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500

0

0 1 2 3 4 5500

Time (s)

Time (s)

500

0

0 1 2 3 4 5500

Time (s)

Local field potential

0 1 2 3 4 550

50

100

0

Spikes

Low-pass filter

High-pass filter

50

0

50

100

150

200

0 20 40 60

50

0

50

100

150

200

0 20 40 60

121 spikes

50

0

50

100

150

200

0 20 40 60

50

0

50

100

150

200

0 20 40 60

42 spikes16,273 spikes 269 spikes

V

oltage

(V)

Voltage

(V)

Data point

Information theory

A mathmatical thoy that

dal with mau of

infomation and thi

application to th tudy of

communication ytm. In

nuocinc it i ud totablih th amount of

infomation about a timulu o

bhaviou that i containd in

th nual pon.

Local field potential

(LFP). A nuophyiological

ignal that i obtaind by

low-pa filting xtacllula

coding. It pnt th

man fild potntial gnatd

by th low componnt of

ynaptic and nual vnt in

th vicinity of th coding

lctod.

cay idicat whth a app a ag is big

s, bt this ctaity might b sd wh thactiity f th s is csidd. I fact, th pp-ati aaysis basd dcdig ifmati thyca a hw th th s s this ambig-ity f xamp, by cdiatig thi fiig t tagpaticay sait ts22,23 by haig ach pst a patica stims fat2426. Scd,pstsyaptic a systms mst say itpta spss btaid i y tia. Bth thifmati-thtic ad th dcdig appachsqatify stims kwdg btaid with th bsa-ti f sig-tia ppati spss, ths pidiga famwk that is cmpatib with th stict timscas

f i bai pcssig. Thid, th stims fats

cdd by th spik tais ca b discd by assss-ig whth th ppati sps ca discimiatdifft stimi ctaiig a patica fat. Fth,it is pssib t systmaticay aat hw difftfats f th spik tais affct th pfmac f adcdig agithm th amt f xtactd if-mati. Fifth, th ifmati gi by difft mas-s f a actiity, sch as spik tais ad localfild potntial (lFPs), ca b aaysd ad cmbid.Athgh ths tw sigas ha y difft chaac-tistics ad siga-t-is atis, ifmati thyad dcdig agithms aw a dict cmpaisbtw lFPs ad spiks bcas thy pjct th tw

Box 1 | Extracellular recordings

Extracellular recordings are usually performed by inserting microwires into the brain1. After amplification, the signal is

low-pass filtered to obtain the local field potential the mean field potential generated by neurons in the vicinity of the

electrode and high-pass filtered to identify the activity of single neurons using spike detection and sorting algorithms.

The example shown in the figure corresponds to a recording of approximately half an hour in the left hippocampus of an

epileptic patient24,44, of which 5 s of continuous data are shown. After high-pass filtering, the firing of nearby neuronsappears as spikes on top of background activity. Spikes are detected using an amplitude threshold (represented by the red

horizontal line). Features of the spike shapes are extracted and the spikes are sorted accordingly. For neurons located

approximately 50100 m from the electrode tip4,122, the signal-to-noise ratio is good enough to distinguish the activity ofeach single unit (inner circle; spikes in red, green and cyan). For more distant neurons, up to approximately 150m from thetip (outer circle), spikes can be detected but the difference in their shapes is masked by the noise and they are grouped

together in a multi-unit cluster (spikes in blue). Spikes from neurons further away from the tip (shown in light grey in the

schematic) cannot be detected and contribute to the background noise.

There are several issues that make spike sorting challenging18. In particular, some neurons fire very sparsely for

example, the neuron shown in cyan in the figure fired only 42 spikes in approximately half an hour, a mean firing rate of less

than 0.05 Hz. These neurons are usually hard to detect. Interestingly, such sparsely firing neurons showed the most

selective and interesting responses in human recordings24.

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Decoding(apple)

Informationtheory (n bits)

1 2 3

Posterior probability

Th potio pobability of a

andom vaiabl i th

conditional pobability

aignd to th vaiabl givn

om vnt. Fo xampl, th

potio pobability P(s|r) i

th conditional pobability that

timulu s wa pntd, givnthat a pon rwa

obvd.

Shannon entropy

A mau of th unctainty

about th valu that might b

takn by a andom vaiabl.

Bit

Th unit ud to mau

duction of unctainty. On

bit copond to a duction

of unctainty by a facto of

two (fo xampl, a coct

anw to a y/no qution).

sigas t a cmm sca. It th bcms pssibt assss whth th lFP adds sm kwdg abtth stims that cat b btaid fm spiks aad ic sa.

Decoding algorithms. Dcdig is th pdicti fwhich stims bhai icits a patica -a sps i a sig tia. A xamp is gi byBaysia dcdig. lt P(s) dt th pbabiity fpstati f stims s (bgig t a st S)adP(r|s)dt th cditia pbabiity f btaiig a ppa-ti sps r(t f a sps st R) wh stims sis pstd. usig Bays thm, w btai27:

P(s|r) =P(r|s) P(s)

P(r) (1)

with

P(r) = sP(r|s) P(s) (2)

eqati 1 gis th potio pobability that, gi asps r, stims s was pstd. Baysia dcdigcacats fm this psti pbabiity distib-ti a sig pdicti f th mst iky stims (sP)(reFs 13,2830) f xamp, by takig: sP = ag max

s

(P(s|r)).Bsids th Baysia appach, th a sa

th mthds t dcd th stims i a gi tia.A thgh discssi f ths dcdig agithmshas b pidd swh10,11,13,14,31, ad a shtdscipti f th mst cmm mthds is gi iBOX 2. Th impmtati f dcdig agithms isistatd i FIG. 2.

