TOWARDSTHE MODELINGOFNEURONALFIRINGBY GAUSSIAN
PROCESSES
E. Di Nardo,A.G.Nobile,E. Pirozzi andL.M.Ricciardi
ReceivedMarch26,2003
Abstract. This pap er fo cuses on the outline of some computational metho ds for
the approximate solution of the integral equations for the neuronal ring
probabil-ity densityandanalgorithmforthegenerationofsample-pathsinordertoconstruct
histograms estimating the ring densities. Our results originate from the study of
non-MarkovstationaryGaussianneuronalmo delswiththeaimtodeterminethe
neu-ron's ringprobability density function. Aparallelalgorithm hasb eenimplemented
in orderto simulate largenumb ersofsample paths ofGaussianpro cesses
character-izedbydamp edoscillatorycovariancesinthepresenceoftimedep endentb oundaries.
Theanalysisbasedonthesimulation pro cedureprovidesanalternativeresearchto ol
whenclosed-form resultsoranalyticevaluationoftheneuronal ringdensitiesarenot
available.
1 Introduction
This contributiondeals with theimplementationof pro ceduresand metho ds, worked out
in our group during the last few years, in order to provide algorithmic solutions to the
problem of determining the rst passagetime (FPT) probability density function (p df)
and its relevant statistics for continuous state-space and continuous parameter Gaussian
pro cessesdescribingthesto chasticmo delingofasingleneuron'sactivity.
In most mo deling approaches, it is customary to assume that aneuron is subject to
inputpulseso ccurringrandomlyintime,(see,forinstance,[13]andreferencestherein). As
aconsequenceof thereceivedstimulations,itreactsbypro ducingaresp onsethatconsists
of aspiketrain. Therepro ductionofthestatistical features of such spiketrainshas b een
thegoalofmanyresearcheswhohavefo cusedtheattentionontheanalysisoftheinterspike
intervals. Indeed,theimp ortanceoftheinterspikeintervalsisduetothegenerallyaccepted
hyp othesis that theinformationtransferredwithin thenervoussystemis usuallyenco ded
bythetimingofo ccurrenceofneuronalspikes.
To describ ethe dynamicsof the neuronal ringwe considerasto chastic pro cessX(t)
representingthechangeintheneuronmembranep otentialb etweentwoconsecutivespikes
(cf., for instance,[9]). In this context, thethreshold voltage is viewedas adeterministic
functionS(t)and theinstantwhenthemembranep otentialcrossesS(t)asaFPTrandom
variable.
Themo delingofasingleneuron'sactivitybymeansofasto chasticpro cesshasb eenthe
object of numerous investigations duringthe last four decades. A milestone contribution
inthis directionis themuch celebratedpap erby Gersteinand Mandelbrot[7]inwhich a
randomwalkanditscontinuousdi usionlimit(theWienerpro cess)wasprop osedwiththe
aim of describing a p ossible, highly schematized, spike generation mechanism. However,
despite the excellent ttingof avariety ofdata, this mo delhasb een thetarget of severe
criticismonthebaseofitsextreme idealizationincontrastwithsomeelectrophysiological
2000MathematicsSubjectClassi cation. 60G15;60G10;92C20;68U20;65C50.
evidence: forexample,this mo deldo esnottakeintoaccountthe sp ontaneousexp onential
decay of the neuron membranep otential. An improved mo del is the socalled
Ornstein-Uhlenb eck(OU) mo del,that emb o diesthepresence of such exp onential decay. However,
theOUmo deldo esnotallowtoobtainanyclosedformexpressionforthe ringp df,except
forsomeveryparticularcasesofnointerestwithintheneuronalmo delingcontext. Rather
cumb ersome computationsarethus required to obtain evaluations of thestatistics of the
ring time. Successively, alternative neuronal mo dels have b een prop osed, that include
morephysiologicallyfeatures. Theliteratureonthissubjectisto ovasttob erecalledhere.
Welimitourselvestomentioningthatareviewof mostsigni cantneuronalmo delscanb e
found in [10], [13] and in the references therein. In particular, in [10] it is presented an
outlineofappropriatemathematicaltechniquesbywhichto approachtheFPTproblemin
theneuronalcontext.
