Some Principles for Decoding Local Neuronal Systems
in the Mammalian Central Nervous System

In the mammalian brain, a typical neuron obtains information from thousands of other neurons through a number of mechanisms which may be both inhibitory and excitatory. The postsynaptic potentials are summed both temporally and spatially on the background of a constantly fluctuating threshold of excitability. When a neuron receives enough excitatory input, it sends information in the form of nerve impulses through long axons that split into thousands of branches reaching other neurons. At the end of each branch, a synaptic ending converts the spike potential into a chemical action involving neurotransmitters that inhibit or excite activity in the connected neurons. Thus, convergence of inputs on a single neuron, and divergence of outputs on a large number of other neurons is an indispensable component in the functioning of neuronal systems. The degree of convergence and divergence is highly dependent upon the type of neuron and its spatial location (Bernard and Wheal, 1994).

This relatively simple computing scheme is complicated by several other modes of neuronal interactions. Besides classical neurotransmission using neurotransmitters released from the synaptic knob and diffusing toward the postsynaptic membrane, the neurotransmitters may also diffuse further, out of the synapse, and react with the extrasynaptic and postsynaptic receptors of several other neurons. The effects of neurotransmitters may be altered by neuromodulators. Also, hormones produced elsewhere in the body and secreted into the circulation may alter the function of those neurons that have specific receptors for them (Arnold, 1992). Transneuronally transported proteins carried from one cell to the next may be involved in neuronal interactions, together with ionic changes in the extracellular space produced by the firing of neighboring neurons. Electrical effects, such as a direct electrotonic transfer from one neuron to the next or from one unmyelinated axon to the next (Meyer et al., 1985), and a spread of small electric fields generated by active cells also modify neuronal excitability and production of nerve impulses (Faber and Korn, 1989). Some cells do not require input from other neurons because they are excited by their own metabolic processes. These are the pacemaker neurons. And, some neurons do not produce action potentials at all and influence the target neurons by graded (analog) changes of their resting potential.

Thus, the description of the function of a single neuron is rather complex. Neurons are not digital switches. Rather, they are analog computer devices integrating many inputs and producing an all-or-nothing digital output. They either produce or do not produce a nerve impulse, or a burst of nerve impulses, when one or more of the described factors cause the neuron to reach its firing threshold. Still, single neurons are usually unable to accomplish any useful sensory, cognitive or motor task.

Meaningful information in the brain may be analyzed and transferred only by neuronal populations. Only very large systems of neurons may execute specialized, collective functions. Biological neuronal systems, where each unit cooperates directly or indirectly with others, are characterized by the unequal roles of individual components; by a presence of excitatory and inhibitory elements; by a dependence on the previous history of the system (Cramer and Sur, 1995); by mutual interactions of many kinds due to temporal and spatial convergence and divergence; by interactions lasting hundreds of milliseconds and longer, and still maintaining high accuracy (Abeles et al., 1994); by multiplication and summation operations (Bugman, 1991); by parallel and serial processing; and by numerous and multiple feedback and feedforward loops with different delays which are able to generate periodic or aperiodic chaotic rhythms (Glass et al., 1988). In a population of cooperating nerve cells, the behavior of individual components is influenced by the properties of the whole network. In all these respects, the biological neural systems differ from the von Neumann machine which may only accomplish one step at a time.

At the present state of knowledge of the principles of global brain function, it is necessary to begin with the analysis of the simplest modules. As a first step, it is necessary to decode local neuronal ensembles at the level of a single cortical column or cooperating group of nerve cells elsewhere in the brain. These local systems represent a fundamental neural processing element (Knopf and Gupta, 1993). The local neuronal systems are the natural building stones, the primary modules, of large-scale neuronal networks involved in cognitive processes and the functioning of the conscious mind. The modules and, at a higher level of integration, the large-scale systems become interdependent and functionally connected by the parallel processing of information (Bressler, 1995).

In most publications of the past thirty years, neuronal spike trains are characterized by their firing rate. This information excludes the exact position of a neuronal spike in the time series. This position, however, contains crucial information about when the neuron produces a spike, and thus, how it was influenced by other activated neurons. Information is, in the neuronal systems, represented in a distributed and collective manner, and although the mean firing rates may provide some insight into the working of the system, decoding the mutual interactions requires, in the first place, an exact positioning of each neuronal spike in timed relation with the others. Such decoding may then be considered a tool for analyzing what a neuronal system is doing. This should be then followed by the building of a model that accounts for the activity of the particular system (Abbott, 1994).

