
of two models characterized by V
I
= − 10 or − 7
mV, m = 1 mV ms
− 1
, s
2
= 0.4 mV ms
− 1
, u = 5 ms while the two curves at the bottom consider
Fig. 3. FPT distributions. From top to bottom V
I
= − 10 mV,
m = 1 mV ms
− 1
; V
I
= − 7 mV, m = 1 mV ms
− 1
; V
I
= − 10
mV, m = 0 mV ms
− 1
; V
I
= − 7 mV, m = 1 mV ms
− 1
. Fig. 5. Different choices for the parameters give rise to the
same FPT probability density a and distribution b.
the same choices of values except that m = 0 mV ms
− 1
. Note that for both sets of parameters the FPT can be ordered but when one introduces a
net positive excitation the distance between the two curves decreases. Hence in this instance the
model is less effected by the values of the rever- sal potential.
The next situation considers the dependence of the ISI distribution on m. Use of 6 in this
instance confirms our intuition. Indeed a larger net excitation implies smaller ISIs if the other
parameters are kept equal. Fig. 4 illustrates this result when u = 5 ms, s
2
= 0.4 mV ms
− 1
, V
I
= −
10 mV and m = − 0.5, 0, 0.5, 1 mV ms
− 1
, respectively.
Afterwards we consider the dependence of FPT on the noise parameter s
2
. Since, following Feller’s classification of boundaries, for m − V
I
u ] s
2
2 the inferior boundary is entrance, in order to keep the realism of the model s
2
must satisfy this inequality. In this instance, if s
1 2
B s
2 2
, the FPT of the first model is faster than the corresponding value for the second one. This
confirms the intuition that noise facilitates the crossings.
In these four scenarios we have tried to un- derstand the effect of changing an individual
parameter. However our ordering results can be used in more complex circumstances when more
than one parameter vary and the intuition from the simple inspection of mean trajectories can
lead to misleading results.
Finally we analyze an application of 12. We consider the following instance: assume we have
determined the value of the infinitesimal vari- ance with a possible error of 10. We want to
understand whether this error can be balanced by a variation in the values of the other
parameters to obtain equal FPTs. Making use of condition 12 it is easy to see that if, corre-
sponding to an increased value of s
2
we in- crease by the same percentage the coefficient m
the boundary value and decrease the reversal potential, the FPT distribu-tion cannot change.
The curve in Fig. 5 can be drawn either with the choice for the parameters s
2
= 0.4 mV, u =
0.5 ms, m = 1 mV ms
− 1
, V
I
= − 10 mV, S = 10
mV and y =
0 mV or with the choice s
2
= 5
mV, u = 5 ms, m = 1 mV ms
− 1
, V
I
= − 11 mV,
S = 11 mV and y =
0 mV. In the same way we could determine many other instances that give
rise to the same shape. Hence we cannot uniquely determine parameters values just from
measures of ISIs.
Fig. 4. FPT distributions. From top to bottom: m = 1 mV ms
− 1
; m = 0.5 mV ms
− 1
; m = 0 mV ms
− 1
; m = 0.5 mV ms
− 1
.
5. Conclusions
