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j ou rna l h o me p ag e :w w w . e l s e v i e r . c o m / l o c a t e / b i o s y s t e m s
Effects
of
Poisson
noise
in
a
IF
model
with
STDP
and
spontaneous
replay
of
periodic
spatiotemporal
patterns,
in
absence
of
cue
stimulation
Silvia
Scarpetta
a,b,∗,
Ferdinando
Giacco
c,
Fabrizio
Lombardi
d,
Antonio
de
Candia
eaDepartmentofPhysics‘E.R.Caianiello’,UniversityofSalerno,84084Fisciano(SA),Italy bINFNUnita’diNapoliGruppocoll.diSalerno,Italy
cDepartmentofMathematicsandPhysics,SecondUniversityofNaples,81100Caserta,Italy dInstituteofComputationalPhysicsforEngineeringMaterials,ETH,Zurich,Switzerland eDepartmentofPhysics,UniversityFedericoII,CNR-SPINandINFN,Napoli,Italy
a
r
t
i
c
l
e
i
n
f
o
Articlehistory:
Received16October2012
Receivedinrevisedform28February2013 Accepted19March2013 Keywords: Neuralcriticality STDP Associativememory
a
b
s
t
r
a
c
t
Weconsideranetworkofleakyintegrateandfireneurons,whoselearningmechanismisbasedonthe Spike-Timing-DependentPlasticity.Thespontaneoustemporaldynamicofthesystemisstudied, includ-ingitsstorageandreplayproperties,whenaPoissoniannoiseisaddedtothepost-synapticpotentialof theunits.Thetemporalpatternsstoredinthenetworkareperiodicspatiotemporalpatternsofspikes. Weobservethat,eveninabsenceofacuestimulation,thespontaneousdynamicsinducedbythenoise isasortofintermittentreplayofthepatternsstoredintheconnectivityandaphasetransitionbetween areplayandnon-replayregimeexistsatacriticalvalueofthespikingthreshold.Wecharacterizethis transitionbymeasuringtheorderparameteranditsfluctuations.
© 2013 Elsevier Ireland Ltd. All rights reserved.
1. Introduction
We study theeffects of noise in the spontaneous collective dynamicsof anetwork of Nleaky Integrateand Fire(LIF) neu-rons, whose learning mechanism is based onthe Spike-Timing DependentPlasticity(STDP).Thetemporalpatternsweconsiderare periodicspike-timingsequences,whosefeaturesareencodedinthe relativephaseshiftsbetweenneurons’spikes.Wereviewthe stor-ageandreplaypropertiesofsuchLIFnetworkswhoseconnections comefromalearningstrategybasedonSTDPwithPphase-coded patternswithrandomphases.
ThenwefocusontheeffectsofanindependentPoissoniannoise addedtothepost-synapticpotentialsduringspontaneous activ-ity.Inpreviousworks(ScarpettaandGiacco,2012;Scarpettaetal., 2010)wefocusedonthedynamicsemergingasaconsequenceof ashortcuestimulation,andweshowedthatinordertotrigger spontaneouspatternsofactivityreminiscentofthosestored dur-ingthelearningstage,afewspikeswiththerightphaserelationship aresufficient.Herewefocusonthespontaneousdynamicswithout anyexternalstimulus,andweshowthatanoisysignaladdedtothe potentialsmayinduceanintermittentreplayofthestoredpatterns.
∗ Correspondingauthorat:DepartmentofPhysics‘E.R.Caianiello’,Universityof Salerno,84084Fisciano(SA),Italy.Tel.:+39089969418;fax:+39089969658.
E-mailaddress:silvia@sa.infn.it(S.Scarpetta).
