Arousal increases the representational capacity of cortical tissue

17 Pages·2009·1.05 MB·English

JComputNeurosci(2009)27:211–227
DOI10.1007/s10827-009-0138-6
Arousal increases the representational capacity
of cortical tissue
Tomer Fekete&Itamar Pitowsky&Amiram Grinvald&
David B. Omer
Received:18June2008/Revised:14January2009/Accepted:21January2009/Publishedonline:27March2009
#SpringerScience+BusinessMedia,LLC2009
Abstract Arousal patently transforms the faculties of com- clusters. Changes of this sort in the structure of cortical
plexorganisms.Althoughtypicalchangesincorticalactivity activity spaces can serve as the basis of the enhanced
such as seen in EEG and LFP measurements are associated capacity to make perceptual/behavioral distinctions brought
withchangeinstateofarousal,itremainsunclearwhatinthe about by arousal. To show the feasibility of these ideas, we
constitution of such state dependent activity enables this analyzedvoltagesensitivedyeimaging(VSDI)dataacquired
profound enhancement of ability. We put forward the fromprimatevisualcortexindisparatestatesofarousal.Our
hypothesisthatarousalmodulatescorticalactivitybyrender- results lend some support to the theory: first as arousal
ing it more fit to represent information. We argue that increased so did the complexity of activity (that is the
representationalcapacityisofa dualnature—it requires not complexity of VSDI movies). Moreover, the complexity of
only that cortical tissue generate complex activity (i.e. structure of activity space (that is VSDI movie space) as
spatiotemporal neuronal events), but also a complex cortical measuredbypersistenthomology—amultiscaletopological
activity space (which is comprised of such spatiotemporal measureofcomplexity—increasedwitharousalaswell.
events). We explain that the topological notion of complex-
ity—homology—is the pertinent measure of the complexity Keywords Opticalimaging.Voltagesensitivedye.
of neuronal activity spaces, as homological structure Arousal.Statesofconsciousness.Statedependentactivity.
indicates not only the degree to which underlying activity Representationalcapacity.Persistenthomology.Complexity
is inherently clustered but also registers the effective
dimensionality of the configurations formed by such
1 Introduction
ActionEditor:AlessandroTreves
Arousal has a notable effect on the behavioral, perceptual
Electronicsupplementarymaterial Theonlineversionofthisarticle and cognitive capacities of complex organisms. What are
(doi:10.1007/s10827-009-0138-6)containssupplementarymaterial,
the changes in cortical activity mirroring these remarkable
whichisavailabletoauthorizedusers.
: : changes brought about by arousal? Plainly put, the
T.Fekete(*) A.Grinvald D.B.Omer
hypothesis we wish to state and provide some evidence
DepartmentofNeurobiology,WeizmannInstituteofScience,
for is the following: arousal increases the capacity of
Rehovot76100,Israel
e-mail:[email protected] cortical systems to serve as information carrying media,
that is, represent information.1 We will begin by formulat-
T.Fekete
TheInterdisciplinaryCenterforNeuralComputation,
TheHebrewUniversity,
Jerusalem91904,Israel 1In what follows we limit our discussion to the continuum of states
ranging from loss of consciousness through dreamless sleep and
I.Pitowsky drowsiness(orquietwakefulness)tofullfledgedalertness(something
DepartmentofCognitivescience, like the peak in the Yerkes–Dodson Law 1908). Within this range
TheHebrewUniversity, statesofarousalseemtolieonacontinuum(i.e.aonemanifold,see
Jerusalem91904,Israel Hobsonetal.2000).
212 JComputNeurosci(2009)27:211–227
ing a working definition of representational capacity. To Tobetterappreciate therelation between the structure of
thisendwewishtointroducethenotionofactivityspace— an activity space, and the complexity of a sample drawn
the space comprised of spatiotemporal neuronal events from it, let us first consider what representational capacity
brought about by a given cortical system (for related ideas an Euclidean convex set possesses;2 a sufficient amount of
see Edelman 2001; Tononi 2004). Experimentally, activity instances drawn from such a space will result in a
spaces can be probed via dense spatiotemporal measure- homogenous distribution of points. Any partitioning of
ments of neuronal activity. The data analysis described such a distribution of points into clusters will be arbitrary.
