Spike train and point processes

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Author: Dr. Jose Pedro Segundo, Department of Neurobiology, University of California, Los Angeles

This communication targets the underpinnings of our viewing Spike Trains as Series of Events and Point Processes. Specifically, it describes and critically discusses how Neuroscientists behold, think about and deal with Spike Trains. Trains are the sequences of spikes, impulses or action potentials generated by an individual nerve cell.

These underpinnings are provided by our Perception of those Spike Trains plus the Conceptual Framework and Core Notions that safeguard the logical coherence of their handling and shaping. This approach has allowed major contributions to understanding neural operations; its critical justification will be reiterated below, even if well-known and summarized elsewhere (e.g., Segundo, 2009). This communication, in spite of brevity, concentrating on trains from a single neuron and setting aside other valid approaches, hopefully meets the goal of explaining such underpinnings.

The first step in the approach is our instinctive Perception of a Spike Train made up by identifiable spikes coming to pass along time. This perception is handled in individual cases with different degrees of precision, from fuzzy to sharp as, say, respectively, intuitive early work to reasoned contemporary elaborations.

Also a characteristic thought process, alluded to as Conceptual Framework, takes place in the Neuroscientist's mind. Set off by the Perception and inseparable from it, the Framework captures that Perception, evaluates it critically and makes sure that its logic will validate subsequent elaborations.

The Conceptual Framework is composed by key sequential Components i-v, the first four almost simultaneous. Component i is the recognition that data, i.e., records, include events that individually match the mental template of entities called "spikes". The defining template's features, well-known both conceptual and practically, will not be discussed here. Figure 1-A-upper portion depicts a Train where one of its spikes is enclosed in an ellipse (dotted, green). Thereafter (ii-v), spike recognition is accepted without qualms. Moreover, data is manipulated as if consisting exclusively of spikes

Component ii is the detection along ongoing time of the instant, the "spike time", when each event happens. Instants are the only spike feature retained and used thereafter (iii-v).

Time is kept by "clocks". Relative to observers, clocks are either inherent in intuitions and internal or instrumental and external. Statements require a time origin, set arbitrarily or relating to trains or referents (stimuli, acts, etc.). Time keeping has different precisions. When fuzzy, origins are identified roughly just by saying "around" some reference (the beginning of stimulus' or train, the generic spike in this or that neuron, etc.); epochs are identified as "before, around or after" relative to origins. When precision is sharp, time {tj} is measured in conventional units relative to the origin (t=0). Units, obviously, must be meaningful for the problem tackled. Units can be no smaller than allowed by how well one recognizes variable events against noisy backgrounds. For, say, mammalian trains milliseconds (microseconds, as occasionally alleged, unrealistically embellish our rigor even if legitimate for ersatz pulses). 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 the superposed ranges within which the true values should lie: ranges are the narrower lower probability one colored red on top, the intermediate yellow one is in between and the broader more likely green one underneath.

Perceptions also tell us that trains come to pass along order. Order is followed by counting spikes and noting how many precede and follow each. Time and order, though intertwined, are not identical. "Counters" recognize order by reckoning and tallying spikes. Counters, as clocks, are internal or external and imply some origin, arbitrary or relating to trains or referents. Precision about order can be fuzzy when saying "among the first", "with lots before and after", "among the last", etc., sharp when providing exact integers, e.g., spike "0", "1",?,"j",? or in between.

Component iii is a pooling of all spikes into their collective, the train. Components i and ii de facto distilled the set of spikes, the train, into the set of spike times. Set sameness remains tacit thereafter: both become indistinguishable, any statement about one necessarily applying to the other. When spike times are expressed precisely, the set {tj j = 0, ? } will be referred to as the train's "timing".

Component iv is the evaluation of Spike Train intensity. "Intensity" is the potency with which spikes are generated, coming across right away as part of our spontaneous Perception. Intensity is appraised dually, using indistinctly either how many spikes there are and how close to one another those spikes arise. Thus, both spike numbers (counts) and interspike intervals appraise intensity. Evaluation precision goes from fuzzy when saying "spikes are few", "far apart", "lots", "tightly packed" to exact when measuring numbers or intervals. When exact, number and interval estimates can be compared legitimately only when samples to be matched start at spikes; otherwise, errors are introduced and must not be ignored.

Also inherent in Frameworks are the "strategies", i.e., the ways whereby trains are grasped when appraising intensity. The strategy called here "overall" grasps the train as a whole, i.e., as a single indivisible object, pulling out a single verdict, vis-a-vis the train large-scale and all-encompassing. Precision goes from fuzzy when saying that spikes are "many or few", "packed or sparse", etc. to exact if stating the overall averages of numbers per unit time or of intervals. Thus, overall strategies place the sample in a restricted region of the one-dimension space of average intensities. Restricted regions go from big and ill defined when verdicts are fuzzy to practically points when exact. Overall strategies are useful when comparing broadly; they are also necessary complements to more discerning interpretations based on strategies specified below.

