A simple method of cross-correlating spike trains demonstrated on primary muscle spindle ending discharges

Neuroscience Letters, 5 (1977) 57--61 © Elsevier/North-Holland Scientific Publishers Ltd.

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A SIMPLE METHOD OF CROSS-CORRELATING SPIKE T R A I N S DEMONSTRATED ON PRIMARY MUSCLE SPINDLE ENDING DISCHARGES U. WINDHORST Physiologisches Institut der Universit~t GiJttingen, Lehrstuhl II, GiJttir, gen (G.F.R.) (Received February 8th, 1977) (Revised version received March 14th, 1977) (Accepted March 14th, 1977)

SUMMARY

A simple method is shown by which spike trains can be correlated. It is based on the Parzen convolution estimation and can easily be implemented with some electronic hardware and an averaging computer. The method is demonstrated on spike trains from primary muscle spindle endings. Advantages and drawbacks are discussed.

In contemporary neurophysiology extensive use is made of auto- and crosscorrelation techniques in the treatment of spike tr~ins from neurones in order to detect any functional relationships between these cells (more recent papers with further references are refs. 1--4; for methodological reviews see refs. 6 and 7). The interaction between two cells can be inferred to a certain extent [4] from the influence which the occurrence of a spike in one train (A) exerts upon the spike density at continuous lags r in the second train (B). A straightforward approach to estimating this 'conditional density' would be to convert the sequence of spikes in the first train A to reference triggers (by passing it through a level detector), with respect to which the (original) analog signal obtained from the second cell is averaged in a common averaging computer. It would soon show up, however, that a vast number of triggered sweeps would have to be evaluated in order to get a somewhat smooth estimate. This is due to the usually small duration of action potentials from nerve cells. It would thus seem reasonable to increase the duration or 'pulse width' of the spikes in train B or, in other words, to convert the spike train into an artificial pulse train in which the duration and height of the pulses can be adjusted arbitrarily. This is easily done by passing the spike train B through a level detector and a subsequent pulse shaping unit. Indeed, this simple method ol cross-correlatingspike trainsworks very well if certain conditions are observed (see below). It has a theoreticalbackground in the so-called

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gfrom e+mch selected ads a r e salected ccepted; triggeringof each subseq'Jent analysisperiod onlybeing possible a£~r the termination of the preceding one, Only a portion of allpossible triggersin C are thus actually used, thisportion being smaller, the higher the mean discharge frequency in A and the longer the anelysis period are. The ensuing reduction of the sample size callsfor a careful choice of the analysis period. This and further practical

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problems are discussed with respect to Fig. 2, which shows auto- and crosscorrelograms for spike trains from two primary muscle spindle endings in an

the peak at the origin of the cross-colrrelograms indicates a more than random probability that the two cells will fi~, ~imultaneously. Assigning significance to such conspicuous features of a cross-correlogram has always been a difficult problem to solve mathematically [3,7], an empL~ical solution usually being more easily implemented. The reference spike train A is replaced by a regular trigger pulse train, generated by a function generator, with approximately the same frequency as the mean frequency of train A. Train B is then correlated with this artificial trigger train. The resultant cross-correlogram should be fiat to within random fluctuations (cf. Fig. 2, 'controls'), The horizontal lines in this figure are drawn through the maximal occurring excursions of the five superimposed correlograms: the probability of any excursions randomly exceeding these 'confidence limits' is very low. It can be seen from the cross-correlol~ams that the significant peak at the origin is reproduced in the computations with two different analysis periods (100 and 200 msec), apart from random fluctuations due to different samples. In general, the analysis period should be chosen as short as is compatible with the requirements of the problem investigated. In most cases, analysis periods can be confined to the order of 100 msec, except when special circumstances prevail. The analysis period and the length of the recording determine the number of evaluable triggers in C. With this number given, the smoothness of the estimates is highly dependent upon the pulse width A t as can be seen from Fig. 2. The choice of A t obviously depends on the characteristics .of the correlated spike trains. The more amdysis periods are actually available and the higher the discharge frequency m B, the smaller At can be chosen to increase the time resolution, but it should be several times longer than the spike represented. For low numbers of analysis periods and low discharge frequencies in B, ~ t should be increased, but must not exceed the shortest interspike intervals in B. Othe]~ise spikes in B would be lost. In this sense, A t ffi 8 msec in Fig. 2 is near the upper limit of possible values whereas At ffi 2 msec is definitely too low. An intermediate value of 5--6 msec would be a good compromise. The main drawback of the described method lies in the reduction of sample size (see above). This is compensated for by the following advantage, however. In order to achieve a certain accuracy of estimation or smoothness, the proposed technique requires a sample of only a fifth t o a tenth in size compared with conventional histogram techniques (see ref° 8). The condition for such an efficiency is an approp~ate choice of z~t. Other advantages are: the de:~,elopment of the correlograms c~n be monitored and documented at all

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stages of the computation. For screening, the computation can be done vnline (except for controls). By use of an averager with multiple input channels, several correlations can be computed simultaneously (cf. Fig. 2), and other analog signals can be averaged with respect to the same reference train. ACKNOWLEDGEMENTS

I am very grateful to Prof. H.-D. Henatsch, Drs. J. Meyer-Lohmann and J. Schmidt for helpful criticism on the manuscript, and to Mr. Howard Schultens for improving the English text. REFERENCES 1 Brillinger, D., Bryant, H.L. and Segundo, J.P., Identification of synaptic interactions, Biol. Cybemet., 22 (1976) 213--228. 2 Bryant. H.L., Ruiz Marcos, A. and Segundo, J.P., Correlations of neuronal spike discharges produced by monosynaptic connections and by common inputs, J. Neurophysiol., 36 (1973) 205--225. 3 Johnson, D.H. and Kiang, N.Y.S., Analysis of discharges recorded simultaneously from pairs of auditory nerve fibers, Biophys. J., 16 (1976) 719--734. 4 Moore, G.P., Segundo, J.P., Perkel, D.H. and Levitan, H., Statistical signs of synaptic interactions in neurons, Biophys. J., 10 (1970) 876--900. 5 Parzen, E., On estimation of a probability density function and mode, Ann. Math. Statist., 33 (1962) 1065--1076. 6 Perkel, D.H., Gerstein, G.L. and Moore, G.P., Neuronal spike trains and stochastic point processes. I. The single spike train, B~ophys. J., 7 (1967) 391--418. 7 Perkel, D.H., Gerstein, G.L. and Moore, G.P., Neuronal spike trains and stochastic point processes. IL Simultaneous spike trains, Biophys. J., 7 (1967) 419--440. 8 Sanderson, A.C. and Kobler, B., Sequential interval histogram analysis of non-stationary neuronal spike traivs, Biol. Cybemet., 22 (1976) 61--71.