When is an Integrate-and-fire Neuron 
like a Poisson Neuron? 
Charles F. Stevens 
Salk Institute MNL/S 
La Jolla, CA 92037 
cfs@salk.edu 
Anthony Zador 
Salk Institute MNL/S 
La Jolla, CA 92037 
zador@salk.edu 
Abstract 
In the Poisson neuron model, the output is a rate-modulated Pois- 
son process (Snyder and Miller, 1991); the time varying rate pa- 
rameter r(t) is an instantaneous function G[.] of the stimulus, 
r(t) - G[s(t)]. In a Poisson neuron, then, r(t) gives the instan- 
taneous firing rate--the instantaneous probability of firing at any 
instant t--and the output is a stochastic function of the input. In 
part because of its great simplicity, this model is widely used (usu- 
ally with the addition of a refractory period), especially in in vivo 
single unit electrophysiological studies, where s(t) is usually taken 
to be the value of some sensory stimulus. In the integrate-and-fire 
neuron model, by contrast, the output is a filtered and thresholded 
function of the input: the input is passed through a low-pass filter 
(determined by the membrane time constant v) and integrated un- 
til the membrane potential v(t) reaches threshold 0, at which point 
v(t) is reset to its initial value. By contrast with the Poisson model, 
in the integrate-and-fire model the ouput is a deterministic function 
of the input. Although the integrate-and-fire model is a caricature 
of real neural dynamics, it captures many of the qualitative fea- 
tures, and is often used as a starting point for conceptualizing the 
biophysical behavior of single neurons. Here we show how a slightly 
modified Poisson model can be derived from the integrate-and-fire 
model with noisy inputs y(t) = s(t) + n(t). In the modified model, 
the transfer function G[.] is a sigmoid (err) whose shape is deter- 
mined by the noise variance a. Understanding the equivalence 
between the dominant in vivo and in vitro simple neuron models 
may help forge links between the two levels. 
104 C.F. STEVENS, A. ZADOR 
I Introduction 
In the Poisson neuron model, the output is a rate-modulated Poisson process; the 
time varying rate parameter r(t) is an instantaneous function (7[.] of the stimu- 
lus, r(t) = G[s(t)]. In a Poisson neuron, then, r(t) gives the instantaneous firing 
rate--the instantaneous probability of firing at any instant t--and the output is a 
stochastic function of the input. In part because of its great simplicity, this model 
is widely used (usually with the addition of a refractory period), especially in in 
vivo single unit electrophysiological studies, where s(t) is usually taken to be the 
value of some sensory stimulus. 
In the integrate-and-fire neuron model, by contrast, the output is a filtered and 
thresholded function of the input: the input is passed through a low-pass filter 
(determined by the membrane time constant v) and integrated until the membrane 
potential v(t) reaches threshold 0, at which point v(t) is reset to its initial value. 
By contrast with the Poisson model, in the integrate-and-fire model the ouput is 
a deterministic function of the input. Although the integrate-and-fire model is a 
caricature of real neural dynamics, it captures many of the qualitative features, and 
is often used as a starting point for conceptualizing the biophysical behavior of single 
neurons (Softky and Koch, 1993; Amit and Tsodyks, 1991; Shadlen and Newsome, 
1995; Shadlen and Newsome, 1994; Softky, 1995; DeWeese, 1995; DeWeese, 1996; 
Zador and Pearlmutter, 1996). 
Here we show how a slightly modified Poisson model can be derived from the 
integrate-and-fire model with noisy inputs y(t) = s(t) + n(t). In the modified 
model, the transfer function 15/[.] is a sigmoid (erf) whose shape is determined by 
the n(ise variance an . Understanding the equivalence between the dominant in vivo 
and in vitro simple neuron models may help forge links between the two levels. 
2 The integrate-and-fire model 
Here we describe the the forgetful leaky integrate-and-fire model. Suppose we add 
a signal s(t) to some noise n(t), 
y(t) = ,(t) + s(t), 
and threshold the sum to produce a spike train 
z(t): :qs(t)+ .(t)], 
where .T is the thresholding functional and z(t) is a list of firing times generated by 
the input. Specifically, suppose the voltage v(t) of the neuron obeys 
v(t) 
= + y(t) (1) 
T 
where r is the membrane time constant. We assume that the noise n(t) has O-mean 
and is white with variance a. Thus y(t) can be thought of as a Gaussian white 
process with variance a and a time-varying mean s(t). If the voltage reaches the 
threshold 00 at some time t, the neuron emits a spike at that time and resets to 
the initial condition v0. This is therefore a 5 parameter model: the membrane 
time constant r, the mean input signal t, the variance of the input signal a2, the 
threshold O, and the reset value v0. Of course, if n(t) - O, we recover a purely 
deterministic integrate-and-fire model. 
