J. P. Segundo

Statistical Signs of Synaptic Interaction in Neurons

The influence of basic open-loop synaptic connections on the firing of simultaneously recorded neurons has been investigated with auto- and cross-correlation histograms, using experimental records and computer simulations. The basic connections examined were direct synaptic excitation, direct synaptic inhibition, and shared synaptic input. Each type of synaptic connection produces certain characteristic features in the cross-correlogram depending on the properties of the synapse and statistical features in the firing pattern of each neuron. Thus, empirically derived cross-correlation measures can be interpreted in terms of the underlying physiological mechanisms. Their potential uses and limitations in the detection and identification of synaptic connections between neurons whose extracellularly recorded spike trains are available are discussed.

Spike probability in neurones: influence of temporal structure in the train of synaptic events.

Zusammenfassung1.Bestimmte Input-Output Beziehungen wurden untersucht durch intraneurale Registrierungen von isolierten visceralen Ganglien von Aplysia californica. Alle Zellen wiesen eine verlängerte Folge von einem einzigen Typ von EPSPs auf, und lösten infolgedessen aufeinanderfolgende Spikes aus. EPSPs traten entweder (i) spontan auf, oder wurden (ii) ausgelöst durch Erregung eines Konnektifs mittels elektrischer Impulse eines Erregers, der durch einen Geigerzähler betrieben wurde. Die präspike Epoche wurde untersucht, und die Vorgänge, welche den postsynaptischen Spike auslösen, wurden probabilistisch identifiziert. Versuche mit durch digitalen Rechner simulierte Neuronen reproduzierten und erweiterten die Ergebnisse von Tierexperimenten.2.Die Spike-Wahrscheinlichkeit spiegelt das Verhalten von EPSPs wieder, welche innerhalb eines begrenzten Zeitraumes (als Integrationsperiode bezeichnet) vorkommen und erfaßt daher nur eine begrenzte Anzahl dieser Potentiale (als beeinflussende EPSPs bezeichnet). Die Wahrscheinlichkeit, daß ein gegebenes EPSP einen Spike auslöst, ist im allgemeinen eine abnehmende Funktion des mittleren zeitlichen Abstands (eine zunehmende Funktion der Durchschnittsrate) einer bestimmten Anzahl von kurz zuvor erfolgten EPSPs. Bei gegebener mittlerer Zeitspanne (Durchschnittsrate) wird sie im allgemeinen größer bei Mustern, in denen sukzessive Intervalle immer kürzer werden. Darüber hinaus wird die Spike-Wahrscheinlichkeit beeinflußt durch kurz zuvor erfolgte postsynaptische Spikes; die Wirksamkeit irgendeiner EPSP-Konfiguration kann durch geeignete Anordnung in bezug auf die vorangegangenen Spikes verbessert werden. Es wird daraus gefolgert, daß eine postsynaptische Zelle die Entscheidung, einen Spike zu erzeugen, unter dem Einfluß einer großen Zahl von EPSPs durch laufende Auswertung der genauen Zeitfolge kurz zuvor erfolgter Input-Ereignisse fällt. Deshalb hängt die Bildung der postsynaptischen Spike-Kette von verschiedenen statistischen Besonderheiten der präsynaptischen Entladungen ab. Die Begrenzung dieser Begriffsbestimmung und ihre Anwendbarkeit auf verschiedene, z.T. komplexere Fälle wird diskutiert.3.Die Arbeitsweise einer synaptischen Verbindung dieses Typs wird gemessen und diskutiert unter besonderer Berücksichtigung der folgenden drei Wahrscheinlichkeitswerte und deren Beziehungen untereinander: (i) die generative Wahrscheinlichkeit, daß eine bestimmte Input-Zeitfolge von präsynaptischen Spikes (oder EPSPs) erfolgt; (ii) die prospektive Wahrscheinlichkeit, daß ein Output-postsynaptischer Spike durch die vorhergehende Input-Zeitfolge ausgelöst wird; (iii) die retrospektive Wahrscheinlichkeit, daß eine bestimmte Input-Zeitfolge erfolgt ist, wenn ein Output-Spike ausgelöst wurde. Jeder dieser Wahrscheinlichkeitswerte (s. Appendix) kann direkt aus den Versuchsergebnissen abgeschätzt werden (a) und hat eine bestimmte physiologische Bedeutung in bezug auf die Eigenschaften der präsynaptischen Neurone, der synaptischen Verbindung, und/oder der postsynaptischen Neurone (b). Die in den prospektiven und retrospektiven Schemata bestehenden Unbestimmtheiten werden gemessen (s. Appendix). Die gefundenen Werte zeigten in welchem Ausmaß die Unbestimmtheit bezüglich Input (Output) reduziert werden kann durch das Bekanntsein von Output (Input) und wie sie verringert werden kann indem die betreffenden Wahrscheinlichkeiten als Funktion der Zeitfolge ausgedrückt werden.

