An Information-Theoretic Approach to 
Deciphering the Hippocampal Code 
William E. Skaggs Bruce L. McNaughton Katalin M. Gothard 
Eton J. Markus 
Center for Neural Systems, Memory, and Aging 
344 Life Sciences North 
University of Arizona 
Tucson AZ 85724 
bill @nsma. arizona. edu 
Abstract 
Information theory is used to derive a simple formula for the 
amount of information conveyed by the firing rate of a neuron about 
any experimentally measured variable or combination of variables 
(e.g. running speed, head direction, location of the animal, etc.). 
The derivation treats the cell as a communication channel whose 
input is the measured variable and whose output is the cell's spike 
train. Applying the formula, we find systematic differences in the 
information content of hippocampal "place cells" in different ex- 
perimental conditions. 
1 INTRODUCTION 
Almost any neuron will respond to some manipulation or other by changing its firing 
rate, and this change in firing can convey information to downstream neurons. The 
aim of this article is to introduce a very simple formula for the average rate at which 
a cell conveys information in this way, and to show how the formula is helpful in 
the study of the firing properties of cells in the rat hippocampus. This is by no 
means the first application of information theory to the study of neural coding; see 
especially Richmond and Optican (1990). The thing that particularly distinguishes 
1030 
An Information-Theoretic Approach to Deciphering the Hippocampal Code 1031 
our approach is its simplemindedness. 
To get the basic idea, imagine we are recording the activity of a neuron in the brain 
of a rat, while the rat is wandering around randomly on a circular platform. Suppose 
we observe that the cell fires only when the rat is on the left half of the platform, 
and that it fires at a constant rate everywhere on the left half; and suppose that on 
the whole the rat spends half of its time on the left half of the platform. In this case, 
if we are prevented from seeing where the rat is, but are informed that the neuron 
has just this very moment fired a spike, we obtain thereby one bit of information 
about the current location of the rat. Suppose we have a second cell, which fires 
only in the southwest quarter of the platform; in this case a spike would give us two 
bits of information. If there were in addition a small amount of background firing, 
the information would be slightly less than two bits. And so on. 
Going back to the cell that fires everywhere on the left half of the platform, suppose 
that when it is active, it fires at a mean rate of 10 spikes per second. Since it is active 
half the time, it fires at an overall mean rate of 5 spikes per second. Since a spike 
conveys one bit of information about the rat's location, the cell's spike train conveys 
information at an average rate of 5 bits per second. This does not mean that if the 
cell is observed for one second, on average 5 bits will be obtained--rather it means 
that if the cell is observed for a sufficiently short time interval At, on average 5At 
bits will be obtained. In 20 milliseconds, for example, the expected information 
conveyed by the cell about the location of the rat will be very nearly 0.1 bits. The 
longer the time interval over which the cell is observed, the more redundancy in the 
spike train, and hence the farther below 5At the total information falls. 
The formula that leads to these numbers is 
(1) 
where I is the information rate of the cell in bits per second, x is spatial location, 
p(x) is the probability density for the rat being at location , A(x) is the mean firing 
rate when the rat is at location x, and ,k - f ,k(z)p(x)dz is the overall mean firing 
rate of the cell. The derivation of this formula appears in the final section. (To our 
knowledge the formula, though very simple, has not previously been published.) 
Note that, as far as the formula is concerned, there is nothing special about spatial 
location: the formula can equally well be used to define the rate at which a cell 
conveys information about any aspect of the rat's state, or any combination of 
aspects. The only mathematical requirement 1 is that the rat's state  and the spike 
train of the cell both be stationary random variables, so that the probability density 
p(x) and the expected firing rate A(x) are well-defined. 
The information rate given by formula (1) is measured in bits per second. If it is 
divided by the overall mean firing rate A of the cell (expressed in spikes per second), 
then a different kind of information rate is obtained, in units of bits per spike .let us 
call it the information per spike. This is a measure of the specificity of the cell: the 
more "grandmotherish" the cell, the more information per spike. For a population 
 Other than obvious requirements of integrability that are sure to be fulfilled in natural 
situations. 
1032 Skaggs, McNaughton, Gothard, and Markus 
of cells, then, a highly distributed representation equates to little information per 
spike. 
Figure 1: "Spot plot" of the activity of a single pyramidal cell in the hippocampus 
of a rat, recorded while the rat foraged for food pellets inside a small cylinder. The 
dots show locations visited by the rat, and the circles show points where the cell 
fired--large circles mean that several spikes occurred within a short time. The lines 
indicate which direction the rat was facing when the cell fired. The plot represents 
29 minutes of data, during which the cell fired at an overall mean rate of 1.319 Hz. 
