Spike train and point processes

This article targets the underpinnings of viewing spike trains as Point Processes. A “train” is a sequences of spikes generated by an individual cell.

## Contents

* 1 Underpinnings
* 2 Perception of the Spike Train
* 3 Conceptual Framework
* 4 Framework Components i-v
* 5 Core Notions
* 6 Justification
* 7 References
* 7.1 Internal references
* 8 Figure legends

## [edit] Underpinnings

“Underpinnings” are the taking in of trains and the safeguarding of logical coherence in their handling and shaping. They are composed jointly by the Perception of the trains, a Conceptual Framework and Core Notions.

A reminder of such underpinnings is opportune, even if brief, restricted to single trains and ignoring other valid approaches. Indeed, handling trains as Point Processes has contributed significantly to understanding neural operations (see below, Segundo, 2009). Handling occurs, obviously, against the background of a specific experimental goal or purpose, affecting decisions throughout and tackled directly as component v.

## [edit] Perception of the Spike Train

The approach’s first step is the spontaneous awareness of facing an entity made up of spikes that appear along time, i.e., the instinctive Perception of the Spike Train.

## [edit] Conceptual Framework

This Perception sets off in the Neuroscientist’s mind and inseparable from it a characteristic thought process called here “Conceptual Framework”. This Framework captures the Perception critically, making sure that a proper logic validates both it and its elaboration.

## [edit] Framework Components i-v

Frameworks are made up by key Components i-v, sequential though almost simultaneous. Component i is spike recognition, i.e., acknowledging that data include events matching the mental template of entities called “spikes”. The template’s features are well-known conceptual and practically, and thus not discussed. Figure 1-A-upper portion depicts a fake train where one spike is enclosed (dotted, green ellipse). Thereafter (ii-v), spike recognitions are accepted without qualms, and data is manipulated as if consisting exclusively of spikes.

Component ii is the detection of spike-times, i.e., the instants along ongoing time when events happen. Spike-times are the only spike feature retained and used thereafter (iii-v).

“Clocks” keep time. Clocks are subjective or instrumental and, relative to observers, internal or external. Timekeeping requires an origin, set either arbitrarily, relating to referents (mandated by specific experimental goals, generic spikes, stimuli, acts), etc.

Time detection is characterized by its degree of precision. When fuzzy, as in intuitive or preliminary observations, detections are rough: for example, origins are “close to the reference” and spike-times “before”, “approximately at” or “after”. When sharp, as in contemporary constructs, origins are {t = 0} and spike-times {tj} in conventional units. Units must be meaningful for the problem tackled, and no smaller than allowed by how well one recognizes changeable spikes against noisy backgrounds. Legitimate for mammalian trains are milliseconds (microseconds, acceptable for ersatz pulses, unrealistically embellish our ability to detect). The lower part of Fig. 1-A shows the {tj} sequence extracted from the train. The unavoidable uncertainty in {tj} identification is represented below by superposed ranges within which the elusive true values should lie: ranges are the green higher probability one containing all, the yellow intermediate probability, and the red lower probability contained in the others.

Together with our perception of trains as spikes along time, is that of spikes along order, achieved by counting spikes and noting how many precede or follow each. Time and order, though closely intertwined, are not identical. Order comes from “counters” that reckon and tally spikes. Counters, as clocks, are subjective or instrumental, implying an origin, arbitrary or referent related. Precision about order is fuzzy when saying “among the first”, “lots before and after”, “among the last”, etc., or sharp when listing the integers {j = 0,…j,…,N}.

Component iii brings together all spikes into their collective, thus pooling spikes into the train. Component ii had de facto distilled spikes into spike-times, so now there is sameness between spike and time sets. These are called “timings” and, when precision is exact, are {tj j= 0, …, j, …, N}. Sameness remains tacit thereafter, both providing the data set. Any statement about one necessarily applies to the other.