Static and dynamic properties of gravity-sensitive receptors in the cat vestibular system.

SummaryThe spike activity of eighth cranial nerve units tonically responsive to head position was recorded in cats anesthetized with pentobarbital sodium, and related with linear accelerations induced by gravity during maintained positions and during dynamic trajectories achieved through rolling around a rostro-caudal axis.The steady-state discharge of 80% of the cells had relatively small coefficients of variation, narrow histograms and periodic autocorrelograms. That of most remaining cells had large coefficients variation, nearly exponential histograms and flat or weakly periodic autocorrelograms.The static relation between head position and discharge showed that each cell had directional sensitivity, i.e. a characteristic change associated with each movement sense. Sixty-six percent of the cells had side-up increases in interval mean and standard deviation, with translation of the histogram to the right and reduction in the average autocorrelogram value: 34% had the opposite relations. Many cells showed multivaluedness, i.e. the interval mean (and other statistics) from different stations at any given position covered a range greater than that at each station. Multivaluedness varied from cell to cell.In the dynamic experiments the discharge was recorded during a continuous motion that involved a single sine wave or a mixture of sinusoids at frequencies up to 0.1 Hz. The spike trains exhibited a continuous mapping of the time varying tilt angle into the instantaneous rate with little or no evidence of multivaluedness. In addition to a tonic part, responses showed a phasic component with the characteristics of a unidirectional rate sensitivity that determined a phase-lead of the response with respect to the stimulus. The relative proportions of tonic and phasic components varied from cell to cell.Based upon anatomical and mechanical considerations (see Appendix) and upon the present results it is suggested that deformations of the trampoline-like membrane occur in a distributed manner. Multivaluedness may be due to forces which, like stiction, prevent complete relaxation of the membrane under static but not under dynamic conditions. The phasic response, whose origin is obscure, argues in favor of the otolithic receptors having a dynamic function, in addition to their role in detecting head positions with respect to gravity.

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.

Presynaptic irregularity and pacemaker inhibition

It is known (e.g., Perkel et al., 1964) that when a pacemaker neuron elicits IPSPs in another, there are domains called “paradoxical segments” where in the steady-state i) faster inhibitory discharges determine faster inhibited ones, and ii) pre- and postsynaptic spikes are “locked” in an invariant forward-and-backward positioning in time, spikes alternating in the ratios 1:1 (1 pere for 1 postsynaptic), 1:2, 2:1..., that are also the slopes of the synaptic rate-transformation. The present project examined the matter further in the inhibitory synapse upon the crayfish tonic stretch receptor neuron, confirming the above. In addition it showed that locking and alternation existed also in the segments interposed between the 1:2, 1:1 and 2:1 paradoxical segments, even though they were not as marked and apparent, and that when tests were close to each other their order became influential and hysteresis-like phenomena appeared. The main finding was that paradoxical rate-relations, locking and alternation persisted when the presynaptic train was irregularized up to interval coefficients of variation of around 0.20 (Figs. 2–5). Therefore, both phenomena may not simply be laboratory curiosities, but also have a role in natural operation where probably a substantial population of neurons exhibits that kind of irregularity. As presynaptic irregularity increased, the paradoxical segment slopes and widths decreased and locking and alternation became less clear-cut. With CVs of about 0.20, only a relatively narrow 1:1 paradoxical segment with about O slope and little locking and alternation remained (Figs. 2b, 3g, 4right, 5third row). With larger CVs, the rate relation decreased monotonically and there was no locking nor alternation (Figs. 2e, 3h, 5bottom row). The postsynaptic discharge was more regular and had fewer changes in the number of presynaptic spikes per post-synaptic interval within paradoxical segments (particularly in their centers) than in segments interposed between them (left vs. right-hand columns in Figs. 5, 6; Fig. 7): the contrast, remarkable for regular stimuli, attenuated as variability increased. The following conclusions are relevant to coding of spike trains across a synapse with IPSPs. i) With fairly regular discharges, the same postsynaptic rate may result from several presynaptic ones (e.g., may result from rates in the 1:1 and 2:1 paradoxical segments and in the interposed one, Fig.2): in some cases but not others, the precise presynaptic rate can be identified on the basis of postsynaptic CVs, interval histograms and cycle slips. ii) A small rate change in a regular presynaptic discharge will have very different postsynaptic consequences depending on where it happens: if across a paradoxical-interposed boundary, for instance, it will cause remarkable rate, pattern and correlation changes. iii) The trans-synaptic mapping of variability involves an increase for the more regular presynaptic discharges and a decrease for the more irregular ones. iv) The postsynaptic discharge was slower with IPSPs than without in most cases; however, when the control discharge was weak or absent, IPSPs accelerated it. Results are relevant also to the operation of periodically performing systems that involve neuronal correlates, indicating that it is necessary in every case to ask whether zigzag relations and locking occur. The “delay function” plots the arrival time of an IPSP (or IPSP burst) relative to the last postsynaptic spike, i.e., the “phase” (Φ in Fig. 1b), against the interval lengthening produced, i.e., the “delay” (δ). In all cases, most points clustered around a straight line (Fig. 8), whose slope and ordinate intercept were in the 0.43–0.87 and the 0.02–0.52 ranges, respectively, for single IPSPs. The slope reflects how the IPSP effectiveness depends on when it arrives in the cycle; the intercept reflects the IPSP effectiveness. Large phases often showed “aberrant” points whose ordinates were either large (and having special formal implications), or very small (perhaps reflecting conduction and synaptic delays), or clustered around a second straight segment with a large negative slope (when spontaneous rates were low) (Fig. 8c). Delay functions for widely separated pairs of IPSPs could be multi-valued, points clustering around 2 or 3 parallel straight lines. A mathematical model of pacemaker inhibitory synaptic interactions (Segundo, 1979) agreed with this embodiment insofar as some postulated properties are concerned (e.g., regular discharge, interval lengthening by IPSPs, linear delay functions with slopes around 0.7) and as to the main aspects of the preparations behavior (i.e., zigzag rate relations and locking), but not in terms of some aspects of the postulates (e.g., interval variability, rebound) or behavior (e.g., segment boundaries, jitter in the locking, and hysteresis). The model was judged to be on the balance satisfactorily realistic.

