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On the moments of the firing interval of the diffusion approximated model neuron
On the Moments of the Firing Interval of the Diffusion Approximated Model Neuron SHUNSUKE Department
SAT0 of Biophysical Engineering, Faculty of Engineering Science
Osaka Universiq,
Toyonaka, Osaka 560, Japan
Received 25 May 1977; revised 3I October 1977
ABSTRACT The nth moments of the firing interval of the diffusion approximated model neuron with a constant threshold level were obtained for n = 1,2,3. The distribution function was also computed numerically, utilizing Durbin’s method. Simple relations on the mean and the variance were obtained for the low and the high threshold level, respectively.
I.
INTRODUCTION
There have been a number of works on modeling the neurophysiologitally observed stochastic property of the intervals between two successive action potentials of a single neuron. Some of these works are based on the idea that the neuron action potential is caused only by the first arrival of the neuron membrane potential at the threshold potential. Provided that the neuron membrane potential in these models fluctuates according to randomly arriving excitatory and inhibitory input impulses, the model can be mathematically described by the so-called one dimensional diffusion equation. The solution process is known as the Ornstein-Uhlenbeck (O-U) process. The problem then occurs of obtaining the probability density function of the time at which a sample path of the given diffusion process first crosses a given boundary as the threshold potential. Unfortunately this problem has not been solved except for special cases. In this paper, we will give exact expressions of up to the third central moments of the probability distribution function of the interspike interval for the diffusion approximated model neuron with a constant threshold, and develop an asymptotic form of the distribution function. Numerical results on the shape of the distribution function are given, utilizing Durbin’s
MA THEMA TICAL BIOSCIENCES
0 Elsevier North-Holland, Inc., 1978
39, 53 -70 ( 1978)
53 0025-5564/78/0039-0053$02.25
SHUNSUKE
54
SAT0
method, which gives numerically the boundary crossing probability for the Brownian motion. Simple relations between the mean and the variance of the firing interval are derived. II.
DIFFUSION
MODEL
FOR NEURON
ACTIVITY
In this section, we will sketch a diffusion model for neuron activity, following mainly Capocelli and Ricciardi [1] (though equivalent formulations have been given by Johannesma [3], Sugiyama et al. [7] and others in slightly different ways), because some of their results are needed in what follows. Denote by a random variable x the state of neuron-i.e., the membrane potential, the value of which is assumed to be only the quantity relevant to its firing. The state variable x of the neuron, if there is no input, decays to its resting value (x=0) with time constant 19: x(t)=x(t,)exp
t- t, ( 1
(1)
.
-T
Denote by ae and (Y~the time rates of the excitatory and inhibitory input impulses, which are assumed to arrive independently. The membrane potential change, for an excitatory input arriving in (t, t + dt), is x(t)-+x(t)+e, for an inhibitory
e>O;
(2)
i
(3)
input, as it is x(t)-+x(t)+i,
Denote the threshold potential for the neuron firing by S. Choose (y,, oi, e and i suitably, i.e., put e+O, i-0, cu,+oo and oi+co in such a way that O=a,e+a,i, 0= cu,e”+ ajin,
y = aee2 + aii2, (4)
n > 3.
Then the transition p.d.f. for the membrane potential to be in (x,x+dx) at the instant t under the condition that it was x,, (S > x0) at to (t > to) is as follows:
af(x~tlxo~4-J a
at
p
=Ze (~f(xh3Jo))+7
limf(x,tlx,,to)=S(X-Xo). t-1,
a’f(XdlXOttO) ax2
’
(5) \ I
55
ON THE MOMENTS OF FIRING INTERVALS
For a rigorous treatment, see Capocelli and Ricciardi [I]. The solution process is known as the Omstein-Uhlenbeck process. Putting t,,=O, we have under the free boundary condition the solution
f(x,tlxo)
(x - x,e-‘/@)2
1 = V%a
exp
-
202
(
I7
e2=___ 5
l-e-2’/9
).
(6)
In other words, the membrane potential varies below the threshold or under the free boundary condition with the density function (6). Let us consider the time interval between two successive firings. We denote by r the time that the membrane potential x (with x=x,, at t =O), first arrives at the constant threshold S, and by g(t,S(x,,) the density function of x. The function g(t,Slx,,) satisfies the following equation, since the process is stationary:
f(x34xo)=
s. s,~~Tg(~,SIXo)f(X,~-71S), x >
The solution of this integral equation is not known. Equation (5), with respect to the transition function membrane potential, is reduced to 1 a?f af --= 2 ax’2
at’
’
lim f’(x’,t’j0)=6
p.d.f. f(x, tlx,) of the
(X’)
f-0
by the transformation
$
x’=
(xe”@ - x0),
l),
t’=q~(w’-
f
f’(x’,t’lO)=f (x,flxrJ)g I
. I
The solution process of Eq. (8) is known as the Wiener process. The first passage time p.d.f. (assuming that the boundary varies with time) is given by
[ t’ q/g g(~9~lxo)=&Tw
(~(t~(l+~))~~-x~}lo]lcl,
where g&t’, S’(t’)lO) is the first passage time p.d.f. of the Wiener process to
56
SHUNSUKE
a boundary
S’(t)
SAT0
and satisfies
sy7Q
[S’(f) -
xq&) exp i
2(t’-7’)
i
*
(11) Ricciardi [4, 51 gave the same expression as (10) except for a difference in the scaling factors of x’ and t’. In fact, if we put .x’-+fi x’ and t’+2t’, we have the same expression as he gave. For an arbitrary S(t) [say S’(t’)],the solution of Eq. (11) is not known. However, Ricciardi [4] showed that in the case of the time dependent threshold S(l) = S,exp( - t/0), we have S’(f)
= f-
$
tso-xo)~
and thus
gw(lf,sf(f)(0)
=
wz:3J2xo) exp( - (2’P)(:-xo)2).
In the case that S(t) = S ( = const), we have S’(t’) x0). This case gives us a problem: Obtain the first Wiener process to a square root boundary [6]. In the case that cy,e+ ~~uii=m#O, the problem p.d.f. of the membrane potential under the free solving the following equation:
af(xJlxo) = _ at
(12)
= m (SW passage time p.d.f. of the of finding boundary
J?&?- ;)f(x,rlxo))+;
the transition is reduced to
ayxo) )
which, by a transformation similar to (9), is transformed into FCq. (8). Similarly the first passage time p.d.f. of the solution process satisfying Eq. (7) to a boundary S(t) is obtained through that of the Wiener process to S’( t’), where
S’(t)=6
((S(~ln(l+$))-~0)j/KL$-
-(x0-&)).