We shall now formally de ne the ringp df for amo delbased ona sto chastic pro cess
X(t)withcontinuoussamplepaths. First,assumeP[X(t
0 )=x 0 ]=1,withx 0 <S(t 0 );i.e.
weview thesamplepathsofX(t)asoriginatingatapreassignedstatex
0 atinitialtimet 0 . Then, T x0 = inf tt 0 t:X(t)>S(t) ; x 0 <S(t 0 )
istheFPTofX(t)throughS(t),and
g[S(t);tjx 0 ;t 0 ]= @ @t P(T x0 <t) isitsp df. Henceforth,theFPTp dfg[S(t);tjx 0 ;t 0
]willb eidenti edwiththe ringp dfofaneuron
whosemembranep otentialismo deledbyX(t) andwhose ringthresholdisS(t).
Throughout this pap er, we shall fo cus our attention on neuronal mo dels ro oted on
di usionandGaussianpro cesses,partiallymotivatedbythegenerallyacceptedhyp othesis
that in numerous instances the neuronal ring is caused by the sup erp osition of a very
large numb er of synaptic input pulses which is suggestive of the generation of Gaussian
distributionsbyvirtueofsomesortofcentrallimittheorems.
Itmustb eexplicitlyp ointedoutthatmo delsbasedondi usionpro cessesare
character-izedbythe\lackofmemory"asaconsequenceoftheunderlyingMarkovprop erty. However,
in the realistic presence of correlated input stimulations, the Markov assumption breaks
downand onefacesthe problemofconsideringmoregeneralsto chasticmo dels, forwhich
the literature on FPT problem app ears scarce and fragmentary. Simulation pro cedures
thenprovidep ossiblealternativeinvestigation to olsesp ecially iftheycanb eimplemented
onparallelcomputers,(see[3]). Thegoalofatypicalsimulationpro cedureistosampleN
valuesoftheFPTbyasuitableconstructionofN time-discretesamplepathsofthepro cess
and thento recordtheinstantswhen suchsamplepaths rst crosstheb oundary. Insuch
a way, one is led to obtainestimates of the ring p df and of its statistics, that may b e
implementedfordata ttingpurp oses.
Theaimofthispap eristooutlinenumericalandtheoreticalmetho dstocharacterizethe
FPTp dfforGaussianpro cesses. Attentionwillb efo cusedonMarkovmo delsinSection2,
andonnon-Markovmo delsinSection3. Finally,Section4willb edevotedtothedescription
ofsomecomputationalresults.
2 Gauss-Markov processes
Westartbrie yreviewingsomeessentialprop ertiesofGauss-Markovpro cesses. LetfX(t);t2
Ig,whereIisacontinuousparameterset,b earealcontinuousGauss-Markovpro cesswith
(i) m(t):=E[X(t)]iscontinuousinI;
(ii) thecovariancec(s;t):=E
[X(s) m(s)][X(t) m(t)]
iscontinuousinII;
(iii) X(t)isnon-singular,exceptp ossiblyattheendp ointsofIwhereitcouldb e equalto
m(t)withprobabilityone.
AGaussianpro cessisMarkovifandonlyifitscovariancesatis es
c(s;u)=
c(s;t)c(t;u)
c(t;t)
8s;t;u2I;stu: (1)
Itiswellknown[8],thatwell-b ehavedsolutionsof(1)areoftheform
c(s;t)=h 1 (s)h 2 (t); st; (2) where r (t):= h 1 (t) h 2 (t) (3)
is a monotonically increasing function by virtue of the Cauchy-Schwarz inequality, and
h
1 (t)h
2
(t)>0b ecauseoftheassumednonsingularityofthepro cessonI. Theconditional
p dff(x;tjx
0 ;t
0
)ofX(t)isanormaldensitycharacterizedresp ectivelybyconditionalmean
andvariance M(tjt 0 ) = m(t)+ h 2 (t) h 2 (t 0 ) [x 0 m(t 0 )] V(tjt 0 ) = h 2 (t) h 1 (t) h 2 (t) h 2 (t 0 ) h 1 (t 0 ) ; with t;t 0 2 I; t 0
< t. It satis es the Fokker-Planckequation and the asso ciated initial
condition @f(x;tjy; ) @t = @ @x [A 1 (x;t)f(x;tjy; )]+ 1 2 @ 2 @x 2 [A 2 (t)f(x;tjy; )]; lim "t f(x;tjy; )=Æ(x y); withA 1 (x;t)andA 2 (t)givenby A 1 (x;t)=m 0 (t)+[x m(t)] h 0 2 (t) h 2 (t) ; A 2 (t)=h 2 2 (t)r 0 (t);
theprimedenotingderivativewithresp ect totheargument.
