Auditory-nerve Action Potentials Form a ... - Semantic Scholar
Auditory-nerve action potentials form a nonrenewal point process over short as well as long time scales Steven B. Lowen and Malvin C. Teich
Department of ElectricalEngineering, ColumbiaUniversity, New York,NewYork10027
(Received25 November1991;acceptedfor publication9 April 1992) The firingpatternsof auditory-nerveactionpotentialsexhibitlong-termfractalfluctuations that do not arisefrom the distributionof the intereventintervals,but rather from the ordering of theseintervals.Usingthe serialinterevent-interval correlationcoefficient,the Fano-factor timecurve,andshufflingof intereventintervals,it is shownthat adjacentintervalsfor spontaneous firingsexhibitsignificantcorrelation.The eventsare thereforenonrenewalover shortaswell aslongtime scales. PACS numbers:43.64.Pg,43.64.Yp, 43.64.Bt
INTRODUCTION
ciardi and Esposito,1966;Prucnaland Teich, 1983;Teich, 1985), whichis a renewalpointprocess, alsohasexponentiallydistributed interevent timesafterthedead-timeperiod, andhasbeenusedasa zeroth-order approximation to model suchdata.Two relatedrenewalpointprocesses that provide
It iswellknownthatin mammals, thepathwayfor the transferof information fromtheinnerearto higherauditory centersin the brain is providedby the VIIIth nerve.The neuralsignaltransmittedon individualfibersof this nerve somewhat better models make use of stochastic dead time hasbeenstudiedbymanyresearchers, withthegoalof gaining insightinto the mechanisms of informationencoding (Teich et al., 1978;Young and Barta, 1986) and relative (Galambosand Davis, 1943, 1948;Tasaki, 1954;Katsuki et al., 1958;Kiang etal., 1962, 1965;Roseet al., 1967, 1971;
deadtime, alsocalledsicktime (Gray, 1967;Teich and Dia-
merit, 1980; Gaumond et al., 1982).
Hind etaL, 1967;Evans,1972,1975;Kiang, 1984;Teichand Khanna,1985;YoungandBarta,1986;Teich,1989,1992). Thissignalcomprises a series of briefelectrical nervespikes, whoseamplitudeandenergyare widelyassumed not to be significant variables. Rather,it isgenerally accepted thatthe timesof occurrences of the spikescarrytheauditoryinfor-
However,despitetheexponentialcharacterof theinterevent-timehistogram,therenowexistscompelling evidence that, over long time scales,auditory-nervespike-traindata are not renewal (Teich, 1989;Teich et aL, 1990a,b;Woo,
mation. Randomnessis involved,sinceensemblesof identi-
1990;Powers,1991;Woo etal., 1992;Teich, 1992), but that
I. LONG-TERM
CORRELATIONS
cal single-fiber experiments lead to differingsequences of nervespikes(Tasaki,1954;Peakeet al., 1962;Rupertet al., 1963;Kiang, 1984).
From a mathematical pointof view,theneuralactivity in a peripheral auditoryfiberisperhaps bestcharacterized as an (unmarked)stochastic pointprocess (Parzen,1962).To takethesimplest case,weconsider auditoryneuralfiringsin the absence of any externalacousticstimulation. Auditory
10-2
neuronsspontaneously fire undersuchconditions,albeitat a
lowerratethanira stimuluswerepresent.Any realisticmodel of neuralspike-traindata mustincludethe effectsof dead time (absolute refractoriness), which limits the rate at whichneuronscan fire. After a relativelybrief (1-2 ms) dead-timeinterval,theneuronisreadyto fireagain,andthe
10-3
timeto thenextfiringeventapproximates an.exponentially distributedrandomvariable.(More precisely,the neuron recoversgradually,over a relativerefractoryperiodthat 10 • O 2O 40 6O laststensof milliseconds, ratherthan abruptly.)Figure 1 provides a histogram of theinterevent timesfor thespontaINTEREVENTTIME (rnsec) neousspiketrain from cat auditory-nerveunit A (which is 1.Semilogarithmic plotof relativefrequency of interevent times(intypical)ona semilogarithmic scale;thecurveapproximates FIG. terevent-time histogram)for unit A. The datacloselyfollowa straightline an exponential distributionafterthe dead-timeperiod.The for intereventtimesgreaterthanabout4 ms.Bin width is0.5 ms.The CF for this unit is 10.2 kHz. dead-timemodifiedPoissonpoint process(DTMP) (Ric4
803
J. Acoust. Sec.Am.92 (2),Pt.1,August 1992 0001-4966192/080803-04500.80•) 1992Acoustical Society ofAmerica
