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arXiv:1107.2984v1 [cs.IT] 15 Jul 2011

An Introductory Review of Information Theory in the Context of

Computational Neuroscience

Mark D. McDonnell

Institute for Telecommunications Research

University of South Australia

Mawson Lakes, SA 5095, Australia

Mark.McDonnell@unisa.edu.au

Shiro Ikeda

The Institute of Statistical Mathematics

Tokyo 190-8562 Japan

shiro@ism.ac.jp

Jonathan H. Manton

The University of Melbourne

Victoria 3010 Australia

jmanton@unimelb.edu.au

July 18, 2011

Abstract

This paper introduces several fundamental con-

cepts in information theory from the perspective

of their origins in engineering. Understanding such

concepts is important in neuroscience for two rea-

sons. Simply applying formulae from information

theory without understanding the assumptions be-

hind their definitions can lead to erroneous results

and conclusions. Furthermore, this century will

see a convergence of information theory and neuro-

science; information theory will expand its founda-

tions to incorporate more comprehensively biolog-

ical processes thereby helping reveal how neuronal

networks achieve their remarkable information pro-

cessing abilities.

1 Introduction

Norbert Wiener, the founder of cybernetics, wrote

that it is the “boundary regions of science which

offer the richest opportunities to the qualified in-

vestigator. They are at the same time the most re-

fractory to the accepted techniques of mass attack

and the division of labor” (Wiener, 1948, p. 2).

He went on to explain that “a proper exploration of

these blank spaces on the map of science could only

be made by a team of scientists, each a specialist

in his own field but each possessing a thoroughly

sound and trained acquaintance with the fields of

his neighbors; all in the habit of working together,

of knowing one another’s intellectual customs, and

of recognizing the significance of a colleague’s new

suggestion before it has taken on a full formal ex-

pression. The mathematician need not have the

skill to conduct a physiological experiment, but he

must have the skill to understand one, to criticize

one, and to suggest one. The physiologist need not

be able to prove a certain mathematical theorem,

but he must be able to grasp its physiological sig-

nificance and to tell the mathematician for what

he should look.”

Indeed, three giants of science, Wiener, von

Neumann and Shannon, realised in the 1940s

the need for understanding the brain in terms

of the fundamental engineering principles applica-

ble to any computational device: energy, entropy

and feedback (Wiener, 1948; von Neumann, 2000;

Shannon and Weaver, 1949). This led to the Macy

conferences (1946–1953) which attracted leading

scientists from across engineering and the physi-

cal and life sciences. The Macy conferences were

one of the earliest organised approaches to trans-

disciplinarity and hailed by some as the most im-

portant event in the history of science after World

War II. They demonstrated the need for, and the

initial difficulties in, establishing a common lan-

guage powerful enough to communicate the intri-

cacies of the relevant fields across the physical and

life sciences and engineering.

While their dream was not realised, this was pri-

marily due to insufficient experimental data. As

time marched on, the barrier to bringing together

the ever more specialised disciplines grew larger.

With tremendous experimental advances having

been made in the past 60 years, it is timely to stand

on the shoulders of these giants and resume their

quest. With this as motivation, the present article

endeavours to whet the appetites of neuroscientists

and information theorists alike for learning more of

each other’s fields.

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1.1Relevance of Information The-

ory (and Feedback)

The human brain is often described as the

most complex structure

verse (Fischbach, 1992). Certainly, it is the most

efficient signal processing device known. Drawing

only 20 watts of power, the brain significantly out-

performs engineered devices at signal processing

tasks such as source separation, feature extraction,

and speech and image recognition (Sarpeshkar,

1998). This is all the more remarkable because sig-

nals within the brain propagate very slowly com-

pared with those in a computer.

This suggests that the brain uses a paradigm for

signal processing very different from any developed

in engineering. Why then should engineering in

general and information theory in particular have

relevance to understanding the brain? The answer

lies partially in the fact that engineers study fun-

damental laws pertinent to any system, including

biological ones (Berger, 2003; Sarpeshkar, 1998).

Indeed, John von Neumann viewed the brain as

a hybrid computer which performs control, com-

munication and computation, and concluded that

information theory is therefore essential for under-

standing its functionality (von Neumann, 2000).

Wiener too recognised that information theory was

essential for a deeper understanding of feedback

and thus life (Wiener, 1948). Anecdotal evidence

suggests Shannon himself, the father of informa-

tion theory, may have been partially motivated by

how his brain processed “information” when per-

forming a complex task such as juggling balls.

A few words on the concept of feedback are in

order. Feedback refers to achieving a task, such as

keeping a car travelling at a constant speed, by re-

peatedly measuring the current state, such as the

car’s speed, and feeding those measurements back

and using them to make the requisite changes at

the input, such as applying more or less pressure to

the accelerator of the car. Feedback is a fundamen-

tal concept in engineering because it can militate

errors caused by imprecisions and external inter-

ference.

The brain too must use feedback to over-

come imprecisions (Burdet et al., 2001; Wiener,

1948; Marko, 1967; Todorov and Jordan, 2002;

Burdet et al., 2006; Franklin et al., 2008); with-

out feedback, we would fall over whenever we at-

tempted to walk. Within the sensory pathways

there are tremendous numbers of feedback paths

connecting regions of higher-level brain function

to regions of lower-level functionality, giving rise

to top-down processing theories of the visual path-

ways and providing a mechanism for selective at-

tention.

Although they started out in different disciplines

— information theory emerged from communica-

inthe known uni-

tion theory while feedback was studied in con-

trol theory — recent years have seen some con-

vergence of feedback and information theory. A

fundamental question is what is the slowest rate

at which information must be fed back for the sys-

tem to work. Scientists have started to consider

how fast the brain must be processing informa-

tion if we are able to walk properly and can move

our hand in a straight line even though random

external forces are impeding its motion in experi-

ments (Burdet et al., 2006). This is an example of

such convergence of two important theories.

By virtue of being introductory, the present pa-

per focuses on discussing information theory in

“one-way” (i.e. feed-forward exclusively) biologi-

cal contexts. A comprehensive account of the brain

in engineering terms would necessarily involve the

marriage of information theory and control theory.

Repeating the words of von Neumann, the brain

must be understood in terms of control (feedback),

communication (information theory) and compu-

tation.

1.2 Information Theory and Com-

munication

It must be recognised that the mathematical disci-

pline of “information theory” does not (and should

not) capture all aspects of how the word “informa-

tion” is used in spoken language. Failing to distin-

guish the two can lead to errors caused by flawed

intuition in one direction, or the inappropriate ap-

plication of information theory in the other1.

Information theory was invented in response to

practical problems faced by the designers of com-

munication systems such as telephones and data

modems. The basic problem is to find an efficient

way of transmitting information from one place to

another, whether it be a probe sending information

from the moon back to earth or a mobile telephone

sending and receiving voice and internet packets.

Indeed, consider the problem of one person trying

to send a series of messages to another person on

the other side of a brick wall; for simplicity, assume

this second person is not allowed to speak or send

any other form of message to the first person such

as an acknowledgement or request for clarification

(or “retransmission”). How should the first person

send each message?

One thing is clear; the louder the first person

shouts, the greater the chance that the second per-

son can understand the message over the back-

ground noise (perhaps the neighbours are mowing

1Blindly applying the mathematical equations defining

entropy or channel capacity does not necessarily endow the

resulting quantities with any meaning or validity; infor-

mation theoretic quantities can be understood only with

respect to the assumptions and limitations of information

theory.

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their lawns). Perhaps a little harder to appreciate

but equally true in the digital world, if the first

person were to speak more slowly the second per-

son would have a greater chance of catching every

word. The third parameter that can be adjusted

is the level of redundancy. When we speak with

a young child we tend to elaborate and use more

words to describe a concept in an attempt to in-

crease the chances of correct reception of the over-

all message.

In information theory, these parameters are re-

ferred to formally as the transmission power, the

transmission rate and coding (or redundancy).