Another problem in the analysis of the local cell assembly is that, in natural neuronal modules within the mammalian nervous system, the input - output characteristics are poorly defined and therefore, it is difficult to define logical functions processed by a local population of cells.

All these characteristics of neuronal systems show that the collective properties of the neuronal networks differ from the sum of properties of the individual components.

For a long time, the activity of the neuronal systems was believed to be stochastic, chance firing of cells with some increase in statistical probability of spike generation due to the input from other areas of the brain or sensory system. Statistically defined relationships between neurons were postulated to produce organized behavior (John, 1972). At that time, this attractive hypothesis was difficult to pursue further for the lack of adequate methods of decoding the informational language of neuronal assemblies, even at their simplest level. The signal has to be distinguished from "noise", but the problem is, what is the signal and what is noise?

Based on chaos, self-organization may contribute to the adequate function of the brain. Self-organization represents the ability of groups of relatively functionally independent neurons to act cooperatively. From chaotic behavior, an organizing process arises which streamlines the activity of the network (Mpitsos, 1989). Molecular events at the synapses may strengthen some connections while suppressing the others. The embryological development of the CNS, during which an abundant number of cells and connections is generated, with a subsequent selection and survival of some and death and disappearance of the others, may support this hypothesis. Also, extreme variability in motor patterns and behaviors and responses of single neurons is probably a consequence of deterministic chaos (Cohan and Mpitsos, 1983).

To our knowledge, there are no proven neuronal systems with a documented fractal or chaotic time series. An important question is whether the neuronal assemblies in the central nervous systems are chaotic at all, or just extremely complex in a predictable way, controlled by a complicated and variable paradigm. Variability is probably an inherent function of many neuronal circuits. This variability is not noise riding on the neural code (Mpitsos 1989). Since the groups of neurons act in cooperation (i.e., they show self-organization), variability may be important for the establishment of their cooperative behavior. The same population of neurons may produce a different pattern of activity, depending on the input and context (Mpitsos, 1989).

An important feature for the development of deterministic chaos is the sensitivity to initial conditions after which the system follows deterministic laws, but still, the behavior of the system is not predictable into the future. For this reason, it is possible that chaotic states of neuronal activity are associated with higher levels of cognitive processes such as generalization and abstraction (Zak 1991). Deterministic chaos could be regarded as a healthy flexibility of the human brain necessary for correct neuronal operations.

On the other hand, if the activity of neurons is chaotic, what is the fidelity of transmission of the signals describing the biological input through such a network? Chaos research indicates that there is no such thing as certainty and predictability of actions. There must however be functional constraints assuring that the system will transmit information optimally.

Second, the interaction of two nerve cells, as studied in most of the literature, is meaningless because the interactions of two nerve cells mean very little in the explanation of even the simplest neuronal circuits. Therefore, it is necessary to study the behavior of a group.

Third, it is necessary not to rely on the average firing rate of each cell, but on the exact position of each neuronal discharge within the time series, at least with the precision of 1 millisecond, and if possible, with a precision of 0.05 to 0.1 ms.

Fourth, because each neuronal module is connected to a large number of other modules, and the stream of nerve impulses may pass through long complex routes outside the recording range of each microelectrode, it is necessary to consider interspike intervals up to the limit of 1000 ms, and if possible, more. In the method of decoding real neuronal systems described here, we met all these conditions.

1. Recording the Neuroelectric Signal
Extracellular neuronal activity is best recorded using microelectrodes with an impedance between one and four megohms. The signal is then filtered with a suitable electronic filter with a bandpass between 300 and 6000 Hz. Pre-amplified neuroelectric signals are then digitized and stored for future evaluation. Action potentials recorded by a single microelectrode are divided into two classes:
a. single cell spikes, produced by a cell nearest to the recording microelectrode. We called this cell "the leading cell" although it was characterized only by the highest amplitude in the record, not by some outstanding functional role.
b. Multiple firing, also called mass activity which is the multiple spiking of a number of cells (or other electrically active structures such as passing axons or dendrites) surrounding the tip of the microelectrode. It is recommended to adjust the output from the preamplifiers to the A/D converter so that the amplitude of the single defined cell spikes between 875 and 1000 mV. Mass activity may then be registered in a selected voltage window set anywhere between 125 and 500 mV. Peaks below 125 mV are assumed to approach the noise level of the equipment and are not used for further evaluation. From the ratio of the average rate of a single cell spiking to the mass activity rate, we estimated that our microelectrodes were recording potentials from up to twenty-five cells

If there were any spikes in between the two studied levels, they were evaluated by K-means cluster analysis (as described later). If they clustered with the leading cell, they were used in subsequent calculations as leading cell spikes. If the middle voltage shapes indicated an existence of another single cell, they could also be used in more complicated calculations with two leading cells.

