Fig. 3. Firing pdf’s for the neuronal model of Section 3, for t
= 0, x
= r = −
70, s
2
= 2, S = − 60, q = 5, l = 14 and
a = 1, 2, 3, 4. Decreasing modes refer to increasing values of a.
parameters mimic the effect of an external neu- ron’s input whose initial strength l exponentially
damps with the time-constant a. When x 5
r ,
which is the interesting case in neurobiology, from Eq. 15 it follows that xt is initially increasing,
to decrease monotonically as t towards the resting potential r, after reaching a maximum.
Note the significant diversity of behavior of xt for l \ 0 and for l = 0, the latter representing the
deterministic version of the OU model. Indeed, if l =
0, xt monotonically tends to the resting potential r for all x
r .
4. Numerical results
In this section, we discuss some features of the Gauss – Markov neuronal model presented above,
on the ground of computational results obtained by making use of the numerical procedure of Di
Nardo et al. 1998b.
Figs. 3 – 6 show some examples of firing pdf’s. Here, we have taken t
= 0, x
= r = −
70 and s
2
= 2, and we have presented separately the four
cases arising when the threshold is S = − 60 or S = − 65, and when q = 5, l = 14, or q = 10,
l = 7. Each figure refers to various choices of a.
In Figs. 3 and 4, we have chosen a = 1, 2, 3, 4, while a = 0.5, 1, 2, 3, 4 in Figs. 5 and 6.
In all cases, the firing pdf’s are unimodal; the magnitudes of the modes appear to be increasing
in a, while their abscissae decrease with a. The latter are listed in Table 1.
The firing pdf’s in Figs. 3 – 6 should be com- pared with those appearing in Figs. 1 and 2 for
the OU model specified by the same choice of the
Fig. 4. As in Fig. 3 but for q = 10 and l = 7.
Fig. 5. As in Fig. 3 but for S = − 65 and a = 0.5, 1, 2, 3, 4. Table 1
Abscissae of the modes of the firing pdf’s plotted in Figs. 3–6 S = −60
S = −65 a
q = 5;
q = 10;
q = 5;
q = 10;
l = 7
l = 14
l = 14
l = 7
0.5 0.99
0.50 5.95
1.56 2.56
0.42 0.86
1 1.22
0.94 2
0.77 0.38
2.11 0.74
0.37 1.89
3 0.87
0.37 1.77
0.83 4
0.72 Fig. 6. As in Fig. 5 but for q = 10 and l = 7.
Fig. 7. Cumulative distribution functions corresponding to the firing pdf’s presented in Fig. 3; from bottom to top it is a = 1,
2, 3, 4.
ron to have fired before t are enhanced by large values of a. Hence, the firing times described by
the random variable T are stochastically ordered in a decreasing fashion see, for instance, Shaked
and Shanthikumar, 1994 as a grows larger. Such a behavior is expected on the grounds of the drift
monotonicity for the Gauss – Markov model; in- deed, from Eq. 14 it follows that A
1
x, t is increasing in a. On the contrary, it could be
shown that the firing times corresponding to dif- ferent a values do not satisfy the stronger hazard-
rate order criterion.
5. Conclusions