The simplest form of coding is to repeat the mes-

sage two or more times. This is known as a repe-

tition code.

Shannon’s pioneering work shattered a long-held

belief that with finite power it was impossible to

be able to transmit a message in such a way as

to guarantee its perfect reception even in the pres-

ence of noise and other interference. Indeed, even

if I shouted at the top of my voice and repeated

myself a hundred times, every so often the inter-

ference (lawn mowers?) will prove too great and

my message will be lost.

The answer lies in coding; repetition codes are

not particularly good codes. Shannon realised that

there exist very clever codes which can ensure that

any two messages are so different from each other

that the receiver can correctly decide which mes-

sage was sent despite the interference. Technically,

perfect reception requires the receiver to listen for-

ever before deciding which message was sent but

the key point is that given any positive but arbi-

trarily small probability of error (such as one mes-

sage being incorrectly received in 1012messages)

then a code can be constructed which achieves this

level of performance in finite time, and more im-

portantly, the transmission power does not need

to be increased. Increasing transmission power to

achieve a particular error rate is grossly inefficient

compared with choosing a better code. In the ex-

ample of one person trying to convey a message to

the other person, the secret is to share a codebook

beforehand, and a different sequence of sounds, one

for each message that may be sent, is written in

it. “It will rain tomorrow” might be encoded as

(a segment of) Beethoven’s 5th symphony while

“It will be sunny tomorrow” might be encoded as

a hard rock song. These two encoded messages

are “sufficiently different from each other” to have

very little chance of being confused. More impor-

tantly, any small fragment of the two messages are

different. This is how interference is overcome. Be-

cause interference itself has limited power (other-

wise the game would not be fair!) then even if there

are times when the interference is particularly bad,

there will be other times when the interference is

back to normal and in the long run, there is no

confusing Beethoven for hard rock.

For the transmission rate to be acceptable the

codebooks would need to contain more than just

two messages. (With two messages, each one en-

coded by a five-minute song, the transmission rate

would be 1 bit per 5 minutes.) In the same way

that the transmission power does not need to go to

infinity, the transmission rate need not go to zero.

Precisely, Shannon discovered a quantity known as

the channel capacity. If the transmission rate is

less than the channel capacity then communica-

tion with any desired level of accuracy is possible

whereas if the transmission rate exceeds the chan-

nel capacity then it becomes impossible to have

arbitrarily good performance in finite time. As is

to be expected, the channel capacity depends on

the interference. The more destructive the inter-

ference the lower the channel capacity.

1.3 Intra-organism communication

Messages are also passed around within an organ-

ism. Information gathered by an organism’s senses

must be communicated for it to have any effect on

the organism. Within the brain and nervous sys-

tem, information is manipulated in at least three

different ways:

• information acquisition (sensory transduc-

tion);

• communication between spatially separated

regions (information transmission);

• memory formation and recall (information

storage (Varshney et al., 2006)).

In brains, each of these are essential for the emer-

gence of broader functions that might be termed

‘computation’. Communication is perhaps most

fundamental, since information from the senses

needs to be communicated in order for it to have

any affect on an organism, while information stor-

age, whether in computers or brain memories, can

be viewed as communication from the past to the

present or future.

Understanding how the brain stores and trans-

mits information is tantamount to understanding

the brain as a whole because if we could “listen

in” to brain messages it would surely be just a

matter of time before we understood the computa-

tional side, too. That said, the possibility that the

brain does not separate out information transmis-

sion from information processing must be consid-

ered. Whereas computer architectures have mech-

anisms known as buses for moving information be-

tween different processing units, the brain might

take a more efficient distributed approach and si-

multaneously process and communicate in a non-

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separable fashion. It has been stated that “com-

putation in the brain always means that informa-

tion is moved from one place to another” (Buzs´ aki,

2006, p. 116). A comprehensive understanding

of the brain’s mechanisms for internal communi-

cation will likely form an integral part of more

advanced theories about how ‘computation’ arises

within brain networks.

Regardless of how the brain actually processes

information, at the end of the day the brain is an

input-output system (we react to what we sense)

and therefore subject to the same laws as any

other input-output system. Information theory is

therefore relevant to understanding how the brain

works, and conversely, it is highly likely that ad-

vances in the field of information theory will be

made in synergy with new discoveries of the com-

puting paradigms used by the brain (Cohen, 2004;

Sarpeshkar, 1998; Berger, 2003). Indeed, informa-

tion theory may have to expand to address new

neurobiologically relevant questions if it is to be

powerful enough to explain all aspects of how the

brain manipulates information.

To demonstrate the relevance of even simply

thinking in information theoretic terms, Landauer

estimated that humans learn information at a rate

of about two bits per second (Landauer, 1986)2.

Taking memory loss into account, a person will ac-

cumulate approximately two billion bits of infor-

mation in a lifetime (or approximately 240MB in

computing terms). Since our brain has many more

synapses than two billion, Landauer concludes that

“possibly we should not be looking for models and

mechanisms that produce storage economies but

rather ones in which marvels are produced by prof-

ligate use of capacity.”

1.4 Outline of Paper

According to Cohen (2004), biological science asks

six kinds of questions about domains ranging from

molecules and cells, up to the biosphere:

1. How is it built? (Structures)

2. How does it work? (Mechanisms)

3. What is it for? (Functions)

4. What goes wrong? (Pathologies)

5. How is it fixed? (Repairs)

6. How did it begin? (Origins)

Utilising information theory in neuroscience is ul-

timately useful only if it can address one or more

of these questions.

2To put this in perspective, a digital camera typically

stores a single photo using around 10,000,000 bits. Clearly

then, we extract only a very small amount of information

from what our senses receive.

In this paper we advocate that information the-

ory 1) can be a useful framework for finding an-

swers to some of these questions; but 2) must be

broadened for its theorems to be directly applica-

ble to neuronal networks. Although information

manipulation can happen at very different levels

of organisation, such as storage of information in

genes, or communication at the level of synaptic

transmission between cells, or at that of spiking

patterns of neurons in a network, in this paper we

will be focusing on examples that involve spiked-

based communication between neurons.

In making these points, it is necessary for us

to introduce the most basic and well-known infor-

mation theoretic concepts in Section 2, before dis-

cussing the challenges of applying the theory mean-

ingfully to questions in neuroscience in Section 3.

Then in Section 4 we summarise a specific ex-

ample which illustrates that information theoretic

approaches depend critically on different assump-

tions that could be made about neural systems.

Finally in Section 5 we conclude the paper with

some closing remarks on the material we cover and

briefly summarise recent developments on informa-

tion theoretic approaches in neuroscience that ex-

tend well beyond the classical ideas we present,

thereby with increasing relevance to neurobiologi-

cal systems.

2 The basics and utility of

Shannon Information The-

ory

This section briefly explains key concepts from

Shannon information theory and hints at possible

contributions in neuroscience. By Shannon infor-

mation theory we are referring to a specific sub-

part of the broader field of information theory.

The latter, by definition, encompasses any mathe-

matical theorems about information, and therefore

is not confined to well-known concepts introduced

by Shannon, such as entropy and mutual informa-

tion. As we discuss later, information theory be-

yond Shannon theory may be very important in

neuroscience.

Shannon’s milestone paper (Shannon, 1948) that

founded the field of information theory showed to

the world that introducing the right kind of redun-

dancy was the key to moving information from one

place to another in an efficient and reliable man-

ner. Since information sources such as spoken voice

or PDF (portable document format) documents

generally contain the wrong kind of redundant in-

formation, Shannon proposed a two-step process:

first remove the existing redundancy by compress-

ing the message to be sent, then introduce the right

kind of redundancy for communicating the message

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through the channel at hand. These two impor-

tant concepts are known as “source coding” and

“channel coding” respectively. They motivate sev-

eral fundamental questions including determining

the maximum amount of compression possible of

an information source. Answers to these questions

are given in terms of quantities such as entropy

and mutual information.

alise that these quantities were given special names

because they serve to answer important questions

for a particular class of problems. It would be a

mistake to assume without additional justification

that they are applicable or even meaningful beyond

the bounds of the original questions for which they

serve as the answers to. See e.g. Johnson (2008)

for more discussion.