When Is an Integrate-and-fire Neuron like a Poisson Neuron? 105 
In order to forge the link between the integrate-and-fire neuron dynamics and the 
Poisson model, we will treat the firing times T probabilistically. That is, we will 
express the output of the neuron to some particular input s(t) as a conditional 
distribution p(TIs(t)) , i.e. the probability of obtaining any firing time T given 
some particular input s(t). 
Under these assumptions, p(T) is given by the first passage time distribution 
(FPTD) of the Ornstein-Uhlenbeck process (Uhlenbeck and Ornstein, 1930; Tuck- 
well, 1988). This means that the time evolution of the voltage prior to reaching 
threshold is given by the Fokker-Planck equation (FPE), 
2 02 0 v(t) 
tg(t , v) - % 
20v 2g(t' v) - vv [(s(t ) v 
- w)g(t,v)], () 
where a N = a. and g(t, v) is the distribution at time t of voltage -cx < v < 00. 
Then the first passage time distribution is related to g(v, t) by 
(3) 
p(T) = - g(t,v)dv. 
The integrand is the fraction of all paths that have not yet crossed threshold. p(T) 
is therefore just the interspike interval (ISI) d{stribution for a given signal s(t). A 
general eigenfunction expansion solution for the ISI distribution is known, but it 
converges slowly and its terms offer little insight into the behavior (at least to us). 
We now derive an expression for the probability of crossing threshold in some very 
short interval At, starting at some v. We begin with the "free" distribution of g 
(Tuckwell, 1988): the probability of the voltage jumping to v' at time t' = t q- At, 
given that it was at v at time t, assuming yon Neumann boundary conditions at 
plus and minus infinity, 
1 exp [ (v' - m(At;ay)) 2 
g(t', v'lt , v) = V/2 r q(At; aN) -- 2 q(At; aN) 
(4) 
with 
and 
qa = o'yv(1 -e-2 at/.) 
re(At) = ve -at/' q- s(t)  v(1 - e-at/'), 
where  denotes convolution. The free distribution is a Gaussian with a time- 
dependent mean re(At) and variance q(At; aN). This expression is valid for all At. 
The probability of making a jump 
in a short interval At << r depends only on Av and At, 
ga(At, Av; %) = 
X/2r qa(ay) 2 qa(ay) ' (5) 
For small At, we expand to get 
= 2zxt, 
which is independent of r, showing that the leak can be neglected for short times. 
106 C.F. STEVENS, A. ZADOR 
Now the probability Pa that the voltage exceeds threshold in some short At, given 
that it started at v, depends on how far v is from threshold; it is 
Pr[v + Av >_ 8] = Pr[Av >_ 8- v]. 
Thus 
Pa -' dvga(At, v;ay) (6) 
--U 
= -erfc (V/ry) ) 
 erfc 
where erfc(x) = 1-  f e-tdt goes from [2  0]. This then is the key result: 
it gives the instantaneous probability of firing as a function of the instantaneous 
voltage v. erfc is sigmoidal with a slope determined by %, so a smaller noise yields 
a steeper (more deterministic) transfer function; in the limit of 0 noise, the transfer 
function is a step and we recover a completely deterministic neuron. 
Note that Pa is actually an instantaneous function of v(t), not the stimulus itself 
s(t). If the noise is large compared with s(t) we must consider the distribution 
g,(v,t;ay) of voltages reached in response to the input s(t): 
i 
ga(at, ay)ct,d7 (7) 
Pt(t) = g(7, t;ay) - 
 1 0-? 
