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On the statistical significance of temporal firing patterns in multi-neuronal spike trains

Repeated occurrences of serial firing sequences of a group of neurons with fixed time delays between neurons are observed in many experiments involving simultaneous recordings from multiple neurons. Such temporal patterns are potentially indicative of underlying microcircuits and it is important to know when a repeatedly occurring pattern is statistically significant. These sequences are typically identified through correlation counts, such as in the two-tape algorithm of Abeles and Gerstein. In this paper we present a method for deciding on the significance of such correlations by characterizing the influence of one neuron on another in terms of conditional probabilities and specifying our null hypothesis in terms of a bound on the conditional probabilities. This method of testing significance of correlation counts is more general than the currently available methods since under our null hypothesis we do not assume that the spiking processes of different neurons are independent. The structure of our null hypothesis also allows us to rank order the detected patterns in terms of the strength of interaction among the neurons constituting the pattern. We demonstrate our method of assessing significance on simulated spike trains involving inhomogeneous Poisson processes with strong interactions, where the correlation counts are obtained using the two-tape algorithm.


Introduction
Detection of temporal ring patterns among groups of neurons is an important task in neuroscience as such patterns may be indicative of functional cell assemblies or microcircuits present in the underlying neural tissue [4,12]. Analysis methods such as the two-tape algorithm of Abeles and Gerstein [1,2] have been developed to discover repeating occurrences of precise ring sequences in simultaneous recordings from multiple neurons. The two-tape algorithm has been used to assess precisely timed activity patterns in vivo [10], in slices in vitro [8], and recently in cultures of dissociated cortical neurons [14]. The two-tape algorithm, as well as alternative methods for identifying spike coincidences such as the unitary event detection algorithm of Grun [6], determines statistical signicance of the discovered patterns under a null hypothesis of independence among the neurons [3]. In this paper, we present a signicance test that allows weak interactions to be included in the null hypothesis by characterizing the strength of inuence among neurons in terms of a conditional probability, which is a very natural way to think about synaptic interactions between neurons).
Let us write a 3-neuron pattern as A T1 → B T2 → C, denoting a ring sequence where A is followed by B after a delay of T 1 and B is followed by C after a delay of T 2 . Most methods count the occurrences of this pattern by essentially nding correlations among time shifted spike trains from A, B, and C.
There are also many methods to determine the statistical signicance of these patterns based on how many times they occur (see [5] and references therein). To assess whether a given number of repetitions of the pattern is signicant, one generally employs a null hypothesis that assumes that all neurons spike as (possibly inhomogeneous) Poisson processes and that dierent neurons are independent. Based on this one can analytically calculate a bound on the number of repetitions required to make a pattern signicant in the sense of being able to reject the null hypothesis at a given level of condence. There are also methods to assess signicance through empirical means [7]. In these methods, one generates many surrogate data streams of spike trains by systematically perturbing the spikes in the original data and then assesses signicance of a pattern by noting the dierence in counts (or in any other statistic derived from such correlation counts) for these patterns in the original data and in the surrogate streams. In these jitter methods, often the implicit null hypothesis is also independence.
When a sequential ring pattern like A T1 → B T2 → C is declared as signicant by any of these methods, the underlying idea is that we can conclude that A, B, and C inuence each other in a sequential fashion because we are able to reject the null hypothesis of independence. However, these methods do not say anything about the strength of inuence among neurons A, B, and C. We present a method for analyzing statistical signicance of sequential ring patterns that also allows rank ordering of signicant patterns in terms of the strength of inuence among participating neurons.
Thus our method extends the currently available techniques of signicance analysis.
We represent the strength of inuence of A on B by the conditional probability that B would re after the delay T 1 given that A has red now, which we denote as e(B|A, T 1 ). Our null hypothesis is then stated in terms of a bound on all such pairwise conditional probabilities and we develop a statistical signicance test for rejecting the null hypothesis. By changing the parameter of the null hypothesis (which is the bound on the conditional probability), we are able to see which patterns are signicant at dierent levels of strength of inuence.
Another interesting consequence of our approach is that since the null hypothesis is stated in terms of bounds on pairwise conditional probabilities, our null hypothesis includes not only all models of independent neurons but also some models of dependent or interacting neurons. This is more general than what is available in current methods. Intuitively, declaring a pattern like A T1 → B T2 → C as signicant should mean that we can conclude that there are strong causative connections from A to B and from B to C with the indicated delays. Hence rejecting a null hypothesis that includes not only independent neuron models but also models of neurons that are weakly dependent is more appropriate. Our method species these weak interactions in terms of bounds on conditional probabilities.
The rest of the paper is organized as follows. In Section 2 we present our signicance test. We rst explain our composite null hypothesis and then develop a test of signicance. In Section 3 we demonstrate the eectiveness of the method through computer simulations. Spike trains are generated by a network of neurons modeled as interdependent inhomogeneous Poisson processes. We show that our method rank orders signicant patterns. Surprisingly, it is also quite eective in situations where some of the assumptions of our theoretical analysis are not valid. We conclude the paper in Section 4 with potential extensions and a discussion of strengths and weaknesses.