Component iv is the evaluation of train intensity. “Intensity” is the abundance with which spikes are generated. Intensity comes across right away in train perceptions, appraised dually by either how many spikes there are and how close to one another they arise: thus, indistinctly by spike numbers (counts) and interspike intervals.

Evaluations go from fuzzy when saying “spikes are few”, “far apart”, “lots”, “tightly packed”, to exact when measuring the train’s numbers or intervals. The experimental context imposes the respective standards. Comparisons of numbers with intervals are legitimate only when matched samples start and end at spikes; otherwise, inaccuracies must be assessed and corrected. Conventionally, when a spike happens at t=0, it is assigned {j=0} and, though not ignored, not counted.

The ways, called here “strategies“, whereby trains are grasped so as to appraise intensity are inherent in this approach. The “overall” strategy grasps the train as a whole, i.e., as a single indivisible object, pulling out a single ruling that, vis-a-vis the train, is large-scale and all-encompassing. Thus, in the one-dimension space of averages (numbers, intervals), it places the train in a restricted region, from big and ill defined when rulings are fuzzy to almost points when exact. Overall strategies by themselves are useful for broad comparisons; moreover (see below), they complement more discerning interpretations.

However, the sum-total of issues affecting spiking rarely is uniform throughout, especially when trains last and have lots of spikes: hence, intensities commonly vary between parts. Inhomogeneity demands more discerning strategies, called “local”, wherein the sample train is broken up into portions, intensity evaluated in each and then matched with intensities in other portions and overall. A “portion” is a part of a train smaller, i.e., shorter span and/or fewer spikes, than the whole.

Local matchings target first if and how much, if slightly, markedly, rarely, frequently, etc., portions differ from one another and the whole: thus train “dispersion”. They target also how consecutive portions evolve, with stationarity, trends, periods, irregularities, etc.: thus, “sequence”. Dispersion and sequence are referred to jointly as “pattern” (Cox and Lewis, 1966; Segundo, 2003).

Trains are equivalent to timings (iii) and, regardless of whether used fuzzily or exactly, are evaluated, be it straightforwardly or after manipulation, in agreement with the experimental goals. Each portion has a magnitude, say, fractions of trains or cycles (intrinsic, imposed by referents, etc.) as well as a place along the train.

When break-ups are straightforward, and regardless of precision, portions often are equal, adjacent and non-overlapping. When fixed are the portion spans, the number of contained spikes can vary; when numbers are fixed, spans vary.

One noteworthy portion break-up uses spans no larger than the smallest interval: these contain no spikes or one and can be represented by, respectively, “0” or “1”. Their set describes trains fully as binary sequences (e.g., 1,0,0,1,0,1, … ,0). Other noteworthy portions have numbers fixed at 1: their spans tj-tj-1are the intervals Tj. Their set {Tj, j = 1, …, N} coincides with the timing {tj , j = 1, … , N}.

Intensity evaluations, performed fuzzily by naked eyes or precisely by instruments, allow form diagnosis (Segundo, 2003). All trains sharing certain features compose a “form”. Forms are, say, “periodic” when intervals are in a restricted number of sizes repeating in the same sequence: numbers can be just 1 (“pacemaker” forms) or more as in respiratory cycles. “Bursting” forms have shorter intervals within clusters (bursts) plus longer ones in between. “Irregular” forms have a wide-range of largely unpredictable intervals: they are “Poisson” if almost a Poisson process (except for the unavoidable refractoriness), unpredictable in the short and long terms; or “messy”, if hard to describe with some preferred intervals in preferred sequences, predictable only in short terms. “Intermittent” trains are almost periodic most of the time but have unexpected, disparate brief irregularizations.

Component v is the accomplishment of experimental goals. Goals often imply recognizing coding rules, i.e., covariations of trains with concurrent referents. Generally, referents can be almost any in or outside the cell itself. “Outside” has the broadest meaning, including other cells, networks, neural and added systems, higher functions, environment, etc. Revealed with different precisions are, say, when cells fire more or less, with shorter or longer intervals, bursts (clusters), preferred orderings, favored frequencies, phases, synchronies, and so forth. Experimental goals lie behind many decisions mentioned above (time-keeping, intensity appraisals, etc.).

