•
•
FIRST PASSAGE TIMES IN BIOLOGY: APPROXIMATIONS
Charles E. Smith
-.
~
Institute of Statistics Mimeo Series '1695
Biomathematics Series '29
August, 1987
NORTH CAROLINA STATE UNIVERSITY
Raleigh, North Carolina
Gft
~~ ~
f1£ t :r (]) 1Z mB~ m6Jf
:at f;l87-10
FIRST PASSAGE TIMES IN BIOLOGY:
APPROXIMATIONS
Charles E. Smith
Department of Statistics
Biomathematics Program
North Carol Ina State University
Raleigh, North Carol Ina 27695-8203, USA
I NTROO1JCT I ON'
The
idea of a first passage time Is easy
mathematically
to
eXPlain,
It Is often very difficult to evaluate.
but
Let X(t)
be a random process In continuous time and let Set) be a determinlstlc time function Set) (an important special case Is when set)
is a constant>.
'want
to
Given that X(t) has a value of Xo at time to' we
know the time,
t(Slx )'
O
that X(t) £lRSI
crosses
the
boundary Set). Since this time varies from one sample path to the
next,
t(S}x )
a
wi I I be a random variable.
Assuming x
a
Seta)'
<
then
The
probability density function,
p.d.f"
of t(Slx )
O
will
be
denoted by g(S,tlx )'
O
The first passage time Is an example of a more general level
crossing
problem.
An
extensive bibliography of level
problems
with emphasis on engineering applications can be
in Blake and Lindsey (1973) and Abrahams (1986).
to
with
the case where X(t) is a diffusion process
continuous
Ricciardi <1977> .
.~-~-----------------~
----
sample
paths) can be found
crossing
found
An Introduction
(rouihly,
In
ohapter
Markov
3
of
~ ~
1 987t.J: 7}'4
In this
processes
report,
with
we primari Iy consider continuous time
either
order discontinuities.
of
the
(SDE),
a continuous state space or
In applications,
with
first
these processes are often
form of an autonomous stochastic
that
random
differential
equation
is,
dx
=
a(x)
dW(t)
( 1 a)
dP(il.,t)
( 1 b)
dt + b(x)
or
=
dx
wet>
where
motion)
trajectory
ments)
dt + b(x)
is a standard Wiener process
and P(A,t)
parameter
a(x)
A
(also
called
is a homogeneous Poisson process
Intutitively,
you can think of X(t)
in an RC circuit
(with,
in general,
Brownian
with
rate
as the voltage
nonlinear
ele-
attached to a noisy current source.
The
biology
above scenario arises
(for reviews,
Tuckwe I I,
19S7),
in many areas of biology:
see Holden,
population
genetics
population biology and harvesting
19S1),
Hanson and Tier,
Kipnis and Newman,
1976;
(e.g.
Yang and
19S5),
psychology
1977,
Ryan and
Hanson,
some other areas such as tumor growth
(e.gol
(e.g.,
1975;
Chen,
(Maruyama,
evolutionary biology
neuro-
(e.g.,
19S3) ,
Lande,
Ratcliff,
19S5;
19S5;
19S0)
and
Smith and Tuckwel',
1973) and DNA sequence analysis.
Since
this
tion methods,
methods.
report concentrates on some heuristic approxima-
a problem in neurobiology
is used
to
illustrate the
The problem is the generation of action potentials
nerve cell
or neuron.
electrical
properties.
A neuron
at a particular spatial
is an animal
in a
cell with special ized
When the voltage across the cel I membrane
location,
2
called the trigger
zone,
ex-
ceeds a voltage threshold,
an
action potential
generated. The noisy
then a brief electrical
or spike
input
(:::100 mV and
is due to
1 msec
central
sion
approximation
when
one
We wi II
passage
to the voltage process.
times
through several
=-
a
(III)
it
e
fi Itered
,
X(t) wi II
to the resting
fi rst
be
level,
the
which
Sato,
1978; Ricciardi
x/7:
dt + a
(2)
dW(t)
pos it i ve constants.
(II) Stein's model
7:,
a
1979)
dx
with
behavior,
hand,
The models are:
and Sa rcedo te,
and
On the other
neural models.
the Ornstein-Uhlenbeck process( e.g.,
7:
Inputs
illustrate some approximation methods for the
is set to zero.
wi th
If many
may be more appropriate.
membrane voltage process referenced
(I)
is
I imit theory type arguments produce a dlffu-
or a few synapses dominate" the
Poisson model
in duration)
inputs from other neurons at
particular places on the neuron cal led synapses.
are present,
pulse known as
(Stein,
1967)
dx = -
x/7: dt + a dP(it
it
and
i
,
a
e
,t)
-
a,dpit,,t)
eel
a,
I
(3)
I
al I positive constants.