Correlation Count
For simplicity of exposition, we rst explain the method for a sequential ring pattern of only two neurons. Consider a pattern A T → B. Suppose we nd the number of repetitions of this pattern in the data using simple correlation as follows. Let t 1 , t 2 , . . . t n denote all time instants at which there is a spike from any neuron in the data. Let where for any neuron x, I x (t) = 1 if there is a spike from x at time t and zero otherwise. Note that f AB is simply a correlation integral which counts the number of spikes from A that are followed by B with a delay of exactly T units, and hence counts the number of repetitions of our pattern. If we want to allow for some small random variations in the delay we can dene the indicator variable I x (t) to take value 1 if there is a spike in a time interval of length ∆T centered around t. (For example, we can take ∆T to be the time resolution in our measurements). From now on we assume that delays are always over some such small intervals.
There are many methods to calculate correlation counts, for example the two-tape algorithm of Gerstein and Abeles [1] and some of its recent variations [17]. Most current methods for detecting serial ring patterns rely on such correlations. Since the focus of this paper is on statistical signicance (and not on computational eciency), we simply assume that one can calculate such counts for pairs of neurons and for various delays T of interest. The question then, is how large should the count be to conclude that the pattern represents a strong inuence of A on B?
Since we want to address this question in a classical hypothesis testing framework, we need to choose a null hypothesis that includes as many models as possible of interdependent neurons without any strong inuences between pairs of neurons. Then, if we can calculate (or bound) the probability under the null hypothesis that f AB is above a threshold, we get a test of statistical signicance. As stated earlier, we want the null hypothesis to contain a parameter to denote the strength of inuence so that we can rank order all signicant patterns.

Strength of inuence as conditional probability 1
We propose that the strength of inuence between any pair of neurons can be characterized in terms of a conditional probability. Let e(B|A, T ) denote the conditional probability that B will re at time T (or more precisely, in a time interval [T − ∆T 2 , T + ∆T 2 ]) given that A has red at time zero. If, for example, there is a strong excitatory connection from A to B, this probability would be large. If, on the other hand, A and B are independent then e(B|A, T ) would be the same as the unconditional probability of B ring in an interval of length ∆T . (For example, if we take that ∆T = 1ms and the average ring rate of B is 20Hz, then this unconditional probability would be about 0.02). We note here that this conditional probability is well dened even if the two neurons are not directly connected through a single synapse. If the pair is directly connected, then T can be a typical mono-synaptic delay; otherwise T can be a multiple of the mono-synaptic delay. In either case, our task is to nd whether a pattern with a specic value for T is signicant.
This conditional probability is a good scale on which to say whether the inuence of A on B is strong. Our main assumption here is that this conditional probability is not time dependent. That is, the probability that B res in an interval [t + T − ∆T 2 , t + T + ∆T 2 ] given that A has red at t is the same for all t for the time period of observations that we are analyzing. Some recent analysis of spike trains from neural cultures [9] suggests that such an assumption is justied. Note that this assumption does not require the ring rates of neurons to not be time-varying. As a matter of fact, one of the main mechanisms by which this conditional probability is realized is by having a spike from A aect the rate of ring by B for a short duration of time. Thus, the neurons would have time-varying ring rates even when the conditional probability is not time-varying. Our assumption is only that every 1 See [16] for the original formalism of using conditional probability as a measure of interaction strength and deriving bounds on counts time A spikes, it has the same chance of eliciting a spike from B after a delay of T , i.e. there are no appreciable changes in synaptic ecacies during the period in which the data is gathered.
When analyzing the signicance of repeating serial patterns, what we are interested in is hypothesizing causative chains. Hence the strength of a pattern should be related to the propensity that a spike from A has on eliciting a spike from B, which can be conveniently represented by the conditional probability of B spiking given that A has spiked. In all serial ring patterns of interest, the constancy of delays in multiple repititions are important. Hence we dened the conditional probability with respect to a specied delay. Capturing inuences among neurons through this conditional probability allows us to formulate an interesting compound null hypothesis in terms of a bound on these probabilities.

