Some thoughts about neural coding and spike trains.

This communication introduces the topic. Foundations: Core concepts: Codings are relations summarized by rules or 'codes'. Special codings are 'neural', 'natural' (in everyday life), 'experimental' (in laboratories), 'conditional' (to partner restrictions). etc. Partial aspects are mechanisms, what partners say about each other, etc. Critical experimental issues: Trains are evaluated by when spikes occur: i.e. as point processes and timings. Trains and point process representations become synonyms. Any code must: (i) be a 'number (rate) code' and an 'interval code'; and (ii) include 'referent, train' covariations involving steady states with overall averages and fluctuations with patterns (dispersions, sequences). Seminal findings. Early data proved trains participated in codings; this is accepted unanimously. Inevitably, though accepted less readily, codings included rates, intervals, averages and patterns. Literature highlights. (1) Confirmed the seminal finding (2.2.) over vast domains; (2) Demonstrated both general and synaptic codings (referents, respectively, sensory, states, etc. and trains in directly connected neurons); (3) Revealed overlap between general and synaptic coding features. Overlap allows train participation in network dynamics; (4) Introduced natural formal contexts. (Point Process Mathematics, Communication. Information and Dynamical Systems Theories); (5) Includes confused opinions: (i) Opposition between rates and intervals; (ii) claims that averages are meaningful but patterns irrelevant. Both, overlooking foundations and evidence, are untenable.

The spike trains of inhibited pacemaker neurons seen through the magnifying glass of nonlinear analyses.

This communication describes the new information that may be obtained by applying nonlinear analytical techniques to neurobiological time-series. Specifically, we consider the sequence of interspike intervals Ti (the "timing") of trains recorded from synaptically inhibited crayfish pacemaker neurons. As reported earlier, different postsynaptic spike train forms (sets of timings with shared properties) are generated by varying the average rate and/or pattern (implying interval dispersions and sequences) of presynaptic spike trains. When the presynaptic train is Poisson (independent exponentially distributed intervals), the form is "Poisson-driven" (unperturbed and lengthened intervals succeed each other irregularly). When presynaptic trains are pacemaker (intervals practically equal), forms are either "p:q locked" (intervals repeat periodically), "intermittent" (mostly almost locked but disrupted irregularly), "phase walk throughs" (intermittencies with briefer regular portions), or "messy" (difficult to predict or describe succinctly). Messy trains are either "erratic" (some intervals natural and others lengthened irregularly) or "stammerings" (intervals are integral multiples of presynaptic intervals). The individual spike train forms were analysed using attractor reconstruction methods based on the lagged coordinates provided by successive intervals from the time-series Ti. Numerous models were evaluated in terms of their predictive performance by a trial-and-error procedure: the most successful model was taken as best reflecting the true nature of the system's attractor. Each form was characterized in terms of its dimensionality, nonlinearity and predictability. (1) The dimensionality of the underlying dynamical attractor was estimated by the minimum number of variables (coordinates Ti) required to model acceptably the system's dynamics, i.e. by the system's degrees of freedom. Each model tested was based on a different number of Ti; the smallest number whose predictions were judged successful provided the best integer approximation of the attractor's true dimension (not necessarily an integer). Dimensionalities from three to five provided acceptable fits. (2) The degree of nonlinearity was estimated by: (i) comparing the correlations between experimental results and data from linear and nonlinear models, and (ii) tuning model nonlinearity via a distance-weighting function and identifying the either local or global neighborhood size. Lockings were compatible with linear models and stammerings were marginal; nonlinear models were best for Poisson-driven, intermittent and erratic forms. (3) Finally, prediction accuracy was plotted against increasingly long sequences of intervals forecast: the accuracies for Poisson-driven, locked and stammering forms were invariant, revealing irregularities due to uncorrelated noise, but those of intermittent and messy erratic forms decayed rapidly, indicating an underlying deterministic process. The excellent reconstructions possible for messy erratic and for some intermittent forms are especially significant because of their relatively low dimensionality (around 4), high degree of nonlinearity and prediction decay with time. This is characteristic of chaotic systems, and provides evidence that nonlinear couplings between relatively few variables are the major source of the apparent complexity seen in these cases. This demonstration of different dimensions, degrees of nonlinearity and predictabilities provides rigorous support for the categorization of different synaptically driven discharge forms proposed earlier on the basis of more heuristic criteria. This has significant implications. (1) It demonstrates that heterogeneous postsynaptic forms can indeed be induced by manipulating a few presynaptic variables. (2) Each presynaptic timing induces a form with characteristic dimensionality, thus breaking up the preparation into subsystems such that the physical variables in each operate as one