(14)
51
ON THE MOMENTS OF FIRING INTERVALS Thus,
Obtain
if S(t)= S (=const), the problem is classified into the following: the first passage time p.d.f. of the Wiener process to
(a) a boundary
if S > m6, (b) a constant
boundary
$
S’( t’) =
(mO-x0)
f if S=mO, (c) a boundary S’(f)=
d-3 $
I+ $ d-7
(me-x0)-(me-S>
if S < m8. In case (b), g(t, S(xO) is easily calculated g(t s,xo)_ ,
to be
VW (me-a
(ewe
)I -
1)
-312
CGe [V%Qme-n0)12 X
2e(*/@)‘exp 2(c
I
’
(z/@)r_ 1)
(15)
I
which is of course equivalent to that obtained by Sugiyama et al. [7]. We have mentioned so far the known results and the facts easily derived from them about this type of the model neuron. III.
ASYMPTOTIC BEHAVIOR PASSAGE TIME p.d.f.
AND
MOMENTS
OF THE
FIRST
In this section, we will discuss the asymptotic behavior and the moments of the first passage time p.d.f. of the O-U process to a constant boundary, using a singular point analysis of the Laplace transform gx(Slxr,). The Laplace transform of the p.d.f. in the case that S > x0 is already known to be I41
D-M(-xom) D_xB(-S~)’
(16)
58
SHUNSUKE SAT0
where
g,(Slx,)=
fmg(t,Slxo)e-A’df
(17)
Jo
and DA(z)is the Weber function. In the case that x0> S >O, from the symmetric property of a sample path of the O-U process, the transform is given as follows:
D-AO(XO~)
(18)
.
D_,,(Sm>
In the latter case, the membrane potential of the modeled neuron is initially set above the threshold potential, and then it decays toward zero level, being affected by randomly arriving input impulses. Thus it is expected that the membrane potential crosses the threshold level (S > 0) earlier than in the former case. Clearly case (c) in Sec. II corresponds to this case. If we know the inverse Laplace transform of gA(Slxo) given by Eq. (16) or Eq. (18) then we have the explicit form of the first passage time p.d.f. g(t,Slx,). Let us consider the singular points of gh(SIxo) first. To do this, it is convenient to use the integral representation of the Weber function: Dh(z)_
e-z2’4/me-zl-t2/2t-hldf, Ix-A)
Re(A) < 0.
where l?(z) is the gamma function. Note that z=O is a regular point. Then Eq. (16), for S > x0 (S >O), becomes
g,(Slxo)=
(19)
0
fme
Jo / me
xd - t=/2$
- 1 dt
St - 12/ZtX
- I dt
Re(A)
singular
>0,
0
where we have put 13= 1 and p= 2 for the sake of simplicity. (18) for x0> S >0 gives
sA(wo)=
-‘O- SI/0
Changing
the variable
,
a,
e
r2/2th
t in the integrals
Similarly,
Eq.
Re(h) > 0.
- I dt
by fi
and executing
them in
59
ON THE MOMENTS OF FIRING INTERVALS
terms of T, we have 9 (h x0)
I
where
1 Iyh/2)
= l+
&#y. n= *
Put X-0. We see that q&z) is analytic where p is a positive integer. If n = 2m, n+A
( 1
?lim
=
r
x+-2P
$ (
(22)
x,>s >o, ’
cp(k-S)
Qa=)
s>xo,
q(h3S) ’ G, - x0)
gA(Slxo)=
(23) around
X=0. Now put A+-
m >P,
0,
1
(-l)“p!
(24)
(pem)!
1
2p,
m
where we have used the relation
-- (-Vk k!
lim (z+k)T(z)-
for a positive integer k.
r--k
If n=2m+l, r
lim
n+h 2
(-1
r m+T) =
(
lim A+-2p
x+-2p
=o.
r$
r(+ >
(
(25)
>
Thus
x~~2p9)(w "P!
IJ (vrzz)2” (-1)”
2
m=O
pm)!
’
(P_m)!
p=o,1,2,*** (26)
Put A-+ -2p - 1. We have in a similar way
h
9_1(~+2P+wfw=
2r(p++)
p
(fiz)*m+I
(-l)m+l
B
2 m=O
(2m+l)!
(p-m)!
p=o,
1,2 )... .
’ (27)
60
SHUNSUKE
SAT0
Thus we have seen that C&Z) has singular points of the first order at X= -2~ - 1, p = 0,1,2,. . . , and is analytic elsewhere. Next we will consider the zero points of cp(X,z). Let X be a real number in the interval ( - l,O). From the above discussion, for any z,
,‘iy_ 9, (A, z) = 1
(28)
and lim X*-l+
(A+ l)cp(X,z)=
-
5
z.
(29)
Ir-
Thus if z > 0, then there exists a real zero point in (- l,O). -2p(r). Moreover, since acp/aX and acp/az exist and are equal to zero, the function -2p(z) of z such that q( continuous, from the implicit function theorem. Let z >0 Equating r
cp(h,z)=l+
1+x (-1 2 r;
(
Denote this by not identically 2p(z),z) =0 is be very small.
(30)
flz+o(z*) >
to zero, we have -2p(z)--1+
;
z.
(31)
i-
Now
(V7 z)” n!+-CO
if
z+co,
because r
n+h 2
(-1 r$ (
Therefore,
l/n!
<0
for
n=1,2 ,...,
-l
>
in order to make cp(A,z) be zero for large z, we must have -2p(t)*O.
(32)
61
ON THE MOMENTS OF FIRING INTERVALS Functionp(S)
TABLE 1 for Numerical Solution A= -2p(S)
s
P(S)
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0
/I61079 .424086 .388997 .355784 .324418 .294867 .267098 .241076 .2 16763 .194119 .I73101 .I53664
.470 .436
.I35759
.144
S
P’(S)
.310 ,254 .203
.119334 .I04337 907083 x IO- ’ .783888 .673151 .574208 .486373
P(S)
.I11 .963x lo- ’ .833
.517
P*(S)
.408935x IO-’ .341163 .2823 15 .231640 .I88384 .I51803 .I21 166 .957670x 1O-2 .749327 .580285 .444671 .337124 .252834 .187556 .I37609 .998542x lO-3 .716603 .508612 .357024 .247869
2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0
.423
of &,S)=O
,435 x 10-l ,302
.129
.618X 1o-2
.271
.790x lo-’
.272
aValues obtained directly from distribution function.
