Biological Cybernetics
Biol. Cybern. 62, 487-493 (1990)
9 Springer-Verlag 1990
Recurring Discharge Patterns in Multiple Spike Trains I Detection
R. D. Frostig, Z. Frostig, and R. M. Harper Neuroscience Program, The Brain Research Institute and Department of Anatomy, University of California at Los Angeles, Los Angeles, CA90024-1763, USA
Abstract. We present a procedure to detect recurring
discharge patterns in multiple spike trains. Such recurring patterns can include many spikes and involve from three to many spike trains. The pattern detection procedure is based on calculating the exact probability of randomly obtaining each individually recurring pattern. The statistical evaluation is based on the use of 2 x 2 contingency tables and the application of Fisher's exact test. Several simulations are applied to evaluate the method. Findings based on applying the procedure to simultaneously recorded spike and event trains are described in a companion paper (Frostig et al. 1990).
Introduction
The temporal structure of neuronal discharge in a single spike train frequently deviates from random sequences (Frostig et al. 1984; Klemm and Sherry 1982; Legendy and Salcman 1985; Levine 1980; Yamamoto and Nakahama 1983). Spike trains produce nonrandom sequences of short and long intervals (Klemm and Sherry 1982) or exhibit repeating epochs of successive interspike interval sequences termed "favored patterns" (Dayhoff and Gerstein 1983a, b). Moreover, repeating complex patterns including nonsuccessive interspike intervals have also been described in single spike trains (Abeles et al. 1983). Such patterns may result from recurring multineuronal interactions reflected in a single spike train. To test this possibility, a procedure for detecting such recurring multiple spike train patterns is needed. The characterization and quantification of recurring patterns between simultaneously recorded spike trains may contribute to our understanding of the rules by which neurons assemble into functional groups. Existing evidence for recurring patterns, obtained using a quantified version of the
"snowflake" technique (Abeles 1983; Perkel et al. 1975), indicates repeating patterns that involve no more than three spike trains. Thus, our aim was to develop a pattern detection method that could handle more than three spike trains and would be able to detect patterns which do not frequently recur. Recently, another method which is capable of detecting such patterns was described (Abeles and Gerstein 1988). The detecting procedure described in this paper is a generalization of a technique that extracts spikes labeled as "information carriers" from spike trains (Frostig et al. 1984). In a companion paper (Frostig et al. 1990), we describe the application of the procedure to simultaneously recorded spike trains obtained from forebrain areas of unrestrained, drug-free cats during different sleep-waking states.
Methods
We will describe here the pattern detecting method for precise (1 ms) patterns of discharge which repeat without addition or omission of spikes. The method uses two basic steps. In step one all spike trains are divided into groups of three spike trains each having one combination of the three. Each group is searched for repeated patterns, and for each recurring pattern a decision based on the statistical probability of obtaining such pattern is made. If the recurrence is statistically significant the pattern is saved for the next step. In step two the patterns found in step one are checked for expansion with spikes in other spike trains using the same statistical decision. This expansion process can progressively include any number of spike trains. The following requirements were imposed for the choice of a proper statistical test for recurring pattern detection: a) to apply the test no prior knowledge about the underlying distribution of the intervals of
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Fig. 1. Simulated example of a recurring pattern involving three spike trains. There are 12 AB intervals of 7 ms (small horizontal arrows under the B train) and 10 AC intervals of 10 ms (large horizontal arrows under the C train). All the spikes in A train (the reference train) that belong to either an AB interval or an AC interval are indicated with vertical arrows. The joint recurrences of both AB and AC, which produce a recurring pattern ABC, are indicated by stars above the A train spikes that belong to these patterns the spikes trains is needed b) the test should be tuned to detect association between spike occurrences c) the test can be applied on small sample sizes. The test of choice was Fisher's exact probability test (Hayes 1981) which is compatible with all three requirements. However, the use of Fisher's exact test has the following disadvantages: a) the pattern search starts with three trains rather than two b) the pattern search is done in a step by step fashion and not on all the spike trains at once. To illustrate the procedure a hypothetical example of three spike trains A, B, and C will be considered (Fig. 1). A spike in train A was followed after 7 ms by a spike in train B (AB interval = 7 ms) and also followed, 10 ms later, by a spike in train C (AC interval = 10 ms). We tested the hypothesis that the observed pattern was randomly produced using Fisher's exact