Obvious too, especially when trains have several spikes or are observed for some time, is some measure of inhomogeneity. This reflects that issues affecting spiking often are not uniform throughout, intensities varying from one part of the train to another. Inhomogeneity demands more discerning strategies, called "local", whereby the sample is broken up into portions, the intensity in each is evaluated and then matched with those in others or the whole. A "portion" is a part of a train smaller than the whole, i.e., shorter and/or with fewer spikes.

Portions stagger along either time or order; in practice (though not necessarily a priori), all are identical, adjacent and non-overlapping. When along time, span is kept constant and the number of spikes contained can vary; when staggering along order, numbers are constant but spans vary. Portion magnitudes (spans, numbers) are chosen according to the particular experimental goals as, say, fractions of the train itself, of periodic referent events (e.g., sounds, chewing), etc.

Some portions are noteworthy. That along time whose fixed span is no larger than the smallest interspike interval contains either no spikes or one, and is thus represented by, respectively, "0" or "1". Their set (e.g., 0,0,0,1,0,0,1, ? ) describes the train fully as a binary sequence. The portion along order whose fixed number of spikes is "1" has as variable span tj-tj-1, the interspike interval Tj: their set {Tj, j = 1, ? }, equivalent to {tj , j = 1, ? } is the train's timing.

Matching of portions targets first if and by how much portions differ, from one another and from the whole: this informs about train "dispersion". A second matching targets how consecutive portions evolve along time or order, i.e., reveals stationarity, trends, periods, irregularities, etc.: this informs about the train's "sequence". Dispersion and sequence are referred to jointly as "pattern" ("variability" is a looser synonym) (Cox and Lewis, 1966; Segundo, 2003).

Patterns recognize "forms", i.e., collections of trains with shared features (Segundo, 2003). Forms are described with different degrees of precision. Some are fuzzy. "Pacemaker" when all intervals look the same; "periodic" when intervals sizes apparently are only in restricted numbers that repeat in the same sequence; numbers can be 1 (as in pacemaker), 2 or many as in, say, respiratory cycles. "Intermittent" forms are periodic much of the time but unexpectedly show non-identical irregularizations. "Irregular" is a general term for when intervals are largely unpredictable: "Poisson", if interval sequence is uncertain except for shorter ones always more likely and unpredictable in the short and long terms; "messy", with some preferred intervals arising in preferred sequences but hard to describe and predictable only in short terms. "Bursting", with short intervals in clusters and longer ones in between.

Form descriptions become precise when based on number or interval statistics proposed by Point Process Theory and Dynamics (Cox and Lewis, 1966; Cox and Isham, 1980; Daley and Vere-Jones, 1988; Segundo, 2003; Segundo et al. 1968). Figs. 2, 3 use "recurrence" plots to illustrate forms seen in a crayfish neuron driven by IPSP's (Segundo 2003). Such plots, by way of its colored entry distribution, satisfactorily reveal essential features to the naked eye; they also have solid formal justifications. In Fig. 2, driving is by "pacemaker" IPSPs: forms are "periodic" with periods 1 ("pacemaker", all white) (a) or 5 (b); "intermittent", mostly pacemaker (white) but with unexpected, non-identical departures (colored) (c); and "irregular" because of noise ("messy stammerings") (f). Two panels depict "phases", i.e., crossed intervals from the postsynaptic spikes to IPSPs: (d) shows the slowly varying progressions of "phase walkthroughs" (intermittent quasi-periodicities) and (e) the vague semblances of some periodicities of "irregular chaotic (erratic)" forms. In Fig. 3, IPSP's (left column) are modulated periodically at 0.03 (A), 0.50 (B) and 2.03 Hz (C): their half-cycles are uniform, smooth, monotonic and symmetric. Postsynaptic plots (right) show first in the long term ("typology") that the driver cycles are imposed postsynaptically, though better with lower frequencies and milder slopes (A vs B vs C). Plots also show locally ("texture") the synaptic coding's clear and complex non-linear distortions for, contrastingly, half-cycles are neither uniform, smooth, monotonic or symmetric.

Patterns include the many features evaluating dispersions (ranges, variances, histograms, etc.) or sequence (by second-order properties like return maps, serial correlograms, auto- and cross-intensity functions, spectra, etc.). Patterns, fully described by Tj/?Tj, have (N-1)) dimensions; plus the average's (1), they complete the timing's N.