Compound null hypothesis
Now we propose the following compound null hypothesis. Any model of interacting neurons is in our null hypothesis if it satises: e(Y |X, T ) ≤ e 0 for all neurons X, Y and a set of specied delays T , where e 0 is a user-chosen constant. The exact mechanism by which a spike output by A aects the spiking of B is immaterial here. Whatever be the mechanism, if the resulting conditional probability is less than e 0 then that model would be included. Thus our compound null hypothesis includes many models of interdependent neurons where whether or not a neuron spikes can depend on actual spikes of other neurons (unless we choose e 0 to be very small). The idea is that we choose an e 0 based on how strong we want the inuences to be before we agree to say there is a causative inuence of A on B.

Signicance Test
To get a test for statistical signicance we need to calculate a bound on the probability that, under this null hypothesis, the count f AB is above a given threshold. For this, consider the following stochastic model. Suppose L is the total time duration of the data and let the random variable N A (L) denote the total number of spikes by neuron A during this time. Dene where X i are independent and identically distributed 0-1 random variables with If we take p = e(B|A, T ), it is easy to see that S AB is a random variable equivalent to f AB since every time there is a spike from A, with probability p a spike from B would follow with the appropriate delay. Also, per our assumption, every time A spikes a spike from B with the appropriate delay occurs with the same probability regardless of B's spike history. This implies that the X i in the denition of S AB can be assumed to be independent and identically distributed. Now we assume that the spiking of A is Poisson. (Note that we are not assuming that spiking by B is Poisson and, more importantly, as per our null model the spiking process of B is not independent of that of A.) 2 Then the random variable S AB is such that we are accumulating a random variable 2 Indeed when spiking by A is Poisson, the spiking of B would not be Poisson if there is sucient inuence of A on B. Also, in Section 3 we show empirically that our hypothesis testing method is eective even in cases when A is not Poisson. X i every time an event of a Poisson process happens. This implies that, since the X i are 0-1 random variables, S AB is also a Poisson random variable [ Let the rate of the Poisson process for A be λ A .
Computing the threshold Let Z be a Poisson random variable with mean λ Z . Suppose the allowed Type I error in our hypothesis test is α. Let M be the smallest number satisfying Given α and λ Z , we can calculate the M needed to satisfy the above using the Poisson distribution.
For a Poisson random variable, the probability on the LHS of Eq. (5) is monotonically increasing with λ Z as long as λ Z < M . We know that S AB is Poisson with mean e(B|A, T )Lλ A . Under our null hypothesis, we have e(B|A, T ) < e 0 . Hence, if we take λ Z = e 0 Lλ A and calculate the M needed to satisfy Eq. (5), then we have, under our null hypothesis, Since, as discussed earlier, the random variable S AB represents the count f AB , the above M is the threshold on the count to reject our null hypothesis and hence conclude that the pattern found is signicant.
The test of statistical signicance is as follows. Let e 0 be the bound on conditional probability that we chose for our null hypothesis. Let α be the allowed Type I error. Let λ A be the rate of ring for the rst neuron in the pattern. Set λ Z = e 0 Lλ A . Using the cumulative distribution of a Poisson random variable with parameter λ Z , we calculate the M needed to satisfy Eq. (5). This M is the threshold on the count of the pattern for us to be able to reject the null hypothesis and declare the pattern to be signicant.
To calculate this threshold, we need λ A . This is easily estimated from the data as the average rate of ring for neuron A. A simple parametric study shows that the threshold M is well-behaved (see