test. Fisher's test was applied to examine the association between the occurrence of the A B interval and the occurrence of the AC interval, given that a spike occurred in train A (the reference train). Thus, each time a spike occurred in train A, a window with a selected width was initiated from such a spike (in the same way as in a crosscorrelogram). The n u m b e r of such windows was equal to the n u m b e r of spikes in train A and was labeled Na. The n u m b e r of windows that included the specific AB interval was counted and the total was labeled Nab, while the n u m b e r of windows including the specific AC interval was labeled Nac. The number of windows containing joint occurrences of both AB and AC intervals was labeled N(abc~ac). Using the four p a r a m eters Na, Nab, Nac, and N(abnac), a 2 x 2 contingency table, typically used in Fisher's exact test, was constructed (Table 1). In the example presented in Fig. 1, spike train A included 600 spikes (Na = 600). Out of the 600 spikes (for clarity, only a subset of A's 600 spikes is shown in Fig. 1), 12 were followed after 7 ms by a spike in train B (Nab = 12) and 10 spikes were followed 10 ms later by a spike in train C (Nac= 10). In 5 of 12 AB occurrences and of 10 AC occurrences, both intervals
Table 1. A typical 2 x 2 contingency table for the detection of patterns involving three spike trains A, B, and C. No: number of spikes in A train; Nab: number of spikes in A followed by AB interval; Nac:number of spikes in A followed by AC interval; N~: number of spikes in A not followed by AB interval; Na~:number of spikes in A not followed by AC interval; N(abn,c):number of spikes in A train followed by AB interval and AC interval; N(ab~): number of spikes in A train followed by AB interval but not by AC interval; N(~n,c): number of spikes in A followed by A C interval but not by AB interval; N(a~a~: number of spikes in A followed by neither AB interval nor AC interval
N(abnac)
N(abc~)
Nab
N~
Na~
N~
Table 2. A numeric example illustrating the use of a contingency table for pattern detection (see text) 5
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5
583
588
10
590
600
appeared jointly in the same window (i.e., related to the same spike in A). This joint appearance of Nab and Nac intervals related to the same spikes in A produced the repeating pattern ABC (N(abc~ac)= 5). Assuming that the occurrence of AB intervals was independent of the occurrence of AC intervals (the null hypothesis), the expected probability for joint occurrence should be the product of the probabilities of the individual events, namely, P(abc~ac)expected=Pab*Pac. Using Fisher's test, one can calculate the exact probability of randomly obtaining the observed joint event N(abc~ac) times or more. F o r this example (the full contingency table is shown in Table 2), the probability of randomly obtaining five or more of these joint events was calculated to be 0.000000298. Depending on the choice
489 Table 3. A typical 2 x 2 contingencytable for the detection of patterns involvingfour spike trains A, B, C, and D N(abcc~aa) a'a-~r~aa) N.a
N(abcna-d)
Nabc
Nt~,~a)
N.~-ff~ N~-b7
N~
N.
of significance level the null hypothesis is accepted or rejected. If the null hypothesis is rejected the alternative hypothesis that a positive association exists between the joint occurrences of the AB interval and the AC interval, producing a recurring ABC pattern, is accepted. If there were a fourth spike train D, we could check whether a detected pattern could be expanded to include more than three trains. Table 3 presents a 2 x 2 contingency table constructed to expand a three train pattern (e.g., ABC) to a four train pattern (e.g., ABCD). For this case of expansion, an association was sought between an already detected pattern between three spike trains and spikes in the fourth train. Thus, Fisher's exact test was adapted to test for a possible association of a simple event (e.g., the occurrence of an AD interval) with a complex event (e.g., the joint occurrence of AB and AC intervals) given that a spike occurred in the reference train (train A in this case). Rejecting the null hypothesis for this case would associate the occurrence of ABC with the occurrences of a specific AD interval, producing a recurring ABCD pattern. Given N spike trains, expansion of patterns could proceed from K to K + 1 trains (K_
Biol. Cybern. 62, 487-493 (1990)
9 Springer-Verlag 1990
Recurring Discharge Patterns in Multiple Spike Trains I Detection
R. D. Frostig, Z. Frostig, and R. M. Harper Neuroscience Program, The Brain Research Institute and Department of Anatomy, University of California at Los Angeles, Los Angeles, CA90024-1763, USA
Abstract. We present a procedure to detect recurring
discharge patterns in multiple spike trains. Such recurring patterns can include many spikes and involve from three to many spike trains. The pattern detection procedure is based on calculating the exact probability of randomly obtaining each individually recurring pattern. The statistical evaluation is based on the use of 2 x 2 contingency tables and the application of Fisher's exact test. Several simulations are applied to evaluate the method. Findings based on applying the procedure to simultaneously recorded spike and event trains are described in a companion paper (Frostig et al. 1990).