Theclass of the Gauss-Markovpro cesses fX(t);t 2 [0;+1)g, such that f(x;tjy; )
f(x;t jy);ischaracterizedbymeansand covariancesofthefollowingtwoforms:
m(t)= 1 t+c; c(s;t)= 2 s+c 1 (0st<+1; 1 ;c2R; c 1 0; 6=0) or m(t)= 1 2 +ce 2t ; c(s;t)=c 1 e 2t c 2 e 2s 2 2c 1 2 e 2s 0st<+1; 1 ;c;c 2 2R; 6=0; c 1 6=0; 2 6=0; c 1 c 2 2 2 0 .
The rsttyp eincludestheWienerpro cess,usedin[7]todescrib etheneuronal ring,while
thesecondtyp e includestheOrnstein{Uhlenb eckpro cessthat hasoftenb een invokedasa
mo delforneuronalactivity(see,forinstance,[13]).
Any Gaussian pro cess with covariance as in (2) can b e represented in terms of the
standardWienerpro cessfW(t);t0gas
X(t)=m(t)+h 2 (t)W r (t) ; (4)
andisthereforeMarkov. ThislastequationsuggeststhewaytoconstructtheFPTp dfofa
Gauss-Markovpro cessX(t)intermsoftheFPTp dfofthestandardWienerpro cessW(t):
Asanexample,from(4)fortheconditionedFPTp dfonehas:
g[S(t);tjx 0 ;t 0 ]= dr (t) dt g W S [r (t)];r (t)jx 0 ;r (t 0 ) ; (5)
wherer (t)isde nedin(3)andg
W [S (#);#jx 0 ;# 0 ]istheFPTp dfofW(#)fromx 0 attime # 0
tothecontinuousb oundaryS (#), with x 0 = x 0 m[r 1 (# 0 )] h 2 [r 1 (# 0 )] ; S (#)= S[r 1 (#)] m[r 1 (#)] h 2 [r 1 (#)] :
Results on the FPT p df for the standard Wiener pro cess can thus in principle b e used
via(5) to obtainthe FPT p dfof any continuousGauss-Markovpro cess. For instance,if
S (#) is linearin #, g W [S (#);#jx 0 ;# 0 ] is known and g[S(t);tjx 0 ;t 0
] canb e obtainedvia
(5). Instead, if g W [S (#);#jx 0 ;# 0
] is not known, anumerical algorithm, ora simulation
pro cedure,shouldb e used forthestandardWienerpro cess and,after that,g[S(t);tjx
0 ;t
0 ]
canb eobtainedviatheindicatedtransformation.
Theab ovepro cedureoftenexhibitstheseriousdrawbackofensuingunacceptabletime
dilations(see[4]). Forinstance,exp onentiallylargetimes areinvolvedwhentransforming
theOrnstein-Uhlenb eckpro cesstotheWienerpro cess,whichmakessuchametho dhardly
viable. Hence,it isdesirableto disp oseof adirectand eÆcientcomputational metho dto
obtainevaluation ofthe ringp df. Alongsucha direction,in[4]it hasb een provedthat
the conditioned FPT density of a Gauss-Markovpro cess can b e obtainedby solving the
non-singularVolterrasecondkindintegralequation
g[S(t);tjx 0 ;t 0 ]= 2 [S(t);tjx 0 ;t 0 ]+2 Z t t0 g[S( ); jx 0 ;t 0 ] [S(t);tjS( ); ]d x 0 <S(t 0 ) (6) withS(t);m(t);h 1 (t);h 2 (t)2C 1 (I)and [S(t);tjy; ]= S 0 (t) m 0 (t) 2 S(t) m(t) 2 h 0 1 (t)h 2 ( ) h 0 2 (t)h 1 ( ) h 1 (t)h 2 ( ) h 2 (t)h 1 ( ) y m( ) 2 h 0 2 (t)h 1 (t) h 2 (t)h 0 1 (t) h 1 (t)h 2 ( ) h 2 (t)h 1 ( ) f[S(t);tjy; ] (7)
wheref[x;tjy; ]isthetransitionp dfofX(t): Closedformsolutionsof(6)areavailablein
[4]fordi erentfamiliesofb oundaries.