Extending to longer patterns
Here we explain how the signcance test can be extended to patterns involving more than two neurons.
Suppose we are considering the pattern A T1 → B T2 → C. We assume we get the count of f ABC by taking 3-point correlations. 4 As before we dene S ABC as a sum of 0-1 random variables X i . Now, we want X i to be 1 if the spike by A is rst followed by B and then by C with the indicated delays. Hence we take 3 An equivalent way of looking at this is to consider the Poisson process of spikes from A and suppose that we classify each spike as type-Y with probability p and as type-N with probability (1 − p). Then it can be shown that the sequence of type-Y spikes (and also the type-N spikes) would constitute a Poisson process. Here, the classication of each A spike is dependent on whether or not there is a spike from B after the appropriate delay and is independent of everything else. 4 In general, taking n-point correlations like this for all possible n-tuples of patterns is computationally expensive. For this reason, such correlation counts are obtained only for patterns of length 3 or 4 in most cases. Here we are only explaining how our test can be extended to assess signicance of longer patterns provided we can get such correlation counts.

Spike train simulator
In this section, we present results from computer simulations to demonstrate the eectiveness of our method. We used a simulator for generating spike data from a network of 25 interconnected neurons labeled A through Y as shown in Fig. 2. There were four chains, each four neurons in length, with 5 This calculation for p is correct if all inuence of A on C comes only through B. This is a reasonable assumption if we want signicant patterns to represent a chain of triggering events. In such a case, S ABC would be same as f ABC . Even if there are other paths for A to inuence C, this value of p would represent a lower bound on the probability of the pattern occurring at any spike from A. Hence, with this p, S ABC would be less than f ABC and hence the threshold on the count calculated using this S ABC would be sucient in a hypothesis testing framework. Note that this means that the ring rate of a neuron (under the inuence of random synapses) varies over the range of λ/2 to 2λ where λ is the nominal background ring rate. The synaptic delays for the random connections were uniformly distributed between 2 and 5 ms. All neurons also had a refractory period of 1ms. Further details on the simulator can be found in Appendix B. since they have a refractory period of 1 ms. However, since this refractory period is small compared to the interspike intervals when ring at 20Hz, the deviations from the Poisson ring due to the refractory period are small). We repeated the simulation 5,000 times obtaining the counts of W and the counts of the pattern W-O-L-V. In Fig. 3 we show the histogram of these counts compared to a Poisson distribution with the rate parameter set as the sample mean for each count. The variance-to-mean ratio, or the Fano factor, for the count of W is 0.97. Since the Fano factor of a Poisson random variable is 1 this indicated that the rst neuron was approximately Poisson, and indeed a χ 2 goodness-of-t test failed to reject a null hypothesis of Poisson (P-value>0.05, test performed following the guidelines in [11]). The theory developed in Section 2 tells us that when the rst neuron in a pattern is approximately Poisson the count of the pattern will also be approximately Poisson, and our empirical results conrmed this (W-O-L-V Fano factor 1.01, χ 2 goodness-of-t test P-value>0.05). Then we repeated the simulations this time allowing random connections between all neurons with 25% connectivity. We can see in Although the theory behind our signicance thresholds assumes the rst neuron spikes according to a Poisson process, we will now demonstrate empirically that even when the rst neuron is not Poisson-distributed (due to random connections present in the network) we are still able to rank order the relative strength of patterns eectively using our signicance thresholds.

Rank ordering of signcant patterns
Even when the pattern counts are not Poisson-distributed (as for the simulations shown in Fig. 4), we are still able to rank order the relative strengths of the patterns as shown in Fig. 5. We also see that the line showing the threshold count corresponding to e 0 = 0.1 is able to distinguish the counts of pattern G-M-R-D from the maximum of random 4-neuron pattern counts not involving any of the neurons in the four chains.