2. Spike Recognition
In records where a single shape of all the cell spikes is visually recognized, and the background firing was distinctly separated in amplitude, just the positions of the high-level and the background spikes (expressed in time units - milliseconds or fractions of a millisecond - from the beginning of the record) were stored in a file together with their amplitudes.

Quite often, a more cautious approach was necessary to discriminate the spikes. In some records, the spikes generated by several nerve cells differed in both amplitude and waveform. Also, the amplitude of the spikes produced by a single cell may fluctuate substantially, even by more than two standard deviations (Schwartz et al., 1976), and the only reliable measure of the spike identification is its shape. After reviewing various identifiers of the spikes, such as height, width, peak-to-peak amplitude, spike area, Fourier analysis, and visual curve fitting (for review see Schmidt, 1984), we have recommended clustering the spikes into categories according to their shape by K-means cluster analysis. This method was previously used and evaluated by Salganicoff et al. (1988), and Sarna et al. (1988).

About one millisecond of digitized voltages was usually sufficient to characterize the spike shape. The software we used for the K-means analysis was the Systat Statistical Package (Wilkinson, 1986). The Euclidean distance calculation was used to detect the shape differences. An important advantage of K-means clustering is that the spikes are classified in an unsupervised manner.

Spikes formed by the summation of the action potentials of two cells firing simultaneously may, due to their irregular shape, form separate clusters which are then excluded from the final calculation. In the resulting list of spikes produced by a single cell, each spike was labeled by its time of appearance in the record. These lists were used for the calculation of the auto- and cross- correlograms.

The peaks forming multiple unit activity do not have to be classified by clustering, since they represent a measure of general neuronal activity and their shape is continuously changing, and only the positions of spikes within a certain amplitude range were stored.

3. Leading Cell and Mass Activity Correlograms
In order to quantify multiple interactions between cells of the recorded system, correlograms of spiking produced by a single cell and by the mass activity were calculated using essentially the method of Gerstein and Perkel (1969).

The hypothetical example depicted in Figure 1 shows the calculation of an auto-correlogram. All of the interspike intervals between the first high spike in the record, and all the subsequent spikes of the same amplitude are measured and stored. In the next iteration, all the interspike intervals between the next high amplitude spike and all the preceding as well as all the subsequent spikes are measured. When all existing interspike intervals in the studied record are listed, they are sorted according to their length, and the frequencies of the presence of each particular length of the interspike intervals are counted and stored. The matrix of all frequencies of all the intervals is the correlogram. The correlogram is, therefore, a frequency histogram of all interspike intervals calculated along the entire record of spike activity.

An upward deflection of the correlogram curve (peak) indicates that a certain interval between two spikes was present in the record more often than the others. Conversely, a downward deflection indicates that a particular interval between two spikes occurred less frequently than other intervals.

Interspike intervals may express causal relationships between spikes. The firing of one cell is induced by a previously firing cell. The time interval between their firing suggests the length of the loop of neurons through which the nerve impulse has to travel before returning to the area of the microelectrode. There could be, however, time intervals between spikes which do not express an interaction. They may merely reflect a random association of events in the record. Several tests have to be employed to distinguish between a causal and a random relationship. It is assumed that the intervals appearing more often than the others indicate the existence of a causal relation between pairs of spikes (Gerstein and Kiang, 1960).

Thus, it is critical to establish statistical significance of the frequency of interspike intervals. In order to accomplish this, the significance of the peaks in the correlogram was determined by calculating confidence limits, as recommended by Abeles (1982).

In a functioning neuronal network, nerve impulses pass from one cell to another in a presumably deterministic but rather chaotic way using a number of polysynaptic paths (Siegel, 1990). Therefore, the search for a causal interaction between two randomly selected units usually shows a rather low connectivity and a single low peak in the cross-correlogram. On the other hand, there can be a high number of significant correlations within a network of several neighboring cells recorded simultaneously. The evaluation of mass firing may thus detect interactions which are missed when a single pair of cells is recorded.