By makinguseofthisresult,in[4]aneÆcientnumericalpro cedurebasedonarep eated
Simpson'srule has b een prop osed to evaluate FPT densities of Gauss-Markovpro cesses,
thatcanb eimplementedtoobtainreliableevaluationsof ringdensitiesforneuronalmo dels
3 Gauss non-Markovprocesses
Themetho dsprop osedinthepreviousSectionrestonthestrongMarkovassumptiononthe
sto chasticpro cessmo delingtheneuron'smembranep otential,whichgrantsthep ossibilityof
makinguseofthementionedanalyticandcomputationalmetho dsforFPTp dfevaluations.
Thisisnotthecasewhenthesto chasticpro cessusedtomo deltheneuron's ringmechanism
isnon-Markov.Hereweshallfo cusourattentiononGaussnon-Markovpro cesses. However,
thusdoingwefacethelackofe ectiveanalyticalmetho dsforobtainingmanageable
closed-formexpressionsfortheFPTp df,althoughsomepreliminaryanalyticalresultshaveb een
obtainedbyRicciardiandSatoin[11] foraclassofstationaryGaussianpro cesses.
Indeed,ifX(t)isone-dimensionalnon-singularstationaryGaussianpro cessmeansquare
di erentiable,aseriesexpansionfortheFPTp dfwasderived(see,[12]). Inthefollowing,
forconvenience,weshalltaket
0
=0asinitialtimeand,withoutlossof generality,assume
that E[X(t)] = 0 and P[X(t
0 ) = x
0
] = 1, with x
0
an arbitrarily sp eci edinitial state.
Furthermore,thecovariancefunctionE[X(t)X( )]:= (t )will b eassumedto b esuch
that (0)=1; (0)_ =0and (0) <0(this last assumptionsb eing equivalentto themean
square di erentiableprop erty). TheFPT p df of X(t) through S(t) is then given bythe
followingexpression g[S(t);tjx 0 ]=W 1 (tjx 0 )+ 1 X i=1 ( 1) i Z t 0 dt 1 Z t t 1 dt 2 Z t t i 1 dt i W i+1 (t 1 ;:::;t i ;tjx 0 ); (8) with W n (t 1 ;::: ;t n jx 0 ) = Z 1 _ S(t1) dz 1 Z 1 _ S(tn) dz n n Y i=1 [z i _ S(t i )]p 2n [S(t 1 );:::;S(t n );z 1 ;:::;z n jx 0 ]; where p 2n (x 1 ;:::;x n ;z 1 ;:::;z n jx 0 ) is the joint p df of fX(t 1 );::: ;X(t n );Z(t 1 ) = _ X(t 1 ); ::: ; Z(t n )= _ X(t n )g conditional up on X(0) = x 0
. Due to the great complexity of the
involvedmultipleintegrals,expression(8)do esnotapp earto b emanageablefor practical
uses, even though it has recently b een shown that it allows to obtain some interesting
asymptotic results[5]. Since(8) is aLeibnitz series foreach xedt >0;estimates ofthe
FPT p dfcaninprincipleb e obtainedsinceitspartialsumof ordernprovidesaloweror
anupp erb oundto gdep endingonwhethernisevenoro dd. However,alsotheevaluation
ofsuchpartialsumsisextremelycumb ersome.
In conclusion, at the present time for this class of Gaussian pro cesses, no e ective
analytical metho ds, nor viable numerical algorithms are available to evaluate the FPT
p df. Asimulationpro cedureseemstob etheonlyresidualwayofapproach.