Introduction
The temporal structure of neuronal discharge in a single spike train frequently deviates from random sequences (Frostig et al. 1984; Klemm and Sherry 1982; Legendy and Salcman 1985; Levine 1980; Yamamoto and Nakahama 1983). Spike trains produce nonrandom sequences of short and long intervals (Klemm and Sherry 1982) or exhibit repeating epochs of successive interspike interval sequences termed "favored patterns" (Dayhoff and Gerstein 1983a, b). Moreover, repeating complex patterns including nonsuccessive interspike intervals have also been described in single spike trains (Abeles et al. 1983). Such patterns may result from recurring multineuronal interactions reflected in a single spike train. To test this possibility, a procedure for detecting such recurring multiple spike train patterns is needed. The characterization and quantification of recurring patterns between simultaneously recorded spike trains may contribute to our understanding of the rules by which neurons assemble into functional groups. Existing evidence for recurring patterns, obtained using a quantified version of the
"snowflake" technique (Abeles 1983; Perkel et al. 1975), indicates repeating patterns that involve no more than three spike trains. Thus, our aim was to develop a pattern detection method that could handle more than three spike trains and would be able to detect patterns which do not frequently recur. Recently, another method which is capable of detecting such patterns was described (Abeles and Gerstein 1988). The detecting procedure described in this paper is a generalization of a technique that extracts spikes labeled as "information carriers" from spike trains (Frostig et al. 1984). In a companion paper (Frostig et al. 1990), we describe the application of the procedure to simultaneously recorded spike trains obtained from forebrain areas of unrestrained, drug-free cats during different sleep-waking states.
Methods
We will describe here the pattern detecting method for precise (1 ms) patterns of discharge which repeat without addition or omission of spikes. The method uses two basic steps. In step one all spike trains are divided into groups of three spike trains each having one combination of the three. Each group is searched for repeated patterns, and for each recurring pattern a decision based on the statistical probability of obtaining such pattern is made. If the recurrence is statistically significant the pattern is saved for the next step. In step two the patterns found in step one are checked for expansion with spikes in other spike trains using the same statistical decision. This expansion process can progressively include any number of spike trains. The following requirements were imposed for the choice of a proper statistical test for recurring pattern detection: a) to apply the test no prior knowledge about the underlying distribution of the intervals of
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Fig. 1. Simulated example of a recurring pattern involving three spike trains. There are 12 AB intervals of 7 ms (small horizontal arrows under the B train) and 10 AC intervals of 10 ms (large horizontal arrows under the C train). All the spikes in A train (the reference train) that belong to either an AB interval or an AC interval are indicated with vertical arrows. The joint recurrences of both AB and AC, which produce a recurring pattern ABC, are indicated by stars above the A train spikes that belong to these patterns the spikes trains is needed b) the test should be tuned to detect association between spike occurrences c) the test can be applied on small sample sizes. The test of choice was Fisher's exact probability test (Hayes 1981) which is compatible with all three requirements. However, the use of Fisher's exact test has the following disadvantages: a) the pattern search starts with three trains rather than two b) the pattern search is done in a step by step fashion and not on all the spike trains at once. To illustrate the procedure a hypothetical example of three spike trains A, B, and C will be considered (Fig. 1). A spike in train A was followed after 7 ms by a spike in train B (AB interval = 7 ms) and also followed, 10 ms later, by a spike in train C (AC interval = 10 ms). We tested the hypothesis that the observed pattern was randomly produced using