below was carried out on voltage sensitive dye imaging In other words, such a space also has little representational
(VSDI, Grinvald and Hildesheim 2004) data. VSDI capacity if at all.3
produces movies—ordered sequences of images which are We see then that the representational capacity of an
proportional to the distribution of membrane potential activity space is contingent on it possessing of a structure
mainly in layers 2–3 of the imaged cortex (Grinvald et al. that will ensure that if the space is sufficiently sampled the
2001; Arieli et al. 1995, 1996; Petersen et al. 2003a, resulting distribution will be far removed from a homoge-
2003b). Thus in our case cortical activity space as a neousdistribution.Apossiblemechanismforsatisfyingthis
measurable entity is given by a VSDI movie space constraint is the topological notion of complexity—homol-
corresponding to a particular experimental setup. ogy. Homology is a construction which counts configu-
Wecontendthattherepresentationalcapacityofanactivity rations of different dimensions existing within a space. For
spacehasadualnature:first,theabilityofinstancesofactivity example,a ring isa 2D structure, while a sphere (such asa
to carry information is limited by the their complexity of basketball) is a 3D one. To reiterate, a sufficient sample
structure—astheinformationtoberepresentedbecomesmore drawn from a space endowed with complex homological
complex,soshouldthestructureofaphysicalprocess,thatis structure, will result in a distribution with complex
an instance of activity, if it is to represent it. However, the structure. Thus, such spaces can be said to be inherently
maxim “representation of complex information requires clustered, as increased homological structure will result in
complex structure” is true not only for instances of activity, increasingly complex configurations of clusters when the
butappliestothestructureofactivityspaceaswell(andsee underlying space is sufficiently sampled.4 The degree to
TononiandEdelman1998; Tononi 2004). which higher dimensional structure exists in a space attests
To begin to appreciate this point, consider a system that
givesrisetoanactivityspacecomprisedofasinglepointby
undergoing a dynamic sequence repeatedly. However com-
2Throughout the discussion we assume that each point in a given
plexthestructureofthesingleinstanceofactivityproduced
activity space is equiprobable. Note however that the foregoing
by this system might be, still, as a whole such a space can discussioniseasilyapplicabletospacesinwhichpointsdifferintheir
represent little if at all. To use an illustrative metaphor, an probability, for example due to inherent noise (that is at times the
system is “inaccurate” while generating activity). Adequate samples
activity space can be thought as attesting as to the
fromsuchaspacewillcontainsubspaceswhichdifferintheirdensity
“vocabulary” available to the system inorder to achieve its
(regions containing points of low probability will give rise to low
computational (and thus behavioral) goals. Therefore, the density regionsin thesample and vice versa). Insuch cases samples
questionthatarisesiswhichnotionofcomplexitypertainsto canbefirstthresholdedbydensity(deSilvaandCarlsson2004),and
thecapacityofsucha“vocabulary”tocarryinformation. thecomputationsdescribedbelowcarriedoutunchanged.
3Our discussion concerns the scenario in which a system is to
Consider a system which is to represent a set of objects
representinformationbyactivityalone.Howeveronemightarguethat
belongingtoseveralcategories.Theminimalrequirementfor
neural systems are actually engaged in coding (in the information
a system to embody the information regarding different theoretic sense); that is in attaching arbitrary codes to objects/
classes of objects is to produce activity which not only parameters/events. Thus given a suitable decoding mechanism what
appears to be noise (i.e. a homogeneous distribution) is actually
differentiates among objects, but does so in a systematic
perfectly matched to the task. There is however strong evidence that
manner enabling the distinction between categories. If
this is not the case in neuronal systems. Sharma et al. (2000) report
similarity is thought of as a metric (in which distance is that when the optic nerve of a ferret is connected to the primary
inversely related to similarity), this can be realized by auditorycortex,orientationmapsaredevelopedinA1,asissight(von
Melchner et al. 2000). In other words the activity of brain stem
producingclusteredactivityadheringtothedictumthatinner
structures contains sufficient information about the nature of stimuli
class similarity (distance) should be greater (smaller) than
forthecortextoadaptwithoutsomehardwireddecodingmechanism
inter class similarity (distance; for related ideas see Shepard resultingfromnaturalselection.
1987;Edelman1998,1999;Gardenfors2000;Tononi2004). 4Notethatactivityspacesseemnottobeclusteredperse—assensory
Wethereforesuggestthatthecomplexityofanactivityspace objects(suchasimagesandsounds)canbesmoothlywarpedintoany
other sensory object. This would indicate that activity spaces consist
(or vocabulary), and thus, in part, its representational
ofsinglecomponents.Howeverasmentioned,higherorderhomolog-
capacity, is contingent on the degree to which activity
ical structure results in inhomogeneous samples, i.e. they possess of
sampled from it clusters in complex configurations. structurewhichcanbelearnedandthereforeactedupon.