Consider, as an example, a typical "place cell" (actually an especially nice place 
cell) from the CA1 layer of the hippocampus of a rat--Figure 1 shows a "spot plot" 
of the activity of the cell as the rat moves around inside a 76 cm diameter cylinder 
with high, opaque walls, foraging for randomly scattered food pellets. This cell, like 
most pyramidal cells in CA1, fires at a relatively high rate (above 10 Hz) when the 
rat is in a specific small portion of the environmentrathe "place field" of the cell. 
but at a much lower rate elsewhere. Different cells have place fields in different 
locations; there are no systematic rules for their arrangement, except that there 
may be a tendency for neighboring cells to have nearby place fields. The activity of 
place cells is known to be related to more than just place: in some circumstances 
it is sensitive to the direction the rat is facing, and it can also be modulated by 
running speed, alertness, or other aspects of behavioral state. The dependence on 
An Information-Theoretic Approach to Deciphering the Hippocampal Code 1033 
head direction has given rise to a certain amount of controversy, because in some 
types of environment it is very strong, while in others it is virtually absent. 
Table 1 gives statistics for the amount of information conveyed by this cell about 
spatial location, head direction, running speed, and combinations of these variables. 
Note that the information conveyed about spatial location and head direction is 
hardly more than the information conveyed about spatial location alonerathe dif- 
ference is well within the error bounds of the calculation. Thus this cell has no 
detectable directionality. This seems to be typical of cells recorded in unstructured 
environments. 
Table 1: Information conveyed by the cell whose activity is plotted in Figure 1. 
VARIABLES 
Location 
Head Direction 
Running Speed 
Location and Head Direction 
Location and Running Speed 
INFO 
2.40 bits/sec 
0.48 bits/see 
0.03 bits/see 
2.53 bits/sec 
2.3 bits/sec 
INFO PER SPIKE 
1.82 bits 
0.37 bits 
0.02 bits 
1.92 bits 
1.79 bits 
The information-rate measure may be helpful in understanding the computations 
performed by neural populations. Consider an example. Cells in the CA3 and 
CA1 regions of the rat hippocampal formation have long been known to convey 
information about a rat's spatial location (this is discussed in more detail below). 
Data from our lab suggest that, in a given environment, an average CA3 cell conveys 
something in the neighborhood of 0.1 bits per second about the rat's positionmsome 
cells convey a good deal more information than this, but many are virtually silent. 
Cells in CA1 receive most of their input from cells in CA3; each gets on the order of 
10,000 such inputs. Question: How long must the integration time of a CA1 cell be 
in order for it to form a good estimate of the rat's location? Answer: With 10,000 
inputs, each conveying on average 0.1 bits per second, the cell receives information 
at a rate of 1000 bits per second, or 1 bit per millisecond, so in 5-10 msec the cell 
receives enough information to form a moderately precise estimate of location. 
2 APPLICATIONS 
We now very briefly describe two experimental studies that have found differences 
in the spatial information content of rat hippocampal activity under different con- 
ditions. The methods used for recording the cells are described in detail in Mc- 
Naughton et al (1989)--to summarize, the cells were recorded with stereotrodes, 
which are twisted pairs of electrodes, separated by about 15 microns at the tips, 
that pick up the extracellular electric fields generated when cells fire. A single 
stereotrode can detect the activity of as many as six or seven distinct hippocampal 
cells; spikes from different cells can be separated on the basis of their amplitudes on 
the two electrodes, as well as other differences in wave shape. The location of the 
rat was tracked using arrays of LEDs attached to their heads and a video camera on 
the ceiling. Spatial firing rate maps for each cell were constructed using an adaptive 
binning technique designed to minimize error (Skaggs and McNaughton, submitted), 
1034 Skaggs, McNaughton, Gothard, and Markus 
and information rates were calculated using these firing rate maps. As a control, 
the spike train was randomly time-shifted relative to the sequence of locations; this 
was done 100 times, and the cell was deemed to have significant spatial dependence 
if its information rate was more than 2.29 standard deviations above the mean of 
the 100 control information rates. 
2.1 EXPERIMENT: PROXIMAL VERSUS DISTAL VISUAL CUES 
In this study (a preliminary account of which appears in Gothard et ai (1/)/)2)), the 
activity of place cells was recorded successively in two environments, the first a 7 cm 
diameter cylinder with four patterned cue-cards on the high, opaque gray wall, the 
second a cylinder of the same shape, but with a low, transparent plexiglass wall 
and four patterned cue-cards on the distant black walls of the recording room. The 
two environments thus had the same shape, and from any given point were visually 
quite similar; the difference is that in one all of the visual cues were proximal to 
the rat, while in the other many of them were distal. 
DISTAL CUES 
======================================= 
PROXIMAL CUES 
=========================================================== 
========================================================= ;;:... 
 :::::::::::::::::::::::::: :!:i:i:i:i:i:::::i: ::::::" :: ..... 
...:.;.:.:.:.:;i:.;.:,:.:.:.   - 
Figure 2: Firing rate maps of four simultaneously recorded cells, in the distal cue 
environment (top) and proximal cue environment (bottom). The scale is identical 
for all plots; black _ 5 Hz. 
Fifty cells were recorded with robust place-dependent firing in one or the other 
cylinder. There was no discernable relationship between place fields in the two 
environments--a cell having a place field in the proximal cue environment might 
be nearly silent in the distal cue environment, and even if it did fire, its place field 
would be in a different location. (Figure 2 shows firing rate maps for four of the 
cells.) A substantially higher fraction of the cells had place fields in the proximal cue 
environment, and overall the average information per second was almost 50% higher 
An Information-Theoretic Approach to Deciphering the Hippocampal Code 1035 
in the proximal cue environment. For the cells possessing fields, the information 
per spike was significantly higher in the proximal cue environment, meaning that 
place fields were more compact. 