Tothisaim,wehaverestoredandup datedanalgorithmduetoFranklin[6]inorderto
constructsamplepathsofastationaryGaussianpro cesswithsp ectraldensityofarational
typ eanddeterministicstartingp oint. Theideaisthefollowing. Letus considerthelinear
lter X(t)= Z 1 0 h(s)W(t s)ds (9)
where h(t) is the impulse resp onse function and W(t) is the input signal. By Fourier
transformation,(9)yields X (!)=jH(!)j 2 W (!) (10)
0
0.5
1
1.5
2
2.5
3
3.5
4
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Figure1: Plotoftheb oundaryS(t)givenin(13)for =0:5andd=0:25;0:50(b ottomtotop).
where
W
(!)and
X
(!)arethesp ectraldensitiesofinputW(t)andoutputX(t),resp
ec-tively, andwhere H(!) denotestheFouriertransformofh(t):Equation(10) is suggestive
of a metho d to construct a Gaussianpro cess X(t) having a preassignedsp ectral density
X
(!) (!). ItisindeedsuÆcienttoconsiderawhitenoise(t);havingsp ectraldensity
W
(!)1;astheinputsignalandthenselecth(t)insuchawaythat jH(!)j 2
= (!). If
thesp ectraldensityof X(t)isofrationaltyp e, namelyif
(!):= Z 1 1 (t)e i!t dt= P(i!) Q(i!) 2 (11)
whereP andQarep olynomialswithdeg(P)<deg(Q);settingH(!)=P(i!)=Q(i!),from
(10)itfollows
X(t)= P(D )
Q(D ) (t)
where D = d=dt: To calculate the output signal X(t) it is thus necessary to solve rst
thedi erentialequationQ(D ) (t)=(t)to obtainthestationary solution (t), andthen
evaluateX(t)=P(D ) (t):Thesimulationpro cedureisdesignedtoconstructsamplepaths
ofthepro cessX(t)attheinstantst=0;t;2t;::: wheretisap ositiveconstanttime
increment. The underlying idea can b e appliedto any Gaussian pro cess havingsp ectral
densitiesof arational typ e and,since thesample pathsof the simulatedpro cess are
gen-erated indep endentlyof one another, this simulationpro cedure is particularlysuited for
implementationonsup ercomputers.
Extensivecomputationshaveb eenp erformedonparallelcomputersto explorethe
dif-ferent shap es of the FPT p df asinduced bythe oscillatory b ehaviors of covariances and
thresholds(cf.,forinstance,[1]and[2]).
4 Numericalcomparisons
0
1
2
3
4
0
0.5
1
1.5
0
1
2
3
4
0
0.5
1
1.5
0
1
2
3
4
0
0.5
1
1.5
0
1
2
3
4
0
0.5
1
1.5
(a)
(b)
(c)
(d)
Figure2: PlotsrefertoFPTp dfg(t)throughtheb oundary(13)with =0:5andd=0:25fora
zero-mean Gaussianpro cess characterized bycorrelation function (12). InFigure2(a)g(t)given
by(14)hasb een plotted. TheestimatedFPT p df ~g (t)with =10 10
isshownin Figure2(b),
with =0:25inFigure2(c)andwith =0:5in Figure2(d).
pro cessesandGaussiannon-Markovpro cesses,inordertoanalyzehowthelackofmemory
a ects the shap e of the density, also with reference to the sp eci ed typ e of correlation
function. Forsimplicity,set x
0
=0and P[X(0)=0]=1;sothatinthefollowingweshall
considertheFPTp dfg(t):=g[S(t);tj0;0].
To b e sp eci c, we consider a stationary Gaussian pro cess X(t) with zero mean and
correlationfunction (t):=e jtj cos( t); 2IR + (12)
which is the simplest typ e of correlation having a concrete engineering signi cance [14].
Whenthecorrelationfunctionisoftyp e(12),X(t)isnotmeansquaredi erentiable,since
_
(0)6=0. Thus theseries expansion(8)do es nothold. However,sp eci c assumptionson
theparameter helpuscharacterizetheshap eoftheFPTp df.