It is important to re-

2.1 Entropy and Source Coding

Living in the digital age, readers will be famil-

iar with compressing files.

send to a friend is an example of lossless compres-

sion. Generally (but not always) the compressed

file will be smaller than the original yet no infor-

mation has been lost; the friend can recover the

original file by decompressing the compressed file

(Fig. 1). For compressing music or photos, signifi-

cantly greater compression can be achieved by us-

ing lossy compression algorithms such as MP3 and

JPEG. As the name suggests, some information is

lost (Berger and Gibson, 1998). The original can

be recoveredsufficiently well for a satisfactory com-

promise to have been reached; a small amount of

quality is sacrificed for a large saving in storage

space.

Zipping up a file to

X

Compression

C

Decompression

˜ X

Figure 1: Source Coding.

The remainder of this section discusses lossless

compression only. Consider the problem of com-

pressing a short message of length 8 bits. A bit

is simply a “0” or a “1” so an 8-bit message

is a sequence of eight zeros and ones, such as

“01011101” or “10101010”3. A calculation (or by

writing out all the possibilities if need be, starting

with “00000000”, “00000001” and continuing un-

til “11111111”) shows that there are precisely 256

different 8-bit messages. Compressing a message

would mean using fewer than 8 bits to store the

message. A simple enumeration shows that this is

impossible as stated; there are only 128 7-bit mes-

sages, not enough to represent all possible 8-bit

3Of course, any other alphabet could have been used.

An 8-character message consisting of a string of 8 letters

of the alphabet might look like “abzpuikq” and the same

reasoning would apply.

messages. How then does a computer compress a

file losslessly?

The secret is that there is often redundancy in

the kinds of information that people are interested

in. Equivalently, it is generally the case that not all

messages have an equal chance of occurring. For

argument’s sake, assume that out of the 256 pos-

sible messages, there are 15 messages which occur

most of the time. To exploit this, we may decide to

use 4 bits to represent each of these messages. Pre-

cisely, “0000” would represent the first message,

“0001” the second, up to “1110” for the 15th mes-

sage. To represent any other message, we would

first write down “1111” to mean “not one of the

15” and then we would write down the original

message using 8 bits. This means that 15 of the

messages can be written down using only 4 bits

but the remaining 256 − 15 = 241 messages now

require 4 + 8 = 12 bits for their storage.

The only way to make this meaningful is to

consider repeating this compression exercise many

times. If we had to store a very large number N of

8-bit messages using this scheme, how many bits

will be required? Assume that K out of the N

messages belong to the set of 15 special messages.

These K messages require 4 bits while the remain-

ing N − K messages require 12 bits, or in total,

4K + 12(N − K) bits are required compared with

8N bits had we not compressed the messages. Pro-

vided K is sufficiently large, we will have succeeded

in compressing the data. For example, if K = 750

and N = 1000 then we would require only 6,000

bits rather than the original 8,000 bits.

The conventional way to describe the above sce-

nario is to work with probabilities.

that the messages we are being asked to compress

are being generated at random and there is no cor-

relation between the message we are being asked

to compress now and the messages we have al-

ready compressed. Mathematically, we represent

the original sequence of messages by a sequence

of independent and identically distributed random

variables {x1,x2,···}, each having a probability

density p(X). In the above example, each xkwould

be an 8-bit message (or equivalently, a number be-

tween 0 and 255 inclusive) and p(X) would be the

probability that a particular message X is cho-

sen. For concreteness, assume that each of the

first fifteen messages have a 5% chance of occur-

rence (meaning there is a 75% chance of a ran-

domly chosen message being one of these 15 and

thereby corresponding with the earlier choice of

K = 750 and N = 1000 ). Then p(00000000) =

p(00000001) = ··· = p(00001110) = 0.05. We

assume that all other messages each have a proba-

bility of 0.25/(256−15) of occurring. The expected

number of bits required to compress a single mes-

sage can then be calculated by summing over i the

We assume

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probability that the i th message occurs multiplied

by the number of bits required to represent the i th

message. When most of the probabilities are the

same the calculation simplifies. With the values

given above, the expected length is calculated to

be 15×0.05×4+(256−15)×

on average, 6N bits would be required to compress

N messages drawn at random if the above scheme

were used.

Is there a better compression scheme, one which

requires fewer than 6 bits per message on average?

In fact, what is the best possible? As elucidated

presently, Shannon was able to answer these ques-

tions. First though, a technicality needs mention-

ing.

Coding each message separately, as was done

above, is inefficient. It is better to concatenate a

series of messages and compress them all at once;

this provides more opportunity for better compres-

sion through the simple fact that there are more

compression schemes to choose from. (It also alle-

viates the wasted space caused by otherwise having

to use an integer number of bits to represent each

message.)

It is therefore quite standard to refer to each

xk as a symbol rather than a message and ask

how many bits per symbol on average must be

used to compress the infinitely long sequence of

independent and identically distributed symbols

{x1,x2,···} if each symbol has a probability p(X)

of occurring4.

When X is a random variable and its distribu-

tion is p(X), its entropy is defined as

0.25

256−15×12 = 6. Thus,

H(X) = −Ep(X)[logp(x)], (1)

where Ep(X)[·] denotes the expectation with re-

spect to p(X). The practical operation is a sum-

mation when X is discrete and an integration when

X is continuous. When 2 is the base of logarithm,

i.e. log2, the units of entropy are bits and H(X)

is precisely the number of bits per symbol required

on average to compress an infinitely long sequence

of symbols when each symbol has probability p(X)

of occurring.

It is for this reason that people endeavour to

explain entropy as quantifying the “ambiguity”

or “uncertainty” about the random variable X.

When X has only one possible state (that must

therefore occur with probability 1), there is no am-

biguity about X and the entropy is 0. However if

X takes one of two states with probability p and

1 − p, respectively, (0 ≤ p ≤ 1), entropy is max-

imised when p = 0.5 and H(X) = log22. This is

exactly 1 bit and implies that a sequence of equally

4It is traditional in probability theory to use an upper

case letter to represent the random variable while the cor-

responding lower case letter represents realised values.

likely zeros and ones cannot be compressed. Note

also that if p = 0, 0log20 = 0.

Shannon’s source coding theorem states that (in

the limit as the number of symbols goes to infin-

ity) it is possible to compress each symbol to H(X)

bits on average (and impossible to do better). It

does not however say how to design such a source

code. Furthermore, the practical construction of

compression and decompression methods is com-

plicated by considerations of algorithmic efficiency

(which affects battery life in portable equipment

such as mobile telephones) and latency (how long

the receiver must wait from the time a symbol is

sent until that symbol can be received and de-

coded). That said, having a target to aim for is

extremely useful and entropy provides that target

for source compression.

The reader may wish to verify that for the ex-

ample introduced in this section, the corresponding

entropy is 5.72 bits per symbol. This represents the

best any compression scheme can hope to achieve,

and indeed, it is lower than the 6 bits per symbol

scheme presented here.

2.2Mutual Information,

Capacity and Channel Coding

Channel

The following example of a binary symmetric chan-

nel will be used to add concreteness to the ensu-

ing introduction of mutual information and chan-

nel capacity. Let {s1,s2,···} denote a binary se-

quence which is to be transmitted to another per-

son or device. It is called the source sequence. The

medium through which a message can be sent from

one person or device to another is called the chan-

nel. Mathematically, a channel takes a sequence

at its input and it generates another sequence at

its output. If the channel were ideal, it would sim-

ply copy its input to its output and communica-

tion would be straightforward. Generally though,

the channel is not ideal. It introduces random er-

rors. If {xi} is the binary input sequence (which is

shorthand notation for {x1,x2,···}) then the bi-

nary output sequence {˜ xi} of a binary symmetric

channel with error probability p is given by the

rules that 1) for each integer i, the output ˜ xi at

time i depends only on the corresponding input xi

at the same time i; and 2) the probability that ˜ xi

differs from xiis p. If p = 0.1 then on average one

in every ten symbols will be corrupted, meaning

either a 0 was sent and a 1 was received, or a 1

was sent and a 0 was received.