Periodically-modulated inhibition of living pacemaker neurons--III. The heterogeneity of the postsynaptic spike trains, and how control parameters affect it.

Codings involving spike trains at synapses with inhibitory postsynaptic potentials on pacemakers were examined in crayfish stretch receptor organs by modulating presynaptic instantaneous rates periodically (triangles or sines; frequencies, slopes and depths under, respectively, 5.0 Hz, 40.0/s/s and 25.0/s). Timings were described by interspike and cross-intervals ("phases"); patterns (dispersions, sequences) and forms (timing classes) were identified using pooled graphs (instant along the cycle when a spike occurs vs preceding interval) and return maps (plots of successive intervals). A remarkable heterogeneity of postsynaptic intervals and phases characterizes each modulation. All cycles separate into the same portions: each contains a particular form and switches abruptly to the next. Forms differ in irregularity and predictability: they are (see text) "p:q alternations", "intermittent", "phase walk-throughs", "messy erratic" and "messy stammering". Postsynaptic cycles are asymmetric (hysteresis). This contrasts with the presynaptic homogeneity, smoothness and symmetry. All control parameters are, individually and jointly, strongly influential. Presynaptic slopes, say, act through a postsynaptic sensitivity to their magnitude and sign; when increasing, hysteresis augments and forms change or disappear. Appropriate noise attenuates between-train contrasts, providing modulations are under 0.5 Hz. Postsynaptic natural intervals impose critical time bases, separating presynaptic intervals (around, above or below them) with dissimilar consequences. Coding rules are numerous and have restricted domains; generalizations are misleading. Modulation-driven forms are trendy pacemaker-driven forms. However, dissimilarities, slight when patterns are almost pacemaker, increase as inhibition departs from pacemaker and incorporate unpredictable features. Physiological significance-(1) Pacemaker-driven forms, simple and ubiquitous, appear to be elementary building blocks of synaptic codings, present always but in each case distorted typically. (2) Synapses are prototype: similar behaviours should be widespread, and networks simulations benefit by nonlinear units generating all forms. (3) Relevant to periodic functions are that few variables need be involved in form selection, that distortions are susceptible to noise levels and, if periods are heterogeneous, that simple input cycles impose heterogeneous outputs. (4) Slow Na inactivations are necessary for obtaining complex forms and hysteresis. Formal significance--(1) Pacemaker-driven forms and presumably their modulation-driven counterparts, pertain to universal periodic, intermittent, quasiperiodic and chaotic categories whose formal properties carry physiological connotations. (2) Only relatively elaborate, nonlinear geometric models show all forms; simpler ones, show only alternations and walk-throughs. (3) Bifurcations resemble those of simple maps that can provide useful guidelines. (4) Heterogeneity poses the unanswered question of whether or not the entire cycle and all portions have the same behaviours: therefore, whether trajectories are continuous or have discontinuities and/or singular points.

Complex responses of living neurons to pacemaker inhibition: a comparison of dynamical models.

A neuron can respond to periodic inhibitory input with a variety of complex behaviors, periodic and aperiodic. We present a succession of models to test hypotheses for mechanisms underlying complex behavior generation. Model comparison using non-linear dynamics techniques indicates that long-duration IPSP aftereffects and spiking behavior are necessary for most of the basic response properties, though not sufficient for some of their more subtle aspects.

Responses to transients in living and simulated neurons.

This paper is concerned with synaptic coding when inputs to a neuron change over time. Experiments were performed on a living and simulated embodiment of a prototypical inhibitory synapse. These were used to test a simple model composed of a fixed delay preceding a nonlinear encoder. Based on these results, we present a qualitative model for phenomena previously observed in the living preparation, including hysteresis and dependence of discharge regularity on rate of change of presynaptic spike rate. As change is the rule rather than the exception in nature, understanding neurons responses to nonstationarity is essential for understanding their function.