Thisspontaneousreactivationinducedonlybynoise,is particu-larlyrelevantconsideringthatrelationshipbetweenspontaneous andevokedcorticalactivityduringdevelopmentistheobjectof manyrecentstudies(Berkesetal.,2011;Hanetal.,2008).Notably westudythephase-transitionwhichappearsinthenoisy sponta-neousdynamicsofourmodel,inabsenceofexternalstimulation. Wecharacterizethecriticalregime,atthetransitionbetweenthe regionofspontaneouspersistentreplayandtheregionofno-replay. Inthiscriticalregimeashorttransientintermittentreplayis initi-atedbynoise.Thereplaylastsforashorttimeandafterawhile anewpatternmaybetransientlyreactivatedspontaneously.To characterizethespontaneousactivityofthenetwork,weevaluate theorderparameteranditsfluctuationsasafunctionofspiking thresholdofthecoupledneurons.Atalowspikingthresholdthe orderparameterishighandfluctuationsarelow,whileatthe criti-calspikingthreshold,wherethereisatransitionbetweenno-replay (highth)andpersistentreplay(lowth), theorderparameter’s
fluctuationsaremaximized,asexpectedfora continuousphase transition.Interestinglymanyrecentexperimentalresults,suchas
Tagliazucchietal.(2012),suggestthatrestingbrainspendsmost ofthetimeneartothecriticalpointofasecondorderphase transi-tion.Animportantresultofthispaperisthestudyofthedynamical criticalpointandofthedifferentdynamicalregimesobservedby changingtheexcitabilityparametersofthenetwork.
Thepaperisorganizedasfollows:inSection2weintroduce theLIFneuronalmodelandtheSTDPlearningruleusedtodesign 0303-2647/$–seefrontmatter © 2013 Elsevier Ireland Ltd. All rights reserved.
theconnections,withabriefillustrationoftheevokedbehavior (Section2.1);inSection3westudytheabilityofthenoisetoinduce thespontaneousrecallofthepatterninabsenceofexternal stimula-tion,weintroduceanorderparameter,andwefocusonthecritical region;summaryanddiscussionareoutlinedinSection4.
2. Themodel
ThespikingmodelhasbeenintroducedinScarpettaandGiacco (2012),Scarpettaetal.(2010).Webrieflyreviewthemodelhere. ThesingleneuronmodelisaLIF.WeusetheSpikeResponseModel (SRM) formulation (Gerstnerand Kistler, 2002; Gerstner et al., 1993)oftheIFmodel.Inthispicture,thepost-synapticmembrane potentialofaneuroniisgivenby:
hi(t)=i(t)+
j Jij ˆtj>ˆti (t− ˆtj), (1)wherei=1,...,N,Nbeingthenumbersofunitsinthenetwork,i(t)
isaPoissonnoise,Jijarethesynapticconnections,
(t)describestheresponsekerneltoincomingspikesonneuroni,andthesumover ˆtjrunsoverallpre-synapticfiringtimesfollowingthelastspike
ofneuroni.Namely,eachpre-synapticspikej,witharrivaltime ˆtj,issupposedtoaddtothemembranepotentialapost-synaptic
potentialoftheformJij
(t− ˆtj),where (t− ˆtj)=K exp −t− ˆtj m −exp −t− ˆtj s (t− ˆtj) (2)wherem isthemembranetimeconstant(here10ms),sisthe
synapsetimeconstant(here5ms),istheHeavisidestep func-tion,andKisamultiplicativeconstantchosensothatthemaximum valueof thekernelis 1. Thesignof thesynapticconnectionJij
setsthe signofthe post-synapticpotential’s change,sothere’s inhibitionfornegativeJijandexcitationforpositiveJij.Whenthe
membranepotentialhi(t)exceedsthespikingthresholdith,aspike
isscheduled,andthemembranepotentialisresettotheresting valuezero.
Wewanttomimicthespontaneousactivityofcorticalnetworks invitro,intheabsenceofexternalstimulation.Therearemany differentpossiblecausesofnoiseinthesystem.Here,wefocuson theeffectsofPoissonnoise,thereforei(t)ismodelledas
i(t)=Jnoise
ˆtnoise>ˆti
(t− ˆtnoise). (3)Thetimes ˆtnoisearerandomlyextractedforeachneuroni,andJnoise
arerandomstrengths,extractedindependentlyforeachneuroni andtime ˆtnoise.Theintervalsbetweentimes ˆtnoiseareextractedfrom
a Poissonian distribution P(ıt)∝e−ıt/(Nnoise), while the strength
JnoiseisextractedfromaGaussiandistributionwithmeanJnoiseand
standarddeviation(Jnoise).Thisdescribesthenoisyenvironment
inwhichournetworkisembedded,intheabsenceofanyexternal stimulation.