We have not succeeded in expressing -2p(z) we give numerical values of p(z) in Table 1. On the contrary, if z < 0,
lim L-1,
(X+l)(p(h,z)=
-
;
form. Instead
z >o,
C
and if z is very small in modulus, l.7
r&z)=
in an analytical
1+
then we have 1+x C-1 2 I-4 (
v2 z + 0 (z2),
(33)
)
from which
adw
ax
-_q &,A
[~(q+(p)]
- 1
(34)
62
SHUNSUKE
SAT0
where, and in what follows, q(z) is the polygamma function. Thus q(x,z) is monotonically decreasing with respect to A. This implies that there is no zero in ( - 1,0) if (zl is very small. However, if z < 0.
Thus Q@,z) has a zero [denoted by -2p(z) and if Iz( is very small, we have
as before] in (-3,
- 1). If z
-2p(z)--1+
3
z,
z
p&z)={
‘z:
t$i:
v&z)={
rz:
Sz$
(36)
d Similarly,
if 1z I-0. lim x4-2p+l_
(37) lim A+-2p-1,
This means that there exists at least one zero point in each interval (-2p-l,-2p+l),p=1,2 ,..., if 1~1-0. As the result, if z >O, then there exists a real solution -2p(z) of q&z)=0 such that - 1 < -2p(z) x0 (S GO) is given by -2p(S). Thus by the inverse Laplace transformation, we have an estimation: g(t,SJxo)--aepZP(S)’
for large t,
(38)
where 0 < 2p(S) < 1, and if S is small, 2p(s)- 1- w S. For the case that x0 > S > 0, we have g (t, S 1x0) -(Ye-’ because - 2p(S) for the case is the second largest pole in modulus. Assume that there is a complex solution - & of y(A, S) =0 such that 0 > Re( - &,) > -2p(S). Then, since - &, is also a solution, we have
which gives us, for large t, g(t,SIxo)-e-“o’cosa,t,
A, = wo +
ia,.
(9
63
ON THE MOMENTS OF FIRING INTERVALS
This contradicts the positivity of g(t, SIX,,). Here, let us consider a special case that x0 > S z 0. The Laplace transform is
gA(olxo)=g,(~~ -x0>.
(41)
As we already know the singular points of q(X, - x0), by the inverse Laplace transformation-namely by the integral
&
f$nm J’+imgA(O[xO)eArdX
for a>0
(42)
o--i&G -we
have g(t,Olxo)=
3
~(-xo)e-(2P+‘)r,
(43)
p=o where 2r(p+9 4(t)=
77
p
(fiz)*m+’
c m=O (2m+l)!
(-,),+I (p-m)!
.
(49
On the other hand, we can give the explicit form of g(z,O1xo) via the first passage time p.d.f. of the Wiener process to the boundary - SV%?’ +x0 EX
0:
s(ezfg(~~olxo)=
l)P3’2e2’exp(
- 2(e2_
1)).
(45)
It is easy to show that the right hand side of (43) coincides with that of (45). In fact, expanding the right hand side of (45) with respect to em*‘, we have
where we have made use of the relation
n!r(n
+ 4) = v’G (2n + 1)!/2*“+ i.
SHUNSUKE SAT0
64
Clearly the last expression gives the right hand side of (43). Let us return to gx(SIx,). Recall that the nth moment of the first passage time p.d.f. is given by
(46) The relation (46) can be easily obtained by differentiating Eq. (17) with respect to X and letting X+0. Let us denote the nth central moment by t5,(Slx,), n=O, 1,2,... . The following relations can be derived:
Po(Slxo>= to(Slxo)> j_~,(SIxa)= t,(SIx,)=mean; I_Lz(SIX~)=~~(SIX~)-t:(SIxO)=variance;
(47)
~L3(~i~g)=f3(~i~O)-3f:(~i~O)t2(~i~O)+2t:(~i~O),
Ildslx0) = skewness. CT3 ...
...
...
Thus we have
~(sIxo)=9)(o,xo)/(P(o~~)=1~ I* =
cp”‘(0, S) - ‘p(‘)(O,x,);
u2= - (p’2’(o,s) + 93(2)(o,x,,) + [ r$“(o, s)12p3(Slx,)
[ v(‘)(O,X,)]~;
= (p’3’(0,S) - @(O,Xe) - 3@“(0, s)(P(2)(0,s)
+
~~1'(0,S)]3-2[cp'l'(Olx,)]3~ ...
+3cp’~‘(O,x,)(~‘~‘(O,x,)+2[ ...
(48)
...
where
P’(O,z) =
T$rp(h,z)
k=1,2,...
7
(49)
h-0
In the above expressions, A)/2)/I(A/2). Then
r~@)(O,z)are obtained
where n(A)=$((n+X)/2)-q(G).
as follows: Rut [(A) = r((n +
Thus,
5‘(h)= exp( f+
‘)(A))
(50)
ON THE MOMENTS
OF FIRING
65
INTERVALS
with T+_‘)(h)= Jq(h)dh. Then according to the Bell’s polynomial on the compound function, we have the general expression for <(k)(A). For instance, for k= 1,2,3, we have
dS@) -
dh d/I*
‘ms (Q
3
=
,rcq77T) + $(A)],
d*m -
1
=
1
(51)
d3KV
-
dX3
1 = 231(~)[1’*‘(X)+3~“‘(h)1l(h)+113(h)l,
where
E(x)=r( qq/r( $), (52)
,(i)(,)=,(O(~)-,(i)(~), Thus, making k= 1,2,3,
use of the properties
i=l,2_
of r(z)
and
I/P(Z),
m
c
+
m=O
+I2 the Kummer
(p’*‘(o,z)=e*
l)!!
& (
)I
2m+2
m=O (m+
with F(a,P,z)
for
2m+2
2” 1)(2m+
(m+
we obtain
1)&I+
I)!!
(53)
&j(
11
function,
m fi
1 c
m=O
(2m+
I)m!
m
+
c
nt-_o
2m+1
P2(m)
2mY2
(
&
(ml
(m+ 1)(2m+ l)!!
