Refractoriness and Neural Precision 
Michael J. Berry II and Markus Meister 
Molecular and Cellular Biology Department 
Harvard University 
Cambridge, MA 02138 
Abstract 
The relationship between a neuron's refractory period and the precision of 
its response to identical stimuli was investigated. We constructed a model of 
a spiking neuron that combines probabilistic firing with a refractory period. 
For realistic refractoriness, the model closely reproduced both the average 
firing rate and the response precision of a retinal ganglion cell. The model is 
based on a "free" firing rate, which exists in the absence of refractoriness. 
This function may be a better description of a spiking neuron's response 
than the peri-stimulus time histogram. 
1 INTRODUCTION 
The response of neurons to repeated stimuli is intrinsically noisy. In order to take this 
trial-to-trial variability into account, the response of a spiking neuron is often described 
by an instantaneous probability for generating an action potential. The response 
variability of such a model is determined by Poisson counting statistics; in particular, the 
variance in the spike count is equal to the mean spike count for any time bin (Rieke, 
1997). However, recent experiments have found far greater precision in the vertebrate 
retina (Berry, 1997) and the H1 interneuron in the fly visual system (de Ruyter, 1997). In 
both cases, the neurons exhibited sharp transitions between silence and nearly maximal 
firing. When a neuron is firing near its maximum rate, refractoriness causes spikes to 
become more regularly spaced than for a Poisson process with the same firing rate. Thus, 
we asked the qu6stion: does the refractory period play an important role in a neuron's 
response precision under these stimulus conditions? 
2 FIRING EVENTS IN RETINAL GANGLION CELLS 
We addressed the role of refractoriness in the precision of light responses for retinal 
ganglion cells. 
2.1 RECORDING AND STIMULATION 
Experiments were performed on the larval tiger salamander. The retina was isolated from 
the eye and superfused with oxygenated Ringer's solution. Action potentials from retinal 
Refractoriness and Neural Precision 111 
ganglion cells were recorded extracellularly with a multi-electrode array, and their spike 
times measured relative to the beginning of each stimulus repeat (Meister, 1994). 
Spatially uniform white light was projected from a computer monitor onto the 
photoreceptor layer. The intensity was flickered by choosing a new value at random from 
a Gaussian distribution (mean/, standard deviation 6/) every 30 ms. The mean light level 
(I = 4' 10 -3 W/m 2) corresponded to photopic (daylight) vision. Contrast C is defined here 
as the temporal standard deviation of the light intensity divided by the mean, C = 61/1. 
Recordings extended over 60 repeats of a 60-sec segment of random flicker. 
The qualitative features of ganglion cell responses to random flicker stimulation at 35 % 
contrast are seen in Fig. 1. First, spike trains had extensive periods in which no spikes 
were seen in 60 repeated trials. Many spike trains were sparse, in that the silent periods 
covered a large fraction of the total stimulus time. Second, during periods of firing, the 
peri-stimulus time histogram (PSTH) rose from zero to the maximum firing rate 
(-200 Hz) on a time scale comparable to the time interval between spikes (-10 ms). We 
have argued that these responses are better viewed as a set of discrete firing "events" than 
as a continuously varying firing rate (Berry, 1997). In general, the firing events were 
bursts containing more than one spike (Fig. lB). Identifiable firing events were seen 
across cell types; similar results were also found in the rabbit retina (Berry, 1997). 
60- 
40- 
20- 
B 
rr 0-- 
43.4 43.5 
43.6 43.7 43.8 
Time (s) 
Figure 1: Response of a salamander ganglion cell to random flicker stimulation. 
(A) Stimulus intensity in units of the mean for a 0.5-s segment, (B) spike rasters 
from 60 trials, and (C) the firing rate r(t). 
2.2 FIRING EVENT PRECISION 
Discrete episodes of ganglion cell firing were recognized from the PSTH as a contiguous 
period of firing bounded by periods of complete silence. To provide a consistent 
demarcation of firing events, we drew the boundaries of a firing event at minima v in the 
PSTH that were significantly lower than neighboring maxima p] and P2, such that 
p4-p%/v > p with 95 % confidence (Berry, 1997). With these boundaries defined, every 
spike in each trial was assigned to exactly one firing event. 
112 M. J. Berry and M. Meister 
Measurements of both timing and number precision can be obtained if the spike train is 
parsed into such firing events. For each firing event i, we accumulated the distribution of 
spike times across trials and calculated several statistics: the average time T i of the first 
spike in the event and its standard deviation 6T i across trials, which quantified the 
temporal jitter of the first spike; similarly, the average number N i of spikes in the event 
and its variance 6Ni 2 across trials, which quantified the precision of spike number. In 
trials that contained zero spikes for event i, no contribution was made to T i or 6Ti, while 
a value of zero was included in the calculation of N i and 6Ni 2. 