These results indicate that in the proximal cue environment, spatial location is 
represented by the hippocampal population more precisely, and by a larger pool of 
cells, than in the distal cue environment. The most likely explanation is that, at 
least in the absence of local cues, the configuration of visual landmarks controls the 
activity of the place cell population. 
2.2 EXPERIMENT: LIGHT VERSUS DARK 
Visual cues have a great deal of influence on place fields, but they are not the 
only important factor; in fact, some hippocampal cells maintain place fields even in 
complete darkness (McNaughton et at., lgSgb; Cuirk et at., lg90). This experiment 
(Markus eZ aI., 1992) was designed to examine how lack of visual cues changes the 
properties of place fields. Rats traversed an eight-arm radial maze for chocolate 
milk reward, with the room lights being turned on and off on alternate trials. (A 
trial consisted of one visit to each of the eight arms of the maze.) Figure 3 shows 
firing rate maps for four simultaneously recorded cells. 
LIGHT 
DARK 
Figure 3: Firing rate maps of four simultaneously recorded cells, with room lights 
turned on (top) and off (bottom). The scale is identical for all plots; black >_ 5 Hz. 
(The loops at the ends of the arms are caused by the rat turning around there.) 
The most salient effect was that a much larger fraction of cells showed significant 
spatially selective firing in the light than in the dark: 35% as opposed to 20%. 
However, the average information per second decreased only by 15% in the dark 
as compared to the light, from 0.326 bits per second in the light to 0.278 bits per 
1036 Skaggs, McNaughton, Gothard, and Markus 
second in the dark. (These are overestimates of the population averages, because 
cells silent in both light and dark were not included in the sample.) 
Interestingly, the drop in information content from light to dark seemed to be much 
smaller than the drop from proximal cues to distal cues in the previous experiment. 
A major difference between the two experiments is that, in the eight-arm maze, 
tactile cues potentially give a great deal of information about spatial location, but 
in a cylinder they serve only to distinguish the center from the wall. While it 
is dangerous to compare the two experiments, which differed methodologically in 
several ways, the results suggest that tactile cues can have a very strong influence 
on hippocampal firing, at least when visual cues are absent. 
3 THEORY 
The information-rate formula (1) is derived by considering a neuron as a "channel" 
(in the information-theoretic sense) whose input is the spatial location of the rat, 
and whose output is the spike train. During a sufficiently short time interval the 
spike train is effectively a binary random variable (i.e. the only possibilities are to 
spike once or not at all), and the probability of spiking is determined by the spatial 
location. The event of spiking may be indicated by a random variable $ whose value 
is 1 if the cell spikes and 0 otherwise. If the environment is partitioned into a set of 
nonoverlapping bins, then spatial location may be represented by an integer-valued 
random variable X giving the index of the currently occupied bin. 
In information theory, the information conveyed by a discrete random variable X 
about another discrete random variable Y, which is identical to the mutual infor- 
mation of X and Y, is given by 
where a: and Yi are the possible values of X and Y, and PO is probability. 
If h i is the mean firing rate when the rat is in bin j, then the probability of a spike 
during a brief time interval At is 
P($--- 1Ix--j) -- At. 
Also, the overall probability of a spike is 
P($= 1) = hat, 
where 
with pi = P(X=j). 
After these expressions are plugged in to the equation for I(YIX ) above, it is a 
mater of straightforward algebra, using power series expansions of logarithms and 
keeping only lower order terms, to derive a discrete approximation of equation (1). 
An Information-Theoretic Approach to Deciphering the Hippocampal Code 1037 
4 DISCUSSION 
In many situations, neurons must decide whether to fire on the basis of relatively 
brief samples of input, often 100 milliseconds or less. A cell cannot get much 
information from a single input in such a short time, so to achieve precision it needs 
to integrate many inputs. Formula (1) provides a measure of how much information 
a single input conveys about a given variable in such a brief time interval. 
The formula can be applied to any type of cell that uses firing rate to convey 
information. The only requirement is to have enough data to get good, stable 
estimates of firing rates. In practice, for a hippocampal cell having a mean firing 
rate of around 0.5 Hz in an environment, twenty minutes of data is adequate for 
measuring position-dependence; and for a "theta cell" (an interneuron, firing at a 
considerably higher rate), very clean measurements are possible. 
We have used the measure in the study of hippocampal place cells, but it might 
actually work better for some other types. The problem with place cells is that 
they fire at low overall rates, so it is time-consuming to get an adequate sample. 
Cortical pyramidal cells often have mean rates at least ten times faster, so it ought 
to be easier to get accurate numbers for them. The information measure might 
naturally be applied to study, for example, the changes in information content of 
visual cortical cells as a visual stimulus is blurred or dimmed. 
Supported by NIMH grant MH46823 
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