803
0.8
long-termpositivecorrelations arepresentin thedata.One measure of this correlation
is the Fano-factor
time curve
(FFC), definedas the variance of the number of eventsin a
specifiedcountingtime T dividedby the mean number of
eventsin thatcountingtime.In general,theFFC varieswith the countingtime T. For a DTMP process,the Fano factor neverexceedsunity for any countingtime, but for the audi0.7 tory-nerve spike-traindata the Fano factor increasesin a power-lawfashionbeyondunity for largecountingtimes. Figure 2 showsthe experimentalFEe for unit A (solid curve). Note that the Fano factor increasessteadily for countingtimesgreaterthanabout100ms,andexceeds10for countingtimes in excessof 30 s. This monotonicincrease indicatesthe presence of a fractalprocess, with fluctuations 0.6 20 40 60 80 100 on manytimescales, ratherthana collectionof a fewsimple processes with singletime constants, suchasbreathingor COUNTINGTIME T (msec) heartbeat.A plot of the FFC alone does not determine whetherthislargeFanofactorarisesfromthedistributionof FIG. 3. Doublylinearplotof Fano-factortimecurve(FFC) for unit A for the intereventtimesor their ordering.This issueis resolved countingtimeslessthan 100ms.The curvesfor the shuffleddataresemble by shuffling(randomly reordering)the intereventintervals eachotherbut are consistently higherthan the curvefor the unshuffled and then replottingthe Fano factorversuscountingtime. data. FFCs constructed from shuffled data yield information aboutthe relativesizesof the intervalsonly; all correlation anddependencies amongthe intervalsaredestroyedby shuffling.The nonsolid curvesin Fig. 2 illustratefivesuccessive shufflings.The shuffled interevent intervals have FFCs whichall approach a valuelessthanunityfor largecounting times,illustratingthatit istheorderingof theintervalsthat givesriseto thegrowthin theFCC. All auditory-nerve-fiber dataexaminedto date(for whichlong-duration spiketrains exist)showthistypeof behavior,bothin thepresence andin the absence of stimulation.
Thus the auditory-nervedata are more clusteredthan the DTMP process.The intereventintervalswill be relatively more similar within a long countingtime T than in the
DTMP process, leadingto a largervariationamongcounting times,and a largerFano factor.The FFC providesa sensitivetest for the presenceof clusters,detectingthem even when they are interleavedand thereforenot readily apparentin visualrepresentations of the auditory-nervefirings. Doubly stochasticmodels,such as the fractal-shotnoise-drivenPoissonpoint process(FSNDP) (Lowen and Teich, 1991) are not renewal,and fit auditory-nerve-fiber data well. The FSNDP modelcontainsinterdigitatedclusters,much as the auditory-nerve-fiber data appearto. For a varietyof statisticalmeasures,the data and simulationsof the FSNDP yield nearly identicalresults,particularlyfor time scales•f 100 ms or larger (Teich et aL, 1990b).
10
II. SHORT-TERM
Aside from the positivecorrelationover long times, there is also a small but significantcorrelationover time scaleson the orderof tensof milliseconds.Figure 3, whichis a magnification of Fig. 2 (but plottedwith linear coordinates), showsthis effect.Again, the FFCs for the shuffled intervalsare repeatablydifferentfrom thoseof the unshuffled intervals,and again they are closerto unity. For this particularunit, and in this rangeof countingtimesT, the FFC increases with shuffling,indicatinganticlusteredintereventintervals.Thus for thisunit, longintervalsare likely to be followedby shortintervalsand Diceoersa;over long time scales,as with all auditory units,the oppositeis true. Many
1
SHUFFLED ; [..•, 10 -4
10--•
10-2
10-1
10 0
CORRELATIONS
101
COUNTINGTIME •' (see)
of the units studied exhibit this short-time
FIG. 2. Doublylogarithmicplotof Fano-factortimecurve(FFC) for unit A. The curvefor theunshuffied dataapproximates a straightlineof positive slopeforcounting timeslargerthanseveral hundredmilliseconds, whereas the curve for the shuffled data does not rise. This indicates that the action
potentialsare clustered.