GLOBAL BIFURCATION STRUCTURE OF A BONHOEFFER-VAN DER POL OSCILLATOR DRIVEN BY PERIODIC PULSE TRAINS - COMPARISON WITH DATA FROM A PERIODICALLY INHIBITED BIOLOGICAL PACEMAKER

The Bonhoeffer-van der Pol (BVP) oscillator is a valuable dynamical system model of pacemaker neurons. Isochrons, phase transition curves (PTC), and two dimensional bifurcation diagrams served to analyze the neurons response to periodic pulse stimuli. Responses are described and explained in terms of the nonlinear dynamical system theory. An important issue in the generation of spikes by pacemaker neurons is the existence of both slow and fast dynamics in the state points trajectory in the phase plane. It is this feature in particular that makes the BVP oscillator a faithful model of living pacemaker neurons. Comparison of the models responses with those of a living pacemaker was based also on return maps of interspike intervals. Analyzed in detail were the complex discharges called ‘stammering’ which involve interspike intervals that arise unpredictably and exhibit histograms with several modes separated by the equal intervals.

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 transients 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 model of excitatory synaptic interactions between pacemakers. Its reality, its generality, and the principles involved

This is a model of the steady-state influence of one pacemaker neuron upon another across a synapse with EPSPs. Its postulates require firstly the spontaneous regularity of both cells, whose intervals are E and N, respectively. In addition, they require a special shortening or negative “delay” of the interspike interval by one or more EPSPs, with a V-shaped dependence of the delay on the position or “phase” of the EPSPs in the interval; the minimum of the delay function corresponds to the earliest EPSP arrival phase (λ) that triggers a spike immediately. Finally, they impose on the variables certain bounds. The models behavior has two main features. The first is a zig-zag relationship with an overall increasing trend between the steady-state pre- and post-synaptic discharge intensities (Fig. 7). The zig-zag is formed predominantly, if not exclusively, by segments with positive slopes that are rational fractions. Passage from one such segment to others is negatively-sloped (“paradoxical”), involving staggered positively-sloped segments whose details are unclear for weak presynaptic discharges and discontinuities for intense discharges. The same postsynaptic intensity may result from several presynaptic ones; the maximum postsynaptic intensity may reflect refractoriness, or the earliest instants of immediate triggering. The second main feature is the “locking” of the discharges in an invariant forward and backward temporal relation. With at most one EPSP per postsynaptic spike, locking is always present. If the presynaptic interval E is in the closed {rN+λ,(r+1)N} range, locking is 1:r+1, either stable at a greater-than-λ phase or unstable at a smaller one; arrivals at integral multiples of N do not affect the postsynaptic intensity. If E is in {rN, rN+λ} (r>0), locking is at other ratios (e.g., 2:3) and less apparent. With more than one EPSP per spike, when E is below bounds that depend on the interspike interval and the point of earliest triggering, locking happens in the simple s′:1 ratio (s′=2,3, ...) and is stable; when E is above those bounds, there are E ranges where locking is in other ratios (e.g., 3:2) and ranges where behavior is unclear. The validity of any model is based jointly upon an a priori judgment as to whether postulates depart reasonably little from nature, and upon an a posteriori experimental comparison of modelled and real behaviors. The models domain of applicability depends on the specific embodiment, each of the latter tolerating characteristically each departure. The present model will be evaluated in the crayfish stretch-receptor neuron (Diez-Martínez et al., in preparation). The model is applicable to any physical system that complies with its postulates, and evidence compatible with this notion is available in many disparate fields. It illustrates the modelling path to a scientific proposition, other paths being inference from experimentation, or deduction from premises acceptable at other approach levels (in this case, for example, from that of synaptic mechanisms). The periodicity postulates set this model within the category of those for oscillators. The notion of an oscillator has a far broader applicability than appears at first sight, since all physically realizable systems have some predominant output frequency, i.e., to a certain extent are oscillators.