T aidat dcdig sts, sm tias ca b sdt ptimiz th dcd (th taiig st) ad th st ttst its pfmac, a pcd cad css-aidati32.It is imptat that tias bgig t th taiig st at sd t aat th dcdig pfmac bcas

this may ad t atificiay high as wig t fit-tig33. Fthm, bth th taiig ad th tstigsts shd b ag gh t aid dstimatig thdcdig pfmac wig t p ptimizati fth dcd i th fist cas ad w statistics f tstigi th att.A cmm pcd is th a--t

aidati, i which ach tia is pdictd basd thdistibti f a th th tias. This has th ada-tag that bth ptimizati ad tstig a basd thagst pssib mb f tias33.

Dcdig sts a say pstd i th fmf cfsi matics (FIG. 2c). Th as a giw i ad cmj f a cfsi matix pst th(maizd) mb f tims that a pstati fstims i is pdictd by th dcd t b stimsj.If th dcdig is pfct, th cfsi matix shdha tis qa t ag th diaga ad zywh s. F qipbab stimi, pfmacat chac s shd b fctd by a matix i whichach ty has qa pbabiity1/K(with Kbig thmb f stimi).

Shannon information theory. Ath pwf way tstdy th actiity f a ppatis is t cacat

th ifmati abt a gi stims bhaictaid i th a spss sig th fma-ism f Sha ifmati thy. As bf, sppsthat a stims s bgig t a st S is pstd with apbabiityP(s). Th shannon ntopyH(S) f th disti-bti f pbabiitis P(s) f ach stims is dfidas15,16,27,34:

H(S) = P(s)log2P(s)

s (3)

This qatifis th ctaity abt which stimsis pstd , csy, th aag amt f if-mati gaid with ach stims pstati. etpyis masd i bit if th gaithm is tak with bas 2

Figure 1 | T man t t atn anay na dng. The common steps for analysing how a

population of neurons encodes information about visual inputs are shown. First, recordings are taken at different sites

with implanted electrodes. Second, the simulated activity of single neurons is extracted from the continuous data using

spike-sorting algorithms. Third, information is inferred from the multiple spike trains with decoding algorithms (which can

predict that the stimulus was an apple), or information theory (which quantifies the knowledge about the stimulus gained

by observing the population response). The population analysis allows the study of the information carried by the different

features of the multiple spike trains. For example, it can be established whether the information of the apple is given by an

increase in firing (neuron in red), by a particular temporal firing pattern (neuron in green) or by the simultaneous firing of a

subset of neurons (neurons in blue and grey). The vertical dotted line marks stimulus onset.

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Unbiased decoder

A dcod i aid to b

unbiad if th xpctd valu

of it dcoding o (th

diffnc btwn th tu

and th timatd timulu

valu) i zo.

(as i qatis 35 ad 1012). Th Sha tpyis z wh th sam stims is pstd ach tim,ad is psiti thwis. It achs a maximm fH(S) = log

2Kwh th pbabiity f pstig ach

fKdifft stimi is qa.If th a ppati sps r ctais

ifmati abt th stims, th its bsatiwi dc th stims ctaity f qati 3. Thsida ctaity abt th stims aft bsigth a sps is cad qicati ad it isgi by th wightd aag tpy f th pstidistibti P(s|r) (s qati 1):

H(S|R) = P(r) P(s|r) log2P(s|r)

s,r

(4)

Fm this, mta ifmati is dfid as thdcti f ctaity ( gaid ifmati) abt

th stims btaid by kwig th a sps.It is gi by th diffc btw th stims tpyH(S) ad th qicati H(S|R):

I(S;R) =P(r)P(s|r)log2s,r

s,r

P(s|r)

P(s)

P(s,r)

P(s) P(r)=P(s,r)log2

(5)

P(s,r) dts th jit pbabiity f bsigth sps R = rtgth with th stims S = s.Ths pbabiitis a t kw a priori bt thy

ca b stimatd mpiicay. F xamp, th ag fspss f ach ca b patitid itMbis,ad th fqcy f bsig a sps i ach biis cmptd (f iws f th stimati mthds,s reFs 35,36). Ifmati, ik tpy, is masdi bits. ey bit f ifmati pidd by th sdcs th a ctaity abt th stims by afact f tw. If th stimi ad th spss a id-pdt, th mta ifmati qas z. othwisit taks psiti as. Pfct kwdg abt thstims fm th a actiity , i th wds,a ss stims cstcti gis a maximmmta ifmati fI(S;R) = H(S).

Atatiy, th kwdg abt th stims c-taid i th a spss ca b masd sigFish ifmati34:

FI= P(r|s) d lnP(r|s)r

2

ds. (6)

Th is f Fish ifmati is a w bdt th ma sqa dcdig btaid with ay

unbiad dcod. Fish ifmati ca b sd ywith ctis stimi ad ths its appicati txpimta scic has maid m imitdtha that f Sha ifmati.