1
(
&
2m+2
11
(54)
66 with Mm)
SHUNSUKE SAT0 = iC/(m + iI + Y, yz(m)=
QP’(O,t)
=
$(m + 1) + Y,
83
(55)
with
where y is the Euler constant and we have put X-+X8, z--+zm again. One may note that expression for the mean p in (48), if the second expression of (53) is substituted for q(I), is identical to that derived by Capocelli and Ricciardi [l]. We omit the expressions for the higher order moments as well as those of cpck)(O,z).However, one will see that the expressions for T(~)(O,Z) [i.e., those for /I&(m) and yk(m)] have close relations to those for 5 ck- i)(X), k = 1,2,. . . . For the case that x0 > S > 0, in order to find the moments, put S-, - S, x,,-+ - x,, in Eqs. (47)-(55). If x0 = 0 and $?? >>S >O, the mean and the variance of the first passage time are approximated as follows:
(56)
note that #(i) + y < 0. Thus the following practice, holds: pao2.
relation,
which may be useful in
(57)
We have not succeeded in giving an expression for g(t,S 1x0) for small t, because of lack of knowledge about the singular points. However we can calculate it numerically. J. Durbin proposed a numerical method of computing the first passage time p.d.f. of the Wiener process to a suitably smooth boundary a(t’). The possibility of making use of Durbin’s method within
ON THE MOMENTS
OF FIRING
INTERVALS
67
the neurophysiological context was first pointed out by Ricciardi [4, 51. However, if one tries to compute the values of gw( t’,u(t’)]O) up to large t’, he will find that it takes much computation time to obtain the results. Even if we make use of the modified method by transforming the Wiener process to the tied-down process as suggested by Durbin [2], we have to take care of the computing error. Thus combining the numerical results on g(t,S]x,,) according to Durbin’s method with the asymptotic evaluation of it, we could obtain the shapes of g(t,S]x,) (see Fig. 1) as well as the means and variances of the first passage times for various values of S >O and x0=0 (see Table 2). To have the moments, the relations
/i=JT,‘““(r,slo)dr+( to+ &)( e2=
l-
@,sIO)),
(58)
I0~%w0)~~+[ (to+&J+( hi’] x l(
‘“g(t,SJO)dt J0
(59)
-(@)* 1
were used, where g(t, S]O)= gw(t’,S’(t’)(0)Jdt’/dtl was computed up to t = to (i.e., the time up to which the asymptotic evaluation held for practical purposes), and p = 2, ~9= 1 were used. It seems from the computation results that the relation P-G
-[2p(s)]-’
for
S>l@
(60)
holds. This may be explained as follows: If S > m, then t,(S(xo) is very large in view of Eq. (48). This means that the “tail” part of g(l, S/x,), which obeys the estimate (ye -2pCS/G , makes the main contribution to the values of the moments. Thus they are approximated from Eq. (58) and Eq. (59) by
fi-[
2~‘( -+)I-‘.
8’-[
2~( +)I-:
(61)
In fact, the last column of Table 2 shows that [2p(s)]-’ is a good approximation of t,(s]x,) for S > a (= fl in the experiment). This means in turn that the above approximation must hold true. Note that the number of impulses produced by the model neuron in a fixed interval is subject to the Poisson distribution in this case because of (60). Estimations (57) and (60) would be useful in a practical sense, because in a neurophysiological experiment the observed data would first be reduced to the mean and the variance. Thus if one or both of the relations (57) and
SHUNSUKE
68
SAT0
2
s =.l -0 cn 25 al
1
FIG. 1.
Three examples of Firing interval distributions obtained by Durbin’s numeri-
cal method. In each case, we have set x,=0.
69
ON THE MOMENT8 OF FIRING INTERVALS TABLE 2 Moments of the Firing Interval
s
Mean’
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0
.I30549 .272478 .427394 .597201 .784163 990991 .122094x 10
Meanb .275
Varianceb ,429
1/2&S) .I08441 x 10 .I 17901
.I28536 ,789
.121x 10
.I40535 .154122
.I69568 .187197 .207403 .230667 .257574 .288849 .325385 .368300 .418992 .479216 .551217 .637846 .742775 .870765 .102802x l@
.147796 .176683
.148x 10
.337
.20934 1 .246490 .28902 1 .338044 .394949 .461487 .539885 .632984 .744442 .878999 .104284x Id .I24410
,208
.556
.332
.118x lOa
449
.202
.611
.359
.100x ld .120
.945 .135x Id
.I75
.292
.415
.167x 104
,879
.760
.208x Id
.428x 16
.197758
.762
.580x 106
,227 x 104
.515x 10’
.266587 .363348 .500730 .697736 .983068 .140047x104 .201719
.149355 .180556 .219945 .270142 .334724 .418614 .528647 .674385 .869316 .113263xld .149184
.198678 .267553 .364358 .501783 .698828 .9841498 .140163x lo’ .201839
.I22269 .146558 .177107 .215852 .265415 .329374 .412657 .522101 667265 .861646 .112443xld .148313
“True means obtained from Eq. (48) with Eq. (53). bComputed directly from the distribution function using Eq. (58), where 2p*(S) were used. cFor comparison.
70
SHUNSUKE SAT0
(60) are recognized remain a mechanism, physiologically
to hold experimentally,
candidate to explain the if the assumptions underlying
the present
diffusion
model
may
neurophysiological background the diffusion approximation are
plausible.
The author thanks Professor R. Suzuki of Osaka University for his kind interest in this study. He also thanks Professor L. M. Ricciardi of Luboratorio di Cibernetica de1 CNR, Naples, Italy, for his valuable suggestions. REFERENCES 1 R. M. Capocelli and L. M. Ricciardi, Diffusion approximation and first passage time problem for a model neuron, Kybemetik 8, Heft 6 (1971), 214-223. 2 J. Durbin, Boundary crossing probabilities for the Brownian motion and Poisson processes and techniques for computing the power of the Kolmogorov-Smimov test, J. Appl. Probability 3
4
5 6 7
8 (1971), 431-453.
P. I. M. Johannesma, Diffusion models for the stochastic activity of neurons, in Neural Networks (E. R. Caianiello, Ed.), Springer, 1968. L. M. Ricciardi, Diffusion approximations to neural activity and to the time dependent threshold problem, in Proc. of the 3rd European Congress on Cybernetics and Systems Research, Vienna, Apr. 1976, to be published. L. M. Riccardi, Diffusion processes and related topics in biology, Lecture Nofes in Biomathematics 14, Springer, 1977. S. Sato, Evaluation of the first passage time probability to a square root boundary for the Wiener process, J. of Appl. Probability, to be published. H. Sugiyama, G. P. Moore, and D. H. Perkel, Solutions for a stochastic model of neuron spike production, Math. Biosci. 8 (1970), 323-341.