Data requirements
To determine how much data is required to rank order we simulated the network with random connections to all neurons (again with 25% connectivity) for various lengths of time, and for each data length compared the counts of the patterns to the thresholds corresponding to e 0 values which are 0.1 greater   we nd that 300 seconds of data is sucient for the weakest pattern (G-M-R-D). We then used this ring rate and data length to demonstrate how our techniques can enhance the signicance analysis of counts obtained using the two-tape algorithm of [1].
Enhanced signicance analysis of two-tape algorithm counts Abeles and Gerstein [1] provide a formula for calculating the expected number of patterns of a particular description that will occur r times in data of length L if the neurons spike according to independent Poisson processes. Based on this formula, if we had 25 independent spike trains and 300 seconds of data we would expect to nd 3.86 patterns of complexity (length) four that repeat at least twice (with λ = 5Hz, ∆T = 1ms, and the total time span between the spikes of the rst and fourth neuron constrained to be 15ms or less). When we ran a simulation with these parameters and then mined the data with the two-tape algorithm we found six 4-neuron patterns that repeated twice or more and satised the temporal constraint. Again following [1], we calculated that this is not a statistically signicant excess of patterns for a Poisson random variable with a mean of 3.86 (P-value>0.05).
However if it had been, since it is a small number of patterns each pattern could be investigated further by the experimenter to determine which particular patterns out of the six are actually of interest. On the other hand, if the number of patterns found is large (and statistically signicant) it is not practical to designate all patterns of that description as candidates for further investigation. To illustrate this, we repeated the simulation but instead of having independent neurons we had chains of connected neurons as shown in Fig. 2, as well as random connections between all neurons with 25% connectivity as described previously. This time when we mined with the two-tape algorithm we found 3,870 4-neuron patterns that repeated at least twice. Here we need some additional criteria to select which of these individual patterns are most likely to be signicant and the best candidates for further analysis. Abeles and Gerstein [1] remarked that this selection process is very important, and called for future research to be conducted in this area to devise selection methods beyond their suggested strategy of repeating their analysis procedure for dierent subgroupings of the patterns (with the hopes of nding a smaller group of patterns that is signicant which can then be investigated further to nd the individual patterns responsible). Our framework of a compound null hypothesis based on conditional probabilities can be very useful as the selection criteria. By having dierent values for e 0 in the null hypothesis, we can ask what patterns are signicant at what value of e 0 and thus rank order patterns according to their relative strength. We demonstrate this in Fig. 7 as our thresholds at various e 0 are able to separate the three strongest patterns (I-S-C-E, W-O-L-V, and P-A-T-K) from the rest based on the counts obtained using the two-tape algorithm. We also see that when mining with a threshold of e 0 = 0.1 there are a considerable number of false positives (i.e. there are other 4-neuron patterns that are more frequent than G-M-R-D, our weakest pattern). This is due to subpatterns (e.g. P-A-T) of the stronger patterns being very frequent, and as a consequence 4-neuron patterns such as P-A-T-X occur frequently even if there is no strong connection between T and X. Future work will address this issue.

Discussion
In this paper we proposed a method of assessing signicance of serial ring patterns using correlation counts as the statistic. There are two attractive features of this method. First, we can rank order signicant patterns in terms of their relative strength. For this we represent the strength of inuence of A on B by the conditional probability that B res after a prescribed delay following A. We state our composite null hypothesis in terms of a parameter e 0 which is an upper bound on all such pairwise conditional probabilities. This allows us to rank order signicant patterns in terms the value of e 0 with the two-tape algorithm. There were 3,870 4-neuron patterns that repeated more than twice.
We plot the logarithm of how often each pattern repeated, and also plot our threshold for statistical signicance of 4-neuron patterns for various e 0 .
at which the pattern (which has a given number of repetitions) is no longer signicant. The second interesting feature of the method follows from this structure of our composite null hypothesis. Since we now include many models of interdependent neurons (as long as all the relevant conditional probabilities are less than e 0 ), rejecting such a null hypothesis is intuitively more satisfying. When we declare a pattern such as A T1 → B T2 → C as signicant, we can conclude that a spike by A has a strong inuence in eliciting a spike from B with delay T 1 and a spike from C after a further delay of T 2 . Here strong would denote that the relevant conditional probability is greater than e 0 . Thus, our idea of casting the null hypothesis in terms of a bound on conditional probabilities allows for a richer level of analysis.