Asshowninmanyrasterplotofinvitrospontaneous dynam-icswithneuronalavalanches,thereisoftenasmallsubsetofunits whichhasahigherspikingratethantheotherunits.Theseunits withspontaneoushigherspikingratearemodelledhereasa sub-setofunitswithlowerspikingthreshold.Theseunitswithlower spikingthresholdarethereforemoresensitivetothePoissonian noisewhichactsonthemembranepotentialsofalltheunitsof thenetwork.Ifsomeoftheselow-thresholdunitsbelongstoone ofthestoredpatternandhaveconsecutivephasesinthispattern, thenitmayhappenthattheseunitsareabletoinitiateacollective replayofthepattern.Tocheckthishypothesis,therefore,we imag-inethatforeachstoredpatternthereisasmallsubsetofZ=N/30 units,withconsecutivephasesinthepattern,thathavelowspiking
Fig.1. PlotofthelearningwindowA()usedinthelearningruletomodelSTDP effects.TheparametersofthefunctionA()aredeterminedbyfittingthe experi-mentaldatareportedinBiandPoo(1998).
thresholdth1.Theselowthresholdunitswillbemoresensitiveto
noise,andtheiractivationmayeventuallytriggerthereplayofthe storedpatterns.WhiletheothersN−PZunitswillhavethreshold th2.
Numericalsimulations of thedynamics are performed for a networkwithPstoredpatterns,whereconnectionsJijare
deter-minedviaalearningrulepreviouslyintroducedinScarpettaand Giacco(2012),Scarpettaetal.(2010,2011,2001,2002),Yoshioka etal.(2007).Theconnectionsvaluesarefrozen,andthecollective dynamicsisstudied.
Therefore,followingScarpettaandGiacco(2012),the connec-tionsJijduetothelearningoftheperiodicpatternissimplygiven
by Jij = ∞
n=−∞ A(tj−ti+nT)= n A j 2 − i 2 + n , (4) wheretj (ti)isthespiketimeofunitj(i)inthepattern.The learn-ingwindowA(=t−t’)isthemeasureofthestrengthofsynaptic changewhenatimedelayoccursbetweenpreandpost-synaptic activity(Fig.1).Thestoredpatternsaredefinedasprecise peri-odicsequence of spikes,i.e. spike-phasecoded patterns,asthe onesshown inFig.2,madeupofonespikepercycle,and each spikehasaphase j randomlychosenfromauniform distribu-tionin[0,2).Thesetoftimingofspikesofunitjinpatternis tj+nT=(j )/(2 )+n/ ,where =1/Tistheoscillation
frequencyoftheneuronsinthestoredpattern.Thus,eachpattern isrepresentedthroughthefrequency andthespecificphases
ofspike j oftheneuronsj=1,...,N.
Whenmultiplephase coded patternsarestored,thelearned connectionsaresimplythesumofthecontributionsfrom individ-ualpatterns,namely
Jij= P
=1
Jij. (5)
AsdiscussedinScarpettaandGiacco(2012),thefunctionA()used satisfiesthebalancecondition
∞
−∞A()d=0.Thisassuresthatin
theconnectionmatrixthesummedexcitation(1/N)
i,JFig.2.Examplesoftwoperiodicspatiotemporal(phase-coded)patternsofspikes thatarestoredinourIFnetwork.Only10units(randomlychosen)areshown.
thesummedinhibition(1/N)
i,Jij<0Jijareequalinthe
thermody-namiclimit.
2.1. Emergingofcollectivepatterns
TherecurrentnetworkwithNLIFunits,withconnectionsfixed tothevaluescalculatedinEq.(4)fordifferentvaluesofPisstudied. FollowingScarpettaandGiacco(2012),theresponseofthesystem toacueexternalstimulation,inabsenceofnoise,isillustratedin
Fig.3.AshortcuewithM=300spikeswiththeproperphase rela-tionshipisabletoinducethepersistentreplayofthestoredpattern inaproperregionofparameters,i.e.forlowspikingthresholds. Whileathighthreshold,suchasinFig.3c,thecuestimulationonly triggerashorttransientreplay,andatevenhigherthresholdno activityistriggeredbythestimulation.
Intheregimeofcorrectreply,showninFig.3aandb,wecheck thatifthepartialcueistakenfrompattern=2,theneuronsphase relationshipsrecallthephaseofpattern=2(uncorrelatedwith pattern=1)(notshown).