JComputNeurosci(2009)27:211–227 213
toitseffectivedimensionality;asclusteredactivityatteststo modulates cortical activity by rendering it more fit to
the distinctions a system is capable of making, the represent information can be stated as follows: as arousal
dimensionality of a configuration of clusters suggests the increases the representational capacity of the activity (sub)
number of dimensions (that is parameters, features, proper- spaces generated by a cortical systemshould increase. This
ties…) along which a given distinction is made. increasewillbemanifestedinanincreaseinthecomplexity
We wish to illustrate this point with a concrete example ofbothinstancesofactivityandthetopologicalcomplexity
(data courtesy of Sharon and Grinvald 2002): an anesthe- (homological structure) of activity space.
tized cat was presented with oriented moving square wave
gratings, and VSDI of V1 was carried out. Each stimulus
producedasingleVSDImovie.Gratingswereofoneofsix 2 Results
evenly spaced orientations spanning the entire range—0°,
60°, 120°, 180°, 240° and 300°. After preprocessing,5 2.1 Level set approximation of VSDI movie space
movies were projected onto the first two principal compo-
nentsofthedata,andcolorcodedbyorientation.Ascanbe Toexploretheseideas,weanalyzedVSDIdataobtainedby
seen in Fig. 1(a), not only do movies cluster according to imaging the primary visual cortex of a primate (for exact
the orientation of the presented stimuli, but these clusters details see Section 4). The imaged cortical area was 6×
are ordered sequentially forming a ring like structure. 6 mm, and sampling rate 200 Hz. Three distinct experi-
Now, the orientation domain has the topology of a ring: mental conditions lying along the continuum of states of
if the orientation of a line is incrementally varied, the line arousalwereinduced:(1)inthefirstconditionimagingwas
will eventually return to the starting position having gone carried out while the primate was under Ketamine anesthe-
through all possible orientations. Therefore, the configura- sia (dose was set to achieve cortical inactivation, see
tion of clusters in Fig. 1(a) would seem to suggest that the Section 4). During imaging the monkeys’ eyes were
imaged cortical area takes part in the representation of covered and the room dark. This condition was to serve
orientation. However, if one takes into consideration the as a model of loss of consciousness. (2) In the second
underlying activity space which was sampled, the question condition, the primate was awake, with his eyes covered
remains as to what form this distribution would take if the and again the room was kept dark. This condition was to
orientation domain would be sampled indefinitely (and not serve as a model of drowsiness. (3) Finally, imaging took
sparsely, both in terms of stimulus orientations and place while the primate was awake and viewing simple
repetitions as in practical experiments). Figure 1(b–c), stimuli presented on a screen while fixating. This last
depicts two possible scenarios, one in which a homoge- condition was to serve as a model of alertness. In all
neous distribution would have resulted, and the other in conditions the primate was seated in a monkey seat.
which the ring structure would remain evident. Clearly, Analyzing the data, we opted to capitalize on the dual
only the latter scenario would indicate that the observed nature of representational capacity, that is, the coupling
structure is an inherent property of the underlying space. between the complexity of activity and the complexity of
In summary, the representational capacity of an activity activity space. The idea was to first construct a function
spaceiscontingentonthecouplingbetweenthecomplexity defined over movie space which was sensitive to the
of activity, the limiting factor on the ability of instances of complexity of structure of instances of activity (which we
activitytocarryinformation,andthecomplexityofactivity will refer to as a state indicator function or SIF)—in this
space, the limiting factor on the systems ability to make case second long VSDI movies.6 Thus, according to our
distinctions (and carry information about relations between hypothesis, such a function should be able to classify
categories/parameters, see Tononi 2004). activity(movies)accordingtostateofarousal(experimental
We can now revisit the original question, namely how condition) by their complexity. Ideally, such a function
the increased behavioral and perceptual ability afforded by couldserveasacontinuousmeasureofarousal.Thus,clear
increase in arousal is reflected in neuronal activity. In light cut states of arousal, for example a given level of
of the discussion above our hypothesis that arousal anesthesia, would be associated with a characteristic value
5Each movie was band-pass filtered in space and time. Spatial 6ThereasonforsegmentingdataintosecondlongmoviesisthatEEG
parameters were chosen according to the spatial frequency of the measurementsindicatethatarousalrelatedphenomenaoccur(thatisto
orientation domainsin theimagedcortex. Temporal parameters were say,canbedetected)atthistimescale.Theexactduration(asecond
chosenaccordingtothetimecourseofthestimulus(asquarewave,in rather than 2, 3 or 5) was chosen as a compromise between the
whichtheupperphaserepresentsthepresenceofastimulus).Thiswas number of resulting points (which decreases as the duration of each
doneinordertosuppressspontaneousactivitycomponentsinthedata, movie is increased) and the ability to detect state dependent
toenablethevisualizationoftheringstructure. differencesintheVSDIdata.
214 JComputNeurosci(2009)27:211–227
Fig. 1 Complexity of structure
(a)
ofactivityspaces.(a)Asetof
0
pointsoriginatingfromVSDI 30
movies(anesthetizedcatV1) 60
90
obtainedbypresentingoriented 120
movinggratingsofsixevenly 150
spacedorientationsspanningthe
circle(0°–300°).Pointswere
projectedontotheplanevia
PCA[each1sresponse(movie)
istakenasonedatapoint]and
colorcodedaccordingtoorien-
tation.Thisresultedinaringof
clustersorderedbyorientation.