804
J. Acoust.Soc.Am.,Vol. 92, No. 2, Pt. 1, August1992
effect.
This behavioris demonstratedin Table I usinganother measure, the serial interevent-interval correlation coeffident (SIICC). The SIICC is the correlationcoefficientcalculatedbetweenadjacentintervals,and is definedby
S.B. Lowonand M. C. Teich:Nonrenewalauditory-nerve action
804
TABLEI. Serial interevent-interval correlation coefficients (SIICCs)fortendatasets recorded fromcatauditory-nerve fibers intheabsence ofstimulation. Theaverage interevent intervals foreachunit,inmilliseconds, areshown incolumn 2,andthetotalnumber ofevents incolumn3.SIICCvalues ar• shown for
databothbefore andaftershuffling, incolumns 4 and5.Alsoshown, incolumns 6 and7,arethelikelihood i thatanSIICCofthatmagnitude orlarger would beobtained bychance fromanidentical number ofuncorrelated intervals. Noneoftheshuffled datasets exhibits significant correlation, whilemost ofthe unshuffied data sets do.
Average interval Unit
{ ms)
Number of events
unshuffled
SIICC
15.3 15.2 13.5 16.4 27.0 344. 28.1 27.3 58.1 15.9
26709 15345 22575 37932 4560 206 2191 2603 1176 4190
-- 0.03732 -- 0.04535 q- 0.00441 - 0.03031 + 0.02223 + 0.02175 + 0.09428 + 0.08855 + 0.08675 -- 0.03828
(N--1)-• •E•=, (%-- (r))2
-+ ---+ + + -+
0.00310 0.00354 0.00731 0.00373 0.00647 0.08745 0.02796 0.00082 0.00157 0.01083
Likelihood unshuffled shuffled !.08 X 10-9 1.93X 10-s 5.08)< 10-• 3.58X 10-• 1.33X 10- • 7.56X 10-• 1.02X 10-• 6.28X 10-6 2.94)< 10-• 1.32X 10-2
6.12X 10- s 6.61X 10 • 2.72X 10-• 4.68X 10- • 6.62X 10- • 2.11X 10-• 1.91X 10-• 9.67X 10-i 9.57X 10- • 4.83X 10-•
lumns 6 and 7 of the: table for unshuffied and shuffled inter-
(N-- 2)-' X•_-j'(•-,-- (r)) (•-i. • -- (r))
P--=
shuffled
'
where N is the number of intervals,
vals, respectively.A relativelylarge probabilityindicates that the correspondingdata set is probably uncorrelated, while a small one indicates the existence of correlation. The
N
i--I
resultsshowthat the ten shuffleddata setsare not significanflycorrelated,as expected,sinceshufflingdestroysany
is the sampleaverageintereventtime,andthe r• represent correlation; however, most of the unshuffleddata setsdo the individualintereventintervals.The SIICC rangesfrom exhibitsignificant correlation. Interestingly,fl•ere is no clear trend in the data. Some - I (perfectanticorrelation)to q- 1 (perfectcorrelation), and is zero for uneorrelated intervals. The SIICC is calculatdatasetsexhibitsignificantpositivecorrelation,somesignificant negativecorrelation,and someno significantcorrelaedfor thedatabeforeandaftershuffling, andispresented in tion at all. The correlation does not seem to bc related to the columns4 and5 of thetable,respectively. Shuffling,i.e.,the randomrearrangement of the intereventintervals,provides averageintereventinterval.Thusnosimplemodelislikelyto sufficein describingthe relationshipbetweennearbyintera usefulcheckon theSIICC computation algorithm.A ranevent intervals. domcollectionof intervalsshouldexhibitnosignificantcorrelation. Similar resultsfor the SIICC (for unshuffleddata The fractalbehaviorunderlyingthe long-termcorrelationsis almostcertainlynot relatedto the short-termcorreonly) havebeenreportedby Cooper(1989). lations.The fractalpart of the Fano factor,asan example, TheseSIICC valuesdo not, by themselves, provethe existence or absence of correlation,sowe developa testof scalesas T ", with 0
Department of ElectricalEngineering, ColumbiaUniversity, New York,NewYork10027
(Received25 November1991;acceptedfor publication9 April 1992) The firingpatternsof auditory-nerveactionpotentialsexhibitlong-termfractalfluctuations that do not arisefrom the distributionof the intereventintervals,but rather from the ordering of theseintervals.Usingthe serialinterevent-interval correlationcoefficient,the Fano-factor timecurve,andshufflingof intereventintervals,it is shownthat adjacentintervalsfor spontaneous firingsexhibitsignificantcorrelation.The eventsare thereforenonrenewalover shortaswell aslongtime scales. PACS numbers:43.64.Pg,43.64.Yp, 43.64.Bt