Component v, the final accomplishment of preset experimental goals, normally follows. It goes without saying that all Components must abide by the need to answer the specific questions that, though unmentioned here, anchor critical decisions. Goals imply covariations, i.e., coding roles, with practically any concurrent referent (environmental, input, output, intrinsic to the neuron, trains in even itself or other neurons, etc., etc.). Revealed more or less precisely are, say, when trains show more or less firings, synchronies, bursts (clusters), preferred orderings, shared frequency components, phases, and so forth. Functionally significant features are detected fuzzily by eyes or ears and subsequently confirmed and extended exactly by significant deviations from null hypothesis in time and frequency domain statistics.

Certain Core Notions always 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 (Cox and Lewis, 1968; Cox and Isham, 1980). A "Series of Events" is composed by identifiable entities on a continuum (e.g., space, time). Each entity is localized at a point; continua can have one or more dimensions. Spike Trains arise along one dimension, time. Fig. 1 represents schematically Point Processes with one (A), two (B) and three (C) dimensions: black dots indicate the experimental estimates of the point positions.

The second Core Notion is that Series of Events are represented precisely by Point Processes, i.e., sets of points on a continuum. Real-world embodiments abound: say, Spike Trains, heart beats, epidemics, earthquakes, particle suspensions, stars, etc. Counting points and assessing nearnesses have meaningful implications in all such embodiments. This widespread relevance motivated and inspired Point Process Theory that, in turn, allowed rigorous descriptions, interpretations and modelings. Figure 1-A,B,C illustrates 1, 2 and 3 dimensions Point Processes. The black dot represents the estimated position; also represented are the respective uncertainty surrounds within which true values should lie: the red narrower lower probability one is contained in the yellow intermediate one and both are contained in the green largest.

The Justification emphasized at the beginning of the present communication derives ultimately from the fact that nervous systems implement numerous essential transactions, codings and covariations by performing as networks of interacting neurons. Such implementations are by way of changes in sets of participating neurons and/or in their ongoing temporal activities. The particular changes expressed by Spike Trains have broad domains of relevance: they reveal, for example, neuronal operations singly or in networks, intrinsic cell or membrane roles, presynaptic terminal features (type, power, number, correlation), circuitry (recurrent, reciprocal, elaborate) and so forth. No less significantly, such studies contribute to examining thoughtful questions posed by Higher Function correlates or arising in the formal constructs of several theories (Communication, Information and Nonlinear Dynamics Theories) (Segundo, 2003).

Thus, the significance and impact of approaching Spike Trains as Series of Events/Point Processes are beyond reasonable doubt and dispute. They has been the mainstay for this work, the bulwark that, unnoticed or deliberate, is present always, in first impressions on the run when simply looking and listening to what goes on and, no less, when appealing to convoluted abstractions or elaborate conceptualizations.

These powerful reasons warrant the assurance that this is the inescapable course of action and modus operandi for Neuroscientists confronting Spike Trains and aspiring to be rational and communicable. Intuition prompts it and commonsense harmonizes it in a domain more elementary than that of participating methodologies. A domain perhaps not more accessible to Mathematicians than to others who, though with purely technical expertises or even laypersons, probably can claim within it no less authorities.

The need for reiterating these underpinnings is reinforced because an unjustified neglect and little preoccupation for its rationale, logic and legitimacy, even if at odds, tend to associate with the justified acceptance. Indeed, underpinnings seemingly too obvious to worry about, are paid little attention to and passed over. Indifference, even if unintentional, is risky, at best implying careless unconcern, at worse leading (or mis-leading) into coarse thinking, equivocations and deceptions.

The above, we feel, justifies this reiteration of the stand's underpinnings plus their validation, as well as the entailed warnings against considering them superfluous.

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) submitted

Figure legends.

• Figure 1. Schematic Point Processes of different dimensionality. A, one dimension. Upper portion Spike Train with one 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: red, narrower lower probability; yellow, intermediate probability; green, broader, more likely. Red contained in yellow contained in green.
• Fig. 2. (Segundo, 2003) Recurrence plots of spike trains in crayfish neurons when driven by periodic pacemaker IPSP's". Forms are "periodic" with periods 1 ("pacemaker") (a) or 5 (b). "Intermittent" (c). "Irregular" because of noise ("messy 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 shifted 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 show also but by their local "texture" the clear and complex non-linear distortions of synaptic coding: contrastingly, half-cycles are neither uniform, smooth, monotonic or symmetric.

Acknowledgement. To José Pedro Segundo Moreno who produced the figures.

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