Four correlograms are computed from each set of records:

a. Autocorrelogram of leading cell spiking (L-L correlogram).
b. Autocorrelogram of background mass activity (mass correlogram, B-B correlogram). The mass correlogram summarizes the interactions of groups of neurons within a small area of the CNS.
c. Cross-correlogram between the leading cell spiking and mass activity (L-B correlogram).
d. Cross-correlogram between the mass activity and the leading cell spiking (B-L correlogram).

It should be noted that the L-B correlogram at positive time intervals equals the B-L correlogram at negative time intervals. But, for the purpose of the subsequent calculations we handled them as independent entities.

4. Constructing the Model of Neuronal Interactions
This is the most crucial step in our method. A model of the complex neuronal interactions within the recorded system of neurons is constructed using the data obtained from the set of four correlograms, as well as from the original record. Pairs of spikes in the original record are selected according to the length of the interval between them. If the time period between the two spikes corresponds to one of the significant interspike intervals, the pair is listed in a separate file. Each pair on the list is characterized by:

a. the position of the first spike in the record, expressed in milliseconds from the beginning of the record;
b. by its class (B or L);
c. by the position of the second spike in the original record, expressed again in ms from the beginning;
d. by the class of the second spike, and
e. by the time interval between the two spikes which has been previously selected as statistically significant.

Such selected pairs are then assembled, end-to-end, into a graphic representation of a sequence of neuronal interactions which reflects how the neurons within the vicinity of the microelectrode interact. The model of neuronal interactions is as long as the original record of neuronal firing (Figure 2).

The purpose of the graphic display of interactions is to demonstrate how the pairs of spikes follow each other, whether there is any regular or even rhythmic pattern, and whether the spike pairs form a continuous system of interactions describing continuing local cooperation between the units. The sequence of the pairs of spikes linked by the significant intervals is a time series of a new kind, and may be further analyzed and evaluated.

5. Testing the Validity of the Method.

5.1. The "Tagging" Test.
The implementation of all correlational methods is associated with a crucial decision: Which of the time intervals in the record are generated as a result of a causal relation between spikes and which intervals arise from a random coincidence of mutually independent spike trains? This decision is complicated when two spike trains recorded simultaneously are evaluated. It is even more difficult when an assembly of neurons is considered. First, we have made an attempt to address this problem by the "tagging" test.

One may assume that neurons located in the tissue side by side generate spikes in mutual cooperation. Thus, it might be possible to assemble neuronal spikes into a coherent system of interactions using a limited number of selected intervals. In such a system, most spikes are functionally dependent on the others or on a common input. The purpose of the "tagging" test is to determine how many spikes in the record are selected based on the significant intervals.

For this reason, pairs of spikes, selected because the interval linking them was statistically significant, were marked by a "tag" stored in the record file together with the characteristics of the particular spike. At the end of the selection process, the number of tagged spikes was counted and expressed as a percentage of the total number of spikes in the record.

This tagging revealed that in each studied area of the brain, the number of tagged spikes was usually very high, and sometimes, almost all the spikes in the studied record were tagged when all significant intervals were used. This finding means that the positions of most spikes in the time series could be explained by a relatively small number of interspike intervals.

5. 2. The Monte Carlo Tests
We applied three additional tests of significance to validate the limits of the method and to determine how significant are the results. These tests are are based on the method of the "shift predictor" as recommended by Gerstein and Perkel (1969). The first step of each test was to randomize the positions of the spikes within each record. The total number of spikes as well as their amplitude was preserved, only their positions were randomly shifted.

In one of the tests, each record was randomized 100 times. Each of the randomized files was then used to calculate the spike interactions as described in the previous section. The following parameters of the correlograms, derived from the randomized files, were evaluated:

a. The positions and amplitudes of the correlogram peaks;
b. The statistical significance of each peak;
c. The number of spike pairs selected according to the criteria described above.
d. The number of tagged spikes as calculated by the "tagging" test.
e. Whether the intervals formed a continuous system of interactions as in the real records, or whether they were scattered throughout the record.

After such evaluation of all 100 files produced by a randomization of a single real record file, we compared the results with the same values calculated from the original record, and found the significance of the differences using confidence limits. The number of pairs derived from the original record was supposed to be significantly higher than the mean number of pairs calculated from the randomized records. The number of the untagged spikes, on the other hand, was supposed to be significantly lower in the original records.

In the second, less time-consuming, and simpler test, the list of statistically significant interspike intervals selected from the original datafile was used. One hundred randomized files were then examined for the occurrence of the spike pairs determined by the intervals. In addition, the number of spurious spikes was counted.