SAT0 of Biophysical Engineering, Faculty of Engineering Science
Osaka Universiq,
Toyonaka, Osaka 560, Japan
Received 25 May 1977; revised 3I October 1977
ABSTRACT The nth moments of the firing interval of the diffusion approximated model neuron with a constant threshold level were obtained for n = 1,2,3. The distribution function was also computed numerically, utilizing Durbin’s method. Simple relations on the mean and the variance were obtained for the low and the high threshold level, respectively.
I.
INTRODUCTION
There have been a number of works on modeling the neurophysiologitally observed stochastic property of the intervals between two successive action potentials of a single neuron. Some of these works are based on the idea that the neuron action potential is caused only by the first arrival of the neuron membrane potential at the threshold potential. Provided that the neuron membrane potential in these models fluctuates according to randomly arriving excitatory and inhibitory input impulses, the model can be mathematically described by the so-called one dimensional diffusion equation. The solution process is known as the Ornstein-Uhlenbeck (O-U) process. The problem then occurs of obtaining the probability density function of the time at which a sample path of the given diffusion process first crosses a given boundary as the threshold potential. Unfortunately this problem has not been solved except for special cases. In this paper, we will give exact expressions of up to the third central moments of the probability distribution function of the interspike interval for the diffusion approximated model neuron with a constant threshold, and develop an asymptotic form of the distribution function. Numerical results on the shape of the distribution function are given, utilizing Durbin’s
MA THEMA TICAL BIOSCIENCES
0 Elsevier North-Holland, Inc., 1978
39, 53 -70 ( 1978)
53 0025-5564/78/0039-0053$02.25
SHUNSUKE
54
SAT0
method, which gives numerically the boundary crossing probability for the Brownian motion. Simple relations between the mean and the variance of the firing interval are derived. II.
DIFFUSION
MODEL
FOR NEURON
ACTIVITY
In this section, we will sketch a diffusion model for neuron activity, following mainly Capocelli and Ricciardi [1] (though equivalent formulations have been given by Johannesma [3], Sugiyama et al. [7] and others in slightly different ways), because some of their results are needed in what follows. Denote by a random variable x the state of neuron-i.e., the membrane potential, the value of which is assumed to be only the quantity relevant to its firing. The state variable x of the neuron, if there is no input, decays to its resting value (x=0) with time constant 19: x(t)=x(t,)exp
t- t, ( 1
(1)
.
-T
Denote by ae and (Y~the time rates of the excitatory and inhibitory input impulses, which are assumed to arrive independently. The membrane potential change, for an excitatory input arriving in (t, t + dt), is x(t)-+x(t)+e, for an inhibitory
e>O;
(2)
i
(3)
input, as it is x(t)-+x(t)+i,
Denote the threshold potential for the neuron firing by S. Choose (y,, oi, e and i suitably, i.e., put e+O, i-0, cu,+oo and oi+co in such a way that O=a,e+a,i, 0= cu,e”+ ajin,
y = aee2 + aii2, (4)
n > 3.
Then the transition p.d.f. for the membrane potential to be in (x,x+dx) at the instant t under the condition that it was x,, (S > x0) at to (t > to) is as follows:
af(x~tlxo~4-J a
at
p
=Ze (~f(xh3Jo))+7
limf(x,tlx,,to)=S(X-Xo). t-1,
a’f(XdlXOttO) ax2
’
(5) \ I
55
ON THE MOMENTS OF FIRING INTERVALS
For a rigorous treatment, see Capocelli and Ricciardi [I]. The solution process is known as the Omstein-Uhlenbeck process. Putting t,,=O, we have under the free boundary condition the solution
f(x,tlxo)
(x - x,e-‘/@)2
1 = V%a
exp
-
202
(
I7
e2=___ 5
l-e-2’/9
).
(6)
In other words, the membrane potential varies below the threshold or under the free boundary condition with the density function (6). Let us consider the time interval between two successive firings. We denote by r the time that the membrane potential x (with x=x,, at t =O), first arrives at the constant threshold S, and by g(t,S(x,,) the density function of x. The function g(t,Slx,,) satisfies the following equation, since the process is stationary:
f(x34xo)=
s. s,~~Tg(~,SIXo)f(X,~-71S), x >
The solution of this integral equation is not known. Equation (5), with respect to the transition function membrane potential, is reduced to 1 a?f af --= 2 ax’2
at’
’
lim f’(x’,t’j0)=6
p.d.f. f(x, tlx,) of the
(X’)
f-0
by the transformation
$
x’=
(xe”@ - x0),
l),
t’=q~(w’-
f
f’(x’,t’lO)=f (x,flxrJ)g I
. I
The solution process of Eq. (8) is known as the Wiener process. The first passage time p.d.f. (assuming that the boundary varies with time) is given by
[ t’ q/g g(~9~lxo)=&Tw
(~(t~(l+~))~~-x~}lo]lcl,
where g&t’, S’(t’)lO) is the first passage time p.d.f. of the Wiener process to
56
SHUNSUKE
a boundary
S’(t)
SAT0
and satisfies
sy7Q
[S’(f) -
xq&) exp i
2(t’-7’)
i
*
(11) Ricciardi [4, 51 gave the same expression as (10) except for a difference in the scaling factors of x’ and t’. In fact, if we put .x’-+fi x’ and t’+2t’, we have the same expression as he gave. For an arbitrary S(t) [say S’(t’)],the solution of Eq. (11) is not known. However, Ricciardi [4] showed that in the case of the time dependent threshold S(l) = S,exp( - t/0), we have S’(f)
= f-
$
tso-xo)~
and thus
gw(lf,sf(f)(0)
=
wz:3J2xo) exp( - (2’P)(:-xo)2).
In the case that S(t) = S ( = const), we have S’(t’) x0). This case gives us a problem: Obtain the first Wiener process to a square root boundary [6]. In the case that cy,e+ ~~uii=m#O, the problem p.d.f. of the membrane potential under the free solving the following equation:
af(xJlxo) = _ at
(12)
= m (SW passage time p.d.f. of the of finding boundary
J?&?- ;)f(x,rlxo))+;
the transition is reduced to
ayxo) )
which, by a transformation similar to (9), is transformed into FCq. (8). Similarly the first passage time p.d.f. of the solution process satisfying Eq. (7) to a boundary S(t) is obtained through that of the Wiener process to S’( t’), where
S’(t)=6
((S(~ln(l+$))-~0)j/KL$-
-(x0-&)).