What sequence {xi} should be sent over the

channel if the ultimate aim is to send {si} reliably

to the receiver, assuming of course that the receiver

can process the output {˜ xi} of the channel before

deciding what it believes the message {si} is? This

is illustrated in Fig. 2 where the operation of gen-

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erating {xi} from {si} is called (channel) encoding

and the operation of generating {˜ si}, the receiver’s

best guess at the original message, is called (chan-

nel) decoding.

S

Encoding

X

Noisy Channel

˜ X

Decoding

˜S

Figure 2: Channel Coding.

For simplicity, often the encoding and decod-

ing processes work on blocks of data. Precisely,

the original source sequence S is divided up into

subsequences of length K. Each of these is en-

coded to a longer binary sequence X with length

N. For example, a simple K = 2, N = 3 block

code would be to add a parity bit (i.e. a bit that

is zero when the sequence has an even number of

zeros, and a one when an odd number) after ev-

ery two symbols, so: “00” becomes “000”; “01”

becomes “011”; “10” becomes “101” and “11” be-

comes “110.” Therefore, the sequence “0111” be-

comes “011110” where the 3rd and 6th bits are the

introduced parity bits.

This coded sequence is transmitted through the

channel. At the other end, the receiver reverses

the process, converting each block of N symbols

back into a block of K symbols. In this particular

case, introducing just a single parity bit does not

allow the receiver to have a better guess at what

the original message is, but it does allow the re-

ceiver to detect if a single bit has been changed.

This is called error detection.

when the receiver is not only able to detect an er-

ror has occurred but can fix the error and therefore

recover the original message, requires more redun-

dancy to be introduced, that is, choosing N to be

larger than K + 1. (If there are too many errors

then error correction would fail, but the key point

is that the probability that several consecutive bits

are wrong is significantly smaller than if a single bit

were wrong, therefore a small increase in redun-

dancy allows a substantial increase in reliability.)

The two lengths K and N together define the

“rate” of the code, which perhaps is better un-

derstood as measuring the decrease in throughput

caused by the introduction of redundancy by the

encoder. Precisely, in the above example, the rate

of the code is R = K/N, meaning that if the chan-

nel can accept encoded symbols at a rate of 1 bit

per second then the source symbols must have a

rate of only R bits per second.

Reducing the rate enables more redundancy to

be introduced which can be used to increase the

chance of the receiver being able to work out what

message was sent. Shannon’s remarkable observa-

Error correction,

tion was that there is a much better way of in-

creasing the chance of correct reception than by

decreasing the rate towards zero. For a fixed rate

R, the block size K can be increased (thereby in-

creasing N according to the formula N = KR ).

This allows a more sophisticated form of redun-

dancy to be introduced (but at the price of intro-

ducing greater latency; the receiver must receive

N symbols before it can work out what the corre-

sponding K message symbols were).

Shannon proved that there exists a rate C, called

the channel capacity, such that for any rate R

strictly less than C and any desired error rate ǫ > 0

(meaning that the probability that the receiver de-

codes a bit incorrectly is less than ǫ, which might

be chosen to be ǫ = 10−9or smaller in practice),

there exists a K (possibly quite large) and a block

encoder and decoder pair such that the receiver

can correctly decode each bit of the source mes-

sage with error probability less than ǫ.

customarily summarised by saying that error-free

communication is possible at rates below the chan-

nel capacity5.

Shannon was able to give a formula for comput-

ing the channel capacity C. When this formula

(described below) is applied to the above exam-

ple of a binary symmetric channel with probabil-

ity of error p, the channel capacity is found to be

C = 1 + plog2p + (1 − p)log2(1 − p), meaning for

example that if the channel can transfer one bit per

second then the source symbols must arrive slower

than C bits per second. If p = 0.1 then C = 0.531

meaning that for every 1,000 source symbols, just

over 1,883 encoded symbols are required for reli-

able communications.

The formula for channel capacity involves a

quantity called mutual information.

the mutual information of the input and the output

of the channel measures how much information the

output provides about the input; the more reliable

the channel the higher the mutual information. It

is therefore reasonable to expect that the larger the

mutual information the greater the channel capac-

ity.

Bearing in mind that “information” is a very

general word and it is therefore not possible to cap-

ture all its nuances in a single mathematical defi-

nition, it is expedient to return to the idea in the

previous section of using asymptotic compressibil-

ity as a measure of information. It turns out that

this is the right definition to use when it comes

to determining the capacity of a channel (which in

itself is an asymptotic measure).

Suppose there are two random variables, X and

This is

Intuitively,

5Shannon also proved the converse, that no scheme (even

non-block-based coding schemes) can achieve arbitrarily

small error rates if their rate is greater than or equal to

C.

7

Page 8

Y , and they are somehow related to each other.

For example, X might denote temperature while

Y denotes humidity. Even simpler, X might rep-

resent the outcome of rolling a 6-sided die while

Y is given the value 0 if the die landed on an even

number, or 1 if odd. Knowing Y gives partial infor-

mation about X; how can we measure how much

information Y tells us about X?

The fact that Y gives partial information about

X is reflected in the fact that if Y is known then X

can be compressed more than if Y were not known.

In the above example, if Y were not known then

it is impossible to compress X because each of the

outcomes is equally likely; we are forced to use one

of six possible symbols (or log26 bits) to store each

sample of X; the entropy of X is H(X) = log26.

If Y is known though then only one of three possi-

ble symbols (or log23 bits) needs to be stored; the

average conditional entropy is H(X|Y ) = log23.

The additional amount of compression possible,

I(X;Y ) = H(X) − H(X|Y ), is called the mutual

information and measures the amount of informa-

tion Y provides about X. It turns out that mutual

information is symmetric — I(X;Y ) = I(Y ;X)

— hence there is no need to specify the order of X

and Y . In the above example, if X is known then

Y is known, therefore no additional bits are re-

quired to store Y if X is known: H(Y |X) = 0.

Since H(Y ) = log23 it is indeed the case that

I(Y ;X) = H(Y ) − H(Y |X) = log23 = I(X;Y )6.

Returning to the channel capacity calculation,

assume that a sequence generated by X is sent

through the channel. The output sequence is it-

self generated by a random variable, call it Y . If

the receiver wants to recover X, it needs at least

an extra H(X|Y ) bits of information (for otherwise

there would be an even more efficient scheme for

compressing X than the best possible, a contra-

diction). Looking at it from another angle though,

this implies that I(X;Y ) = H(X)−H(X|Y ) bits of

information have somehow been transmitted suc-

cessfully with each use of the channel (since with

an extra H(X|Y ) bits of carefully chosen informa-

tion it is theoretically possible to recover X). For

the case of the binary symmetric channel with error

probability p, a reasonably straightforward calcu-

lation shows that if the input X takes the value 1

with probability q and the value 0 with probability

1 − q then the mutual information of the input X

and the output Y is I(X;Y ) = H(q)−H(p) where

H(θ) = −θlog2θ − (1 − θ)log2(1 − θ) is the num-

ber of bits required to compress a binary sequence

6The reason for this symmetry is that if we were to com-

press X first then compress Y , or if we were to compress

Y first then compress X, we end up either way with hav-

ing compressed optimally the joint sequence generated by

X and Y . Mathematically, H(X,Y ) = H(X) + H(Y |X) =

H(Y ) + H(X|Y ) from which it follows immediately that

I(X;Y ) = I(Y ;X).

taking the value 1 with probability θ and the value

0 with probability 1 − θ.