Periodically modulated inhibition and its postsynaptic consequences--I. General features. Influence of modulation frequency.

Our aim was to examine the relation, or "synaptic coding", between spike trains across a synapse with inhibitory postsynaptic potentials when the presynaptic rate is modulated periodically and the postsynaptic cell is a pacemaker. Experiments were on the synapse in crayfish stretch receptor organs. Spike trains were considered point processes along time; the time series of corresponding pre- and postsynaptic intervals were extracted. Analyses used displays of intervals along time and order ("basic graphs", and "rasters", respectively), displays of differences between intervals along order ("recurrence plots"), cycle histograms (as such and as Lissajous diagrams with presynaptic and postsynaptic on the abscissae and ordinate, respectively), and correlation histograms. Cycle histograms and correlation histograms demonstrated that all presynaptic modulation frequencies (1/60-10 Hz) are reflected postsynaptically; novel frequencies may arise, not always relating simply to the pre- or postsynaptic ones. The transferred frequency domain is broad and physiologically meaningful. Indeed, vitally important functions have strong periodicities in all portions of the explored domain, and so do the discharges of participating neurons. Overall, pre- and postsynaptic discharges change oppositely, one accelerating while the other slows. Locally, however, pre- and postsynaptic discharges contrast clearly in other ways. The presynaptic evolution is everywhere smooth and orderly, half-cycles usually are symmetric, and there is a single kind of discharge, as expected because the presynaptic axon follows well the controlling stimuli. The postsynaptic cycle shows marked local distortions. These involve presynaptic domains called "congruent portions" where changes are in the same sense (e.g., joint accelerations), "saturated" domains where postsynaptic discharges are arrested, and asymmetric sensitivities to presynaptic change with hysteretic loops in the Lissajous diagrams; the postsynaptic discharge is heterogeneous showing dissimilar forms in succession. Congruent portions are either "positive segments" with pre- to postsynaptic rate ratios practically 1:1, 2:1, 1:1, or parts of Lissajous loops. Different modulation frequencies have different postsynaptic consequences. Differences involve the width and number of positive segments, the proportion of the cycle with saturation, the sense, magnitude and lead-lag characteristics of the hysteretic loops, etc. Because their consequences are separable, frequencies are classified within categories labelled "low" (under 0.5 Hz), "high" (between 0.5 and 5.0 Hz) and "very high" (over 5.0 Hz). Categories arise widely but each prevails in different biological functions (e.g., low or high in, respectively, respiration or vibratory sensitivity). The refactoriness of the inhibitory fibre affects how it can be modulated: consequently, the very high category resembles pacemaker discharges and was not analysed.(ABSTRACT TRUNCATED AT 250 WORDS)

Periodically modulated inhibition and its postsynaptic consequences--II. Influence of modulation slope, depth, range, noise and of postsynaptic natural discharges.