Stein's model with a reversal
potential
(e.9.,
'Tuckwell,
1979; Smith and Smith, 1984)
dx = -
x/7: dt + a (I-xIV) dP(it ,t) + a (I-xIV) dP(it.,t)
e
e
e
i i i
with the parameters as
(IV)
in model
II
and V
e
> 0 and Vi
a stochastic afterhyperpolarization model
3
(4)
< O.
used to model
the
firing of vestibular nerve fibers (Smith and Goldberg, 1986)
where
V and V are positive(synaptic)
k
s
equi I ibrium
potentials
respectively.
and
The
negative(potassium)
g'S
are
normal ized
membrane conductances with gk(t) being a decaying exponential and
g
being
s
impulse
a
shot noise process produced by
train
through
response (FIR) filter.
Models
II,
I,
(1), while model
a
shot
circuit
conductances:
causal,
a
Poisson
finite
impulse
I I I fit into the SDE framework of
equation
IV is a timevarying, nonlinear transformation of
Model
IV arises from a more
model of the trigger site
with
compl icated
three
parallel
a leakage conductance, a synaptic conductance, and
a potassium channel conductance.
satisfies
rectangular
a
The leakage conductance has a ralue of 1.
noise process.
lumped
a
passing
differential
The voltage across the membrane
equation,
which is
approximated
by
equation (5) when the membrane time constant is short compared to
the
time
constant of gk(t)
(see Smith and
Goldberg,
1986
for
details and biological justification for the approximation>.
The
approximation methods for fi rst passage times
classified
of
the
wi II
according to whether or not the mean voltage,
process
X(t)
crosses
the
threshold
S(t).
The
be
.(.t(t),
term
deteLmlclstlc crossing wi I I be used when the mean voltage crosses
S(t)
S(t).
cases,
and ooodetermlclstjc crossing when the mean
doesn't
cross
For the deterministic crossing, we further distinguish two
long correlation time and short correlation time relative
t o t he s t and a r d de v i a t ion
0
f t (S I x0)'
4
For
sma I I
flu c t ua t Ion s
about the mean voltage,
transformation
the long correlation time case becomes a
of the voltage process at the time of determinis-
tic crossing (Stein,
1967),
whi Ie the short correlation case is
approximated by a Wiener process with drift producing locally
inverse
Gaussian
(Lerche,
1986).
crossing
become rare events as the threshold level becomes
large.
In
process.
For
this
exponential
distribution
for
the
first
nondeterministic crossings,
case the first passage times
distribution
When Set)
passage
a
which is characteristic of
is not a constant,
time
the times
have
an
of
very
limiting
a
Poisson
the threshold can be sub-
tracted from the mean voltage to give an "effective" mean voltage
which can be thought of as a mean postspike recovery process. The
problem now becomes when does the effective voltage process reach
zero.
The
three
approximation cases are now examined
in
more
de ta i I .
CASE
1~
determlolstlc crosslog6 lopg correlatloo
In
tl~
this scenario the fluctuations in the first passage time
(FPT> are basically small perturbations about the
for
t*,
simple general ization of the method used by Stein (1967,p
model
mentioned
passage
at
mean voltage, ,Lt(t), crosses Set). Our description is
which the
a
time,
II.
Let rCt) = Set) - ,Lt (t) be the recovery
above,
(j
t
be the standard deviation
time distribution,
and
(j
* be
of
the
53)
process
first
our approximated standard
deviation. The FPT p.d.f. g(S,tlx ) can be approximated, at least
O
locally, by a transformation of the marginal distribution of X(t)
evaluated
change
its
at
t
* when:
(1)
the
voltage
"*
shape drastically near t ,
5
(2)
distribution
(j
tis
doesn't
considerably
*
than the correlation time of X(t) around t ,
less
rC t)
(3)
is
invertible and sufficiently smooth. Let h be the inverse function
of
r(t),
that
is h(r(t»=t.
f(x)/Idh(x)/dxl
evaluated
X(t).
the
This
is
g(~,tlxO)
Then
is
approximately
at t* ,where 'f(x)' is the marginal
usual
Jacobian
variables. For example if f(x)
transformation
is gaussian and r(t)
of
of
random
is a decaying
exponential, then g(S,tlx ) is approximatery lognormal.