Although we must have the channel input X rep-

resent the source message S in some way, there

is otherwise arbitrary freedom in how to choose

X. Why not choose X to maximise the mutual

information? The largest value H(q) can take is

1 (which occurs when q = 1/2 ). Therefore, the

largest number of bits we can ever expect to trans-

mit reliably through the binary symmetric channel

is 1 − H(p) bits per usage of channel. If p = 0.1

then 1 − H(p) = 0.469. In this case, at most ev-

ery 0.469 bits of the source message must be ex-

panded to 1 bit (since the channel transmits 1 bit

per usage), or in other words, we must have the

rate of the code (see above) satisfy K/N < 0.469.

Remarkably, Shannon proved that this bound is

achievable; whenever the rate is less than the max-

imum of the mutual information, (as close as you

like to) error-free communication is possible7.

It is of interest to note that while here we have

considered an example where X is a discrete ran-

dom variable, the most well known case of a chan-

nel for which the capacity achieving input distribu-

tion is known, is the additive Gaussian noise chan-

nel, with a power constraint on the input. In this

case, the capacity achieving input distribution is in

fact continuous, i.e. a Gaussian distribution. As

we discuss later though, it is far more common for

the capacity achieving input to be discrete.

We now summarise and precisely define the im-

portant information theoretic terms that we have

introduced and discussed above without stating

their formal definitions. Each of these are defined

mathematically as follows. We already introduced

entropy, in Eqn. (3). The average conditional en-

tropy requires a double expectation:

H(Y |X) = −Ep(X)

?Ep(Y |x)[logp(y|x)]?.(2)

As an aid to intuition, consider a single outcome of

the random variable X. The entropy of Y given X

can be calculated from Eqn. (2) by calculating the

expectation with respect to the conditional distri-

bution of Y given X = x . If this is carried out for

all possible outcomes of X, the result is a function

of x. This function can then be averaged with re-

spect to the distribution of x, and by definition, the

result is the average conditional entropy, H(Y |X).

As mentioned above, the mutual information can

be expressed as I(X;Y ) = H(X) − H(X|Y ) =

H(Y ) −H(Y |X) . In what follows below we write

mutual information in a different form based on

another entity called relative entropy or Kullback-

7Note that Shannon proved “it is achievable” in the limit

when N and K goes to infinity, but did not show “how to

achieve it.” In order to be close to the bound, we generally

need a good error correction code and N and K must be

very large.

8

Page 9

Leibler divergence. This is defined as

D(p(X)||q(X)) = Ep(X)

?

logp(x)

q(x)

?

,

where p(X) and q(X) are two distributions of the

same random variable X. Note that the relative

entropy is positive and is equal to 0 if p(X) is

identical to q(X). Mutual information is defined

as the relative entropy between the joint distribu-

tion of X and Y , and the product of the marginal

distributions of X and Y :

I(X;Y ) = D(P(X,Y )||P(X)P(Y ))

= Ep(X,Y )

?

log

p(x,y)

p(x)p(y)

?

.

(3)

It is straightforward using p(x,y) = p(y|x)p(x) =

p(x|y)p(y) to obtain the above stated relationships

between mutual information and entropy. The def-

initions as written here hold for both discrete and

continuous distributions of X and Y . In this sec-

tion we have considered only a simple discrete case,

where X and Y are both binary. In general they

can have any number of states, or be continuous,

as is the case below. In full generality, the channel

capacity is defined as

C = sup

P(X)

I(X;Y ).

3 Challenges of Utilising In-

formation Theory in Neuro-

science

In this section, some of the challenges of integrating

information theory into neuroscience are touched

upon. In particular, we must make assumptions

about the way in which information is represented

in the brain, whereas in engineering this is specified

by the designer. Ultimately it will be necessary to

extend the frontiers of information theory if it is to

encompass in its entirety the information process-

ing techniques of neuronal networks. Such an ex-

pansion would involve in part the greater integra-

tion into information theory of systems and control

theory from engineering and the theory of com-

putation from mathematics.

aim to keep separate communication circuitry from

computation circuitry so as to simplify the design

and analysis of engineered systems, there is no rea-

son why nature should maintain such a separation.

Evolution tends to find efficient designs and not

necessarily “simple” designs.

It would be counter-productive though to as-

sume that information theory in its current form

could not be applied usefully in computational neu-

roscience. One place it is immediately applicable

is the early sensory pathways where information is

Whereas engineers

primarily flowing in one direction. Considerably

extra care must be taken when feedback loops are

present. This is especially the case because (in ex-

periments) we have control over the input signal

itself and hence can investigate how a known sig-

nal is communicated from one neuron to another.

The complication though is that it appears the in-

formation is being processed at the same time it is

being communicated.

The brain heavily compresses the information it

receives from its sensory systems. Since the en-

tropy of a signal determines precisely how much

(lossless) compression is possible, it sets fundamen-

tal limits which must be respected by any system,

including biological systems. It is no surprise then

that the estimation of entropy of neural signals

based on experimental data is an active research

area (Borst and Theunissen, 1999; Panzeri et al.,

2007; Vu et al., 2009).

In the brain, neurons communicate with each

other and transfer information.

means of communication are the spikes of each

neuron (Rieke et al., 1997), and it is parsimo-

nious to model their occurrences as depending

randomly on the neuron’s input (Poggio, 1964;

Mainen and Sejnowski, 1995). Thus, neurons com-

municate through a noisy channel, and mutual in-

formation should therefore play an important role

in understanding the nervous system and brain

In order to consider a neuron as a communica-

tion channel, we need to consider what we mean by

“communication” in the specific context of biologi-

cal neurons. There are several important concepts

to consider before we can begin to discuss a specific

example of the application of information theory in

neuroscience.

The primary

3.1Communication Channels and

Modulation

A definition of communication requires the exis-

tence of a physical medium that allows propagation

of energy from one place (an “energy source”) to

another place where that energy has some causal

effect (an “energy sink”)8. We also need to define a

means by which some property of the source can be

altered in a way that results in an observable dif-

ference at the sink after propagation through the

channel.In communications engineering theory,

the energy propagation is called “transmission,”

the source is known as a “transmitter” and the

sink as a “receiver.” These concepts are not suf-

ficient for communication. There also needs to be

an “information source” that is initially observable

8Communication can also take place from the past to

the future, in a fixed location, such as when writing to a

memory device then reading it back again at a later date,

but here we focus on place-to-place communication.

9

Page 10

at the transmitter’s location, but not at the re-

ceiver’s. Communication requires the transmitter

to alter the energy source in a manner that re-

flects the information source, and that can subse-

quently be observed at the receiver after propaga-

tion. This conversion from information source to

energy source is known as “modulation.”

A familiar example where each of these concepts

is readily identified is analog AM or FM radio

transmission, in which recorded sound signals are

communicated, and then reproduced via a speaker.

In this example, the transmission medium can be

a vacuum9or air, the propagating energy source

is electromagnetic radiation, and the transmitter

modulates the electromagnetic waves in a manner

that reflects the recorded sound signal. AM is am-

plitude modulation, and means that a single fre-

quency sinusoidal wave of E-M (electromagnetic)

radiation has its amplitude changed over time. FM

is frequency modulation, which means the ampli-

tude remains constant, while the carrier frequency

is changed over time.

3.2Neuronal Spikes and Spike In-

terval Coding

Modulation of the energy source can be thought

of as a code, since it requires a conversion from

one kind of information representation to another.

Indeed, in neuroscience, modulation has a more

general meaning than in communications engineer-

ing, and the conversion from an information source

to variations in a parameter of the energy source

is instead known as a “code.” This is largely in

contrast with communications engineering, where

“code” instead refers to conversion between differ-

ent representations of the information source prior

to transmission at the source, for example “source

coding” and “error correction coding.”