This paper examines the relation, or "synaptic coding", between the discharges of inhibitory fibres whose instantaneous firing rate is modulated periodically and pacemaker postsynaptic neurons using crayfish synapses and point process statistics. Several control parameters were varied individually, and the other maintained constant as far as possible: it extends the preceding publication that described the general features and varied only the modulation frequency [Segundo et al. (1995) Neuroscience 68, 657-692]. Statistics were mainly cycle histograms and Lissajous diagrams (with presynaptic and post-synaptic histograms on the abscissae and ordinate, respectively), complemented occasionally by displays of intervals along time and of interval differences along order ("basic graphs" and "recurrence plots", respectively). The postsynaptic influence of modulated inhibitory discharges is characteristically sensitive to all control parameters examined. (1) The frequency was reported in the companion paper [Segundo et al. (1995) Neuroscience 68, 657-692]. (2) The average slope per half-cycle, controlled via either frequency or depth, acts by way of its magnitude and sign in ways revealed by hysteretic loops. Hysteresis increases and varies as the modulation's steepness increases: it is minor and with a single clockwise loop at small slopes, but major and multi-looped at the larger ones. Slopes, because of their different postsynaptic consequences, were separated into the categories of "steep", "gentle" and "abrupt" if around, respectively, 1.0, 30.0 and 150.0 s-2. The influence of slopes in restricted portions of the cycle depends on their position on the inhibitory rate scale. (3) The modulation's range acts by way of its depth and of its position on the inhibitory rate scale. Deeper ranges, when compared with the shallower ones they contain, induce effects similar to those of shallower modulations with their central portion, plus effects peculiar to them at extreme rates. Changes in range position from the centre to the extremes of the inhibitory rate scale are influential (e.g., saturations appear). Changes within the centre can be highly influential, particularly when ranges are narrow and close to the postsynaptic natural rate, and modulation frequencies are low: relations between corresponding rates can be linear increasing, linear decreasing or piecewise linear. Changes around extreme rates are negligible, however, and saturations are present. (4) The usual modulations whose individual cycles did not differ from the cycle histogram were compared to others with the same cycle histograms but whose individual cycles had an unpredictable fast variability referred to as "noise".(ABSTRACT TRUNCATED AT 400 WORDS)

Transients in the inhibitory driving of neurons and their postsynaptic consequences.

The presynaptic fiber at an inhibitory synapse on a pacemaker neuron was forced to generate transients, defined here as spike trains with a trend, unceasingly accelerating or slowing. Experiments were on isolated crayfish stretch receptor organs. Spike train analyses used tools and notions from conventional point processes and from non-linear dynamics. Pre- and postsynaptic discharges contrasted clearly in terms of rates and interspike intervals. The inhibitory train evolved monotonically and smoothly, following tightly the simple prescribed curves; it was uniform, exhibiting throughout a single and simple discharge form (i.e. interval patterning). The inhibited postsynaptic train alternately accelerated and slowed, not following tightly any simple curve; it was heterogeneous, exhibiting in succession several different and often complex discharge forms, and switching abruptly from one to another. The inhibited trains depended on the inhibitory transient's span, range and average slope. Accordingly, transients separated (not cuttingly) into categories with prolonged spans (over 1 s) and slow slopes (around 1/s2) and those with short spans (under 1 s) and fast slopes (around 30/s2). Special transients elicited postsynaptic discharges that reproduced it faithfully, e.g. accelerated with the transient and proportionately; no transient elicited postsynaptic discharges faithful to its mirror image. Crayfish synapses are prototypes, so these findings should be expected in any other junction, as working hypotheses at least. Implications involve the operation of neural networks, including the role of distortions and their compensation, and the underlying mechanisms. Transients have received little attention, most work on synaptic coding concentrating on stationary discharges. Transients are inherent to the changing situations that pervade everyday life, however, and their biological importance is self-evident. The different discharges encountered during a transient had strong similarities to the stationary forms reported for different pacemaker drivings that are called locking, intermittency, erratic and stammering; they were, in fact, trendy versions of these. Such forms appear with several synaptic drivings in the same order along the presynaptic rate scale; they may constitute basic building blocks for synaptic operation. In terms of non-linear science, it is as if the attractors postulated for stationary drivings remained strongly influential during the transients, though affected by the rate of change.

A modified radial isochron clock with slow and fast dynamics as a model of pacemaker neurons. Global bifurcation structure when driven by periodic pulse trains.

A simple mathematical model of living pacemaker neurons is proposed. The model has a unit circle limit cycle and radial isochrons, and the state point moves slowly in one region and fast in the remaining region; regions can correspond to the subthreshold activity and to the action potentials of pacemaker neurons, respectively. The global bifurcation structure when driven by periodic pulse trains was investigated using one-dimensional maps (PTC), two-dimensional bifurcation diagrams, and skeletons involving stimulus period and intensity. The existence of both the slow and the fast dynamics has a critical influence on the global bifurcation structure of the oscillator when stimulated periodically.

A Bonhoeffer-van der Pol oscillator model of locked and non-locked behaviors of living pacemaker neurons.