O
In
many
moments
cases we may only be interested in the
of the FPT.
series about r(t*).
first
The function h is now expanded in a
few
Tavlor
The approximations for the mean and variance
are given below with y = rCt*) and J1,
n
is the nth central
moment
of X(t*).
E(t)
= t*
h"
+
(y) J1,2/2 + h'"
Var(t) - (h'J1, )2 + h'h"J1,
2
where
prime
Recall
denotes
3
(y) J1,3/6 + ...
(6)
- (h"J1, /2 + h"'J1, /6)2 + ...
2'
differentiation with respect
to
that the first derivative of an inverse function
expressed
as
(7)
3
the reciprocal of the derivative of
the
voltage.
can
be
original
function, so that
h'
=
1/ (dr(t)/dt)
1/(dS(t)/dt - dJ1, (t)/dt)
(8)
which is simply the derivative of the recovery process at t*.
Stein's
equations
original approximation method was the first term of
(6) and
(7)
and can be illustrated graphically
Figure 1. Note that increasing h"
as
In
decreases the approximate mean
interval, so Stein required for condition (3) that 'ret) be nearly
I inear in the neighborhood of t*.
6
While we might expect such a procedure to work for diffusion
processes,
Figure 2 {model
I I, excitatory inputs only )and Table
1 (model
I I I, excitatory inputs only) show that this approximation
method
also works for Poisson driven systems.
coefficient of variation,
In figure 2,
the
cv, (standard deviation divided by the
mean) of the FPT is plotted against the cv of the voltage process
for
the
case
where the mean voltage
keeping the product a it
e
e
trajectory
is
fixed
by
fixed at a constant value of 8 mV/msec.
The sol id I ine is that predicted from the above equations and has
a slope of 0.92 and is a good approximation over this range.
correlation
time
The
of the process is in the worst case more
than
twice the resultant at'
A
simi lar
procedure is followed in Table 1 for
model
III
with the mean voltage being held constant in the same manner. Two
different mean voltages were used (A,B) and (C,O) with the
of
a
e
differing
by a factor of 16 within
each
pair.
value
Further
details can be found in Smith and Smith, 1984.
What happens if the three conditions above are not
particular
what if the correlation time is quite short
met,
in
compared
to a t' This brings us to our next method.
CASE 2: deterministic crossicgL short correlation time.
Figure 1 is sti I I the framework we are considering,
we
want
variance
the
of
correlation
time to be quite short
X{t) to be increasing I inearly locally
and
for
the
around
t*.
Then X{t) wi II be approximated as a Wiener process with a
drift.
process
The
at
linear
t*
drift
is simply the slope of
and the slope of the variance
7
of
but now
linear
the
recovery
the
membrane
potential
gives the intensity or scale parameter of
the
Wiener
*
process. Said another way, around t , we have
X(t) - a t + b W(t)
where
to
W(t)
(9)
is a standard Wiener process.
The fi rst passage time
a constant threshold for a Wiener process with
has
an
analytic
solution which gives the
wei I
I inear
known
drift
inverse
gaussian distribution for g(S,tlx )' Generalizations of this idea
O
and a rigorous development of the so-cal led tangent method can be
found in the recent monograph by Lerche(1986).
tion
Parameter estima-
in the context of neural models for this situation has been
examined by Lansky (1983).
What
clues can we get from an empirical first passage
distribution
gaussian
this
distribution
increases
times
that
the
situation
might
apply?
skewed
to the
right
is
with increasing cv,
cv
when
skew
The
and
in particular the skew
is measured as
the
time
inverse
the
skew
is
square
three
root
of
Pearson's betal, i.e.
In figure 3, this
simulation
point
series in model
represents
starting
relation~hip
with
between skew and cv is shown for a
IV with a
constant
threshold.
2000 simulated intervals and different
the leftmost correspond to an decreasing
values of the rate parameter that drives the shot noise
All
points
set
of
process.
other parameters were fixed except one which controlled
serial
of
Each
correlation between interspike intervals.
this parameter are denoted by x and boxes,
large
effect
on the plot (see Smith and
8
Chen,
the
The two values
and don't have
1986
for
a
more
details).