If we wish to consider communication between

neurons, we need to identify the transmission

medium, the form of energy propagation, and a

modulation mechanism. From now on we will use

neuroscience terminology, and refer to modulation

as the “code.” Further, we will refer to the infor-

mation source as the “input,” and the observable

effect at the receiver that results from the input as

the “output.”

Although over longer time scales the plasticity of

neurons can encode/carry information, in shorter

time scales the primary physical medium for com-

munication seems to be the axons of neurons, and

the energy propagation is a pulse-like wave of volt-

age that travels along an axon where it may be

received by other neurons at synaptic junctions.

These pulses are known as action potentials, or

9Vacuum is thought of a transmission medium for elec-

tromagnetic waves.

spikes. Typical cortical neurons transmit spikes to

many other neurons, and receive spikes from many

neurons.

While there are a number of different “communi-

cation channels” in neuronal circuitry — including

segments of the dendritic tree which carry post-

synaptic potentials towards the soma of the cell —

we choose to focus on action potentials because it

is one of the most important communication mech-

anisms between two neurons.

The other concept we must also attempt to iden-

tify is the way in which spikes are coded (modu-

lated) in order to communicate information. Two

possibilities are the height and the width of each

spike. However, these are observed to be close to

identical in most cases, and do not seem to be in-

formation carrying parameters. Instead, it is the

interval between spikes (ISI: inter-spike interval)

that is thought to play an important role in carry-

ing information through a neuronal channel.

Given this, how do ISIs represent information?

In neuroscience there are mainly two different

ideas. One idea is that the ISI itself (see for exam-

ple MacKay and McCulloch (1952)) carries infor-

mation. This is called “temporal coding” (Fig. 3a).

The other is that the number of spikes in a

fixed time interval (see Stein (1967); Lansky et al.

(2004)) carries information. This is called “rate

coding” (Fig. 3b).

????

t (spike interval)

(a) Temporal coding.

time

time interval (∆msec)

r (number of spikes)

time

(b) Rate coding.

Figure 3: Two forms of spike codes.

So far we have only stated that an input can be

communicated to a receiver output. If this is a

perfectly repeatable process, the rate at which in-

formation can be transmitted depends on the rate

at which the input is updated, and — in line with

Section 2 — also depends on the probability distri-

bution of the input, via its entropy. The transmis-

sion is usually not perfect, and noise is introduced.

This fact leads us to consider the information the-

oretic concepts of mutual information and channel

capacity.

Information theory is not concerned with the

type of modulation. It requires an abstraction that

specifies only what the observable output variable

should be. Since we are not designing a system, we

must make some guesses about aspects of the input

and output for a neuronal communication channel,

and then proceed to calculations of mutual infor-

mation.

Therefore, in Section 4 where we consider the

channel capacity of a neuron model, we necessarily

begin by specifically defining the input and output

10

Page 11

of the channel, and state a model for the channel

noise.

4Example:

ity of a Neuron

Channel Capac-

In this section we present some of our results on the

channel capacity for simple neuron models. One

reason for providing this example, is to illustrate

that there is no simple single formula for channel

capacity, and hence assumptions about the under-

lying model are very important. If these assump-

tions change, the channel capacity also changes.

As we have seen, channel capacity is the max-

imum amount of information that can be trans-

ferred through a noisy channel in a unit time. It

may be much larger than the actual information

transmission rate. This brings us to a natural ques-

tion, that is, why do we need to know the capacity?

Channel capacity is something similar to the

maximum speed indicated in the speedometer of

an automobile. While you will likely never drive

with that speed, the maximum speed is useful be-

cause it tells you the potential of the automobile,

even though you drive with moderate speed. Chan-

nel capacity provides not only the upper limit of

the possible information transmission rate, but also

describes how good the channel is.

Although there is much interest in the quantity

in neurophysiology (Borst and Theunissen, 1999),

theoretical work is rare (MacKay and McCulloch,

1952; Stein,1967;

Suksompong and Berger, 2010). We have obtained

some interesting results on the capacity from two

different viewpoints. The details will be given be-

low.

Johnson,2010;

4.1 Inputs and Noise of Channel

We consider here a single spiking neuron, and as-

sume that the input to the neuron controls the

expectation of the neuron’s output ISIs.

the terminology introduced above, the information

source modulates the ISI. We introduce channel

noise to the picture by assuming that the ISI is a

gamma-distributed random variable, when the in-

put to the neuron remains constant.

Biologically, each cortical neuron receives inputs

from a lot of (pre-synaptic) neurons and each sen-

sory neuron receives physical stimuli. The above

assumption is to model all the inputs to the neuron

as a single parameter θ. Although this assumption

may seem too simple, θ is a time varying function

and is able to represent a a lot of possible func-

tions. In the gamma ISI model, the expectation of

the ISI is given by κθ. Because κ is fixed, θ is the

input to the neuron.

Using

Due to refractoriness, a neuron cannot fire too

fast; therefore the ISI cannot be 0 but must be

larger than a few milliseconds. On the other hand,

if the ISI is too large, it means the neuron is not

working. Thus we assume the input to the neuron

is trying to control the ISI in a fixed range of time.

The average ISI, which depends on θ and κ, is

limited between a0and b0, that is,

a0≤ T = κθ ≤ b0, where 0 < a0< b0< ∞.

Thus, θ is bounded in Ω(κ) = {θ | a0/κ ≤ θ ≤

b0/κ}.

4.2Channel Capacity of a Single

Neuron

For a noisy channel, one important fundamental

problem is to compute the capacity C. Another

problem is to obtain the capacity achieving distri-

bution.

The family of all the possible distributions π(θ)

of inputs P is defined as

P =

?

π

??π(θ) ≥ 0 for θ ∈ Ω(κ),otherwise 0

The mutual information and the capacity depends

on the choice of an output variable.

called “coding” in computational neuroscience, but

“modulation” is an appropriate term in informa-

tion theory. Traditionally, two types of modula-

tions have been considered in computational neuro-

science. One is “temporal coding” and the other is

“rate coding” (see Fig. 3). Note that both tempo-

ral and rate coding may be used in the brain. For

example, binaural sound localisation needs phase

information and temporal coding seems natural

while rate coding is appropriate for a motor neuron

because muscles react according to the rate.

We provide some results on the capacity of tem-

poral and rate coding in the following.

?

.

This is

Temporal Coding

In temporal coding, the received information is T.

For a π ∈ P, we define the marginal distribution

as

p(t;π,κ) = Eπ(Θ)[p(t|θ;κ)].

The mutual information between T and Θ is de-

fined as

I(Θ;T) = Eπ(Θ)

?

Ep(T|θ;κ)

?

logp(t|θ;κ)

p(t;π,κ)

??

.

The capacity per spike is defined as

CT= sup

π∈PI(Θ;T).

11

Page 12

0 10 2030 4050

0

0.1

0.2

0.3

0.4

0.5

expected time interval [msec]

probability

(a) Temporal coding (κ = 3).

0 10 203040 50

0

0.1

0.2

0.3

0.4

0.5

expected time interval [msec]

probability

(b) Rate coding (κ = 3).

Figure 4: Capacity achieving distributions for tem-

poral and rate coding. For both coding types, the

optimal input distribution is discrete, with a finite

number of probability mass points.

This optimisation problem cannot be solved an-

alytically. However, it has been proven that

the capacity CT is achieved by a discrete dis-

tribution with a finite number of mass points

(see Ikeda and Manton (2009) for the details).

Since the optimal distribution is a discrete dis-

tribution with a finite number of mass points, the

optimisation problem becomes simple, and we can

compute the capacity and the capacity achieving

distribution numerically. Figure 4a shows the ca-

pacity achieving probability distribution for κ = 3.

The channel capacity CT is 34.68bps (bit ber sec-

ond) (See Ikeda and Manton (2009) for further re-

sults).

Figure 4a shows that the capacity is achieved

when the input is a discrete memoryless distribu-

tion10with 3 states. This does not imply that the

brain is using discrete states. It is more plausi-

ble that the brain is using continuous states; it is

likely that the actual information transmission rate

in the brain is less than the numerically computed

capacity.