## [edit] Core Notions

Core Notions lie at the heart of this Framework, regardless of whether approaches are spontaneous and intuitive or precise and discerning. One is that trains are Series of Events along time (Cox and Lewis, 1968; Cox and Isham, 1980). A “Series of Events” is composed by identifiable entities on a one or more dimensional continuum (e.g., space, time). Real-world embodiments abound: spike trains, heart beats, epidemics, earthquakes, particle suspensions, stars, etc. Meaningful in all is counting events and assessing nearnesses.

Such features and widespread relevance motivated and inspired the idea that Series of Events can be represented by assigning to each event one point on a continuum, thus creating a Point Process. Therefore, the resulting formalization, Point Process Theory becomes another core notion, one that allows rigorous descriptions, interpretations and modelings (Cox and Lewis, 1966; Cox and Isham, 1980; Daley and Vere-Jones, 1988; Segundo, 2003; Segundo et al. 1968).

Fig. 1 represents Point Processes schematically with one dimension (as trains, A), two (B) or three (C): black dots indicate the experimental estimates of event positions. Also represented are the uncertainty surrounds within which true positions should lie: the respective probabilities are highest in the green containing all, intermediate in the yellow containing the red, and lowest in the red.

The train itself, the collective of N spikes {0, …., j… , N} with a {Tj, j = 1, … N} timing, has the [N] dimensions of all the intervals expressed as times. Timings can be broken up into [1] dimension, the overall average, plus [N-1] dimensions of the pattern. Indeed, patterns are described by {Tj/∑Tj j = 1, …, N}. Apparently with [N] dimensions, pattern terms always add to unity so one always is redundant and should be omitted (e.g., j = 1, …, N-1): this validates the claimed [N-1] dimensions.

Point Process Theory, with many number and interval statistics, allows exact train descriptions. Statistics include averages, ranges, variances, histograms, second-order properties (return maps, serial correlograms, auto-intensity functions, spectra, etc.). Statistics are applied either simply or after fine-tunings guided by experimental questions, network implications, functions targeted, formal theories (see below), etc.

Statistics allow accurate form descriptions. “Recurrence plots”, for example, quantify forms rigorously and, when color-coded, reveal them to naked eyes (Segundo 2003). Figs. 2, 3 come from IPSP-driven crayfish neurons. In Fig. 2, drivings are periodic with periods 1 (pacemaker) and displays illustrate driven interval forms: panel (a) with period 1 (all intervals white); (b) with period 5 (regularly repeated colored departures); (c) intermittent, mostly pacemaker (white) with unexpected, dissimilar departures (colored); (f) irregular because of noise (disorganized entries). Two panels depict “phases”, i.e., cross-intervals from postsynaptic to presynaptic spikes: (d) shows a “phase walkthrough” with its slowly varying progression; (e) shows an “erratic” form with vague semblances of several periodicities. In Fig. 3, driving is by periodic trains (left column) at 0.03 (A), 0.50 (B) and 2.03 Hz (C). Their periodicity appear as the repetition of the same broad area; moreover, in such cycles in its long-term “typology” are uniform, smooth and with monotonic, symmetric halves, coming through by a simple local “texture”. Postsynaptic trains (right) show that presynaptic cycles are transferred, better with lower frequencies and milder slopes (A vs B vs C). However, the postsynaptic cycles are neither uniform, smooth, monotonic and symmetric, revealing the complex non-linear distortions imposed by synaptic codings.

## [edit] Justification

Justification for this article derives ultimately from the fact that nervous systems implement essential transactions by performing as networks of spiking neurons. Performances involve fluctuating sets of participating neurons and/or ongoing temporal activities apparent as trains and timings. These reveal essential aspects of neural functions at many levels. Moreover, they contribute to answering thoughtful questions posed, vis a vis neural functions, by formal theories (Communication, Information, Dynamics, etc.) (Segundo, 2003).