Notethat,aftertheshortcuepresentation,theretrieval dynam-ics hasthesame phase relationship amongunits as thestored pattern,butthereplaymayhappenonatimescaledifferentfrom thescaleusedtostorethepattern,andthecollectivespontaneous dynamicsisatimecompressed(ordilated) replayofthestored pattern. The time scale of replay changes slightly with spiking threshold.IntheexampleofFig.3aandb,thetimescaleofthe retrievaldynamicsisfasteratlowerthreshold,andisdifferentthan thetimescaleusedtolearnthepatterns.Thetimescaleofreplay activityalsodependson ,asstudiedin ScarpettaandGiacco (2012).Herewefocusonthecase =3Hzbecausethe
correspond-ingtimescaleoftheretrievaldynamicsissuchthattheoscillations frequencyisintherangeofthetaoscillations.Thestorage capac-ity,definedasthemaximumnumberofencodedandsuccessfully retrievedpatterns,hasbeenstudiedinScarpettaandGiacco(2012)
whenthenetworkrespondtoashortcuestimulation.
Thestoragecapacity,asdefinedinScarpettaandGiacco(2012), isshownasafunctionofthespikingthresholdth,inFig.4a,when
M=N/10spikesareusedascue,and =3Hz.Fortheproper
val-uesofthresholdandforPlowerthanPmax,theevokedretrieval
issuccessful(i.e.cuewithafewspikes(M=N/10)isableto selec-tivelyactivatetheself-sustainedreplayofthestoredpattern).We alsoshow,fortheregionofcorrect evokedreplay,thevalueof
theoscillationfrequencyofcollectiveactivityduringthereplayin
Fig.4b.Atspikingthresholdhigherthanacriticalvaluecrit th no
per-sistentreplaywaspossibleforanyvalueofPinresponsetothecue stimulation.FollowingresultsofScarpettaandGiacco(2012),and
Fig.4,weestimatethiscriticalpointwhenN=3000, =3Hztobe aboutcrit
th 95.Atthresholdsclosetothiscriticalvaluethe
net-workrespondstocuewithatransientactivitythatisashortreplay ofthestoredpattern,butnotwithapersistentreplay.
Thecriticalregime,nearthephasetransition,isinvestigated here,butintheabsenceofanyexternalcuestimulation(M=0). Whilein previousworks(ScarpettaandGiacco,2012;Scarpetta etal.,2010;Yoshiokaetal.,2007)westudiedthedynamics trig-geredbyashortstimulationcue,inthefollowingweinvestigate thespontaneous dynamics withoutanycue stimulation, in the presenceofPoissoniannoiseonly.
Whileintheregionofpersistentreplaytheevokeddynamicsis robustwrtnoise,intheregionnearthecriticalpointthesystemis moresensitivetonoise,thereforeitisimportanttofocusoneffects ofnoiseinthisregionintheabsenceofcuestimulation.Notablyin thiscriticalregionitisthenoiseitselfwhichmayinitiateashort replay,withoutanyexternaltrigger.
InthefollowingthereforewestudyifaPoissoniannoise,ina networkwiththeproperconnectivityasours,isabletoinduce tran-sientpatternreplay,intheabsenceonanycuestimulation.Thisis motivatedalsobyrecentliteratureontheintriguingrelationship betweenevokedandspontaneouscorticalactivity,andonthe pat-ternreplayobservedinthepost-tasksleep.Indeedduringsleep thereactivationofpatternsstoredin theconnectivityisclearly nottriggeredbysensorystimulationbutinducedbynoiseitself spontaneously.