Rate Coding

In rate coding, a time window is set and the num-

ber of spikes in this interval is counted. Let us

denote the interval and the rate as ∆ and R,

respectively, and define the distribution of R as

p(r|θ;κ,∆). The form of the distribution of R is

shown in Ikeda and Manton (2009). For π ∈ P, let

us define the following marginal distribution

p(r;π,κ,∆) = Eπ(Θ)[p(r|θ;κ,∆)].

The mutual information of R and θ is defined as

I(Θ;R) = Eπ(Θ)

?

Ep(R|θ;κ,∆)

?

logp(r|θ;κ,∆)

p(r;π,κ,∆)

??

.

Hence, the capacity per channel use or equivalently

per ∆ is defined as

CR= sup

π∈PI(Θ;R).

10The input is chosen from the three states independently

at each time according to the probability distribution shown

in Fig. 4a.

This optimisation problem cannot be solved

analytically either, but the capacity CR has

been proven to be achievable by a discrete

distributionwitha finite

points (Ikeda and Manton, 2009). Figure 4b shows

the capacity achieving distribution for κ = 3.

The channel capacity CRis 44.95 bits per second

(See Ikeda and Manton (2009) for further results).

number ofmass

4.3Tuning Curves

The definition of channel capacity requires a max-

imisation over all possible input probability distri-

butions. This definition arose in an engineering

context, where a system designer is assumed to

have control over the inputs to the channel, but not

the channel itself. A different optimisation prob-

lem results if the input to the channel is assumed to

be fixed, but some control over the channel is pos-

sible. This idea is particularly relevant for studies

of biological sensory transduction. In this context,

an external stimulus that cannot be controlled by

the sensing organism must be transduced and en-

coded into action potentials for communication to

the brain. This stimulus can be thought of as an

input to a communication channel.

Given internal noise in the transduction mecha-

nisms, the encoded stimulus received by the brain

is also noisy. Since we introduced mutual infor-

mation in the context of the channel coding the-

orem and digital data, and here our channel in-

put is a sensory stimulus, mutual information may

not seem relevant. However there are other rea-

sons why it can be useful to ensure mutual in-

formation is as large as possible (Berger, 2003;

Johnson and Goodman, 2008) and we therefore are

interested in how the channel might be altered to

maximise mutual information.

But what can be optimised when the stimulus

cannot be controlled? The only other variable that

can alter the mutual information is the conditional

distribution of the channel output given its input.

In the biological context this is the distribution of

a neural response for a given stimulus. Optimising

this distribution means seeking to find the com-

munication channel that it best suited to a fixed

stimulus distribution. Without some constraints

on the form of the conditional distribution, this

would not be a meaningful task. One such con-

straint is to consider a fixed form for the condi-

tional distribution that has some parameters that

can be optimised. Clearly the optimal parameter

set may change for different input distributions.

One reason for considering such an optimisation

might be to assess whether neuronal mechanisms

exist for adaptively altering the conditional distri-

bution to match non-stationary stimuli. Another

equally intriguing reason is the idea that evolu-

12

Page 13

tion might have enabled neural systems to change

parameters over eons with the end result that

those parameters are information theoretically op-

timal. In this scenario, the underlying fixed form

of the probability distribution must be what it is

due to unavoidable constraints, or perhaps gov-

erned by criteria that are not information theo-

retic, e.g. energy considerations (Laughlin et al.,

1998; Sarpeshkar, 1998; Levy and Baxter, 2002;

Laughlin and Sejnowski, 2003; Berger and Levy,

2010).

There are several potentially important sets of

parameters that might be chosen.

other contexts there has been much interest in

determining the optimal form of neuronal tuning

curves, and this is our sole focus here.

mentally produced tuning curves are plots of the

mean response of a neural system, as a function

of a stimulus parameter (Dayan and Abbott, 2001;

Lansky et al., 2008; McDonnell and Stocks, 2008).

Classic examples of a stimulus parameter include

the angle of a moving bar of light relative to the

receptive field of visual cells, or the sound pres-

sure level of a single frequency (pure tone) sound

played by a speaker. In these examples the av-

erage response as a function of the stimulus de-

fines the tuning curve. The former kind of tun-

ing curve typically has a bell-shape, meaning that

there is a stimulus that produces a maximal re-

sponse, while more than one stimulus can produce

the same lesser response. The latter kind has a

sigmoidal shape, meaning that the mean firing rate

monotonically increases with stimulus, and here we

focus only on this case.

We therefore wish to find the sigmoidal tun-

ing curve that maximises mutual information, for

a given fixed form of a conditional distribution.

While this is generally a difficult optimisation

problem, a simple solution exists for channels

where the capacity achieving input distribution is

discrete, like those considered in this paper. In

fact, the mutual information maximising tuning

curve can be derived for an arbitrary stimulus dis-

tribution, if the capacity achieving input distribu-

tion has been calculated first. The reason for this

is explained in the following.

We consider the same noisy channel as Sec-

tion 4.2, that is the ISIs are governed by a gamma

distribution. We now make a slight generalisation

to the setup of Sections 4.1 and 4.2 and consider

the expectation of the ISIs to be governed by a ran-

dom variable X with a known distribution, such

that Θ is an arbitrary function of X.

We therefore write Θ = f(X). For the gamma

ISI channel, the tuning curve is defined as the con-

ditional expectation of the response variable (ei-

ther T or R) given a specified outcome of X. For

timing coding or rate coding respectively, we write

However, in

Experi-

these expectations as

Ep(T|x;κ)[t] = κf(x)

and

Ep(R|x;κ,∆)[r] =

∆

κf(x).

Since Θ = f(X), when the capacity achieving dis-

tribution for Θ is discrete with say M states, the

tuning curve will also consist of M unique values.

Lets call these µ1,..,µM. While this discontinuous

tuning curve achieves the largest possible mutual

information for the channel, it is not unique; other

tuning curves are equally good. Suppose X is a

continuous random variable, and that the tuning

curve maps large intervals of X to the same M

values µ1, ..., µM. This tuning curve provides the

same M possibilities for the conditional expected

ISI. In order for it to provide the same mutual in-

formation achieved by the original discrete capac-

ity achieving distribution, it is necessary that the

probability with which each µmoccurs is the same

in both cases. It has been proven for a special case

of the gamma distribution and rate coding that

this can be achieved by appropriate choice for the

ranges of X that are mapped to each µm. The

resulting optimal discrete tuning curve is then de-

pendent only on the probability density function

of X (Nikitin et al., 2009).

Consequently, the capacity achieving input dis-

tributions derived in Section 4.3 can be converted

to an optimal tuning curve for any choice of the

stimulus distribution. An example of the capacity

achieving tuning curve is shown in Figure 5, for

the special case of κ = 1, which means the chan-

nel is equivalent to a Poisson neuron (Nikitin et al.,

2009). The maximum rate is restricted to 30 spikes

per input sample.

Although such a result holds exactly only for

discrete input distributions, similar derivations of

information theoretically optimal continuous tun-

ing curves have been made, which hold only in

the low noise limit (McDonnell and Stocks, 2008).

See Kostal and Lansky (2010); Kostal (2010) for

related work in the high noise limit.

4.4Discussion and Interpretation

We have shown our results on neuron channel ca-

pacity from two very different viewpoints. Inter-

estingly, both show that the capacity is achieved

by a discrete distribution. The numerically com-

puted capacities are similar to the range indicated

by some biologically measured results of sensory

neurons (Borst and Theunissen, 1999). The chan-

nel capacity depends on various factors, and we

consider some of them below.

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0 0.2 0.40.60.81

0

5

10

15

20

25

30

35

Input stimulus, x

Mean response, Ep(R|x;κ,∆)[r]

← µ1= 7.56

← µ2= 10.80

← µ3= 30.00

Figure 5: Channel capacity achieving tuning curve

for a Poisson rate-coding neuron (κ = 1), and an

input x ∈ [0,1], with a maximum spike-rate of 30

spikes per input sample—see Nikitin et al. (2009)

for further examples.