INTRODUCTION
ciardi and Esposito,1966;Prucnaland Teich, 1983;Teich, 1985), whichis a renewalpointprocess, alsohasexponentiallydistributed interevent timesafterthedead-timeperiod, andhasbeenusedasa zeroth-order approximation to model suchdata.Two relatedrenewalpointprocesses that provide
It iswellknownthatin mammals, thepathwayfor the transferof information fromtheinnerearto higherauditory centersin the brain is providedby the VIIIth nerve.The neuralsignaltransmittedon individualfibersof this nerve somewhat better models make use of stochastic dead time hasbeenstudiedbymanyresearchers, withthegoalof gaining insightinto the mechanisms of informationencoding (Teich et al., 1978;Young and Barta, 1986) and relative (Galambosand Davis, 1943, 1948;Tasaki, 1954;Katsuki et al., 1958;Kiang etal., 1962, 1965;Roseet al., 1967, 1971;
deadtime, alsocalledsicktime (Gray, 1967;Teich and Dia-
merit, 1980; Gaumond et al., 1982).
Hind etaL, 1967;Evans,1972,1975;Kiang, 1984;Teichand Khanna,1985;YoungandBarta,1986;Teich,1989,1992). Thissignalcomprises a series of briefelectrical nervespikes, whoseamplitudeandenergyare widelyassumed not to be significant variables. Rather,it isgenerally accepted thatthe timesof occurrences of the spikescarrytheauditoryinfor-
However,despitetheexponentialcharacterof theinterevent-timehistogram,therenowexistscompelling evidence that, over long time scales,auditory-nervespike-traindata are not renewal (Teich, 1989;Teich et aL, 1990a,b;Woo,
mation. Randomnessis involved,sinceensemblesof identi-
1990;Powers,1991;Woo etal., 1992;Teich, 1992), but that
I. LONG-TERM
CORRELATIONS
cal single-fiber experiments lead to differingsequences of nervespikes(Tasaki,1954;Peakeet al., 1962;Rupertet al., 1963;Kiang, 1984).
From a mathematical pointof view,theneuralactivity in a peripheral auditoryfiberisperhaps bestcharacterized as an (unmarked)stochastic pointprocess (Parzen,1962).To takethesimplest case,weconsider auditoryneuralfiringsin the absence of any externalacousticstimulation. Auditory
10-2
neuronsspontaneously fire undersuchconditions,albeitat a
lowerratethanira stimuluswerepresent.Any realisticmodel of neuralspike-traindata mustincludethe effectsof dead time (absolute refractoriness), which limits the rate at whichneuronscan fire. After a relativelybrief (1-2 ms) dead-timeinterval,theneuronisreadyto fireagain,andthe
10-3
timeto thenextfiringeventapproximates an.exponentially distributedrandomvariable.(More precisely,the neuron recoversgradually,over a relativerefractoryperiodthat 10 • O 2O 40 6O laststensof milliseconds, ratherthan abruptly.)Figure 1 provides a histogram of theinterevent timesfor thespontaINTEREVENTTIME (rnsec) neousspiketrain from cat auditory-nerveunit A (which is 1.Semilogarithmic plotof relativefrequency of interevent times(intypical)ona semilogarithmic scale;thecurveapproximates FIG. terevent-time histogram)for unit A. The datacloselyfollowa straightline an exponential distributionafterthe dead-timeperiod.The for intereventtimesgreaterthanabout4 ms.Bin width is0.5 ms.The CF for this unit is 10.2 kHz. dead-timemodifiedPoissonpoint process(DTMP) (Ric4
803
J. Acoust. Sec.Am.92 (2),Pt.1,August 1992 0001-4966192/080803-04500.80•) 1992Acoustical Society ofAmerica
803
0.8
long-termpositivecorrelations arepresentin thedata.One measure of this correlation
is the Fano-factor
time curve
(FFC), definedas the variance of the number of eventsin a
specifiedcountingtime T dividedby the mean number of