3 Ensemble of Signals 
What if the inputs s(t) are themselves drawn from an ensemble? If their distribution 
is also Gaussian and white with mean/ and variance a, and if the firing rate is 
low (E[T] >> v), then the output spike train is Poisson. Why is firing Poisson only 
in the slow firing limit? The reason is that, by assumption, immediately following 
a spike the membrane potential resets to 0; it must then rise (assuming/ > 0) to 
some asymptotic level that is independent of the initial conditions. During this rise 
the firing rate is lower than the asymptotic rate, because on average the membrane 
is farther from threshold, and its variance is lower. The rate at which the asymptote 
is achieved depends on r. In the limit as t >> r, some asymptotic distribution of 
voltage qoo(v), is attained. Note that if we make the reset v0 stochastic, with a 
distribution given by qoo(v), then the firing probability would be the same even 
immediately after spiking, and firing would be Poisson for all firing rates. 
A Poisson process is characterized by its mean alone. We therefore solve the FPE 
(eq. 2) for the steady-state by setting g(t, v) -- 0 (we consider only threshold 
crossings from initial values t >> v; neglecting the early events results in only a 
small error, since we have assumed E{T} >> v). Thus with the absorbing boundary 
When Is an Integrate-and-fire Neuron like a Poisson Neuron? 10 7 
at 0 the distribution at time t >> r (given here for/ = O) is 
g(v; ay) - kl I -- k. erfi v exp , (8) 
2 2 erfi(z) -ierf(iz), kl determines the normalization (the sign 
where r = o- s + rn, = 
of k determines whether the solution extends to positive or negative infinity) and 
= 1/erfi(0/(yv)) is determined by the boundary. The instantaneous Potsson 
rate parameter is then obtained through eq. (7), 
d? 
(9) 
Fig. 1 tests the validity of the exponential approximation. The top graph shows 
the ISI distribution near the "balance point", when the excitation is in balance with 
the inhibition and the membrane potential hovers just subthreshold. The bottom 
curves show the ISI distribution far below the balance point. In both cases, the 
exponential distribution provides a good approximation for t >> r. 
4 Discussion 
The main point of this paper is to make explicit the relation between the Potsson 
and integrate-and-fire models of neuronal acitivity. The key difference between 
them is that the former is stochastic while the latter is deterministic. That is, given 
exactly the same stimulus, the Potsson neuron produces different spike trains on 
different trials, while the integrate-and-fire neuron produces exactly the same spike 
train each time. It is therefore clear that if some degree of stochastictry is to be 
obtained in the integrate-and-fire model, it must arise from noise in the stimulus 
itself. 
The relation we have derived here is purely formal; we have intentionally remained 
agnostic about the deep issues of what is signal and what is noise in the inputs to a 
neuron. We observe nevertheless that although we derive a limit (eq. 9) where the 
spike train of an integrate-and-fire neuron is a Potsson process--/.e. the probability 
of obtaining a spike in any interval is independent of obtaining a spike in any other 
interval (except for very short intervals)--from the point of view of information 
processing it is a very different process from the purely stochastic rate-modulated 
Potsson neuron. In fact, in this limit the spike train is deterministically Potsson 
if  = as, i.e. when n(t) = 0; in this case the output is a purely deterministic 
function of the input, but the ISI distribution is exponential. 
108 C.F. STEVENS, A. ZADOR 
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When Is an Integrate-and-fire Neuron like a Potsson Neuron? 109 
0.02 
ISI distributions at balance point and the exponential limit 
,_0.015 
0.01 
0.005 
0 
0 
50 1 O0 150 200 250 300 
Time (msec) 
X 10 -3 
I I I 
350 400 450 500 
I I ! I I I I I I 
0 200 400 600 800 1000 1200 1400 
ISl (msec) 
I I 
1600 1800 2000 
Figure 1: ISI distributions. (A; top) ISI distribution for leaky integrate-and-fire 
model at the balance point, where the asymptotic membrane potential is just sub- 
threshold, for two values of the signal variance a2. Increasing a2 shifts the distribu- 
tion to the left. For the left curve, the parameters were chosen so that E{T}  r, 
giving a nearly exponential distribution; for the right curve, the distribution would 
be hard to distinguish experimentally from an exponential distribution with a re- 
fractory period. (r = 50 msec; left: E{T} = 166 msec; right: E{T} = 57 msec). 
(B; bottom) In the subthreshold regime, the ISI distribution (solid) is nearly expo- 
nential (dashed) for intervals greater than the membrane time constant. (r = 50 
msec; E{T} = 500 msec) 