4.5 Input and Output

We first discuss the input and output of the neuron

channel in this subsection.

Let us start with the input. Although each neu-

ron receives information from many neurons, we

have only considered a single input θ. This may

seem too simple. We assumed that the single input

θ summarises all the inputs to the neuron. More-

over, θ has been assumed to be memoryless and can

have any distribution within the support. Consid-

ering the biological system, this is far from real-

istic. The net input of a neuron may not change

quickly, that is, it has memory. Moreover, a neu-

ron’s input is a collection of many neurons’ noisy

outputs, therefore, it may follow a particular prob-

ability distribution.This implies that we have

computed capacity under less restrictive assump-

tions, and the biologically achievable rate should

be smaller than the capacity obtained in the nu-

merical studies.A better understanding of the

constraints on the input of a neuron would lead

to a more accurate calculation of neuronal channel

capacity.

Next, we discuss the output of a neuron from

two viewpoints, decoding and demodulation.

In order to achieve channel capacity, the re-

ceiver must act as an optimal decoder, meaning

that when˜ X is observed, the receiver must com-

pute the posterior distribution of the input X as

p(x|˜ x). When x takes discrete values x1,··· ,xL,

it becomes p(xi|˜ x). This is a real number for each

xi. In engineering, this type of decoding is called

“soft decoding.” It seems unlikely that a neuronal

mechanisms for carrying this out could exist, since

a computation of the posterior distribution is nec-

essary.

Another standard decoding technique is “hard

decoding,” that is, only a single value of xiis cho-

sen by the decoder.The optimal hard decoder

chooses the one which maximises the posterior dis-

tribution, that is ˆ xi= argmaxxip(xi|˜ x). It is pos-

sible to implement the optimal hard decoder with-

out requiring the online computation of the pos-

terior distribution. Indeed, the output space (the

space where˜ X lies) can be divided into L sub-

spaces in advance, such that the k th subspace

contains all the points ˜ X such that xk has the

largest posterior probability given˜ X. Therefore,

the optimal hard decoder can be implemented by a

“quantisation” or “thresholding” algorithm which

simply checks to see which subspace˜ X lies in. Such

an algorithm is often computationally simpler than

computing first the posterior distribution.

When we consider information processing in the

brain, a naive soft decoding seems difficult, at

least for a single neuron.

idea seems more natural. However, the informa-

tion transmission rate of the best hard decoding is

less than the best soft decoding. We should keep

this point in mind. Further discussion is found

in Ikeda and Manton (2009).

The hard decoding

4.6Discreteness

Under the assumptions made, we have shown that

the capacity of a neuron is achieved by a dis-

crete distribution with a finite number of proba-

bility mass points. In information theory, there

are many other known types of channels for which

channel capacity is achieved by a discrete distri-

bution (Huang and Meyn, 2005).

although the capacity of an average power con-

strained additive Gaussian white noise channels is

achieved by continuously valued Gaussian inputs,

simply placing a constraint on the maximum am-

plitude of the channel means capacity is achieved

by a discrete distribution (Smith, 1971).

Our results do not imply that “neuron signals

are only using discrete levels.” On the contrary,

we believe many neurons are using continuous lev-

els. What is implied by our results is that those

neurons cannot achieve the capacity and the ac-

tual rate of information transfer is therefore less

than the capacity. Another implication is for the

measurement of information capacity in neuro-

science (Borst and Theunissen, 1999). Our result

implies that only a small number of discrete ranges

are sufficient for the input distribution to measure

the information capacity of neural coding.

Information theory provides a way of quantifying

whether discrete or continuously distributed sig-

nals are better for any given communication chan-

nel corrupted by random fluctuations. Many neu-

roscience studies quantify information using Shan-

non’s famous information capacity formula relat-

ing mutual information to signal-to-noise ratio.

This formula is correct only for Gaussian additive

noise channels (Cover and Thomas, 2006) — it re-

lies on many assumptions, and if any are false it

For example,

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can significantly under- or over-estimate the true

capacity (Berger and Gibson, 1998).

4.7Further Points

Guessing the channel model is not straightforward.

In particular, feedback is prevalent in many parts

of the brain, which makes it much more diffi-

cult to relate changes in responses to inputs11.

One important example where it is known that

the circuitry is solely feedforward is that of reti-

nal cells (Jacobs et al., 2009). This example has

been used to demonstrate that it is possible to

rule out certain guesses of neural codes. Because

this is based on optimal Bayesian decoding of a

forced binary choice, it would be interesting to ex-

tend (Jacobs et al., 2009) beyond the binary limi-

tation to that where the signals being coded may

have many possibilities

What can we say about more complicated sit-

uations? For example, information processing of

cortical neurons are not strictly feedforward and

the information is shared by many neurons. If we

assume every neuron is performing the same com-

putation, and each neuron encodes and decodes

information in the same way, it seems possible to

extend our results. However, when different neu-

rons encode information differently, the problem

becomes very different.

brain works in information theoretic terms is one

of the grand challenges of this century.

Understanding how the

5 Conclusions and Future Di-

rections

Most information theory research to date has

been predicated on an engineering viewpoint.

The main thrust has been to design compres-

sion/decompression methods that compress source

information to the limits given by its entropy

(source coding), or to design error correction

schemes and encoding/decoding methods that al-

low communication close to capacity.

On the other hand, the goal of neuroscience is in-

stead to understand information processing in the

brain.

This does not mean that information theory has

no place in computational neuroscience. Informa-

tion theory provides a way to measure informa-

tion and to understand the limits of compression

(namely, entropy) and communication (namely,

mutual information and channel capacity).

A common “information theoretic” method in

computational neuroscience is to obtain quantita-

tive estimates of the mutual information between

observed sets of data. Since various methods for

11Channels with feedback can have larger capacity.

the (difficult) problem of accurately estimating

mutual information exist, the bigger difficulty is

with using the result to say something about how

the system works. Indeed, the actual goal of com-

putational neuroscience is that of “system identifi-

cation,” as engineers would call it.

If the brain uses spikes to transmit information

(which on faster time-scales appears to be the case)

then understanding the neural code — how the

brain encodes information before sending it across

the “channel” — is tantamount to understanding

how the brain works. Indeed, if we would listen in

to the messages as they are sent from one neuron

to another, it would be relatively straightforward

to determine what each neuron is doing.

Although system identification is not the forte

of traditional information theory (Johnson, 2008),

this is merely for historical reasons. With compu-

tational neuroscience as a main motivator, we pre-

dict that the next decade will see the expansion of

information theory to include more powerful tech-

niques for system identification, and possibly even

an integration of control, computation and infor-

mation theory into a unified framework.

Some recent information theoretic approaches

in neuroscience that go beyond standard Shannon

theory are summarized in the following list.

• While it is traditional in engineering to sepa-

rately code an information source, and then

channel code it for communication across

a channel, it has been shown for some

simple but instructive examples, that non-

separation of these two aspects can achieve

an optimal communication system with a

vastly reduced complexity compared to sep-

aration (Gastpar et al., 2003).

potentially important for neurobiological sys-

tems, where separation mechanisms seem im-

plausible.

This fact is

• Many studies have investigated whether the

brainmight

mechanisms for implementing Bayesian algo-

rithms during decision making, prediction and

pattern recognition (Rao and Ballard, 1999;

S.Lee and Mumford, 2003; Knill and Pouget,

2004; George and Hawkins, 2009).

have

• The possibility of analog cortical error correct-

ing codes has been proposed by Fiete et al.

(2008).

• One limitation of Shannon theory is that

its measures say nothing about directional-

ity or causality. However, directed informa-

tion theory has also been developed (Granger,

1969; Marko, 1973; Rissanen and Wax, 1987;

Massey, 1990; Tatikonda and Mitter, 2009)

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