(b,c)Schematicdepictionof (b) (c)
twopossiblescenarioswhich
couldresultiftheunderlying
corticalactivityspacewould
havebeensampleddensely(i.e.
moresamplesandmoresampled
orientations).(b)Theapparent
ringstructurein(a)disappears
(c)theapparentringstructurein
(a)persists
of the SIF.7 By the same token, deviance from the char- to the SIF. Such terms can insure that points on a level set
acteristic value would indicate a shift in state of arousal. shareallthestatisticalandmetricpropertiesapparentinthe
We suggest that the level set8 associated with such a original data set.
characteristic value can serve as a model for an activity The SIF was constructed in several stages (see Section 4
(sub) space associated with a given state of arousal, forexactdetails):first,twotypesoffeatureswerecomputed;
provided that a suitable SIF is used: features measuring either the statistical or geometrical
First, if the SIF satisfies certain conditions (that is, be complexity of the structure of activity and features pertain-
smooth and nonsingular), its level sets will be smooth ingtothetypicalstructureofactivitygivenastateofarousal
manifolds (Lee 2003). The manifold property of the level (Gervasoni et al. 2004). Next, these features were used to
sets in this case insures that they will allow a metric build a classifier. Finally, both metric and statistical con-
structure, which is as noted above a requirement a straintswereaddedtotheclassifierasmultiplicativeterms.
representational system should fulfill (for a discussion on The first class of features, that is measures pertaining to
the benefits of smooth representation see Edelman 1999). thecomplexityofinstancesofactivity,weretheC measure
0
Second, by definition, points on this manifold possess the of randomness (Cao et al. 2007), the β parameter of a
characteristic values associated with activity within such a random Markovian field which is sensitive to spatiotempo-
distinct stateofarousal, andthus willpossessthedegree of ral coherence (Llinas et al. 1997; Contreras and Llinas
complexity typical of a state. Finally, an additional 2001; Leznik et al. 2002, and see Section 4 and Electronic
requirement should be met—although activity produced in supplementary material Fig. 2) and both the temporal and
disparate states of arousal certainly differs in important spatial profiles of correlation (Fiser et al. 2004). Both the
regards, having originated from the same cortical tissue it C andβparametersprovedespeciallyindicativeofstateof
0
also shares substantial common aspects. This commonality arousal, as can be seen in Fig. 2. In general, these
in structure should be encoded in the SIF, and thus be measurements indicated that movies became less random,
imparted to the level sets in question. As is shown below more coherent in space and time, and showed enhanced
this requirement is readily met by adding additional terms correlation in space and time as arousal increased (Fig. 2).9
Both EEG and local field potential exhibit state
dependent regularities (Gervasoni et al. 2004)—that is
7ThecharacteristicvaluewastakenasthemeanSIFvalueassociated
withinstancesofactivityrecordedwhilesubjectswereinadesignated typicaldistributioninthepowerofthesesignalsindifferent
stateofarousal.
8GivenafunctionF:M→N,thelevelsetforc2FðMÞisalltheset 9NotethattheeffectsofKetaminearedosedependent(Longneckeret
of all points x2M for which F(x)=c. In our case M is movie al. 2007)—thus it is possible to achieve anesthesia without inducing
spaceandNisR,the realline. slowcorticalwaves(seeElectronicsupplementarymaterialFig.1).
JComputNeurosci(2009)27:211–227 215
Fig. 2 State dependent charac- (a) (b)
teristicsofVSDImovies.(a)
Decayofcorrelationinspace ***
(mean±SE).Notethatthe *** ***
resultsaresimilarto(Fiseretal.