Complementarities of decoding and information theory.Bth dcdig ad ifmati thy xtact qati-tati ifmati fm th ppati spss, byqatifyig th kwdg abt th stims that isgaid fm bsig a a ppati sps a sig tia. Hw, thy ach qatify a diff-t aspct f this kwdg. Dcdig agithms p-dict th stims that casd th sig-tia aspss, ad thi pfmac is typicay masdby th pctag f cct pdictis (BOX 2). Bt with a ptima dcd, th amt f xtactdifmati may b ss tha th ifmati aaiabi th a spss. Ifmati thy qatifisth a kwdg abt th pstd stims thatis gaid with th sig-tia a spss. Thisdisticti is imptat bcas s ca cyifmati by mas th tha jst tig which is thmst iky stims3739: f xamp, thy ca pidifmati abt th stimi that a iky(FIG. 3).Th cmpmtaitis btw dcdig ad if-mati thy a xpicit wh csidig Baysiadcds, bcas i this cas bth dcdig ad if-

mati thy a jst tw difft cmptatis th psti pbabiityP(s|r) f qati 1: Baysiadcds gi th mst iky stims (ag max P(s|r)),whas ifmati thy gis a smth itgatif ifmati th wh psti pbabiityP(s|r). Th difft amts f ifmati xtactdby dcdig agithms ad ifmati thy adtaid i BOX 3.

oy a fw stdis ha qatifid th ifmatithat is st wh sig dcdig agithms. o f thmaatd th ifmati that cd b gaid fm thppati actiity f had-dicti cs (s thatcd th dicti f th aimas had) i th pimat

Box 2 | Decoding algorithms

Several decoding algorithms have been developed. Nearest-neighbour decoders

assign a given trial to the class of its nearest neighbour31,61. Fisher linear discriminant

algorithms introduce a dimensionality reduction (by projecting the original space

where the classification is performed on to a line with a direction that maximizes the

ratio of between-class to within-class distances) that optimally separates the samples

of each class31,44. Support vector machines project the data into a high-dimensional

space in which it is possible to find a hyperplane (a high-dimensional plane) thatoptimally separates the data123. This is particularly useful if points corresponding to two

given classes are separated by a nonlinear region. Bayesian decoders assign a given

trial to the class that minimizes the probability of an error, given the prior probabilities

of each class and the posterior probabilities (see equation 1)11,28,29. Decoding can also

be performed by training an artificial neural network65.

Decoding performances are usually quantified by the relative number of hits (that is,

the average of the diagonal in the confusion matrix). As the outcomes of the

predictions of each stimulus can be regarded as a sequence of Bernoulli trials

(independent trials with two possible outcomes: success and failure), the probability of

successes in a sequence of trials follows the Binomial distribution124. Given a probability

of getting a hit by chance ( = 1/K, in which Kis the number of stimuli), the probability of

getting k hits by chance in n trials is given by

(7)

, where

(8)

is the number of possible ways of having k hits in n trials. From this it is possible to

assess statistical significance and calculate a -value by adding up the probabilities

of getting k or more hits by chance44:

(9)

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5

10

15

20

30

25

5 10 15 20 25 30

1.0

0.5

0

P

icture

presented

Picture predicted

c

20

10

0

0 20 40

b

a

Neuron 1

Neuron 2

Firing

neuron

2

Firing neuron 1

ad ptd that a Baysia dcdig agithm p-idd ~95% f th tta ifmati aaiab fm tha spss40. This ifmati ss was highwith pdictis sig th ppati ct mthd(s bw)41, i agmt with pis fidigs shw-ig that Baysia dcds pfmd btt tha thppati ct13,28,29,42.

Th is idc that aimas stimat psti dis-tibtis P(s|r) f stims aiabs43; hw, th fiabhaia tcm f mst a cacatis is adcisi abt what is th mst pbab stims thmst apppiat bhai. Ths, dcdig (idtify-ig th mst iky stimi) is w sitd f cmpaiga ad bhaia pfmac. Ifmatithy has th adatag f smmaizig a th ifma-ti ctaid i th ppati spss it a sig,maigf mb. Th mta ifmati btwth stimi ad a spss gis a pp bdt th amt f kwdg that ca b pidd by adcdig agithm. This maks ifmati thya stg t with which t aat th cmptatia

capabiitis f a cds, as it ca csid ways ftasmittig ifmati that might t b ad bydcdig agithms.

A staightfwad way t ik ifmati thyad dcdig is t cmpt th mta ifmatibtw th acta ad th pdictd stimi fm thdcdig tcms that is, th mta ifmatiI(S;SP) btw th ws ad cms f th cfsimatix15,34,40,4446(BOX 3). This ca gi m ifmatitha jst th ati mb f hits (cct pdictis(BOX 2)). F a gi mb f hits m ifmatica b btaid if icct pdictis a cc-tatd it csts ad th cct stims38,47. Fxamp, th ati mb f hits i FIG. 2 ts sthat th pstatis f pict 29 w ft wgydcdd, bt t that thy w maiy cfsd withpict 26 ad t jst pdictd at adm, as wdha ccd if th s had ifmati abtth stims44. Th cmbid aaysis f dcdig adifmati thy ca a sch systmatic s ith dcd.