The number of spike pairs selected from the original datafile was again supposed to be significantly higher than the average number of spike pairs from the one hundred randomized datafiles, and the number of untagged spikes was expected to be significantly lower. The real datafiles, as well as the neuronal systems derived from them, were only considered non-random if and only if these conditions were met.

The previous two tests determined the validity of the model on a general, global scale where the limit of reliability was determined for the whole correlogram in a single test. Since one objection to this method of the construction of neuronal interactions might be that there are still pairs of spikes with a significant interval between them that are not causally linked, but are associated due to a random distribution of spikes in the record, it is necessary to determine the statistical significance of each individual peak in the correlogram separately by a third Monte Carlo test. In this test, we again randomized the studied record one hundred times. Then, we tested the randomized records for the presence of the pairs of spikes linked by each individual statistically significant interval obtained from the original real correlogram. If the frequency of the occurrence of the interval in the real file was above the confidence limit of the mean frequency of randomly generated pairs in one hundred random records, the interval was accepted into the model. This test gives a very conservative measure of significance of each individual interval.

In order to determine the statistical significance of the frequency of the identical pairs of intervals, the spatial point pattern of the distribution of interval pairs was evaluated using Poisson distribution as a null hypothesis (Diggle, 1983, Pielou, 1974). For a Poisson distribution, the variance of the sample is equal to its mean. Several methods based on Poisson distribution have been used previously to determine of the distribution of animals, birds or insects in a forest, stars in the sky and bacterial colonies or growing cells in tissue cultures.

From several indexes of the distribution pattern described in Pielou (1974), such as Lloyd's index of patchiness, Morisita's index, true aggregation index, spatial autocorrelation index or trend test as described by Thioulouse, we selected the "index of dispersion".

The two-dimensional plot of the pairs of interspike intervals was divided into quadrats with a side of 1 ms, and the index of dispersion was calculated:

(x - ) 2 n x2 - ( x)2 I = ---------------------- = --------------------------------- x

where n is the number of quadrats, x is the number of pairs in each quadrat and is the mean number of pairs in each quadrat. If the pattern was random, then the index of dispersion was within the limits

__ ____ Pr (- 1.96 2I - 2n-1 + 1.96) = 0.95

If the value obtained was lower than -1.96, the data showed a regular distribution, if it was over +1.96, the pairs were clustered at the p<0.05 probability level.

B. The next studied area was the rostral ventromedial medulla of Long-Evans rats (McGaraughty et al., 1992, 1993, 1993a, 1995). The rostral ventromedial medulla is a key area modulating ascending pain input. There are three types of cells there, ON cells, OFF cells and NEUTRAL cells. Four types of correlograms were found: "flat" correlograms with a number of statistically significant peaks distributed throughout the first 200 ms of the correlogram; "peak" corelograms with a single peak at the zero point, indicating a prevalence of very short interspike intervals (Figure 7); "inhibition" correlograms with an absence of short interspike intervals, upto 40 ms in duration (Figure 8); and a few "rhythmic" correlograms indicating the presence of rhythmic cell firing with a period of about 100 ms or longer (Figure 9). The "inhibition" correlograms were present only in animals anaesthetized either with Pentobarbital or with Ketamine, and are therefore an indication of the anaesthesia. In the rostral ventromedial medulla, anaesthesia therefore prolongs the refractory period of the cell, and the periods when the cells cannot be excited therefore contribute to the block of transmission through this area. All these correlograms were found in the ON, OFF and NEUTRAL systems.

C. In the ventroposterior nuclei of the thalamus of Long-Evans rats, which is the thalamic area transmitting the somatosensory information arriving from the periphery of the body toward the cerebral cortex, some neurons are also excited by painful stimulation (Tsoukatos et al., in press). Four shapes of single-cell and mass correlograms were found again, similar to those from the rostral ventromedial medulla: the peak correlogram, the flat correlograms, the rhythmic correlograms, and a small number of systems producing inhibition correlograms. The proportion of these four correlogram types in the thalamus differs from those in the rostral ventromedial medulla. Both in this area and in the rostral ventromedial medulla, the administration of morphine, naloxone or combination of morphine and naloxone changed the firing rate of most cells, but not the shape of the correlogram. A painful stimulation, although changing the rate of neuronal firing, also did not alter the shape of the correlograms. The "inhibition" correlograms depended again on the presence of anaesthesia.