Eyes coverd
β200p4a)r.a(mbe)tTerheof(sapMatiaortkeomvpiaonral) Visual stimulation β2
2
randomfield(mean±SD(1
denotesperfectcoherence—see
Section4).(c)The3DC
0
randomnessindex(1denotes
randomness)(mean±SD).Note (c) *** *** ***
thatcoherenceandcorrelation
increasewitharousalwhileran-
domnessdecreases
C
0
frequency bands. Therefore similar measures were comput- function. The mean to incorporate different constraints into
ed in order to capture state dependent regularities in our the level set structure is through taking the product of the
data. As VSDI movies also include substantial spatial classifierwithseveralterms—oneforeachconstraint.Such
information,relativespatialandspatiotemporalpowerwere termsinsurethataninstanceofactivity(aVSDImovie)can
computed,inadditiontorelativetemporalpower.10Asmall achieve a typical value (that is the SIF value representative
subset of features was obtained by selecting power in of a given state of arousal such as drowsiness) of the SIF
severalspatiotemporalbands,whichprovedtobeindicative only if it meets some requirement. A constraint term is
of state. implemented as a composition of two functions: a smooth
Next, the derived features (a total of 13, see Section 4 inequality term,12 and a measurement which is to be
and Electronic supplementary material Fig. 3) were com- constrained. For example, one constraint enforced was that
bined through employing a quadraticFisher discriminant.11 the Mahalanobis distance of feature vectors (that is movies
The resulting discriminant classified movies according to after the feature transformation) will not exceed the values
state very robustly. We confirmed that this classification seen in the empirical distribution. Thus, the Mahalanobis
was not a mere result of high dimensionality—when labels distance of a feature vector is composed with a smooth
(indicating experimental condition) were shuffled, and a approximation of a step function. This function had its
discriminant computed using the exact same procedure support in the interval [0 δ] (i.e. took the value of 1 for
chance level classification was achieved (see Fig. 3). valuessmallerthanδand0forvaluesinfinitesimallylarger
Finally, to transform the classifier into the required SIF, than δ), the range of the empirical distribution. Similarly,
two constraints were incorporated into the discriminant another constraint incorporated into the level set represen-
tation of state was that the distance between the empirical
10It is likely that typical distribution of energy in different data and the level set model would not exceed the average
spatiotemporal bands attests to the complexity of activity as well. nearest neighbor distance of original data. A flowchart
The efficacy of readout of activity, that is, the activity of neurons depicting this construction is shown in Fig. 4 (and see
that form synapses with the neurons whose activity is measured,
Section 4 for details).
critically depends on the spatiotemporal makeup of the data. Given
To summarize, data were fitted with level set models.
the pattern of connectivity, and distribution of cell types in the
readoutnetwork,somespatiotemporalregimeswillbemoreeffective These were smooth manifolds by construction. Each point
than others, that is will be better suited to drive the network, and in such a model was a movie lying in the linear space
thus more effective in their ability to carry information. However
spannedbytheempiricalmoviesofagivenstateofarousal.
without detailed anatomical and functional analysis, such claims
Moreover,eachpointinsuchalevelsetexhibitedthesame
cannotbesubstantiated,andthuswecannotclaimthatthesefeatures
are actually legitimate measures of complexity as conceived by us state dependant statistical characteristics which were mea-
throughout this discussion. suredinthedata(ElectronicsupplementarymaterialFig.4).
11These features were more than necessary to achieve perfect
classification. However all the computed features that significantly
differentiatedamonganypairofconditionswereretained inorderto 12Either a smooth step function (implementing a one sided inequal-
preserve all the state dependent information afforded by the data ity), or a smooth bump function (implementing a two sided
whichwewereabletodetect. inequality).
216 JComputNeurosci(2009)27:211–227
Fig. 3 Classification of VSDI
moviesaccordingtostateof Eyes coverd
arousal.(a)first,datawasseg- (a) Visual stimulation
mentedinto1smovies.Next,
severalmeasuresofcomplexity
ofstructurewerecomputed(see
text).AquadraticFisherdis-
criminantwasfittedtothefea-
tures,resultingnotonlyin
perfectseparation,butalsoin
veryrobustinter-classbound-
aries.(b)AquadraticFisher
discriminantwasfittedtothe (b)
samefeaturevectorswhile
shufflingthelabelsdesignating
classmembership(experimental
condition).Thisresultsinnegli-
gibleseparationoffeaturevec-
torsby“class”.Note—
discriminantscoreshavebeen
placedonthreedifferentaxes
(theydimension)byclassfor
easeofviewing(c)Thesame
(c)
quadraticFisherdiscriminant
wasfittedtorandomvectors.
Notethattheresultisindistin-
guishablefrom(b),indicating
thatclassificationinBisat
chancelevel
Arbitrary units
2.2 Assessing the topological complexity of movie for technical reasonsasa1 (dimensional) hole. Similarly,a
space—computing persistent homology spherecouldbefilledoutwithavolume,thatis,itcontains
a 2 hole. By the same token if a complex comprises of
Equipped with the SIF level sets serving as models of state
dependent activity spaces we could turn to assess their
discriminant
topologicalcomplexity(foradifferentnotionofthecomplex-
ityofactivityspacessee(Spornsetal.2000; Tononi 2004)). . .
ker( ) Fd(..)
In order to compute the homology of a set, a construction
ff
called a simplicialcomplex has tobedefined overthe set in 11
question.Simplystated,asimplicialcomplexwillresultifan … ×
SIF (x )
iterative procedure iscarried out: Firstsomeofthe pointsin
tchheosoernigliinnessedtoarneotcoinntneerscetecdt. Iwfitthhelipnreesv,iopursovsitdeepdretshualttsthine fk )x . m Θ ( ..._c1)
constraints
triangles,someofthesetrianglesarefilledwithsurfaces.Ifthe
previousstepresultsintetrahedra,someofthesetetrahedraare x d(.. .,Xi) Θ ( ..._c )
filled with volumes, and so on (points, lines, triangles and 2
tetrahedra are simplexes of dimensions 0 to 3 respectively
Fig.4 TheSIF(stateindicatorfunction)construction.