It has b shw that ppatis f s cat y cmpt th mst iky stims th pti-ma sps bt as stimat pbabiity distib-tis4850. Th cmbiati f ifmati thy addcdig cd gi a aab mas t caify hwppatis f ssy s ca pt th pdictd

stims ad th at ifmati, sch as thctaity f ths pdictis5153. This ca b achidby cmpaig th ifmati btw stimi ad -a spss, I(S;R), with th ifmati btaidfm th cfsi matix aft dcdig, I(S;SP). Thdiffc btw ths tw qatitis gis th amtf ifmati aaiab i th a spss thatcd b gaid by mas th tha dcdig th mstiky stims. It is pssib t xtd this cacatiby cmptig f ach tia th pdictd stims adath spcific aspct Uf its ctaity (f xamp,th aiac f th pdicti, th ati ikihdf th bst ad scd bst stimi) ad th aat

Figure 2 | Ddng anay. The decoding analysis is

illustrated with the prediction of which picture presentation

elicited single-cell responses recorded from human subjects

implanted with intracranial electrodes for clinical reasons.

a | For simplicity, the responses of two out of a population of

Nneurons to two out ofKstimuli (a picture of a spider and a

picture of the Tower of Pisa) are shown. b | Each trial is

represented as a point in an N-dimensional space. Trials

in which the spider was shown are marked in red; those in

which the Tower of Pisa was shown are marked in blue. Anew trial to be decoded (shown in grey) can, for example, be

assigned to the class of its nearest neighbour. | The

outcome of the decoding algorithm, in the form of a

confusion matrix, for all 32 pictures that elicited responses

in this particular recording session. The colour code shows

the relative number of times that a presentation of picture i

(in theyaxis) was predicted to be picturej (in thexaxis). A

perfect decoding should have all entries in the diagonal. In

this case, the overall percentage of hits the average along

the diagonal was 35.4%, which was significantly better

than chance (1/32 = 3.1%) with p

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0 50 100 150 200

Poststimulus time (ms)

0 20 40 60 80 100

r1

r2

Pear AppleBanana

Spike rate (Hz)

Probability

Window 1 Window 2c

Trial

b

a

Spiketimes

Binnedspikes

Spikecounts

0 1 1

1 1 0

2

2

whth th ifmati gi by th cmbiati fth stims pdicti ad its ctaity, I(S;SPU), isgat tha I(S;SP) ad hw mch it accts f a thifmati I(S;R) caid by th a sps.

Limitations of decoding and information theory. Thmai imitati f dcdig is that it ds t c-sid a th pttia ways t tasmit ifmati.M, dcdig agithms may fai t dcd stim-i wig t a high-dimsia sps spac ths f icct assmptis abt th cdd data.

I sch cicmstacs it may thf b dagst t a cadidat a cd y bcas itgis a a-chac pfmac with a gi dcdigagithm.

o disadatag f ifmati-thtic massis that high ifmati as i a a cd mightt b bigicay at bcas a systmsmight t b capab f xpitig a f this ifmati.This iss ca b addssd by aaysig th pfm-ac f dcdig agithms that icpat sm fth imitatis f th pstsyaptic a cicity.

Ath pbm f ifmati-thtic aayss(ad, t a ss dg, f dcdig agithms) is that

th stimssps pbabiitis mst b stimatdmpiicay. This ca as ad t imitd sampigbiass that is, pcd systmatic s casdby imitd amts f data36,54. Takig th spik ctsfm a ppati fNs ad patitiig thsps f ach it Mbis gis MN ps-sib as. Th cadiaity f th st f a pssibspss R icass xptiay with th mbf s ad bcms ag if w as c-sid tmpa fiig patts. I a xpimts, Rmst b sampd i a fiit mb f tias ad t a

pssib sps tcms ca b sfficity sam-pd. Dspit ct pgss t cct f sch biass(by sig agithms that aat ad m thssystmatic s), wh csidig simtasycdd s th miimm mb f tias pstims that is dd t btai a biasd ifma-ti cacati appximaty qas th cadiaityf th sps st R36. This dimsiaity cs p-

ts th appicati f ifmati thy t yag ppatis. As dcdig agithms a mdata-bst ad ca b asiy appid t cdigs fag ppatis, a pactica way t appximat thifmati caid by ag a ppatis is t

Figure 3 | s nmatn . a | A first source of information loss in the analysis of spike trains is given by a

simplification of the responses for example, when the original temporal precision is reduced (left), when the spikes are

binned in time (centre) or when the total spike counts are used (right). b | The use of decoding algorithms gives another

source of information loss, because decoders miss out information about unlikely stimuli. Given the response distributions

P(r|s) of a (simulated) visual neuron responding to apple, pear and banana stimuli, when the neuron fires a responser1

just

above the average response to pear, a pear will be decoded.However, this particular neural response not only informs us

that the pear is the most likely stimulus, it also informs us that a banana is very unlikely. Similarly, when the neuron fires a

response r2

just below the average response to pears, a decoder will again predict that a pear was presented, but it will

miss the information that it is very unlikely that an apple was presented. | More generally, information is lost when

mistakes about the true neural response probabilities are made. Shown are the spike rasters of a simulated single neuron

to the three different stimuli. If the observer has to decode the responses in time window 1, but does not know the precise

post-stimulus time at which the considered responses were emitted, it might set the decoder using the wrong response

probabilities (for example, those corresponding to window 2 rather than window 1). The stimulus reconstruction will then

be flawed and information will be lost.