A recent investigation of the influence of periodic inhibitory trains on a crayfish pacemaker neuron showed not only well-known locked periodic responses but also intermittent, messy, and hopping responses. This communication studies the responses of the Bonhoeffer-van der Pol (BVP) model with self-sustained oscillation when exposed to periodic pulse trains inputs. The analysis is similar to that used in crayfish and reveals interesting features, both comparable and complementary to those seen in the living preparation.

Effects of stimulus timing on transmitter release and postsynaptic membrane potential at crayfish neuromuscular junctions.

Different synaptic terminals of the single excitor axon to the opener muscle of crayfish (Procambarus clarkii) often release transmitter in a very different manner when stimulated with the same equal-interval, doublet, or triplet patterns. Compared to synapses that show little facilitation (low Fe synapses), highly facilitating (high Fe) synapses show greater percentage increases in several measures of synaptic efficacy when stimulated with any of these patterns. Low Fe synapses usually show the greater absolute changes in these measures of synaptic efficacy. Changes in the span and pattern of doublets and triplets can independently affect both pre- and postsynaptic measures of synaptic efficacy at either low Fe or high Fe synapses.

Locking, intermittency, and bifurcations in a periodically driven pacemaker neuron: Poincaré maps and biological implications.

Slowly adapting stretch receptor (SAO) pacemaker neurons, driven with periodic tugs, were analyzed by way of Poincaré mappings (Appendix). Two behaviors were apparent. i) Intermittency characterized previously unclear situations: discharges shifted irregularly between prolonged epochs where spike phases (relative to tugs) and intervals barely changed (slid), and brief bursts with marked variations (skipped). ii) Locking was well-known: phases and intervals remained almost fixed, regardless of the initiation. Changing frequencies, map domains with locking (ordered according to spikes/tugs ratios), alternated with intermittent ones. The best fit for any experimental map was a curve, not straight but certainly unidimensional, continuous and monotonic; it varied characteristically with frequency. This suggested relations called diffeomorphisms, implying periodicity and quasi-periodicity. Outcomes, expanding previous knowledge and meaningful biologically, were i) a precise, exhaustive behavior list (including between behavior transitions) and ii) a thorough understanding or model. This, in turn, provides norms for more specific models (single-variable ones suffice), constraints upon basic mechanisms (one variable, reflecting several real ones combined, should behave as the phase), and forecasts for future experimentation (e.g., unexamined tug frequencies and amplitudes).

Testing a model of excitatory interactions between oscillators.

The experimentally observed influence of regularly arriving tugs upon the AP discharge of the slowly-adapting stretch receptor organ (SAO) of crayfish was compared to a model of pacemaker excitatory synaptic interactions (Segundo and Kohn 1981). Criteria for compliance referred to facets as A) the excitation, B) the postulates, and C) the behavior. A) Excitation was implied primarily by the tug initially increasing the AP rate (it subsequently decreased it). B) The pacemaker AP discharges, and with more reason the electronically driven tugs, were considered acceptably regular sequence (postulate i). Tugs advanced the next AP (postulate ii); the "delay function" plots of delays vs. phases, i.e. interval shortenings vs. the time from the last AP to the tug, were close to the V of postulate iii, even though the shortest phases tended to postpone the next AP and the longest ones did not trigger immediately but with an around 5 ms latency. These effects were displayed also as "old phase vs. new phase" plots. The interval following that with the tug tended to be lengthened, but the pre-tug timing was not recovered. C) Behavior during a train of excitatory events, both in model and experiments, went through very similar initial settlings and eventual steady-states. The latter were characterized in the model by 1. an average excitatory vs. excited rate display formed by an endless number of segments with all positive rational slopes separated by negative-going discontinuities, 2. locking in the sense of preferential phases, and 3. periodic repetition of the same phases and inter-AP intervals. Experimental results were compatible with this. Such behavior was absent when the tug sequence was highly irregular. The initial settling, in the SAO as in the model, depended jointly on the first phase phi 1 and the intertug interval E. If the former was under lambda, it went through one or two monotonic phase-decreasing stages (one smaller, the other larger, than lambda), or through a single increasing one, depending on E being smaller or greater than, respectively, an estimated but never actually observed E leading to unstable lockings. If the initial phase was greater than lambda, settling with E's under rN + lambda involved jumps between larger than and smaller than lambda phases; with E's over rn + lambda, it involved an intermediate stable locking with phi = E-rN.(ABSTRACT TRUNCATED AT 400 WORDS)

Dynamic and static hysteresis in crayfish stretch receptors.