For the ganglion cell shown in Fig. 1, the temporal jitter 6T of the first spike in an event 
was very small (1 to 10 ms). Thus, repeated trials of the same stimulus typically elicit 
action potentials with a timing uncertainty of a few milliseconds. The temporal jitter of 
all firing events was distilled into a single number  by taking the median o"er all events. 
The variance 6N  in the spike count was remarkably low as well: it often approached the 
lower bound imposed by the fact that individual trials necessarily produce integer spike 
counts. Because 6N  << N for all events, ganglion cell spike trains cannot be completely 
characterized by their firing rate (Berry, 1997). The spike number precision of a cell was 
assessed by computing tae average variance over events and dividing by the average 
spike count: F = {cSN2)/(N). This quantity, also known as the Fano factor, has a value 
of one for a Poisson process with no refractoriness. 
3 PROBABILISTIC MODELS OF A SPIKE TRAIN 
We start by reviewing one of the simplest probabilistic models of a spike train, the 
inhomogeneous Poisson model. Here, the measured spike times {t i  are used to estimate 
the instantaneous rate r(t) of spike generation during a time At. This can be written 
formally as 
r(t) =  
M At 
where M is the number of repeated stimulus trials and O(x) is the Heaviside function 
1 x_>O} 
O(x) = o x < o 
We can randomly generate a sequence of spike trains from a set of random numbers 
between zero and one: {o i } with a i e (0,1]. If there is a spike at time ti, then the next 
spike time ti+ 1 is found by numerically solving the equation 
t,+ 
-lnai+ 1 = J'r(t)dt 
3.1 INCLUDING AN ABSOLUTE REFRACTORY PERIOD 
In order to add refractoriness to the Poisson spike-generator, we expressed the firing rate 
as the product of a "free" firing rate q(t), which obtains when the neuron is not 
refractory, and a recovery function w(t), which describes how the neuron recovers from 
refractoriness (Johnson, 1983; Miller, 1985). When the recovery function is zero, spiking 
is not possible; and when it is one, spiking is not affected. The modified rule for 
selecting spikes then becomes 
-In[z,+, = J' q(t)w(t-ti)dt 
For an absolute refractory period of time St, the weight function is zero for times between 
0 and St and one otherwise 
Refractoriness and Neural Precision 113 
= 1- 
Because the refractory period may exclude spiking in a given time bin, the probability of 
firing a spike when not prevented by the refractory period is higher than predicted by 
r(t). This free firing rate q(t ;/.t) can be estimated by excluding thais where the neuron is 
unable to fire due to refractoriness 
q(t;/.t) = r(t) 
- Z', [1- w(t- 
The sum is restricted to spike times t i nearest to the time bin on a given trial. This 
restriction follows from the assumption that the recovery function only depends on the 
time since the last action potential. Notice that this new probability obeys the inequality 
q(t) > p(t) and also that it depends upon the refractory period p. 
N 4.5- 
' 4.3- 
 '- 4.1 - 
I 
1.00-  
u.. 0.75- 
o 
( 0.50- 
o 
( 0.25- 
0.00 - 
I I I I I I 
0 1 2 3 4 5 
Refractory Period (ms) 
Figure 2: Results for model spike trains with an absolute refractory period. 
(A) Mean firing rate averaged over a 60-s segment (circles), (B) Fano factor 
F, a measure of spike number precision in an event (triangles), and (C) 
temporal jitter r (diamonds) plotted versus the absolute refractory period/.t. 
Shown in dotted in each panel is the value for the real data. 
With this definition of the free firing rate, we can now generate spike trains with the same 
first order statistics (i.e., the average firing rate) for a range of values of the refractory 
period/.t. For each value of/.t, we can then compare the second order statistics (i.e., the 
precision) of the model spike trains to the real data. To this end, the free rate q(t) was 
114 M. J. Berry and M. Meister 
calculated for a 60-s segment of the response to random flicker of the salamander 
ganglion cell shown in Fig. 1. Then, q(t) was used to generate 60 spike trains. Firing 
events were identified in the set of model spike trains, and their precision was calculated. 
Finally, this procedure was repeated 10 times for each value of the refractory period. 
Figure 2A plots the firing rate (circles) generated by the model, averaged over the entire 
60-s segment of random flicker with error bars equal to the standard deviation of the rate 
among the 10 repeated sets. The firing rate of the model matches the actual firing rate for 
the real ganglion cell (dashed) up to refractory periods of t -- 4 ms, although the 
deviation for larger refractory periods is still quite small. For large enough values of the 
absolute refractory period, there will be inter-spike intervals in the real data that are 
shorter than t. In this case, the free firing rate q(t) cannot be enhanced enough to match 
the observed firing rate. 
While the mean fhSng rate is approximately constant for refractory periods up to 5 ms, the 
precision changes dramatically. Figure 2B shows that the Fano factor F (triangles) has the 
expected value of 1 for no refractory period, but drops to - 0.2 for the largest refractory 
period. In Fig. 2C, the temporal jitter r (diamonds) also decreases as refractoriness is 
added, although the effect is not as large as for the precision of spike number. The 
sharpening of temporal precision is due to the fact that the probability q(t) rises more 
steeply than r(t) (see Fig. 4), so that the first spike occurs over a narrower range of times. 