3. Effectsofnoiseinthespontaneouscriticalregime,in absenceofanycuestimulation
Weinvestigatehereif,eveninabsenceofexternalcue stimula-tion,collectivepatternsofactivity,reminiscentofstoredpatterns, emergespontaneously,inducedbynoise.Wethereforestudythe networkspontaneousdynamicsinthecriticalregime,inabsence ofanycuestimulation,inthepresenceofaPoissoniannoise.We focushereonthecriticalbehaviorobservednearthedynamical phase-transition,betweentheregionofpermanentreplayandthe regionofno-replay.Figs.5–7showthespontaneousdynamicsofa networkofN=3000units,P=2patternsstoredintheconnectivity, andnoisegivenbyEq.(3)withnoise=1ms,Jnoise=0,(Jnoise)=5,
intheabsenceofexternalstimulation.AsubsetofN1=200units
havealowspikingthresholdth1=26(tomodeltheheterogeneity
observedininvitrodynamics),andtheotherN−N1=2800units
haveth2=80,90,105,respectively,inFigs.5–7.Differentvaluesof
th1=20,22,24,26,28,30havebeeninvestigated,aswelldifferent
valuesofnoisevariance,andthephenomenaobservedaresimilar butlesspronounced.Sowefixthesevaluesofth1=26,(Jnoise)=5
andweshowthespontaneousbehaviorasafunctionofthreshold ofthemajorityofunitsth2.
Atlowerthreshold,th2=80 inFig.5thecollectivereplayof
thepattern=1,inabsenceofanycuestimulation,istriggered spontaneouslyandlastsforaverylongtime.Therasterplotofthe networkdynamicsisshownwithadifferentsortingofunitson theverticalaxes,showingthatpattern=1isreplayed.Thereplay isinitiatedsincethenoisyunitshavetriggeredit,andthereplay ispermanentandveryrobustinthisregime.Fig.6showsthatat th2=th2c =90,fromtimetotime,thereisashorttransientreplay
ofthepatterns.Eachreplayeventlastsforalimitedintervaloftime andthenfadesaway.InFig.6a,theunitsaresortedaccordingto increasingvaluesofstoredphasesinpattern=1,whileinFig.6b unitsaresortedaccordingto=2.Whenpattern=1isrecalled,
a
b
c
Fig.3. Networkdynamicsatdifferentvaluesofspikingthresholdth.Inthesesimulationsacueexternalstimulation(Mspikes,showningreen)isused,andasinglevalue ofspikingthresholdisusedforallunitsofthenetwork,andthenoiseisabsent.Atpropervalueofthespikingthresholdthecueinitiatesapersistentretrievalofthe spatiotemporalpattern,whileattoohighthresholdthenetworkissilentorwithonlyashorttransientretrieval.Exampleofselectivesuccessfulevokedretrieval(a,b)and exampleoffailure(c)areshown.Therasterplotof50units(randomlychosen)isshownsortedontheverticalaxisaccordingtoincreasingvaluesofphase 1
i ofthefirst storedpattern=1.ThenetworkhasN=3000IFneurons,th=75,90,105,respectively,ina,bandc.ConnectionsaregivenbyEq.(5)withP=2storedpatternsat =3Hz.
Fig.4. (a)Storagecapacity˛c=Pmax/N,definedasinScarpettaandGiacco(2012),isshownat =3Hzasafunctionofspikingthreshold,whenM=N/10spikesareusedas cuestimulation.AsalwaysinthispaperthenumberofunitsisN=3000.Thefigureshowsthatnearcrit
th 95thereisatransitionfromaregionofevokedpersistentreplay toaregionofno-replay.(b)Frequencyofthecollectiveoscillationsofthenetworkdynamicsduringevokedreplay,asafunctionofspikingthreshold,forP=Pmax.
Fig.5.Spontaneousdynamics,withoutanycuestimulation,inanoisyenvironment(noise=1ms,Jnoise=0,(Jnoise)=5)withth2=80,th1=26,N=3000, =3Hz.Spikesare shownwithunitssortedontheverticalaxesaccordingtoorderofunitsinthefirst(a)orsecond(b)storedpattern.Unitswiththresholdth1=26areshowningreen,the otherinblack.Permanentreplayoffirstpatternisobserved,initiatedbythenoisyunits(lowthreshold,moresensitivetonoise).
Fig.6. Spontaneousdynamicswithoutanycuestimulationinanoisyenvironment(noise=1ms,Jnoise=0,(Jnoise)=5)withth2=90,th1=26,N=3000, =3Hz.Spikesare shownwithunitssortedontheverticalaxesaccordingtoorderofunitsinthefirst(a)orsecond(b)storedpattern.Unitswiththresholdth1=26areshowningreen,the otherinblack.Shorttransientreplaysofthetwopatternsareinitiated,fromtimetotime.