Two
other two parameter distributions that are
posi-
tively skewed are shown for comparison, the gamma (GA) and lognormal
(LN). For the middle range of values of cv, which correspond
to deterministic crossings and short correlation times, the simulated
values bounce around the Inverse gaussian
larger
values of cv,
curve(IG).
For
we no longer have deterministic crossings,
which brings us to our final approxImation method.
CASE
3~
cocdetermlclstlc crosslcQSL large tbtesbold
Intutitively what happens in this case is that the
asympto-
tic value of the mean of the process is below the threshold,
the
threshold,
when measured in units of the standard deviation
of the steady state voltage process,
state
mean voltage.
roughly
a
. Poisson
sage
parameter
This
the
Crossings of the process wi I I be
process.
The time between
rare
arrivals
result
wi I I have an exponential distribution with
being the reciprocal of the mean first
for some diffu$ion processes in
a weak convergence result in 1966.
of
diffusion processes,
and
for
a
so the fi rst pasthe
rate
passage
idea has been around for quite a whi le,with Newell
giving
types
Poisson
is far away from the steady
process is exponentially distributed,
time
and
time.
showing
and
Keilson
Recently for
several
1962,
Ricciardi and colleagues
(1985,
1986) have establ ished this result and asymptotic series for
the
rate
Sato
and
for
non-
I) and a
con-
parameter
Ricciardi
of
the
()983,)984,)986)
exponential
have
distribution.
asymptotic
results
constant boundaries.
Using
stant
the
Ornstein Uhlenbeck process (model
threshold as an example,
we wi! I present a flavor of
9
the
type
for
of asymptotic results available and note
obtaining
behavior
of
other
The results are more general
models
<III and IV) are
approaches
in
seen
that
to
the
be
well
by the OU process in the lim j t of large thresholds.
approximated
The
them.
other
scenario for the last two figures is as
follows:
The
mean
fir s t passage time for the OU process is tabulated and for thresholds
time
more
than 3
a
v
away are we II approximated by
equal
compare
by the steady state standard deviation.
We
these processes to the normal ized OU process.
shot
noise model
the neural model
Note
nor
can
now
OnlY
the
FPT is considered since it is the only parameter needed
III
to
In figure 4,
is seen to be wei I approximated by the
OU
In figure
5,
process if the seal ing and dead time are included.
and
X(t)
and voltages are
characterize the I imiting exponential distribution.
the
of
time is measured in units
of the steady state process's correlation time,
normal ized
so we add a dead
to the time at which the mean and variance
reaches 95% of its steady state level,
mean
fir s t
of the asymptotic expansion when the process starts from O.
Models I I I and IV are not stationary processes,
time
the
is normal ized to the standardized OU process
again produces good agreement as the threshold is increased.
that neither of these process have continuous sample
gaussian
correlation
marginal distributions.
What is
similar
structure for smal I time differences.
Model
paths
is
I"
the
is
similar to the OU process in that its linear drift term dominates
at large threshold values.
Since
Seigert
the OU process is a diffusion process the
outl ined in Ricciardi
method
(1977 chapter 3) can be used.
10
of
The
starting value wi I I be taken to be O.
is normal
(2),
a~.
dividing by
(including
leading
,c.f.
the threshold)
From Keilson and Ross
in the asymptotic series as S --.
term
equation
by
The stationary distribution Is now gaussian
zero mean and unit variance.
the
a 2 r:/2
with zero mean and a variance of
so we normalize the voltages
with
The stationary distribution
(1975),
for
00
the
mean FPT is
( 10)
where S* is
series
by
in
the
the normalized threshold.
(10)
rest of the asymptotic
can be obtained by multiplying the
generalized
1/(2S*2».
The
The
hypergeometric
leading
right hand side
function
term agrees with the actual
result within
10% for S* > 3.
The
method
leading term can be obtained
stochastic
of
(Williams,1982;
dx
our
case
boundary
= -
is
tute
the
zero,
for
=
b(x)
dt +
xl
r:.
e
-~
O.
The ex i t t i me
For
0
is given
series
( 11)
dW(t)
ut
0
au
the
(Wi II iams,
au
function
and an expression for
1982,
process,
pI51).
(x) :.
the
and
the
When we substileading term
e
(10)
is obtained. Since
letting
the normalized
threshold goes to
infinity.