hence the name of this procedure). Rendering of objects in xisaVSDImovie(a50×50×200vector)
computergraphicsisacommonexampleofthisconstruction. f1tofkdenotefeaturetransformations
_xdenotesafeaturevector(13×1)
As mentioned above, homology (among other things) dð(cid:2);XiÞ denotes the Euclidean distance to the set of VSDI movies
counts configurations of different dimensions existing in a acquiredintheexperimentalconditioniX
i
simplicial complex. A complex contains a configuration of k(cid:2)km denotestheMa(cid:1)halanobisdistancekerdenotesthequadratick(cid:3)ernel
agivendimension,onlyifpartofthecomplexis“missing", i.e.½x1:::xn(cid:3)1xn(cid:4)!7 x21x1x1x2:::x1xn:::x2n(cid:3)1xn(cid:3)1xn(cid:3)1xnx2nxn
FdistheFisherdiscriminantcomputedforempiricaldata
that is it contains a hole. For example, a ring structure will Θisasmoothstepfunction
disappear if it is filled out with a surface, and is referred to c areestimatedstatistics
1,2
JComputNeurosci(2009)27:211–227 217
several pieces (components) they can be connected by persistent homology see (Robins 2000, 2002; Edelsbrunner
lines, filling out the 0 holes. et al. 2002; Zomorodian and Carlsson 2005)).
The homology of a simplicial complex is directly Toillustratethisprocessletusrevisittheexampleshown
computable by algebraic manipulation which is akin to in Fig. 1(a). In Fig. 5 several stages of the process of
Gaussian elimination (Munkres 1984). The information of growingtheoriginalpoints[i.e.Fig.1(a)]intothemaximal
interest for our purposes can be expressed in terms of the complex (the Delaunay triangulation which is convex by
Betti numbers—an integer vector containing the counts of definition) are shown, each corresponding to a different
holes (and thus configurations) for each dimension. value of the scale parameter. Recall that for a given value
Accordingly, a simplicial complex gives rise to a unique all edges in the triangulation smaller or equal in length to
vector of Betti numbers—½b b ::: b (cid:4), β counting thisvalueareincludedintheassociatedcomplex,asarethe
0 1 n 0
components(separate pieces ofthecomplexor0holes),β surfacesfillingoutanytrianglesformedoutbytheseedges.
1
counting 2D configurations (1 holes), and so forth. For Thus, (5g) is obtained from (5h) by pruning all the edges
exampleacomputermodelofaclosedpicnicbasketwould longer than 0.359, and with them the surfaces enclosed by
give rise to the vector [1 1 1], the first number indicating these edges (the three middle triangular patches). The
that the basket consists of one piece (that is contains no 0 topologicalfeatureofinterestinthiscaseistheexistenceof
holes), the second that there is an opening in the handle the ring structure within the complex. To assess its
(a one hole) and the final number that the basket is significance its persistence interval (the range of scales in
hollow (a two hole). which it exists) should be observed.
However,givenasetofpointsX,thenumberofpossible Note that the relevant range of scale in this case begins
complexeswhichcouldbedefinedonXisexponentialwith with the minimal scale in which the complex will
thenumberofpoints.Therefore,thequestionwhichsprings comprise a single component (i.e. 5c). This is true for
to mind is which of these simplicial complexes should be two reasons: first, in order to assess global structure the
selected. The solution suggested by (Robins 2000, 2002; scale has to be sufficiently large. Secondly, we would
Edelsbrunner et al. 2002; Zomorodian and Carlsson 2005) expect that neuronal responses to a change in a (percep-
is to use a metric criterion to generate complexes: given a tually) continuous parameter acquired in experimental
threshold ɛ, first triangulate all points (for example by conditions in which the context was held constant to be
computingtheDelaunaytriangulation).Thisresultsinalist continuous as well [that is lie on a continuous (i.e. single
of simplexes. Now prune all edges (1 simplexes) in the component) manifold]. Naturally, the region of interest
complex which are longer than ɛ (for which the points in ends in the scale in which the maximal complex is attained
the target set are further apart than ɛ). This might result in (after which no additional homological information is to
elimination of higher order simplices—for example if an be found).
edge of a filled triangle is eliminated so will the surface in As can be seen in Fig. 5, during most of the above
order for the complex to abide by the definition and so on. mentioned interval (97% of the interval separating 5c and
Inotherwords,thepair(X,ɛ)resultsinauniquesimplicial 5g—i.e. the interval 5d to5g) thering structure isapparent
complex,and consequentlyinauniqueBettivector. Thus,ɛ indicating that it is a robust feature of the data set.