Fig.7.Spontaneousdynamicswithoutanycuestimulationinanoisyenvironment(noise=1ms,Jnoise=0,(Jnoise)=5)withth2=105,th1=26,N=3000, =3Hz.Spikesare shownwithunitssortedontheverticalaxesaccordingtoorderofunitsinthefirst(a)orsecond(b)storedpattern.Unitswiththresholdth1=26areshowningreen,the otherinblack.Noreplayisobserved.
ashortsortedsequenceofspikesappearsinFig.6a,whilewhen pattern=2isretrieved,ashortsortedsequenceofspikesappears inFig.6b.Athigherthreshold,suchasth2=105inFig.7noreplay
isobserved,andonlynoisyactivityisobserved.Itisnotablythat, atintermediatevalueth2=90thisintermittentreplayisobserved
here,inabsenceofanyexternalcuestimulation.Weobserveasort ofspontaneousreactivationofallstoredpatterns,astheonethat seemstohappenduringsleep,usefulformemoryconsolidation.
3.1. Thecriticalpoint
In orderto characterizethis transition betweena regime of spontaneouspermanentretrieval(intheabsenceofcue stimula-tion)toaregimeofno-retrieval,wemeasurethevarianceandthe meanvaluem(Tw)oftheorderparameter.
In analogywith theHopfield model, we introduce an order parametertoestimatetheoverlapbetweenthenetworkcollective activityduringthespontaneousdynamicsandthestored phase-codedpattern.Thisquantityis1whenthephases jofneuronsj
coincideswiththestoredphases j,andisclosetozerowhenthe phasesareuncorrelatedwiththestoredones.
Therefore, we consider the following time-dependent dot product|M(t,Tw)|=< (t)| >where isthevectorhaving
componentsei j ,namely: |M(t,Tw)|=
1 N j=1,...,N t<t∗j <t+Tw e−i2tj∗/Twei j
(6)
wheretj∗isthespiketimingofneuronjduringthespontaneous dynamics,andTwisanestimationoftheperiodofthecollective
spontaneousperiodicdynamics.
Then,weconsiderthemeanvalueoftheorderparameter: m(Tw)=N1
s|M(t,T
Fig.8. Meanvaluem(TW)oftheorderparameterforpattern=1(circles)andits fluctuations(stars)asafunctionofthechosenwindowTw,fordifferentth2.N=3000, th1=26and =3HzasinFigs.5–7.
wheretheaverage···isdoneonthestartingtimetofthewindow, andNsistheaveragenumberofspikesonawindowoftimeTw.
Thefluctuationsoftheorderparameteraredefinedby 2(|M(t,Tw)|)= 1
Ns2
[|M(t,Tw)|2−|M(t,
Tw)|2
]. (8)
Theorderparameter|M(Tw)|,attheoptimaltime-windowTw,
i.e.withTw whichmaximizes|M(Tw)|,measuresthesimilarity
inthesequenceofspikingneuronsandinthephaselagbetween thespikes.Thisisasuitablechoiceespeciallywhenthereplayofa spatiotemporalpatternhastobedetectedindependentlyfromthe compressionofthetimescale.Notethatifwehaveaspiketrain thatisnotperiodic,wecannotdefinetheperiod,howeverwecan definetheorderparameterlookingatthetime-windowTwwhich
maximizes|M(Tw)|.
InFig.8starsarefluctuationswhilecirclesarethemeanvalue m(Tw),forthreevaluesof
th2=80,90,100,anddifferentvalues
ofthetime-windowTw.Wesimulate
th1=20,22,24,26,28,30,
andweplotresultsforth1=26(whichisthevaluewhichgives
higherorderparameter)inFig.8.Soweevaluatethemeanvalue andthefluctuationsoftheorderparameterforthevalueofTw
whichmaximizesit.
InFig.9themeanvaluemoforderparameterandits
fluctu-ations2(|M|)areshownasafunctionofspikingthreshold th2,
whenth1=26andfortheoptimalTw.Atlow spikingthreshold
(th2=80)mishighand2(|M|)islow,indicatingthat,asshown
inFig.5thenoiseisabletoinitiateasuccessfulretrieval(persistent replay)ofthestoredpattern.Athighthreshold(th2=100)either
themeanvalueandthefluctuationsoftheorderparameterarelow. Atthecriticalpoint(th2=90)betweenthetworegimes,the
fluc-tuationsoftheorderparameteraremaximized,asexpectedina continuousphasetransition.