(14)
51
ON THE MOMENTS OF FIRING INTERVALS Thus,
Obtain
if S(t)= S (=const), the problem is classified into the following: the first passage time p.d.f. of the Wiener process to
(a) a boundary
if S > m6, (b) a constant
boundary
$
S’( t’) =
(mO-x0)
f if S=mO, (c) a boundary S’(f)=
d-3 $
I+ $ d-7
(me-x0)-(me-S>
if S < m8. In case (b), g(t, S(xO) is easily calculated g(t s,xo)_ ,
to be
VW (me-a
(ewe
)I -
1)
-312
CGe [V%Qme-n0)12 X
2e(*/@)‘exp 2(c
I
’
(z/@)r_ 1)
(15)
I
which is of course equivalent to that obtained by Sugiyama et al. [7]. We have mentioned so far the known results and the facts easily derived from them about this type of the model neuron. III.
ASYMPTOTIC BEHAVIOR PASSAGE TIME p.d.f.
AND
MOMENTS
OF THE
FIRST
In this section, we will discuss the asymptotic behavior and the moments of the first passage time p.d.f. of the O-U process to a constant boundary, using a singular point analysis of the Laplace transform gx(Slxr,). The Laplace transform of the p.d.f. in the case that S > x0 is already known to be I41
D-M(-xom) D_xB(-S~)’
(16)
58
SHUNSUKE SAT0
where
g,(Slx,)=
fmg(t,Slxo)e-A’df
(17)
Jo
and DA(z)is the Weber function. In the case that x0> S >O, from the symmetric property of a sample path of the O-U process, the transform is given as follows:
D-AO(XO~)
(18)
.
D_,,(Sm>
In the latter case, the membrane potential of the modeled neuron is initially set above the threshold potential, and then it decays toward zero level, being affected by randomly arriving input impulses. Thus it is expected that the membrane potential crosses the threshold level (S > 0) earlier than in the former case. Clearly case (c) in Sec. II corresponds to this case. If we know the inverse Laplace transform of gA(Slxo) given by Eq. (16) or Eq. (18) then we have the explicit form of the first passage time p.d.f. g(t,Slx,). Let us consider the singular points of gh(SIxo) first. To do this, it is convenient to use the integral representation of the Weber function: Dh(z)_
e-z2’4/me-zl-t2/2t-hldf, Ix-A)
Re(A) < 0.
where l?(z) is the gamma function. Note that z=O is a regular point. Then Eq. (16), for S > x0 (S >O), becomes
g,(Slxo)=
(19)
0
fme
Jo / me
xd - t=/2$
- 1 dt
St - 12/ZtX
- I dt
Re(A)
singular
>0,
0
where we have put 13= 1 and p= 2 for the sake of simplicity. (18) for x0> S >0 gives
sA(wo)=
-‘O- SI/0
Changing
the variable
,
a,
e
r2/2th
t in the integrals
Similarly,
Eq.
Re(h) > 0.
- I dt
by fi
and executing
them in
59
ON THE MOMENTS OF FIRING INTERVALS
terms of T, we have 9 (h x0)
I
where
1 Iyh/2)
= l+
&#y. n= *
Put X-0. We see that q&z) is analytic where p is a positive integer. If n = 2m, n+A
( 1
?lim
=
r
x+-2P
$ (
(22)
x,>s >o, ’
cp(k-S)
Qa=)
s>xo,
q(h3S) ’ G, - x0)
gA(Slxo)=
(23) around
X=0. Now put A+-
m >P,
0,
1
(-l)“p!
(24)
(pem)!
1
2p,
m
where we have used the relation
-- (-Vk k!
lim (z+k)T(z)-
for a positive integer k.
r--k
If n=2m+l, r
lim
n+h 2
(-1
r m+T) =
(
lim A+-2p
x+-2p
=o.
r$
r(+ >
(
(25)
>
Thus
x~~2p9)(w "P!
IJ (vrzz)2” (-1)”
2
m=O
pm)!
’
(P_m)!
p=o,1,2,*** (26)
Put A-+ -2p - 1. We have in a similar way
h
9_1(~+2P+wfw=
2r(p++)
p
(fiz)*m+I
(-l)m+l
B
2 m=O
(2m+l)!
(p-m)!
p=o,
1,2 )... .
’ (27)
60
SHUNSUKE
SAT0
Thus we have seen that C&Z) has singular points of the first order at X= -2~ - 1, p = 0,1,2,. . . , and is analytic elsewhere. Next we will consider the zero points of cp(X,z). Let X be a real number in the interval ( - l,O). From the above discussion, for any z,
,‘iy_ 9, (A, z) = 1
(28)
and lim X*-l+
(A+ l)cp(X,z)=
-
5
z.
(29)
Ir-
Thus if z > 0, then there exists a real zero point in (- l,O). -2p(r). Moreover, since acp/aX and acp/az exist and are equal to zero, the function -2p(z) of z such that q( continuous, from the implicit function theorem. Let z >0 Equating r
cp(h,z)=l+
1+x (-1 2 r;
(
Denote this by not identically 2p(z),z) =0 is be very small.
(30)
flz+o(z*) >
to zero, we have -2p(z)--1+
;
z.
(31)
i-
Now
(V7 z)” n!+-CO
if
z+co,
because r
n+h 2
(-1 r$ (
Therefore,
l/n!
<0
for
n=1,2 ,...,
-l
>
in order to make cp(A,z) be zero for large z, we must have -2p(t)*O.
(32)
61
ON THE MOMENTS OF FIRING INTERVALS Functionp(S)
TABLE 1 for Numerical Solution A= -2p(S)
s
P(S)
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0
/I61079 .424086 .388997 .355784 .324418 .294867 .267098 .241076 .2 16763 .194119 .I73101 .I53664
.470 .436
.I35759
.144
S
P’(S)
.310 ,254 .203
.119334 .I04337 907083 x IO- ’ .783888 .673151 .574208 .486373
P(S)
.I11 .963x lo- ’ .833
.517
P*(S)
.408935x IO-’ .341163 .2823 15 .231640 .I88384 .I51803 .I21 166 .957670x 1O-2 .749327 .580285 .444671 .337124 .252834 .187556 .I37609 .998542x lO-3 .716603 .508612 .357024 .247869
2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0
.423
of &,S)=O
,435 x 10-l ,302
.129
.618X 1o-2
.271
.790x lo-’
.272
aValues obtained directly from distribution function.