The number precision of the model matches the real data for /a = 4 to 4.5 ms and the 
timing precision matches for g -- 4 ms. Therefore, a probabilistic spike generator with an 
absolute refractory period can match both the average firing rate and the precision of a 
retinal ganglion cell's spike train with roughly the same value of one free parameter. 
3.2 USING A RELATIVE REFRACTORY PERIOD 
Salamander ganglion cells typically have a relative refractory period that lasts beyond 
their absolute refractory period. This can be seen in Fig. 3A from the distribution of inter- 
spike intervals P(A) for the ganglion cell shown above - the absolute refractory period 
lasts for only 2 ms, while relative refractoriness extends to - 5 ms. We can include the 
effects of relative refractoriness by using weight values in w(t) that are between zero and 
one. Figure 3 illustrates a parameter-free method for determining this weight function. If 
there were no refractoriness and a neuron had a constant firing rate q, then the inter-spike 
interval distribution would drop exponentially. This behavior is seen from the curve fit in 
Fig. 3A for intervals in the range 5 to 10 ms. The recovery function w(t) can then be 
found from the inter-spike interval distribution (Berry, 1998) 
w(t) = 
1 P(t) 
q 1- o t P(A)dA 
Notice in Fig. 3B, that the recovery function w(t) is zero out to 3 ms, rises almost 
linearly between 3 and 5 ms, and then reaches unity beyond 5 ms. 
Using the weight function shown in Fig. 3B, the free firing rate q(t) was calculated and 
10 sets of 60 spike trains were generated. The results, summarized in Table 1, give very 
close agreement with the real data: 
Table 1: Results for a Relative Refractory Period 
QUANTITY 
REAL DATA 
MODEL 
STD. DEV. 
Firing Rate 
Timing Precision r 
Number Precision F 
4.43 Hz 4.44 Hz 0.017 Hz 
3.20 ms 2.95 ms 0.09 ms 
0.250 0.266 0.004 
Ref ractoriness and Neural Precision 115 
Thus, a Poisson spike generator with a relative refractory period reproduces the measured 
precision. A similar test, performed over a population of ganglion cells, also yielded close 
agreement (Berry, 1998). 
1000 - 
100 - 
10- 
o 
I 
0 
1.0 m 
0.51 
OOOOO_ 
00 00000000 
2 4 6 8 10 12 
Inter-Spike Interval (ms) 
B 
0.0 ! I I I I I 
0 2 4 6 8 10 12 
Time (ms) 
Figure 3: Determination of the relative refractory period. (A) The inter- 
spike interval distribution (diamonds) is fit by an exponential curve (solid), 
resulting in (B) the recovery function. 
Not only is the average fh'ing rate well-matched by the model, but the firing rate in each 
time bin is also very similar. Figure 4A compares the firing rate for the real neuron to that 
generated by the model. The mean-squared error between the two is 4 %, while the 
counting noise, estimated as the variance of the standard error divided by the variance of 
r(t), is also 4 %. Thus, the agreement is limited by the finite number of repeated trials. 
Figure 4B compares the free firing rate q(t) to the observed rate firing r(t). q(t) is equal 
to r(t) at the beginning of a firing event, but becomes much larger after several spikes 
have occurred. In addition, q(t) is generally smoother than r(t), because there is a 
greater enhancement in q(t) at times following a peak in r(t). 
In summary, the free firing rate q(t) can be calculated from the raw spike train with no 
more computational difficulty than r(t), and thus can be used for any spiking neuron. 
Furthermore, q(t) has some advantages over r(t): 1) in conjunction with a refractory 
spike-generator, it produces the correct response precision; 2) it does not saturate at high 
firing rates, so that it can continue to distinguish gradations in the neuron's response. 
Thus, q(t) may prove useful for constructing models of the input-output relationship of a 
spiking neuron (Berry, 1998). 
Acknowledgments 
We would like to thank Mike DeWeese for many useful conversations. One of us, MJB, 
acknowledges the support of the National Eye Institute. The other, MM, acknowledges 
the support of the National Science Foundation. 
116 M. J. Berry and M. Meister 
400- A ,'. Neuron 
I AA .... ' 
 200---I ! .7 '.'.:  
o 
3000 - 
B 
0 
26.74 
--- q(t) 
r(O 
I 
26.75 26.76 26.77 26.78 
Time (s) 
Figure 4: Illustration of the free firing rate. (A) The observed firing rate r(t) 
for real data (solid) is compared to that from the model (dotted). (B) The 
free rate q(t) (thick) is shown on the same scale as r(t) (thin). All rates 
used a time bin of 0.25 ms and boxcar smoothing'over 9 bins. 
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