Pacemaker synaptic interactions: Modelled locking and paradoxical features

This communication describes a model for two “pacemaker” (i.e., regularly firing) nerve cells, such that one elicits IPSPs in the other. The assumptions involve essentially a linear dependence (“delay function”) of the postsynaptic interval lengthening (or “delay”) produced by the IPSPs on the position (or “phase”) with respect to the preceding spike of the latters arrival. When the number of IPSPs in an interval increases, both the slope and intercept of the delay function increase, the former remaining under 2 and the latter unboundedly. Assumptions are more or less close to the actual biological reality, or are made for convenience. A recurrence equation for the phase can be calculated, as well as an expression for the “locking phase” (see below). Plots of postsynaptic vs presynaptic firing intensity averaged over steady conditions, e.g. of mean rates or intervals, are formed by a sequence of relatively broad“paradoxical” segments exhibiting positive slopes 1, 2, 1/2, 3, 1/3, ..., indicating that “inhibited” discharges are made more intense by those increases in “inhibitory” arrivals. These segments are separated by narrower “intercalated” segments where behavior is unclear except for a large overall negative slope, indicating that “inhibited” discharges are weakened markedly by other increases in inhibitory arrivals. Across the successive paradoxical segments that correspond to more and more intense presynaptic discharges (i.e., to higher rates or shorter intervals), postsynaptic intensities, though overlapping in part, become weaker and weaker. At the extremes, when the presynaptic discharge is very weak, or very intense, the postsynaptic cell tends to its natural undisturbed firing, or to not firing at all, respectively. The pre- and postsynaptic discharges inevitably achieve eventually an invariant relation, i.e., will “lock” at a constant phase, regardless of the phase of the first IPSP arrival. The characteristics of this behavior (e.g., the rate bounds of the paradoxical segments, or the magnitude of the locked phase) depend on such givens as presynaptic and postsynaptic pacemaker rates or intervals, and as the slope or intercept of the delay function.

Behavior of a single neuron in a recurrent excitatory loop

A recurrent excitation loop was constructed by enabling each impulse from the slowly adapting stretch receptor organ SAO (crayfish) to trigger through an electronic circuit a brief stretch, or “tug,” of the receptor. When applied independently, each tug influenced the discharge as would an EPSP. Recurrent excitation led to characteristic discharge timings; hence, even an isolated neuron can have intrinsic mechanisms that prevent positive feedback from freezing it in an extreme non-operational state. Such timings depended critically on the “phase”, i.e., on the time elapsed between an SAO impulse and the tug. When the control discharge was stationary (because the SAO length remained invariant), phases of a few ms simply changed the pattern to one of doublets, and affected little the average rate. As the phase increased, bursts appeared, bursts and interburst intervals became more prolonged, and average rates increased. With the largest phases examined (40 ms), the discharge consisted of a slow alternation of high rate bursts, separated by long intervals. When the discharge was modulated (by 0.2/s sinusoidal length variation) with recurrent excitation, the peak-to-peak rate swing, i.e., the sensitivity, and the proportion of the cycle without afferent discharges increased, and the rate vs. length display was distorted even though remaining “loop-plus-extension.” Changes were phase-dependent: for example, loops could have a sharp high peak at one phase and be flat-topped at another. When the interspike interval variability was exaggerated (by a length jitter superimposed upon either invariant or sinusoidally varying lengths), recurrent excitation exerted fewer, weaker and somewhat different effects: e.g., it reduced the overall intensity of the invariant cases and the peak-to-peak swing in the modulated one. The precise mechanisms of these results can only be conjectured at but are likely to involve an electrogenic pump, electromechanical interactions, topographical issues, as well as their interplays. The functional implications involve, for instance, the modulation of the intensity, duration and occurrence of the bursting patterns in oscillating functions (e.g., breathing, chewing, etc.).