The transition is then characterized by a dynamical order parameter,intermsofadynamicalphasetransition,betweentwo differentdynamicalbehaviors(allintheabsenceofanyexternalcue stimulation).Acriticalbehavior,withmaximizationoffluctuation oforderparameteratthecriticalthreshold,isobservedhereasin continuousphasetransitions.Notablyrecentresults(Tagliazucchi etal.,2012)show,usinglargescalefMRImeasures,thatthebrain spentmosttimenearacriticalpoint.Futureworkswillfocuson
Fig.9.Meanvaluem(TW)oftheorderparameter,atoptimalTw,anditsfluctuations asafunctionofth2,forN=3000,th1=26and =3HzasinFigs.5–7.
thespontaneousactivityatthecriticalthresholdfromthepointof viewofneuralavalanches.
4. Discussion
Here, using an STDP-based learning process (Scarpetta and Giacco,2012;Scarpettaetal.,2011,2011,2009;STDP,2008),we storeintheconnectivityofaLIFnetwork,severalspatiotemporal spikepatterns,andwefindthat,dependingontheexcitabilityof thenetwork,differentworkingregimesarepossible,with collec-tive,transientorpersistent,replayactivityinducedsimplybynoise. Inourmodelnotonlytheorderofactivationofthesequenceis pre-served,butalsotheprecisephaserelationshipamongunitsofthe periodicspatiotemporalpattern.
Whileintheregionofpersistentreplaythesystemisrobust w.r.t.noise, asdiscussed inScarpettaand Giacco(2012),inthe regionnearthecriticalpointthesystemismoresensitivetonoise, asshownhere.Inthecriticalregimeindeed,thedifferentsequences storedintheconnectivitymaybereactivatedtransiently,fromtime totime,duetonoise,andinabsenceofanycuestimulation.
Weshowthatatthecriticalpointthefluctuationsoftheorder parameteraremaximized,sincethecollectiveactivityismadeof differentsequencesofdifferentlengths,whichgiveriseto corre-latedactivityonthetopofanoisyuncorrelatedactivity.
Recentlythereisrenewedinterestinreverberatoryactivity(Lau andBi,2005)andincorticalspontaneousactivity(Ringach,2009; Pastalkovaetal.,2008)whosespatiotemporalstructureseemsto reflecttheunderlyingconnectivity,whichinturnmaybetheresult ofthepastexperiencestoredintheconnectivity.
Similaritybetweenspontaneousandevokedcorticalactivities hasbeenshowntoincreasewithage(Berkesetal.,2011),andwith repetitivepresentationofthestimulus(Hanetal.,2008). Interest-ingly, inourIFmodel, inorder toinducespontaneouspatterns ofactivityreminiscentofthosestoredduringthelearningstage, alimitednumberofspikeswiththerightphaserelationshipare sufficient,andmoreimportantly,eveninabsenceofsensory stim-ulus,anoisewiththerightphaserelationshipsisabletoinducea patternofactivityreminiscentofastoredpattern.Therefore,by adapting thenetwork connectivitytothephase-codedpatterns observedduringthelearningmode,thenetworkdynamicsbuildsa
representation of the environment and then, during proper conditions, suchas sleepor rest, thedynamical attractors cor-responding to the stored patterns are transiently activated by noise.
Regardingplacecellsforexample,apossiblescenarioisthat,the patternactivatedrepeatedlyduringexperienceisstoredinthe con-nectivity,andthenactivatedduringsleepwhenthenetworkisnear criticalpointandnoiseisabletoinitiateshortreplaysequences,in absenceofsensorystimulation.Duringexploration,whenthe ani-malvisitadjacentplacefields,theevokedactivityoftheplacecells isa sequenceofspikeswiththeconsecutiveactivationofplace cells,thenalsoinsideasinglethetacyclethecellsareactivatedin therightsequence.Duringsleep,intheabsenceofexternalinputs, therole ofrecurrentconnections increases,probablydue toan increaseofexcitabilityvianeuromodulatorsorothermechanisms, andthespontaneousactivityofthenetworkshowstemporarily shortreplayofstoredpatterns,probablyinitiatedsimplybythe noise,asproposedinthiswork.
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