Westartassuming =0;sothatthecorrelationfunction(12) canb efactorizedas
(t)=e e (t ) 2R + ;0< <t:
Hence,cho osingh
1 (t)=e t andh 2 (t)=e t
in(2), X(t) b ecomesGauss-Markov.
0
0.5
1
1.5
0
1
2
3
4
5
0
0.5
1
1.5
0
1
2
3
4
5
0
0.5
1
1.5
0
1
2
3
4
5
0
0.5
1
1.5
0
1
2
3
4
5
(a)
(b)
(c) (d)
Figure3: SameasinFigura2withd=0:5.
integralequation(6). Inthefollowing,weconsiderb oundariesoftheform
S(t)=de t ( 1 e 2 t 1 2d 2 ln " 1 4 + 1 4 s 1+8exp 4d 2 e 2 t 1 #) ; (13)
with d > 0. It is evident that lim
t!0
S(t) = d and that S(t) tends to 0 ast increases.
Furthermore,asddecreases,theb oundaryb ecomes atter. InFigure1,theb oundaryS(t)
given in(13) isplottedfor =0:5 and fortwochoicesof the parameterd, i.e. d=0:25
andd=0:5.
As provedin[4]forb oundariesofthe form(13), theFPTp dfg(t) ofaGauss-Markov
pro cessadmitsthefollowingclosedform:
g(t)= 4d e t e 2 t 1 s 1+8exp 4d 2 e 2 t 1 1+ s 1+8exp 4d 2 e 2 t 1 f[S(t);tj0;0]; (14)
wheref[S(t);tj0;0]isthetransitionp dfoftheGauss-Markovpro cessX(t):
Fora zero-meanGauss-Markovpro cess characterized by the correlation function(12)
with = 0:5 and = 0, the FPT p df g(t) given by (14) through the b oundary (13) is
plottedinFigure2(a)ford=0:25and inFigure3(a) ford=0:5. Notethatasdincreases
Setting 6=0in(12),theGaussianpro cess X(t) isno longerMarkovandits sp ectral densityisgivenby (!)= 2 (! 2 + 2 + 2 ) ! 4 +2! 2 ( 2 2 )+( 2 + 2 ) 2 ; (15)
thus b eing of arational typ e. Since in(15) thedegree of the numerator is lessthan the
degree of the denominator, it is p ossibleto apply the simulationalgorithm describ ed in
Section 3inordertoestimatetheFPTp df~g(t)ofthepro cess.
Thesimulationpro cedurehasb eenimplementedbyaparallelFORTRAN90co deona
128-pro cessorIBMSP4sup ercomputer,basedonMPIlanguageforparallelpro cessing.The
numb erofsimulatedsamplepathshasb eenset equalto10 7
. TheestimatedFPTp df~g (t)
throughtheb oundary(13)with =0:5andd=0:25areplottedinFigures2(b)2(d)for
Gaussianpro cesseswith correlationfunction(12) having =10 10
;0:25;0:5, resp ectively.
Forthesamepro cesses, Figures3(b)3(d) showthe estimated FPTp df~g(t)throughthe
b oundary(13) with =0:5 andd=0:5. Note thatas increases,theshap e ofthe FPT
p df~g(t)b ecomes atterandtherelatedmo deincreases. Furthermore,asFigures2(a)-2(b)
andFigures3(a)-3(b)show,g(t)~ isverysimilartog(t)forsmallvaluesof .
Acknowledgement Workp erformedwithinajointco op erationagreementb etweenJapan
Science and Technology Corp oration (JST) and Universita di Nap oli Federico I I, under
partialsupp ortbyINdAM(G.N.C.S).
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E. Di Nardo: Dipartimento di Matematica,
Universit
a dellaBasilicata,
Con-trada MacchiaRomana, Potenza,Italy
A.G.Nobile: DipartimentodiMatematicaeInformatica,
Universit
adiSalerno,
ViaS. Allende,I-84081Baronissi(SA), Italy
E. Pirozzi and L.M. Ricciardi: Dipartimento di Matematica e Applicazioni,
Uni-versit