eventsin thatcountingtime.In general,theFFC varieswith the countingtime T. For a DTMP process,the Fano factor neverexceedsunity for any countingtime, but for the audi0.7 tory-nerve spike-traindata the Fano factor increasesin a power-lawfashionbeyondunity for largecountingtimes. Figure 2 showsthe experimentalFEe for unit A (solid curve). Note that the Fano factor increasessteadily for countingtimesgreaterthanabout100ms,andexceeds10for countingtimes in excessof 30 s. This monotonicincrease indicatesthe presence of a fractalprocess, with fluctuations 0.6 20 40 60 80 100 on manytimescales, ratherthana collectionof a fewsimple processes with singletime constants, suchasbreathingor COUNTINGTIME T (msec) heartbeat.A plot of the FFC alone does not determine whetherthislargeFanofactorarisesfromthedistributionof FIG. 3. Doublylinearplotof Fano-factortimecurve(FFC) for unit A for the intereventtimesor their ordering.This issueis resolved countingtimeslessthan 100ms.The curvesfor the shuffleddataresemble by shuffling(randomly reordering)the intereventintervals eachotherbut are consistently higherthan the curvefor the unshuffled and then replottingthe Fano factorversuscountingtime. data. FFCs constructed from shuffled data yield information aboutthe relativesizesof the intervalsonly; all correlation anddependencies amongthe intervalsaredestroyedby shuffling.The nonsolid curvesin Fig. 2 illustratefivesuccessive shufflings.The shuffled interevent intervals have FFCs whichall approach a valuelessthanunityfor largecounting times,illustratingthatit istheorderingof theintervalsthat givesriseto thegrowthin theFCC. All auditory-nerve-fiber dataexaminedto date(for whichlong-duration spiketrains exist)showthistypeof behavior,bothin thepresence andin the absence of stimulation.
Thus the auditory-nervedata are more clusteredthan the DTMP process.The intereventintervalswill be relatively more similar within a long countingtime T than in the
DTMP process, leadingto a largervariationamongcounting times,and a largerFano factor.The FFC providesa sensitivetest for the presenceof clusters,detectingthem even when they are interleavedand thereforenot readily apparentin visualrepresentations of the auditory-nervefirings. Doubly stochasticmodels,such as the fractal-shotnoise-drivenPoissonpoint process(FSNDP) (Lowen and Teich, 1991) are not renewal,and fit auditory-nerve-fiber data well. The FSNDP modelcontainsinterdigitatedclusters,much as the auditory-nerve-fiber data appearto. For a varietyof statisticalmeasures,the data and simulationsof the FSNDP yield nearly identicalresults,particularlyfor time scales•f 100 ms or larger (Teich et aL, 1990b).
10
II. SHORT-TERM
Aside from the positivecorrelationover long times, there is also a small but significantcorrelationover time scaleson the orderof tensof milliseconds.Figure 3, whichis a magnification of Fig. 2 (but plottedwith linear coordinates), showsthis effect.Again, the FFCs for the shuffled intervalsare repeatablydifferentfrom thoseof the unshuffled intervals,and again they are closerto unity. For this particularunit, and in this rangeof countingtimesT, the FFC increases with shuffling,indicatinganticlusteredintereventintervals.Thus for thisunit, longintervalsare likely to be followedby shortintervalsand Diceoersa;over long time scales,as with all auditory units,the oppositeis true. Many
1
SHUFFLED ; [..•, 10 -4
10--•
10-2
10-1
10 0
CORRELATIONS
101
COUNTINGTIME •' (see)
of the units studied exhibit this short-time
FIG. 2. Doublylogarithmicplotof Fano-factortimecurve(FFC) for unit A. The curvefor theunshuffied dataapproximates a straightlineof positive slopeforcounting timeslargerthanseveral hundredmilliseconds, whereas the curve for the shuffled data does not rise. This indicates that the action
potentialsare clustered.