Fisher's exact test. Fisher's test was applied to examine the association between the occurrence of the A B interval and the occurrence of the AC interval, given that a spike occurred in train A (the reference train). Thus, each time a spike occurred in train A, a window with a selected width was initiated from such a spike (in the same way as in a crosscorrelogram). The n u m b e r of such windows was equal to the n u m b e r of spikes in train A and was labeled Na. The n u m b e r of windows that included the specific AB interval was counted and the total was labeled Nab, while the n u m b e r of windows including the specific AC interval was labeled Nac. The number of windows containing joint occurrences of both AB and AC intervals was labeled N(abc~ac). Using the four p a r a m eters Na, Nab, Nac, and N(abnac), a 2 x 2 contingency table, typically used in Fisher's exact test, was constructed (Table 1). In the example presented in Fig. 1, spike train A included 600 spikes (Na = 600). Out of the 600 spikes (for clarity, only a subset of A's 600 spikes is shown in Fig. 1), 12 were followed after 7 ms by a spike in train B (Nab = 12) and 10 spikes were followed 10 ms later by a spike in train C (Nac= 10). In 5 of 12 AB occurrences and of 10 AC occurrences, both intervals
Table 1. A typical 2 x 2 contingency table for the detection of patterns involving three spike trains A, B, and C. No: number of spikes in A train; Nab: number of spikes in A followed by AB interval; Nac:number of spikes in A followed by AC interval; N~: number of spikes in A not followed by AB interval; Na~:number of spikes in A not followed by AC interval; N(abn,c):number of spikes in A train followed by AB interval and AC interval; N(ab~): number of spikes in A train followed by AB interval but not by AC interval; N(~n,c): number of spikes in A followed by A C interval but not by AB interval; N(a~a~: number of spikes in A followed by neither AB interval nor AC interval
N(abnac)
N(abc~)
Nab
N~
Na~
N~
Table 2. A numeric example illustrating the use of a contingency table for pattern detection (see text) 5
7
12
5
583
588
10
590
600
appeared jointly in the same window (i.e., related to the same spike in A). This joint appearance of Nab and Nac intervals related to the same spikes in A produced the repeating pattern ABC (N(abc~ac)= 5). Assuming that the occurrence of AB intervals was independent of the occurrence of AC intervals (the null hypothesis), the expected probability for joint occurrence should be the product of the probabilities of the individual events, namely, P(abc~ac)expected=Pab*Pac. Using Fisher's test, one can calculate the exact probability of randomly obtaining the observed joint event N(abc~ac) times or more. F o r this example (the full contingency table is shown in Table 2), the probability of randomly obtaining five or more of these joint events was calculated to be 0.000000298. Depending on the choice
489 Table 3. A typical 2 x 2 contingencytable for the detection of patterns involvingfour spike trains A, B, C, and D N(abcc~aa) a'a-~r~aa) N.a
N(abcna-d)
Nabc
Nt~,~a)
N.~-ff~ N~-b7
N~
N.
of significance level the null hypothesis is accepted or rejected. If the null hypothesis is rejected the alternative hypothesis that a positive association exists between the joint occurrences of the AB interval and the AC interval, producing a recurring ABC pattern, is accepted. If there were a fourth spike train D, we could check whether a detected pattern could be expanded to include more than three trains. Table 3 presents a 2 x 2 contingency table constructed to expand a three train pattern (e.g., ABC) to a four train pattern (e.g., ABCD). For this case of expansion, an association was sought between an already detected pattern between three spike trains and spikes in the fourth train. Thus, Fisher's exact test was adapted to test for a possible association of a simple event (e.g., the occurrence of an AD interval) with a complex event (e.g., the joint occurrence of AB and AC intervals) given that a spike occurred in the reference train (train A in this case). Rejecting the null hypothesis for this case would associate the occurrence of ABC with the occurrences of a specific AD interval, producing a recurring ABCD pattern. Given N spike trains, expansion of patterns could proceed from K to K + 1 trains (K_