D. In the auditory pathway (Weiss and Reinis, 1992; Weiss, 1992), represented by the recordings from the lateral lemniscal nuclei, the lateral superior olive, the trapezoid body and the inferior colliculus of Long-Evans rats, the mass correlograms during the resting state were of the "level" type. During sound stimulation, there was a significant dip at the zero point of the correlograms, among the shortest intervals, indicating an inhibitory period following each neuronal spike, and then, a number of peaks was observed which corresponded to multiples of the frequency of the administered sound. The sound frequencies of upto 5000 Hz were studied. Since a single nerve cell or a nerve fiber cannot respond with a frequency of nerve impulses over 1000 Hz due to the restraints given by the limits of the excitability of the neurons, several neighboring cells or fibers fire successively, one after another, so that the final number of neuronal discharges corresponds to the frequency of the tone. This is the so-called volley principle (Wever and Bray, 1936), and these waves of the correlograms represent the first direct experimental evidence for this principle (Figure 10).

E. In the hippocampus, a brain area involved in the orientation in space and also in the processes of learning and memory, correlograms were constructed before and after the stimulation of the perforant pathway which changes the excitability of the neuronal systems in this area (Reinis et al., submitted). This stimulation altered the firing rate of neurons in this area, but not the shape of the correlograms. The most common interspike intervals were either very short, or widely distributed throughout the correlogram upto the cut-off limit of 1000 ms (Figure 6). These significant long intervals indicate either the dependence of the intervals on long postsynaptic potentials, or an existence of very long loops involving chains of hundreds of cells.

F. In a very recent study published only in the form of a defended dissertation, our method was used for the exploration of the effect of visual stimulation on neuronal systems in the superior colliculus of Mongolian gerbils (Bigel, 1996). The superior colliculus is an area responding to visual input and involved in many reflex motor actions. The correlograms in this particular system did not resemble anythingthat we had seen before. The correlograms usually have a peak near zero, and the frequencies of interspike intervals gradually decreased toward statistically insignificant levels (Figure 11). The correlogram therefore forms a slope from the shortest interspike intervals to the longer ones. We called this type of correlogarm "oblique".

First, the times of generation of most spikes recorded by a single microelectrode from a local neuronal system anywhere in the brain is determined by a relatively small number of selected statistically significant interspike intervals. Thus, the interactions of the neighboring cells are controlled by a relatively simple pattern of interactions. This indicates that within a local system of neurons, the nerve impulses run through a limited number of pathways.

Second, some sequences of interspike intervals are distributed nonrandomly in the record, and identical groups of interspike intervals, which we called "words" of the neuronal language, are repeatedly found in these neuronal systems.

Third, in the individual areas of the brain studied, the single cell autocorrelograms as well as mass correlograms show a pattern and distribution of neuronal interactions which are typical for that particular area. Several types of correlograms were described and distinguished, and they probably show the peculiarities of neuronal interactions typical for each area.

Fourth, in some areas of the central nervous system, the shape of correlograms did not change substantially when the function of the area was altered, e.g., by painful stimulation, by an injection of morphine or by the stimulation of the perforant pathway, which alters the firing rate in the pyramidal layer of the hippocampus. On the other hand, Ketamine or pentobarbital anaesthesia substantially altered the correlograms of single cells and groups of cells in the rostral ventromedial medulla. The oscillating correlograms of motion-detecting cells were also altered by visual stimulation.

Fifth, in some local systems, the spikes are connected by short interspike intervals. But, in the pyramidal layer of the hippocampus, the significant interspike intervals are long, even approaching 1000 ms. The long intervals may be explained by long postsynaptic potentials and/or by long paths through which the nerve impulses must pass before they return back to the vicinity of the recording electrode. These long intervals are typical for the hippocampus.

All these data indicate that the activity of single neurons, although seemingly stochastic and chaotic, may be rather tightly controlled. The nerve impulses activating the neurons may pass through a limited number of paths, and short sequences of neuronal spikes may occur repeatedly. The small number of interspike intervals indicates that local neuronal systems, although complicated, are formed by an arrangement of a limited number of local neuronal circuits representing the wiring diagram of the local system. However, the graphs of the neuronal interactions derived from all records in all studied areas display a chaotic arrangement of the neuronal interactions.

Our method, therefore, allows at least a partial glimpse into the local interneuronal relations, how the cells in one small brain area interact and work together. This method of the evaluation of multiple spike trains may describe mutual neuronal interactions in a global way only, because individual cells generating the neuronal spikes cannot be identified. Thus, a more detailed description of the inner computational logic of the working system of neurons cannot be provided as yet.

The most important outstanding question is, what does all this mean; what computational tasks do these collective interactions of neurons accomplish?

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