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cmpt th mta ifmati fm th cfsimatics. This dcs th dimsiaity fR fmMN(th st f a pssib spss fNs) t K(th mb f stimi). Hw, this mthd it-dcs pttiay s ifmati sss (BOX 3) adits sts shd b itptd with ca.

Practical applications of population analyses

A piig stdy41 dfid a ppati ct thwightd sm f th pfd tig f ach t shw that athgh s i th pimat mtctx a bady td t mmt dictis, thactiity f a ppati f ths s cd b sdt accaty pdict th dicti f am mmts.Sbsqt stdis ha shw that, i difft systmsad cditis, it is pssib t if ifmati abta stims bhai sig ppati cts thm cmphsi dcdig appachs dscibdi th pis sctis10,14. W dscib a fw f ths

stdis, which istat hw m ifmati ca bbtaid fm a ppati aaysis tha fm th stdyf sig cs i isati.

Attention or intention in the posterior parietal cortex?eidc has shw that s i th psti pai-ta ctx a id i mmt paig55. rachad day saccad tasks (FIG. 4) shwd that s ith ata itapaita aa (lIP) a id i sac-cad pas5658 ad that s i th paita achgi (Prr) a id i paig am achs55,58.Hw, it has as b agd that ths s at id i mmt paig ad a xcsiydi by attti t th tagt cati59,60, ad thatchags i th spss t achs ad saccads may bd t th difft attti ads f ach task.

This ctsy is diffict t s bcas thifmati gi by sig s is ambigs55,59.Hw, a dcdig aaysis with a ppati f Prrad lIP s ad that s i ths aasdid cd mmt pas. lIP s pdictd

saccads sigificaty btt tha achs, whas -s i th Prr pdictd achs sigificaty btttha saccads61. Fthm, pdictis f mmtittis w sigificaty btt tha pdictis fth cs f attti. I cas was a ach cfsdwith a saccad ic sa, ad pdictis f bthachs ad saccads w ay pfct. oth std-is ga fth idc f th pssibiity f pdict-ig mmt pas fm th actiity f s i thpsti paita ctx6264. Ths pdictis mightb sf i dpig psthtic dics f paayzdpatits ampts3,6567.

Odour identity and concentration in the locust.Fthidc f th a f th ppati aaysis cmsfm ath stdy that xamid whth d id-tity ad itsity cd b dtmid fm th fiigpatt f a ppati f ata b pjctis i th cst68. Sig-c sts w ic-csi bcas difft d cctatis chagdth fiig patts f ths s. Hw, wh thwh a ppati was csidd it was ps-sib t distag this ambigity ad stabish stimsidtity f difft cctatis.

Extracting the stimulus features encoded by the neuro-

nal population. Ath xamp f th s f dcdig

agithms t assss what stims fats a cddi th a ppati cms fm th aaysis fth spss f sig cs i th hma mdia tm-pa b t pict pstatis. ns i this aaha b sggstd t cd th idtity f th p-s bjct icitig spss, ath tha basic isafats24. FIGUre 2 shws th cfsi matix btaidwh a dcdig agithm pdictd th pstati f32 difft picts basd th actiity f 19 simta-sy cdd its. It was pssib t pdict whichps bjct was shw ach tim (FIG. 2c), bt it wast pssib t distigish btw difft picts fth sam ps bjct44.

Box 3 | Data-processing inequality and sources of information loss

There are several sources of information loss when processing neural responses. The

first is given by a simplification of the neural responses for example, by dividing the

spike trains into small consecutive time bins or by taking the spike count (FIG. 3a). These

operations can be formalized as a transformation of the neural response: rf(r). By

analogy with equation 5, the Shannon information about the stimulus carried by the

processed responsef(r)can be defined as:

(10)

The data-processing inequality states that any transformationf(r) can only decrease

the Shannon information that was accessible from the original responses; that is,

I(S;f(R))I(S;R)34. This information loss can, for example, result from ignoring temporal

patterns in the spike trains or from ignoring information given by the precise

coincident firing of different neurons.

Decoding can be formalized as a second transformation of the neural responses:

rg(f(r)) =sp, where spdenotes stimulus prediction. By analogy with equation 10, the

Shannon information about the stimulus carried by sp is defined as

(11)

This is the mutual information between the rows and columns of the confusion matrix

(FIG. 2). The data-processing inequality states that the stimulus prediction usingdecoding algorithms gives a second source of information loss that is:

I(S,Sp)I(S,f(R))I(S,R), for s = g(f(r)). An example of this information loss is given in

FIG. 3b (for a formal proof, see reFs 37,39).

In general, information losses occur when the observer of the neural responses (the

information-extracting algorithm) uses an incorrect probability of stimuli and

responses Q(r,s) rather than the correct one p(r,s). For example, the observer may not

know the precise post-stimulus time at which current responses were emitted (FIG. 3c).