11
a
pro c e s s ,
f a d 0 ma i nO, wh i chi n
is shown using generating
the parameters of the
means
1984). Consider the stoc-
.....;e a (x)
layer theory to be exponential
asymptotic
systems
of
equation
(_OO,S),
rate parameter
dynamic
perturbation
and con sid e r what hap pen s as
and b ( x)
using the
Freidlin and Wentzell,
hastic differential
I E: 'a
in another way,
gO
in
to
However
the
results from stochastic perturbation theory apply to a wider
class of problems provided we scale the threshold
The
appropriately.
restrictions on b(x) and o"(x) are continuity and smoothness
(e.g.
Lipschitz)
deterministic
and
that x = 0 be the only attractor
(no noise here) system.
The problem can
of
the
also
be
viewed as a particle in a potential well. Kipnis and Newman(1985)
have
extended
the
potential wells,
include
exit time results to
bi- and
multi
stable
and Day (1987> has extended earlier results
nonsmooth
quasipotentials.
This is just a sampling
to
of
recent work in these active area.
Finally we mention a third way to obtain the leading term in
equation
(10)
which
using
the
Poisson
nature
of
the
rare
crossings, namely extreme value theory of statistics. Using quite
different arguments than the two above methods Leadbetter et
(1983,p
meter
236)
obtain the leading term in (10) as the rate
al.
para-
for Poisson process of so-called "S" upcrossings for
the
au process.
DLSCUSSION:
We
neural
have
illustrated,
models,
the
in terms of four
relatively
three ranges of behavior of
first
simple
passage
times where heuristic approximation methods can be useful.
the
Whi Ie
approximation methods are no substitute for careful mathema-
tical
analysis and for simulations of models too compl icated
to
be fully analyzed mathematically, they can be used as a guide for
prel iminary
model's
work
structure.
and
in ascertaining
the
robustness
Even with current computers,
some first passage time problems require
12
8
of
the
simulations Qt
prohibitive amount
of
time.
CPU
The prel iminary use of approximations here is
almost
mandatory.
The plot of skew vs coefficient of variation was seen to be a
diagnostic
for
locating
regions
exponential behavior of the p.d.f.
This plot,
of inverse
Gaussian
of the intersplke
with standard errors included,
and
of
intervals.
can also be used as a
guide to how long of a data record is needed to distinguish among
different models.
For
the
the
asymptotic behaviour of the first passage time
nondeterministic crossing case,
parameter
of the I imiting exponential distribution has
obtained in three ways for the
interplay
it was noted that the
between
the
rate
been
Ornstein Uhlenbeck process.
three approaches (asymptotic
in
be
This
series
in
Seigert's approach, perturbation of dynamic systems via WentzellFreidl in method,
value
theory)
(Smith and Sato,
and Poisson nature of upcrossings from
extreme
is richer and more extensive than presented
in preparation).
Furthet study of the relation
between recent results in extreme value theory and in exit
for
one
dimensional
diffusion processes,
Newell's work (1962), seems warranted.
13
here
along the
times
I ines
of
Acknowledgments: Support for this work was furnished by Osaka
University and Office of Naval Research Contract NOOOI4-85-K-Ol05.
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z.a.;
15
t'
Vy
I.O-l-----~-+_7"'c::..,....J
64 , '
,
"
,'" " ,'.\.,"
,"
f," ", :,,
, " ,,:6·
"..~ i
>-
,' ,,
,, ,,
::- 0.5
',//"
,
I
.l-o-.
,,'
"
,,'
0.5
LO
0
0.5
l
~O
tiT
Elgyl:~ 1.... Schematic illustration of deterministic crossing case. Left
pane I sho....s a samp Ie pa th and the mean J.L ( t) 0 { the prucess. The
voltage is normalized by the value of the constant threshold V and
time is normal ized by the time of deterministic crossing, T. ntght
panel sho....s three trajectories. The middle one is the mean voltage and
the other t....o are the mean +/- a ~tandard deviation. The projection of
the times uf intersection of the lhree curves are indicated by
vertical I ines. The tangent approximation to the standard deviation of
lhe crossing time, a, is given·by a divided by the slope, dJ.L/dt,
evaluated at the time of deterministi~ crossing •
. O.j
0.2
>
u
.=
~
=>
.5
u:
O. i
Membrane POlenlial cv
Illustration of Stein's apprOXimation method for model 11.