We have not succeeded in expressing -2p(z) we give numerical values of p(z) in Table 1. On the contrary, if z < 0,
lim L-1,
(X+l)(p(h,z)=
-
;
form. Instead
z >o,
C
and if z is very small in modulus, l.7
r&z)=
in an analytical
1+
then we have 1+x C-1 2 I-4 (
v2 z + 0 (z2),
(33)
)
from which
adw
ax
-_q &,A
[~(q+(p)]
- 1
(34)
62
SHUNSUKE
SAT0
where, and in what follows, q(z) is the polygamma function. Thus q(x,z) is monotonically decreasing with respect to A. This implies that there is no zero in ( - 1,0) if (zl is very small. However, if z < 0.
Thus Q@,z) has a zero [denoted by -2p(z) and if Iz( is very small, we have
as before] in (-3,
- 1). If z
-2p(z)--1+
3
z,
z
p&z)={
‘z:
t$i:
v&z)={
rz:
Sz$
(36)
d Similarly,
if 1z I-0. lim x4-2p+l_
(37) lim A+-2p-1,
This means that there exists at least one zero point in each interval (-2p-l,-2p+l),p=1,2 ,..., if 1~1-0. As the result, if z >O, then there exists a real solution -2p(z) of q&z)=0 such that - 1 < -2p(z)
for large t,
(38)
where 0 < 2p(S) < 1, and if S is small, 2p(s)- 1- w S. For the case that x0 > S > 0, we have g (t, S 1x0) -(Ye-’ because - 2p(S) for the case is the second largest pole in modulus. Assume that there is a complex solution - & of y(A, S) =0 such that 0 > Re( - &,) > -2p(S). Then, since - &, is also a solution, we have
which gives us, for large t, g(t,SIxo)-e-“o’cosa,t,
A, = wo +
ia,.
(9
63
ON THE MOMENTS OF FIRING INTERVALS
This contradicts the positivity of g(t, SIX,,). Here, let us consider a special case that x0 > S z 0. The Laplace transform is
gA(olxo)=g,(~~ -x0>.
(41)
As we already know the singular points of q(X, - x0), by the inverse Laplace transformation-namely by the integral
&
f$nm J’+imgA(O[xO)eArdX
for a>0
(42)
o--i&G -we
have g(t,Olxo)=
3
~(-xo)e-(2P+‘)r,
(43)
p=o where 2r(p+9 4(t)=
77
p
(fiz)*m+’
c m=O (2m+l)!
(-,),+I (p-m)!
.
(49
On the other hand, we can give the explicit form of g(z,O1xo) via the first passage time p.d.f. of the Wiener process to the boundary - SV%?’ +x0 EX
0:
s(ezfg(~~olxo)=
l)P3’2e2’exp(
- 2(e2_
1)).
(45)
It is easy to show that the right hand side of (43) coincides with that of (45). In fact, expanding the right hand side of (45) with respect to em*‘, we have
where we have made use of the relation
n!r(n
+ 4) = v’G (2n + 1)!/2*“+ i.
SHUNSUKE SAT0
64
Clearly the last expression gives the right hand side of (43). Let us return to gx(SIx,). Recall that the nth moment of the first passage time p.d.f. is given by
(46) The relation (46) can be easily obtained by differentiating Eq. (17) with respect to X and letting X+0. Let us denote the nth central moment by t5,(Slx,), n=O, 1,2,... . The following relations can be derived:
Po(Slxo>= to(Slxo)> j_~,(SIxa)= t,(SIx,)=mean; I_Lz(SIX~)=~~(SIX~)-t:(SIxO)=variance;
(47)
~L3(~i~g)=f3(~i~O)-3f:(~i~O)t2(~i~O)+2t:(~i~O),
Ildslx0) = skewness. CT3 ...
...
...
Thus we have
~(sIxo)=9)(o,xo)/(P(o~~)=1~ I* =
cp”‘(0, S) - ‘p(‘)(O,x,);
u2= - (p’2’(o,s) + 93(2)(o,x,,) + [ r$“(o, s)12p3(Slx,)
[ v(‘)(O,X,)]~;
= (p’3’(0,S) - @(O,Xe) - 3@“(0, s)(P(2)(0,s)
+
~~1'(0,S)]3-2[cp'l'(Olx,)]3~ ...
+3cp’~‘(O,x,)(~‘~‘(O,x,)+2[ ...
(48)
...
where
P’(O,z) =
T$rp(h,z)
k=1,2,...
7
(49)
h-0
In the above expressions, A)/2)/I(A/2). Then
r~@)(O,z)are obtained
where n(A)=$((n+X)/2)-q(G).
as follows: Rut [(A) = r((n +
Thus,
5‘(h)= exp( f+
‘)(A))
(50)
ON THE MOMENTS
OF FIRING
65
INTERVALS
with T+_‘)(h)= Jq(h)dh. Then according to the Bell’s polynomial on the compound function, we have the general expression for <(k)(A). For instance, for k= 1,2,3, we have
dS@) -
dh d/I*
‘ms (Q
3
=
,rcq77T) + $(A)],
d*m -
1
=
1
(51)
d3KV
-
dX3
1 = 231(~)[1’*‘(X)+3~“‘(h)1l(h)+113(h)l,
where
E(x)=r( qq/r( $), (52)
,(i)(,)=,(O(~)-,(i)(~), Thus, making k= 1,2,3,
use of the properties
i=l,2_
of r(z)
and
I/P(Z),
m
c
+
m=O
+I2 the Kummer
(p’*‘(o,z)=e*
l)!!
& (
)I
2m+2
m=O (m+
with F(a,P,z)
for
2m+2
2” 1)(2m+
(m+
we obtain
1)&I+
I)!!
(53)
&j(
11
function,
m fi
1 c
m=O
(2m+
I)m!
m
+
c
nt-_o
2m+1
P2(m)
2mY2
(
&
(ml
(m+ 1)(2m+ l)!!
