the effect of a parameter value on the ISI distribu- tion. On the other hand, this approach cannot be
employed in the case of the Feller model since the complexity of the underlying mathematical process
limits to a few instances the use of this method. Indeed, as we illustrate in Section 2 by means of
some examples, the use of mean trajectory plots is useful in understanding the dependences of the
Feller model on the parameters only if spikes correspond to the so called ‘deterministic crossings’
cf. Smith, 1992.
Here we propose a new approach to the study of the parameter dependence of ISI distributions. We
make use of a theorem proved in Sacerdote and Smith, 1999 to order first passage times FPT
corresponding to different processes. In Section 3 we re-state this theorem in the particular instance
of the comparison of first passage times for two Feller processes characterized by two different set
of values of the parameters.
In Section 4 we analyze, with the help of the aforementioned theorem, the dependence of ISI
intervals on the values of the inhibitory reversal potential, the membrane spontaneous decay time
constant, the noise characterizing the model and the net excitation impinging upon the neuron.
Furthermore we determine different sets of values for the parameters corresponding to the same ISI
distribution. For example we can prove in this way that a percent change of the coefficient describing
the noise of the membrane can be balanced by a corresponding percent variation of the values of the
boundary, of the reversal potential and of the net excitation.
2. The model
Classical Stein’s model cf. Stein, 1965 for the description of the membrane potential, adapted to
consider inhibitory reversal potentials, is given by the stochastic differential equation SDE cf. La´n-
sky´ and La´nska´, 1987:
dY
t
= −
Y
t
t dt + a dN
t +
+ oY
t
− V
I
+ j Y
t
− V
I
dN
t −
; Y
= y
\ V
I
1 where
t \ 0,
a \ 0,
− 1 B o B 0,
V
I
B are
constants and N
+
= {N
+
t, t ] 0}, N
−
= {N
−
t, t ] 0} are two independent homogeneous Pois- son processes with intensities l,v respectively. The
constant V
I
represents the inhibitory reversal po- tential, a is the size of the excitatory postsynaptic
potential EPSP and o characterizes the size of the inhibitory postsynaptic potential IPSP.
j is a random variable, independent of N
+
, N
−
, defined in an appropriate interval in order to avoid
a jump to values below V
I
, and for which Ej = 0. The corresponding diffusion model, known as the
Feller model cf. La´nsky´ et al., 1995, is defined by the SDE:
dY
t
= −
Y
t
u + m
dt + s Y
t
− V
I
dW Y
= y
\ V
I
2 where
u = t
1 − ovt B
t ,
3 m
,s \ 0 are new constants which follow from a suitable limiting procedure performed on a se-
quence of models Eq. 1 and W = {Wt, t ] 0} is a Wiener process. The mean trajectory of the model
is
EY
t
Y =
0 = mu1 − e
− tu
4 and the variance of the trajectories is
VarY
t
Y =
= s
2
u V
I
1−e
− 2tu
2 +
s
2
u
2
m 1 − e
− tu
2
2 .
5 The distribution of ISI intervals, that corre-
sponds mathematically to the first passage times FPT of the process 2 through a boundary,
cannot be written in closed form and the model is generally studied with the help of numerical proce-
dures cf. Giorno et al., 1988. The first moments of the FPT can be written in closed form cf. Giorno
et al., 1988 but the complexity of these expressions makes it difficult to understand any functional
relationship between FPTs and parameters values.
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