The abscisca is the cv of the membrane polential process and the
ordinate is the cv of the firing limes. The 7 points are from
simulations (1000 inlerspike intervals each) of model II .... i th
excitatory inputs only. The membrane time constant is 5 msec. The mean
voltage trajectory is the same for all seven cases since the product
A a = 8 mV/msec for each case, .... ith a taking on values of 4, 2, I,
O.5,e O•25 , 0.125 and 0;0625 mY. The sol1d line is that predicted from
the approximation method and has a slope of 0.92. The threshold
function is 5(t) = 10 + 100 exp(- t/200) mV and lime is In msec.
ElgYl:~ ~
16
TABLE
Firing
Rate
(sp/sec)
Standard
Deviation
(msec)
/'leon
Interval
(msec)
Case
~~~-------------------------------~-------------------
cC·
D
D·
----------------
.0215
.0214
.09117
.0860
.0374
.0394
.1538
.1582
150.11
150.3
150.7
150.3
.11l3
.142
.628
.572
.397
.418 .
6.65
6.66
·6.63
6.66
10.62
10.59
10.26
0.59
cv
9/1.2
S4.4
97.5
94.4
I.fin
1.675
•--------------------------------------------------------------------(opprox.)
IAUl.!:: 1: Illustration uf Stein's approximation method for Model III.
SKEW·YS CY
.8
3.a
2.6
2.a
s
K
£
\.S
\I
La
a.s
a.a
I.a
a.2
a.a
a.~
1.8
1.1
cv
a:
The relation between skew and cv for simulations of model
IV. Skew is measured as Pearson's square root of betal. Each symbol
represents 2000 simulated intervals. Different points correspond to
different values of the Poisson release rate that drives the shol
noise process. The higher rates produce smaller cv values. The three
reference curves are the corresponding relalions for the following two
parameter distributions: lognormal (LN), inverse gaussian (lG), and
gamma (GA). Note that an expunential distribution has a cv of 1 and a
skew of 2.
flgy~
17
curv.e5,gkO=O.5,tk=2.357,p= l,a= 1.00,n=2000
TD-Z.Z:iTl( TCoo8.8
1811.lK
f:
It
E
.R
It
I
31.621
I
It
T
E
R
V 18.1111.
A
L
3.162""roo~~...,...~~...,...~~~"""'~""""~_"""""""~"""'''''-''''''''''"T
0.11
B.:5
1.0
1.:5
2.11
2.:5
3.11
3.:5
TIiRESHOl.D
Eilmrg, 1: Asymptotic Ornstein Uhlenbeck approximation to model IV.
Abscissa is normalized threshold and o-rdinate is mean interval. A dead
time of 2.25 times the time constant is used and voltages are
normalized by the standard deviation of the steady state voltage
process. The solid curve is the relationship for the OU process and
the x's art: from simulations of model IV.
Elgy[~~: Asymptotic Ornstein Uhlenbeck approximation to model III.
Abscissa and ordinate al'e as in figure 4. The dead time Is the time
for the mean voltage to reach 90X of its asymptotic level. The points
represent simulations of model I II .and the sol id I ine the OU process.
The OU process curve in figures 4 and look different due to the
ordinate being the time scale of the OU process (fig.S) or of the
simulated process (fig.4). Values below a thr~lold of zero represent
deterministic crossings.
18
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REPORT DOCUMENTATION PAGE
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11. TIllE (Inc/ud~ S~cu.,ty CI.usrf,cMion)
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FIRST PASSAGE TIMES IN BIOLOGY: APPROXIMATIONS (UNCLASSIFIED)
PERSONAL AUTHOR(S)
1·13~. TYPE
Charles E. Smith
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18
.
16. SUPPLEMENTARY NOTATION
17.
(OSA TI (ODES
FiElD
GROUP
19 ABS TRACT
I
(Continu~
18 SUBJECT TERMS
(Cont;nu~
SUB·GROUP
on f,verse If nf!c,SJoJr)' oJnd
;d~nt;fy
by blo<lc
.
on
fevefI~
if necesJlry ,nd idl'ntdy by blo<lc num~r)
num~r)
Three approximation methods are examined for the first passage time
(FPT's) of continuous time stochastic processes. The methods are illustrated
through four stochastic neural models for the generation of action potentials.
The deterministic crossing, long correlation time case approximates the FPT
distribution by a transformation of the marginal distribution at the time of
deterministic crossing. The deterministic crossing) short correlation time
case approximates the process locally by a Wiener process with drift, producing
an inverse Gaussian distribution. The final case is the nondeterministic
crossing, large threshold case which produces exponential FPT's in the limit.
Three ways to produce this result for the Ornstein Uhlenbeck process are noted.
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