Hence, for much work the notions presented here are the mainstay, the bulwark that, unnoticed or deliberate, is present always, from first impressions on the run when simply looking and listening to what goes on to, no less, when appealing to convoluted abstract conceptualizations.

Well warranted is the assurance that Point Process assimilation is an inescapable course of action and modus operandi for neuroscientists who confront Spike Trains and aspire to be rational and communicable. Intuition prompts it and commonsense harmonizes it in a domain more elementary than that of detailed methodologies. A domain no more accessible to mathematicians than to others who, though with less technical expertise, can claim comparable authority.

The need for underpinning reiteration is strengthened by the frequent association, but at odds with it, of the approach’s warranted acceptance and an unjustified neglect of its rationale, logic and legitimacy. Underpinnings are considered superfluous, too obvious to worry about, paid little attention to and passed over. Indifference, even unintentional, is risky, at best implying careless unconcern, at worse leading (or mis-leading) into coarse thinking, equivocations and deceptions.

## [edit] References

Cox D.R., and Isham, V. (1980). Point Processes. (London and New York: Chapman and Hill).

Cox D.R., and Lewis, P.A.W. (1966). The statistical analysis of series of events. (New York: John Wiley and Sons).

Daley, D.J., & Vere-Jones, D. (1988). An introduction to the theory of point processes. (New York: Springer).

Segundo J.P. (2003). Nonlinear Dynamics of Point Process Systems and Data, International Journal of Bifurcation and Chaos, 13,2035-2116.

Segundo, J.P. (2009) A history of Spike Trains as Point Processes in Neural coding. J. Physiol. (Paris) in the press.

### [edit] Internal references

Thomas Kreuz (2011) Measures of spike train synchrony. Scholarpedia, 6(10):11934.

## [edit] Figure legends

Fig. 1. Schematic Point Processes of different dimensionality. A, [1] dimension. Upper portion Spike Train with a spike enclosed in dotted, green ellipse. Lower part {tj} sequence. Uncertainty in {tj} identification illustrated by the superposed ranges within which the true values should lie: green, broader, more likely; yellow, intermediate probability; red, narrower lower probability. Green contains yellow containing red.

Fig. 2. (Segundo, 2003) Recurrence plots of spike trains in crayfish neurons driven by periodic pacemaker IPSP’s”. Forms are “periodic” with periods 1 (“pacemaker”) (a) or 5 (b). “Intermittent” (c). “Irregular” because of noise (“stammerings”) (f). Phases are displayed in panels illustrating “intermittent quasi-periodic phase walkthroughs” (d) and “irregular chaotic (erratic)” forms (e). For Figs. 2, 3. Interval order {i} in column, shift {j} in row. The entry {i,j} is the differences between the ith interval Tj and that Tj+j lagged by j; differences are color-coded according to scales on left (white 0, positive up, negative down). Numbers in panels (e.g., 2801-2851 and 1-10) are orders {i} and lags {j}.

Fig. 3 (Segundo, 2003). Recurrence plots of spike trains in crayfish neurons when driven by periodically modulated IPSP’s. Presynaptic driver trains left: modulations 0.03cps (A), 0.50cps (B) and 2.03cps (C): half-cycles are uniform, smooth, monotonic and symmetric. Postsynaptic driven trains right. Plots show first by their long term “typology” that driver cycles are imposed postsynaptically, better with the lower frequencies and milder slopes (A vs B vs C). Plots also show but by their local “texture” that, contrastingly, half-cycles are neither uniform, smooth, monotonic or symmetric, thus revealing the clear and complex non-linear distortions of synaptic codings.

Acknowledgement. To José Pedro Segundo Moreno who produced the figures. To Dr. Oscar Scremin, MD, who prepared the manuscript for submission.