This report calls attention to the magnitude and pervasiveness of hysteresis in the coding from length to afferent discharges in crayfish stretch receptor organs (SRO's). The influence of previous lengths on the rate that corresponded to a particular length L was manifest by a substantial excess of that encountered when L was arrived at from a shorter value over that when arrived at from a longer one. Hysteretic loops were present under dynamic conditions when length was modulated quasi-sinusoidally in the length vs. rate Lissajous plots of both the slowly and the fast-adapting organs (SAO, FAO), either not perturbed or perturbed. Loops became narrower with increasing frequency (except for when 1 to 1 locking appeared, Diez Martínez and Segundo, 1983). Hysteretic loops were present under static conditions when length changes were step-like, and fully adapted rates were noted in the SAO and in the perturbed FAO. Earlier reports suggest that hysteresis reflects jointly at least mechanical and electrogenic factors in the "length-to-local dendritic effects" and in the "generator potential to discharge" stages. Several models, either mechanical or mathematical, reveal hysteretic behavior. Detailed analysis has not been performed except for one instance (Chua and Bass, 1972) where, for example, loop-narrowing at higher frequencies occurs only with certain weighting functions whose physiological significance is as yet obscure. Hysteresis may be more widespread than suspected in sensory (and perhaps other) systems: it involves a multi-valuedness that raises the issue of how central mechanisms infer stimulus magnitude retrospectively from the discharge.

Pervasive locking, saturation, asymmetric rate sensitivity and double-valuedness in crayfish stretch receptors.

The correspondence between afferent discharges and sinusoidal length modulations (0.2--10 cps, under 10% of the natural length variations) was studied in isolated fast-adapting stretch receptor organs (FAO) of crayfish, largely using average displays of rate vs. length (or derivatives) along the cycle. Rate modulations were greatest during early cycles and then stabilized, an initial adjustment remindful of mechanical preconditioning. Responses to stimulation in the FAO, as in the slowly-adapting organs (SAO) and possibly other receptors, exhibit the following features, all striking because of their magnitude and ubiquity. i) A zig-zag overall afferent rate vs. stimulus frequency graph with positively and negatively sloped segments. This precludes the straightforward use of Bode plots. ii) Marked non-linearities as an obvious stimulus-response locking in the positively sloped segments, a double-valuedness with one rate while stretching and another while shortening, a lower-limit saturation with the receptor silent for more than half a cycle, and an asymmetric rate sensitivity. iii) Clear-cut discharge leads relative to the stimulus at low frequencies and lags at high ones. The FAO responds worse than the SAO to low frequencies, and better to high ones; it is locked 1-to-1 in a much broader range (e.g., 3--100 vs. 1--3 cps). All features were strongly frequency-dependent. With higher frequencies: i) the number of impulses per cycle fell from several to just one and finally to one every several cycles at higher values; ii) the two values of each length approached one another usually but not always; iii) the silent proportion of the cycle increased; and iv) the rate sensitivity changed. Each feature can arise in principle at any of the transduction stages from length to discharge: the mechanical transduction from length to dendritic deformation, an the encoder one from generator potentials to discharges are particularly likely candidates.

Empirical examination of the threshold model of neuron firing.

An elementary model of neuronal activity involves temporal and spatial summation of postsynaptic currents that are elicited by presynaptic spikes and that, in turn, elicit postsynaptic potentials at a trigger zone; when the potential at the trigger zone exceeds a "threshold" level, a postsynaptic spike is generated. This paper describes three methods of estimating the "summation function", that is, the function of time that converts the synaptic current into potential at the trigger zone: namely, maximum likelihood, cross-correlation analysis and cross-spectral analysis. All three methods, when applied to input-output data collected on various neurons of Aplysia californica, give comparable results. As estimated, the summation function involved in the explored cells has an early positive-going swing that is large and brief. In the cell L5, but not in R2, there was also a late negative-going swing of longer duration.