1
(
&
2m+2
11
(54)
66 with Mm)
SHUNSUKE SAT0 = iC/(m + iI + Y, yz(m)=
QP’(O,t)
=
$(m + 1) + Y,
83
(55)
with
where y is the Euler constant and we have put X-+X8, z--+zm again. One may note that expression for the mean p in (48), if the second expression of (53) is substituted for q(I), is identical to that derived by Capocelli and Ricciardi [l]. We omit the expressions for the higher order moments as well as those of cpck)(O,z).However, one will see that the expressions for T(~)(O,Z) [i.e., those for /I&(m) and yk(m)] have close relations to those for 5 ck- i)(X), k = 1,2,. . . . For the case that x0 > S > 0, in order to find the moments, put S-, - S, x,,-+ - x,, in Eqs. (47)-(55). If x0 = 0 and $?? >>S >O, the mean and the variance of the first passage time are approximated as follows:
(56)
note that #(i) + y < 0. Thus the following practice, holds: pao2.
relation,
which may be useful in
(57)
We have not succeeded in giving an expression for g(t,S 1x0) for small t, because of lack of knowledge about the singular points. However we can calculate it numerically. J. Durbin proposed a numerical method of computing the first passage time p.d.f. of the Wiener process to a suitably smooth boundary a(t’). The possibility of making use of Durbin’s method within
ON THE MOMENTS
OF FIRING
INTERVALS
67
the neurophysiological context was first pointed out by Ricciardi [4, 51. However, if one tries to compute the values of gw( t’,u(t’)]O) up to large t’, he will find that it takes much computation time to obtain the results. Even if we make use of the modified method by transforming the Wiener process to the tied-down process as suggested by Durbin [2], we have to take care of the computing error. Thus combining the numerical results on g(t,S]x,,) according to Durbin’s method with the asymptotic evaluation of it, we could obtain the shapes of g(t,S]x,) (see Fig. 1) as well as the means and variances of the first passage times for various values of S >O and x0=0 (see Table 2). To have the moments, the relations
/i=JT,‘““(r,slo)dr+( to+ &)( e2=
l-
@,sIO)),
(58)
I0~%w0)~~+[ (to+&J+( hi’] x l(
‘“g(t,SJO)dt J0
(59)
-(@)* 1
were used, where g(t, S]O)= gw(t’,S’(t’)(0)Jdt’/dtl was computed up to t = to (i.e., the time up to which the asymptotic evaluation held for practical purposes), and p = 2, ~9= 1 were used. It seems from the computation results that the relation P-G
-[2p(s)]-’
for
S>l@
(60)
holds. This may be explained as follows: If S > m, then t,(S(xo) is very large in view of Eq. (48). This means that the “tail” part of g(l, S/x,), which obeys the estimate (ye -2pCS/G , makes the main contribution to the values of the moments. Thus they are approximated from Eq. (58) and Eq. (59) by
fi-[
2~‘( -+)I-‘.
8’-[
2~( +)I-:
(61)
In fact, the last column of Table 2 shows that [2p(s)]-’ is a good approximation of t,(s]x,) for S > a (= fl in the experiment). This means in turn that the above approximation must hold true. Note that the number of impulses produced by the model neuron in a fixed interval is subject to the Poisson distribution in this case because of (60). Estimations (57) and (60) would be useful in a practical sense, because in a neurophysiological experiment the observed data would first be reduced to the mean and the variance. Thus if one or both of the relations (57) and
SHUNSUKE
68
SAT0
2
s =.l -0 cn 25 al
1
FIG. 1.
Three examples of Firing interval distributions obtained by Durbin’s numeri-
cal method. In each case, we have set x,=0.
69
ON THE MOMENT8 OF FIRING INTERVALS TABLE 2 Moments of the Firing Interval
s
Mean’
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0
.I30549 .272478 .427394 .597201 .784163 990991 .122094x 10
Meanb .275
Varianceb ,429
1/2&S) .I08441 x 10 .I 17901
.I28536 ,789
.121x 10
.I40535 .154122
.I69568 .187197 .207403 .230667 .257574 .288849 .325385 .368300 .418992 .479216 .551217 .637846 .742775 .870765 .102802x l@
.147796 .176683
.148x 10
.337
.20934 1 .246490 .28902 1 .338044 .394949 .461487 .539885 .632984 .744442 .878999 .104284x Id .I24410
,208
.556
.332
.118x lOa
449
.202
.611
.359
.100x ld .120
.945 .135x Id
.I75
.292
.415
.167x 104
,879
.760
.208x Id
.428x 16
.197758
.762
.580x 106
,227 x 104
.515x 10’
.266587 .363348 .500730 .697736 .983068 .140047x104 .201719
.149355 .180556 .219945 .270142 .334724 .418614 .528647 .674385 .869316 .113263xld .149184
.198678 .267553 .364358 .501783 .698828 .9841498 .140163x lo’ .201839
.I22269 .146558 .177107 .215852 .265415 .329374 .412657 .522101 667265 .861646 .112443xld .148313
“True means obtained from Eq. (48) with Eq. (53). bComputed directly from the distribution function using Eq. (58), where 2p*(S) were used. cFor comparison.
70
SHUNSUKE SAT0
(60) are recognized remain a mechanism, physiologically
to hold experimentally,
candidate to explain the if the assumptions underlying
the present
diffusion
model
may
neurophysiological background the diffusion approximation are
plausible.
The author thanks Professor R. Suzuki of Osaka University for his kind interest in this study. He also thanks Professor L. M. Ricciardi of Luboratorio di Cibernetica de1 CNR, Naples, Italy, for his valuable suggestions. REFERENCES 1 R. M. Capocelli and L. M. Ricciardi, Diffusion approximation and first passage time problem for a model neuron, Kybemetik 8, Heft 6 (1971), 214-223. 2 J. Durbin, Boundary crossing probabilities for the Brownian motion and Poisson processes and techniques for computing the power of the Kolmogorov-Smimov test, J. Appl. Probability 3
4
5 6 7
8 (1971), 431-453.
P. I. M. Johannesma, Diffusion models for the stochastic activity of neurons, in Neural Networks (E. R. Caianiello, Ed.), Springer, 1968. L. M. Ricciardi, Diffusion approximations to neural activity and to the time dependent threshold problem, in Proc. of the 3rd European Congress on Cybernetics and Systems Research, Vienna, Apr. 1976, to be published. L. M. Riccardi, Diffusion processes and related topics in biology, Lecture Nofes in Biomathematics 14, Springer, 1977. S. Sato, Evaluation of the first passage time probability to a square root boundary for the Wiener process, J. of Appl. Probability, to be published. H. Sugiyama, G. P. Moore, and D. H. Perkel, Solutions for a stochastic model of neuron spike production, Math. Biosci. 8 (1970), 323-341.