Alternatively, the observer may not know that neurons are correlated and may

incorrectly treat them as independent95.

The information I(S;R) that can be extracted by an algorithm using the wrong

probability Q(r,s) is less than or equal too the information that is encoded by the neural

responses: I(S;R)I(S;R). A lower bound to I(S;R) is given by:

(12)

A formulation similar to equation 12 was first explored in reF. 32 and later developed

in reF. 125. If the error in the probability distribution is made only because of a

response simplification rf(r) (that is: Q(s,r) = p(s,f(r)); Q(r) = p(f(r))), then equation 12 is

an equality and becomes identical to equation 10. Similarly, if this simplification is due

to decoding (that is: Q(s,r) = p(s,sp); Q(r) = p(sp)), then equation 12 is equal to equation 11.

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Preferred direction Non-preferred direction

PRR

LIP

Time (s)

Rate

(Hz)

Rate

(Hz)

Rate

(Hz)

Rate

(Hz)

0

20

40

60

0

20

40

60

0 0.5 1

0 0.5 1 0 0.5 1

Time (s)

Time (s) Time (s)

0 0.5 1

0

50

100

0

50

100

Reach directions Saccade directions

Decoded reaches Decoded saccades

True

saccades

True

rea

ches

1

0

2

4

6

8

10

12

14

16

2 4 6 8 10 12 14 16

1

2

3 5 8

7

4 6 9

10

11 13 16

15

12 14

Go!Memorize target Wait

Reaches

Saccades

a

b c

Brainmachine interface

A dict communication link

btwn a bain (human o

animal) and an xtnal dvic,

uch a a pothtic limb o a

ning dvic.

Th pssibiity f dcdig ifmati fm hmaa ppati cdigs has ac f th

dpmt fbainmachin intfac66,67, ad schdcdig pids ifmati that is t appat fmth aaysis f sig-c cdigs. I xpimts ikths shw i FIG. 2 it was fd that: fist, spik stigsigificaty impd pdictis44, stssig th af ptima pcssig f th cdigs (BOX 1); scd,dcdig pfmacs w th sam wh a s y ths that pdcd sigificat spss wcsidd, shwig that spsi s did tcay at ifmati; thid, ach spsi - caid a aag f 0.25 bits f ifmati abtth stims; ad fth, mst f th ifmati abt thpicts was gi by a aag f y 4 spiks fid

btw 300 ad 600 ms aft th stims st i aatiy sma mb f s.

Th fats that a cdd by a ppati caas b assssd with th ifmati-thy appach.F xamp, by gpig stims fats ad cacat-ig th amt f ifmati btaid i ach cas itwas fd that s i th at ba ctx cdda cjcti f th stims fqcy ad ampitd(th pdct f bth, ad t ach f ths paamtsidpdty) wh stimatd with sisida whisk

ibatis69.

Temporal coding of sensory information. Tw hypth-ss ha b ppsd t xpai hw s cdifmati. Th spik-ct hypthsis stats that

Figure 4 | Ddng am a and aad t gt dnt dtn. a | In a delay reach and saccade task a

monkey is shown a briefly flashed target in one of eight possible locations and, after a go signal (the disappearance of the

initial fixation point), has to perform either a reach or a saccade depending on the colour of the target (green or red,respectively).b | Average firing rates of a neuron in the parietal reach region (PRR) and a neuron in the lateral intraparietal

area (LIP) to delay reaches and delay saccades in their preferred and non-preferred directions. The neuron in PRR is tuned

to the preferred direction for reaches and has a lower firing rate for saccades. Conversely, the neuron in LIP is tuned to the

preferred direction for saccades and has a lower firing rate for reaches. | Decoding of all 8 directions for reaches and

saccades using the whole population of 47 PRR and 32 LIP neurons with a nearest-neighbour algorithm. The colour code

in the confusion matrix shows the relative number of times that a reach or saccade i (along theyaxis) was predicted to bej

(along thex axis). In no case was a reach confused with a saccade or vice versa. Predictions of both reaches and saccades

were nearly perfect. The only exception was for saccades to the left, as recordings were made in the left hemisphere and

saccades are usually encoded in the contralateral field. Partsb and are modified, with permission, from reF. 61 (2006)

Society for Neuroscience.

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Principal-component

analysis

A lina tanfomation that

pojct th data on to an

othogonal ba, in which th

gatt vaianc of th data

li on th fit coodinat (th

fit pincipal componnt),

th cond gatt vaiancon th cond coodinat, and

o on. It i uually ud to

duc th dimnionality of

complx data.

Spike afterpotential

A tanint hyppolaization

of a nuon following th

fiing of an action potntial. It

i caud by K+ channl,

which opn duing th pik

and clo a fw millicond

aft th nual mmban

potntial go back to it

ting valu.

s pst ifmati y by chagig thifiig ats sm at tim widw12,70. Thtmpa-cdig hypthsis stats that th pcis timigf spiks adds imptat ifmati t th ifmatigi by th spik cts45,7173.