Computational considerations
We have given a simple test of statistical signicance for deciding whether or not to reject the null (under a given condence level) based on the counts calculated through simple multi-point correlations.
As said earlier, the motivation is that such correlations are what are presently used for detecting such patterns.
At this point one may wonder whether there is any need for the test of signicance that we presented, given that we formulate our null hypothesis in terms of conditional probabilities. The correlations counts f AB as dened here would directly lead to an estimate of the conditional probability, e(B|A, T ). Hence, one can estimate the conditional probability and check whether it is less than e 0 . While it is true that we can directly get an estimate of the conditional probability, to get the required condence intervals on the estimate, we once again need to use similar kind of assumptions as here and hence, theoretically, the testing procedure is not irrelevant. But there are other reasons why this approach is better than estimating all conditional probabilities.
First, our test will directly give the threshold needed for the count, given any pattern. Thus, we need not actually obtain the true correlation count which is required if we want to estimate the conditional probability. We only need to ascertain whether a pattern occurs more than some number of times. Many of the algorithms for detecting patterns use the correlations in this way and it leads to better computational eciency.
There is a second and more interesting reason why our approach could be benecial. In general, obtaining correlation counts or ascertaining whether a pattern occurs a given number of times is computationally intensive. If we want to look for long patterns, the number of candidate patterns increases exponentially and, furthermore, the multi-point correlation is dicult to compute. However, there may be other more appealing ways to count what may be called the frequency of a pattern.
The correlation count we considered here counts all occurrences of the relevant pattern. Suppose we want to count only those occurrences such that the time span of one occurrence does not overlap with that of any other occurrence. Let us call such occurrences non-overlapped occurrences of the pattern.
(This means, e.g., if the spike sequence is AABB, then we count only one occurrence rather than two).
There are very ecient algorithms based on data mining techniques for obtaining all patterns whose counts in terms of maximum possible number of non-overlapped occurrences are above a threshold [13]. These algorithms are also computationally ecient in discovering very long patterns involving more than ten neurons [13]. It appears possible to extend this type of statistical signicance analysis to such counts also. Also, even when nding correlations, it is possible to tackle the combinatorial explosion in candidates (when we are looking for long patterns) by using similar data mining methods if we can put a bound on the count and decide that we are interested in only those patterns above this count. We will be addressing these issues in our future work. It is in terms of such generality that the approach presented here is interesting.

Strengths and weaknesses of the method
The strength of the approach is that we can accommodate dependence between neurons and conclude that some pattern is signicant only if it represents strong inuences among the set of neurons. Since this strength of inuence is controlled by a parameter in the null hypothesis, we can rank order dierent signicant patterns by varying this parameter.
However, the weakness of the specic test proposed here is that we need to assume that the rst neuron in the chain is Poisson. The assumption was needed to conclude that the pattern counts would be Poisson. In most of the currently available methods, one also assumes Poisson processes in the null hypothesis. In general, if the variations in the rate of ring of a neuron are small (which would be the case if all synapses into the neuron are very weak) then the Poisson assumption is likely to be a good approximation. Thus, the assumption is not restrictive if we know that some neuron is necessarily the rst in a chain. However, it is not always possible to have such knowledge. In spite of the assumption of Poissonness of the rst neuron, we feel that the approach presented here is interesting and useful.