804
J. Acoust.Soc.Am.,Vol. 92, No. 2, Pt. 1, August1992
effect.
This behavioris demonstratedin Table I usinganother measure, the serial interevent-interval correlation coeffident (SIICC). The SIICC is the correlationcoefficientcalculatedbetweenadjacentintervals,and is definedby
S.B. Lowonand M. C. Teich:Nonrenewalauditory-nerve action
804
TABLEI. Serial interevent-interval correlation coefficients (SIICCs)fortendatasets recorded fromcatauditory-nerve fibers intheabsence ofstimulation. Theaverage interevent intervals foreachunit,inmilliseconds, areshown incolumn 2,andthetotalnumber ofevents incolumn3.SIICCvalues ar• shown for
databothbefore andaftershuffling, incolumns 4 and5.Alsoshown, incolumns 6 and7,arethelikelihood i thatanSIICCofthatmagnitude orlarger would beobtained bychance fromanidentical number ofuncorrelated intervals. Noneoftheshuffled datasets exhibits significant correlation, whilemost ofthe unshuffied data sets do.
Average interval Unit
{ ms)
Number of events
unshuffled
SIICC
15.3 15.2 13.5 16.4 27.0 344. 28.1 27.3 58.1 15.9
26709 15345 22575 37932 4560 206 2191 2603 1176 4190
-- 0.03732 -- 0.04535 q- 0.00441 - 0.03031 + 0.02223 + 0.02175 + 0.09428 + 0.08855 + 0.08675 -- 0.03828
(N--1)-• •E•=, (%-- (r))2
-+ ---+ + + -+
0.00310 0.00354 0.00731 0.00373 0.00647 0.08745 0.02796 0.00082 0.00157 0.01083
Likelihood unshuffled shuffled !.08 X 10-9 1.93X 10-s 5.08)< 10-• 3.58X 10-• 1.33X 10- • 7.56X 10-• 1.02X 10-• 6.28X 10-6 2.94)< 10-• 1.32X 10-2
6.12X 10- s 6.61X 10 • 2.72X 10-• 4.68X 10- • 6.62X 10- • 2.11X 10-• 1.91X 10-• 9.67X 10-i 9.57X 10- • 4.83X 10-•
lumns 6 and 7 of the: table for unshuffied and shuffled inter-
(N-- 2)-' X•_-j'(•-,-- (r)) (•-i. • -- (r))
P--=
shuffled
'
where N is the number of intervals,
vals, respectively.A relativelylarge probabilityindicates that the correspondingdata set is probably uncorrelated, while a small one indicates the existence of correlation. The
N
i--I
resultsshowthat the ten shuffleddata setsare not significanflycorrelated,as expected,sinceshufflingdestroysany
is the sampleaverageintereventtime,andthe r• represent correlation; however, most of the unshuffleddata setsdo the individualintereventintervals.The SIICC rangesfrom exhibitsignificant correlation. Interestingly,fl•ere is no clear trend in the data. Some - I (perfectanticorrelation)to q- 1 (perfectcorrelation), and is zero for uneorrelated intervals. The SIICC is calculatdatasetsexhibitsignificantpositivecorrelation,somesignificant negativecorrelation,and someno significantcorrelaedfor thedatabeforeandaftershuffling, andispresented in tion at all. The correlation does not seem to bc related to the columns4 and5 of thetable,respectively. Shuffling,i.e.,the randomrearrangement of the intereventintervals,provides averageintereventinterval.Thusnosimplemodelislikelyto sufficein describingthe relationshipbetweennearbyintera usefulcheckon theSIICC computation algorithm.A ranevent intervals. domcollectionof intervalsshouldexhibitnosignificantcorrelation. Similar resultsfor the SIICC (for unshuffleddata The fractalbehaviorunderlyingthe long-termcorrelationsis almostcertainlynot relatedto the short-termcorreonly) havebeenreportedby Cooper(1989). lations.The fractalpart of the Fano factor,asan example, TheseSIICC valuesdo not, by themselves, provethe existence or absence of correlation,sowe developa testof scalesas T ", with 0