canbeconsideredasascaleparameterwhichgivesrisetoan The computation of persistent homology via a construc-
ordered series of simplicial complexes. Such a series of tionofafiltration (asequenceofcomplexesorderedbythe
complexes is referred to as a filtration. By the same token, scale parameter) can be improved on both in terms of
the scale parameter gives rise to a series of graphs of Betti computational efficiency and robustness to noise by using
numbers as a function of scale—one for each dimension. an approximation procedure—building witness complexes
This enables to assess the stability, or persistence, of (de Silva 2003; Carlsson and de Silva 2004; de Silva and
topological features as a function of scale.13 For this reason Carlsson 2004). To this end, instead of considering all
this construction is referred to as persistent homology (for points in X, only a small subset L is used as landmark
the exact definitions of persistent Betti numbers and points, while the entire set of points serves as a source for
potential witness points (existence of witness points in X is
a geometrical constraint equivalent to that used in the
Delaunay triangulation for determining which simplices
13An illustrative example is the analysis of an image at different
will be included in a triangulation; de Silva 2003). The
resolutions. Consider a HD picture of a face, in which case one can
zoom in and track pours in the skin (“holes” in our metaphor). filtration is generated by analyzing the (n×l) partial
However, if the image is downsampled such features are no longer distance matrix between the landmarks and points in the
discernable (or they do not persist at this scale). Further down-
set in general, rather than the full pairwise distance matrix.
sampling will cause other features to disappear—such as nostrils.
This results in a profound reduction in the amount of
Finally, additional downsampling will result in a blob (a smooth
gradientofcolor). simplexesgenerated,andthusincomputationtimerequired
218 JComputNeurosci(2009)27:211–227
to obtain homological information. Another advantage of Fig. 5 Generating a filtration from a set of points (a–h). Several„
stages of growing a maximal complex from a set of points (i.e.
this approach is that as selection of landmark points has a
producingafiltration).Numeralatbottomofeachfigureindicatesthe
random aspect to it, this procedure can be carried out scale parameter ɛ (arbitrary units). The set of points originates from
several times, and the results subjected to statistical VSDI movies (anesthetized cat V1—see Sharon and Grinvald 2002
analysis. This helps overcome biases resulting from for details) obtained by presenting oriented moving gratings of six
evenly spaced orientations spanning the entire range (0°, 60°, 180°,
contingencies of the sample under consideration in a
240°, 300°). Points were projected onto the first two principle
manner which is akin to bootstrapping. This method (i.e.
componentsofthedataandcolorcodedaccordingtoorientation.This
constructing a witness complex) is bound to recover the resulted in a ring of clusters ordered by orientation. The points were
homological type of a space, provided that the space at triangulatedusingtheDelaunaymethod,resultinginaconvexsimplicial
complex.Next,aseriesofsimplicialcomplexes(afiltration)wasobtained
hand is sampled densely enough, and that it is triangulable
byapplyingametricthreshold.Startingwiththeinitialcomplex,thesetof
(Carlsson and de Silva 2004).
points (vertices), the scale parameter was incrementally increased. At
Another attractive property of computation of homology various stages along this progression, the scale parameter equaled the
using witness complexes is that it alleviates in part the lengthofanedgeconnecting two verticesinthe triangulation.Atthis
“curse of dimensionality”, the reason being that a witness point this edge was added to the complex. As soon as all edges
comprisingatrianglewereaddedtothecomplex,sowasitsface(thearea
complex can be extended into any desired dimension—the ofthetriangle).Thisresultedinaseriesofcomplexes—oneforeachvalue
addition of simplexes of dimension n to a witness complex ofthethreshold(scale)inwhichasimplex(edgeorsurface)wasaddedto
depends only on the lower dimensional simplexes in the thecomplex.Theregionoftopologicalintereststartsthemomentpoints
form a single component (i.e. c in which ɛ=0.199, see text for
complex and not on the dimensionality of the data.
explanation) and continues up until the maximal complex is formed
Therefore, as the Betti number of rank k depends only on (i.e.h—ɛ=0.397).Notethatduringmostofthisintervalthecomplexes
the k+1 and k simplexes in the complex, it is thus possible exhibit a ring structure—the length of the region of interest is 0.397–
to extract the low dimensional properties of spaces of high 0.199=0.198;theringstructurefirstappearsatɛ=0.204(i.e.d),andis
persists along the interval 0.397–0.204=0.193 (i.e. d–h); thusthe ring
dimensionality.
structurepersistsin97%oftheinterestinterval(i.e.0.193/0.198)
To summarize, computation of persistent homology can
be thought of as a multiscale analysis of the topological
complexity of a data set, as Persistent homology records dimensional disk punctured by five one-holes. Figure 6
two attributes of a data set: (1) its inherent clustering, that shows that the measured Betti profiles for the two-
is to what degree points tend to form clusters (i.e. deviate dimensional case (top row), and three-dimensional case
from a uniform distribution), (2) the effective dimensional- (bottomrow)conformtoexpectations.
ity of the data set, that is the dimensionality of the Nextwecarriedouttheexactsamecomputationsforthe
configurations of clusters found in the data. In terms of levelsetmodels(i.e.selectionof50landmarksfollowedby
persistent homology, our hypothesis concerning the com- averaging100runs).Notethatscalewasnormalized(bythe
plexity of state associated activity spaces was that increase maximal distance of the landmarks points to the witness
in arousal will result in larger Betti numbers which persist points) in each run to avoid possible influence of differ-
across longer ranges of scale. ences in signal variance between experimental conditions.