Dtmiig which cd bst psts ifmatiis aags t aatig th tmpa pcisi withwhich spiks pid kwdg abt stimi. Hightmpa pcisi fas tmpa cdig, whasgh tmpa pcisi fas spik-ct cds.Th tmpa pcisi ca b stabishd by takigtasfmatis f th igia sps rf(r) thatimiat ay ifmati pttiay caid i thtmpa patt f th spik tais (BOX 3). This ca bachid by biig th a spss (FIG. 3a) by admy jittig th spik tims withi sm timag. Th, sig qati 10, w ca mas hwifmati, dcdig pfmac, chags withth timig pcisi. Tmpa ifmati ca as bidtifid by cmpaig th ifmati gi by thtta spik cts with that ctaid i th tmpa

wafm f th spss dtmid, f xamp,sig pincipal-componnt analyi71,74.

Stdis i sbctica stcts ad pimay s-sy ctics ha ptd that th timig f spiksadds ifmati t that gi by spik cts, dw t a timig pcisi f th d f a fw mii-scds72,7579. eidc f tmpa cdig is ssstabishd i high aas80 (bt s reFs 71,8183 adFIG. 5d,). Mst stdis tmpa cdig ha cc-tatd sig bai aas ad y a fw ha ptdhw timig cds a tasfmd acss difft stagsf a ssy pathway8488. o sch stdy csiddhw th tmpa cdig f ataistic whisk stimiis tasfmd acss th at smatssy pathway,ad fd that miiscd-pcis timig cydhigh amts f ifmati bth i th gagi adi th smatssy ctx89. I th gagi th tm-pa cd was spptd by pcis itspik itas,whas i th ctx itspik itas w aiabad th tmpa cd was basd th tims f ta-sit spss masd fm th tim f th whiskdfctis. This atcy cd has b ptd widyat th ctica 76,9092. Hw, it is ca whth hw th atcy cd is sd by th bai, gi thatth masmt f ths sps atcis qiskwdg f th stims tim.

Th xtt t which ifmati i sps atcis

is accssib t th bai ca b addssd with ifma-ti thy, by csidig hw mch ifmati cab xtactd with a impcis kwdg f th stim-s tim. F this, ca s a appximat stimssps distibti btaid by shiftig th spik taiswithi th ag f th tim ctaity f th stims,ths gatig a ss f ifmati if stims tim-ig is ccia (s qati 12 ad FIG. 3c). I th stdydscibd ab89, ifmati fm th spik timig ith ctx qid a pcis kwdg f th stimstim, whas ifmati fm th gagi did t.This fidig ca b itptd by tig that th c-tx might stimat stims tim sig th tpt f th

mt systm93 xpitig th adatags f atcycds, amy sca iaiac ad simpicity f dw-stam cmptatis73 whas gagi cs adthi dict tagts a iky t ci stims timifmati ad may b fcd t di ifmatii th ways.

Assessing the importance of correlations. Gi th tch-ica diffictis i acqiig ad pcssig simta-sy cdd data fm a ag mb f s(BOX 1), it is cmm t btai th ppati actiityby cdig aft ath d xacty thsam xpimta paadigm. Hw, this assmsthat s fi idpdty ad igs th fcatd fiig i a ifmati pcssig.It is thf imptat t stabish hw aistic thifmati btaid by cdig c at a tim is,cmpad with that btaid by cdig th ssimtasy. This iss has b cd i dtaiswh13,9497.

With simtas cdigs, th f ca-

tis ca b assssd by cmpaig th ifmatiabt th stimi fm th cdd ppati ad thatbtaid aft imiatig th catis acss cs.Th att is cmptd thgh qati 5 by pacigth t sps pbabiity, P(r,s), with P

sh(r,s), which

is btaid by shffig th tias cspdig t achstims idpdty f ach . This pc-d has shw that catis ca ith dc icas th a ifmati ctt f a ppati.Shffig th a spss typicay icass thifmati wh psitiy catd s thsthat td t icas dcas thi fiig tgth with simia stims sctiitis a csidd13,94,98.Atatiy, mig catis by shffig caas dcas th ifmati gi by a a pp-ati wh, f xamp, s a psitiy c-atd bt ha ppsit stims sctiitis whcatis chag fm stims t stims t tag pa-tica ssy fats99. F dcdig, th pfmaci th absc f catis ca b as stimatd byshffig bth taiig ad tst data sts44.

I pactic, stdis sma ppatis idicatthat is catis mak a sma qatitati dif-fc i stims cdig44,95,100102 (bt s reF. 103 fa xcpti). Hw, aaytica xtapatis sg-gst that this ffct may b imptat f ag appatis95,98. Tstig ths scaig pdictis i a

data qis th dpmt f impd ifmati-thy dcdig agithms that a ptimizd fag ppatis.

Merging information from different neuronal signals.Mass f a actiity sch as lFPs fct amb f sbthshd itgati pcsss thata t fctd i spik tais, icdig ipt aditactica ppati syaptic pttias104,105, pikaftpotntial ad tag-dpdt mmba scia-tis106,107. Ths, th simtas cdig ad aaysisf spiks ad lFPs (BOX 1) may pid cmpmtayifmati abt th a twk actiity.