Summary
In this paper we suggested an analytical method to assess the statistical signicance of sequential ring patterns with constant delays between successive neurons. The main methods of detecting such patterns depend on multi-point correlations. Our method can be used to nd thresholds on such correlation counts for deciding on the signicance of the patterns. Our main motivation is to have a method that can rank order signicant patterns in terms of the strength of inuence among the neurons constituting the pattern. For this we suggested that the inuence of A on B can be denoted in terms of the conditional probability of B ring after a prescribed delay given that A has red. Our compound null hypothesis is then stated in terms of an upper bound on all such pairwise conditional probabilities.
This upper bound is a parameter of the null hypothesis and by varying it we can compare dierent signicant patterns in terms of the strength they represent. This feature is very novel in relation to the current methods of signicance analysis. Another important consequence of our approach is that the null hypothesis now admits many models of interdependent neurons also in addition to the usual case of independence. Hence our approach to signicance analysis is more general.
Through extensive simulation experiments we demonstrated the eectiveness of the method. The method is seen to work well and is seen to be able to rank order dierent patterns in terms of their strengths even when our assumption in the theoretical analysis, namely that the rst neuron in the chain is Poisson, is not valid.
The method presented can assess signicance of sequential ring patterns only when the underlying inuences are excitatory. This is because the signicance test is stated in terms of a lower bound on the correlation count. Using a similar null hypothesis where we assume that the conditional probability is much smaller than the case under independence, it may be possible to nd how low the correlation count should be for us to conclude that there are signicant inhibitory inuences. This needs further investigation.
As we have pointed out there are some weaknesses in the approach. One is the assumption that the rst neuron in the sequence is Poisson. The other is the computational problems involved in nding correlation counts when one wants to detect interactions among a large group of neurons. There are some ecient algorithms based on data mining techniques which nd somewhat dierent counts but are computationally very ecient for discovering patterns involving large numbers of neurons [13]. We will be addressing the issue of extending the analysis presented here to such counts in our future work.  Here we describe the simulator used for generating spike data from a network of interconnected neurons.
The spiking of each neuron is an inhomogeneous Poisson process whose rate of ring is updated at time intervals of ∆T . The neurons are interconnected by synapses and each synapse is characterized by a delay (which is in integral multiples of ∆T ) and a weight which is a real number. All neurons also have a refractory period. The rate of the Poisson process is varied with time as follows: where λ j (k) is the ring rate of j th neuron at time k∆T , and K j , d j are two parameters. I j (k) is the total input into j th neuron at time k∆T and it is given by where O i (k) is the output of i th neuron (as seen by the j th neuron) at time k∆T and w ij is the weight of synapse from i th to j th neuron. O i (k) is taken to be the number of spikes by the i th neuron in the time interval ( (k − h ij − 1)∆T, (k − h ij )∆T ] where h ij represents the synaptic delay (in units of ∆T ) for the synapse from i to j.
We build the network in the following manner. The parameter K j is chosen based on the dynamic range of ring rates that we need to span. The parameter d j is determined by specifying the background spiking rate. This is the ring rate of the neuron under zero input. (We normally keep the same background ring rate for all neurons). Specifying this rate xes d j by using (7). The network has many random interconnections with low weight values and a few strong interconnections with large weight values. For the random connections we connect each neuron to some percentage of all the other neurons randomly. The weight values for these random connections are uniformly distributed over a suitable range. We specify all weights in terms of the conditional probabilities they represent. Given a conditional probability, we rst calculate the needed instantaneous ring rate so that probability of at least one spike in the ∆T interval is equal to the specied conditional probability. Then using (7) and (8) we calculate the value of w ij needed so that the receiving neuron (j) reaches this instantaneous rate given that the sending neuron (i) spikes once in the appropriate interval and assuming that input into the receiving neurons from all other neurons is zero. In our simulations, we specify the range of random weight values as well as the values of strong weights in terms of the equivalent conditional probabilities.
We then generate a spike train by simulating all the inhomogeneous Poisson processes where rates are updated every ∆T time instants. We also have a xed refractory period for all neurons, so that once a neuron is red we will not let it re until the refractory period is over.
We note here that the background ring rate as well as the eective conditional probabilities in our system would have some small random variations. As said above, we x d j so that on zero input the neuron would have the background ring rate. However, all neurons would have synapses with randomly selected other neurons and the weights of these synapses are also random. Hence, even in the absence of any strong connections, the ring rates of dierent neurons keep uctuating around the background rate that is specied. Since we choose random weights from a zero mean distribution, in an expected sense we can assume the input into a neuron to be zero and hence the average rate of spiking would be the background rate specied. We also note that the way we calculate the eective weight for a given conditional probability is also approximate and we chose it for simplicity. If we specify a conditional probability for the connection from A to B, then, the method stated earlier xes the weight of connection so that the probability of B ring at least once in an appropriate interval given that A has red is equal to this conditional probability when all other input into B is zero. But since B would be getting small random input from other neurons also, the eective conditional probability would also be uctuating around the nominal value specied. Further, even if the random weights have zero mean, the uctuations in the conditional probability may not have zero mean due to the nonlinear sigmoidal relationship in (7). The nominal conditional probability value determines where we operate on this sigmoid curve and that determines the bias in the excursions in conditional probability for equal uctuations in either directions in the random input into the neurons. We consider this as some more noise in the system and have shown through simulation that our method of signicance analysis is still eective.