The computation of persistent homology was carried out It was found that increased arousal indeed resulted in
usingthePLEXsoftware(ParryanddeSilva,www.comptop. increased homological structure: for the first to third Betti
stanford.edu/programs/). We first tested the performance of numbers larger numbers were found on average across all
PLEX on a few simple artificial data sets. For each set of scalesasarousalincreased(seeFig.7).Forallscalesshownthe
points the Betti numbers from orders 0 to 3 (the practical zeroBettinumberβ0was1(indicatingthatgiventhecovering
limit)werecomputedafterselecting50landmarks(viamin– bylandmarksthedatasetconsistedofasinglecomponent)14.
max method; Carlsson and de Silva 2004; de Silva and Toevaluatethedifferencesbetweenconditions,statistics
Carlsson 2004). For each of the conditions, 100 PLEX runs were carried out on the integral of the Betti graphs. In all
were carried out and averaged. Some results of the analysis three cases differences were significant (p<<0.001 using
of artificial data are shown in Fig. 6. In each artificial data ANOVA;posthocanalysisp<<0.001 using Mann–Whitney
set the number of points generated was of thesameorder
ofmagnitudeofthenumberofpointsinthelevelsetmodels 14The difference in shape between the first and second Betti curves
(~104). Data sets were generated by first randomly and the third curve results (in part) from landmarking. Without
suitable landmarking rather than remaining constant the zero Betti
sampling points from an n dimensional ball, and then
graphwouldhavedecayedfromn,thenumberofpoints,to1,andasa
puncturing each such ball with k n−1-dimensional holes. result the first and second graphs would resemble the third one in
Such configurations are topologically equivalent to gluing shape. The reason is that the probability of finding pairs of points
k n−1 spheres.TheBettinumbersofsuchconfigurationsare whose distance is below a given threshold is larger than the
probability of finding triplets for smaller scales (up until saturation
½b b ::: b b (cid:4)¼½ 1 0 ::: 0 k (cid:4). In Fig. 6
0 1 n(cid:3)2 n(cid:3)1 thatis).Thesameistrueifquartetsandtripletsarecomparedandsee
(middlerow)onesuchdatasetisshown—inthiscaseatwo- (Robins2000,2002,2006).
JComputNeurosci(2009)27:211–227 219
(a) (b)
(c) (d)
(e) (f)
(g) (h)
220 JComputNeurosci(2009)27:211–227
β0 β1 β2
β1 β2 β3
Fig.6 Thepersistenthomologyofartificialdatasets.Datasetswere analyzedabove—i.e.a2Dball(adisc)puncturedwithfiveone-holes.
generated by first randomly sampling points from an n dimensional Middlerow,thezero,firstandsecondBetticurvesforadisc(atwo-
ball, and then puncturing each such ball with k n−1-dimensional dimensionalball)puncturedwithfiveone-holes.ThethirdBetticurve
holes.Suchconfigurationsaretopologicallyequivalenttogluingkn− was identically 0 (not shown). Bottom row, the first to third Betti
1 spheres. All curves shown are an average of 100 computations of graphs fora3D ballpunctured with15two-holes holes(voids). The
Betti numbers as a function of normalized scale, after choosing 50 zeroBetticurvewasidenticallyone(notshown)
landmark points using the PLEX software. Top row, the point cloud
for all nine comparisons). As this increase of topological measured instances of activity. Note that such measure-
complexity was accompanied by an increase in the mentsarefarfrombeingexternalbynature—theactivityof
complexity of instances of activity this hints at an increase a cortical system must be readily apprehended by cortical
inrepresentationalcapacityfollowingincreaseinarousal. mechanisms to produce rapid adaptive behavior. Thus,
homogenously distributed activity reflects failure of the
3 Discussion underlying system to make distinctions concerning the
environment. Such distinctions should manifest though
We presented the hypothesis that arousal increases the activity clustering by category. Homological structure
capacity of cortical systems to represent information. We enables departure form such uninformative structure, as it
argue that representational capacity has a dual nature: gives rise to activity which is inherently clustered.
complex activity can result in increased capacity, as can Exploiting the dual nature of representational capacity,
complex structure of cortical activity spaces provided that we were able to explore the effect of arousal on the
these two types of complexity exist in concert. The homological structure of the space of cortical activity
structure of activity space determines the distribution of brought about by primate V1. This was achieved by