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Delta(sdu)

10

20

30

Trials

Trials

10

20

30

36 38 40 42 44 46 48Time (s)

36 38 40 42 44 46 48Time (s)

a

b

c

d

e

3/2

/2

2

0

180 0

Firing-phase band

Meanheadingand

positionerror(degrees)

0

0

15

30

45

60

75

90

105

60 120 180 240 300

P

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Voxel

In MrI ach, a voxl f

to th mallt maud

volum unit, analogou to a

th-dimnional pixl. In

functional MrI tudi th

a typically of th od of

30 mm3, although much

mall voxl volum hav

bn achivd in mo cnt

wok.

Th f lFP phass i th at hippcamps hasas b dbatd. Sig-c stdis sig th gbastatistica atiship btw lFP phas ad spikfiig cd idtify a patia dissciati f phasad fiig at i sps t th aimas cati adspd f mmt82 bt cd t dtmi whthths sigas cd cmpmtay ifmati abtspaat xta fats. oth stdis ha as shwthat bth thta phas ad at fct th ampitd f thcs ipt ad might cy ddat spatia ifma-ti110112. Itstigy, a ct ppati stdy sigdcdig tchiqs81 cay dmstatd that thphas f fiig f hippcampa pac cs cds fa-ts atd t th aimas spatia aigati that at cdd by spik ats (FIG. 5d,).

Th fidig f difft tigs f lFPs ad spikshas as b ptd f mdia tmpa b si hma sbjcts113 ad iftmpa s i thmacaq ctx114. Itstigy, i th ctica aas,sch as i th psti paita ctx i mkys, itwas fd that lFPs ad spiks cay simia ifma-

ti abt mmt pas63,64. M, th lFPs ca-id m ifmati tha spiks abt th tim f thpad mmt ad th bhaia stat. This hassigificat impicatis f th dpmt f baimachi itfacs, gi that lFPs a mch m sta-b ad asi t cd f ag pids f tim thasig-c actiity115.

Conclusions

W ha dscibd th adatags f a sig-tia pp-ati aaysis taditia sig-c stdis f tia-aagd spss. I patica, w ha shw hwmch m kwdg ca b xtactd sig dcdigad ifmati-thy mthdgis ad hw if-mati that is ambigs at th sig-c casmtims b cay itptd wh csidig thwh ppati.

Dcdig ad ifmati thy dscib cmp-mtay aspcts f kwdg xtacti. Dcdig hasth mai adatag f pdcig a tpt (th stim-s pdicti) that is asy t itpt ad is cs tbhaia chics, whas mta ifmati gisa m cmphsi qatificati f th ifmatictaid i a a ppati, by aatig thdcti f ctaity abt th stimi that ca bbtaid fm th a spss. Th cmpm-tay kwdg ffd by ths tw appachs is y

w bgiig t b istigatd i scic47,54. Am systmatic jit appicati f bth mthdgismay ff additia isights.

A itstig qsti that cd bfit fm scha jit appach is cmpaig th ifmati gi by

th pdicti f th stims ad th ctaity fths pdictis with th tta ifmati ctaidi th ppati. rpstig ctaity is imptatf dcisis iig isks ad may b fdamta fa cmptatis that tak pac i th ps-c f bth ssy ad a is5153. As discssdab, cmbiig ifmati-thtic ad dcdigappachs pids a igs way t qatify hw mchf th ifmati pidd by a ppatis c-cs th pdicti f th stims, ad hw mch f thisifmati is abt spcific aspcts f th ctaity fths pdictis. A cmbiati f ths appachs cathf pid pcis qatitati asws abt hwth bai das with itisicay isy sigas.

Ifmati-thtic cacatis a diffict witha ppatis bcas f th cs f dim-siaity. Athgh ti cty it was thght t bpbmatic t cmpt accat ifmati mas-s fm th actiity f m tha 9, ctpgss w pmits th cmptati f th if-mati caid by ppatis f p t t s36.

This abs s t xami th dtais f ifmatipcssig i ca twks, ad ft wk may stthis ba high. A maj ad imptat chagf statisticias is t fid ways t fth xtdth fasibiity f pfmig ifmati cmptatiswith ag ppatis. Tw dictis a paticaypmisig. Fist, it is imptat t xp th s fptima dimsiaity-dcti tchiqs that dis-cad dimsis t cayig ay at ifmati.Scd, th ifmati-bias pbm f ag ppa-tis is xacbatd by th fact that th actiity f -s is ft catd. This pbm ca b dimiishdby simpifyig th cati stct f xamp, bycsidig pai-wis catis btw s adigig high-d itactis116.

A scitific ti has b tiggd byth s f imagig tchiqs sch as fctia MrI.Hw fctia MrI sigas cat with th actiityf sig s ad lFPs is fa fm cmpty d-std117. Hw, a ths sigas gi cmpmtayifmati, with difft cag ad difft tm-pa ad spatia stis. Dcdig ad ifma-ti thy ff gat ways t cmbi a f thssigas ad qatify thi ifmati ad ddacy.Itstigy, a fw stdis ha aady shw th af dcdig f xtactig ifmati fm ppatis fvoxl118,119, ad th stdis ha cmpad th if-

mati gi by spiks ad lFPs. It mais t b shw mch m ifmati ca b xtactd fm asig-tia cmbid aaysis f th physigi-ca sigas (spiks ad lFPs) ad th x actiity, atpis that has y cty b iitiatd120,121.

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