ALBERT

Notes: Abstract This is a model of the steady-state influence of one pacemaker neuron upon another across a synapse with EPSP's. 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 EPSP's, with a V-shaped dependence of the delay on the position or “phase” of the EPSP's 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 model's 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 model's 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.

Notes: Abstract This communication compares the well known phenomenon of respiratory driving by the respiratory pump through the Breuer Hering reflex with a model (Segundo, 1979) of a neuronal pacemaker (i.e. regularly firing) interactions via IPSP's. The assumption involves a linear dependence (“delay function”) of the postsynaptic interval lengthening (or “delay”) produced by the IPSP's on the position (or “phase”) with respect to the preceding spike of the latter's arrival. Cats anesthetized and paralized with gallamine were artificially ventilated using a computer driven respirator. The pump period and the respiratory period (identified by the phrenic discharge) corresponded to the model's presynaptic pacemaker interval and to that of the post synaptic one, respectively. The delay of the respiratory period by an inflation was related linearly and in increasing manner to the latter's phase with respect to the inspiration onset. In the steady state, plots of average respiratory period versus pump period consisted in a succession of broad “paradoxical” segments with positive slopes 3, 2, 1, 1/2, 1/3 in which respiration was locked with the pump as predicted by the model. The locked condition were less easy to reach when FA CO 2increased or when level of anaesthesia decreased. The limits of paradoxical segments were different when measured using pump periods that increased or using periods that decreased. There was therefore a hysteresis as if the delay function parameters changed, a behavior that was not part of the model that assumed fixed characteristics. These modifications were related to dependency of each cycle on the preceding one. The model proposed for simple neuronal pacemaker interactions can thus be applied satisfactorily to the drive of the complex respiratory neuronal oscillator by the respiratory pump through Breuer Hering reflex, providing nevertheless some additonal assumptions concerning respiratory cycle interdependency are introduced to account for the hysteresis phenomenon. The Breuer Hering reflex could be considered as the equivalent of IPSP acting on the central respiratory oscillator. FA CO 2increase and anesthesia level decrease produced the same effect as the addition of noise to the model's presynaptic pacemaker, thus leading to the hypothesis that they act by adding noise (e.g., randomly distributed excitatory input) at the level of Breuer Hering reflex inhibition.

Notes: Abstract This communication describes a model for two “pacemaker” (i.e., regularly firing) nerve cells, such that one elicits IPSP's in the other. The assumptions involve essentially a linear dependence (“delay function”) of the postsynaptic interval lengthening (or “delay”) produced by the IPSP's on the position (or “phase”) with respect to the preceding spike of the latter's arrival. When the number of IPSP's 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.

Notes: Abstract The correspondence between afferent discharges and sinusoidal length modulations (at 0.2–10 cps) was evaluated by average Lissajous displays of rate vs. length in isolated fast-adapting stretch receptor organs (FAO) of crayfish. Frequency effects (Diez Martínez et al., 1983) were compared using modulation depths and background lengths of, respectively, 1–10% of and 0.3–40% over resting length, well within physiological ranges. The receptor remained silent with shallow and slow stimuli. With greater depths, more frequencies became effective, discharges occupied more of each cycle, and peak and overall rates increased. At large depths, increases were smaller or were substituted by decreases (overstretch). With greater background lengths at 0.2 and 1 cps, individual and peak Lissajous rates increased, as did the overall value; eventually, the flat extension disappeared. At 0.2 cps, Lissajous plots became ellipses; at 1 cps, loops became counter-clockwise. At 3 and 10 cps, rates augmented only with small increases; beyond, peak rates remained unchanged or dropped. In short, each characteristic of the transduction is apparent only within a domain defined by the stimulus features as well as by the way stimuli and afferent discharges are evaluated. Each domain overlaps partially with others but falls short of that used in nature. For example: the stimulus-response relation was linear with low frequencies, large background lengths and moderate depths when observing average Lissajous plots; saturations were present with the low frequency, shallow modulations without discharges; clockwise loops occurred with particular frequencies, background lengths and depths, but not with others. Hence, any simple and succinct description is only a local rule that holds exclusively within restrictive constraints. Any generalization beyond this domain constitutes a misleading oversimplication.

Notes: Abstract 1. This communication describes, in the tonic stretch receptor organ (RM1) of crayfish, the inhibitory fibre's influence upon sensory modulated discharges. Periodic trapezoidal length changes were imposed, and rate plots of the afferent discharges were compared without inhibition (C) and with inhibition, either irregular (P) or regular (R), and at different rates. 2. Inhibition changed all sensory response components. Changes were dependent on issues as inhibitory rates and patterns, RM 1 discharge modulations, and pre-post-synaptic rate ratios. The most common effect was rate reduction, usually non-uniform along the cycle and with little evidence of proportionality. Saturation, i.e., inefficacy of shifts around extreme inhibitory rates was apparent. Rate increases occurred also. Accelerations were manifest by increased phasic lengthening response slopes and heights, by “faster inhibitor-faster inhibited” relations, or by postinhibitory rebounds. 3. Irregular inhibitory discharges (i) favoured variability along individual and average cycles, (ii) favoured monotonicity, (iii) rarely silenced the RM 1, and (iv) reduced without eliminating nonproportionality and saturation. 4. Regular inhibitory discharges showed the most clear-cut nonmonotonicities and saturations silencing the RM 1 effectively. Furthermore, they included characteristic epochs where (i) the RM 1 spike slid across the invariant interinhibition intervals, (ii) intervals recurred in stereotyped sequences and (iii) rate ratios had special values (e.g., 1:1, 1:2). Thus, the gradually decaying slope of the control adaptation after the lengthening transient was changed into a staircase profile or a sudden drop to a constant plateau. 5. Inhibition changed phasic and tonic sensitivities, usually refucing them (phasic decreasing less than tonic); increases, joint or individual, occurred also. The fidelity with which the rate plot reproduced the sensory stimulus was modified in many ways (e.g., by conversion into a phasic prototype, or into a system with perfect reproduction, etc.). Changes depended on the inhibitory discharge, and on the stimulus features. 6. These experiments have implications in two fields. In that of synaptic rate effects, (i) they confirm that the inhibition repertory includes slowings (predominant here) and accelerations, plus special effects, and (ii) they demonstrate extensively their dependence on the post-synaptic features. In the field of sensory control, they note sensory-synaptic interactions that, in intact animals, must arise but whose characteristics and roles can only be conjectured about.

Notes: Abstract 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.

Notes: Abstract Stability properties of gravity receptor transduction mechanisms were investigated by recording action potentials from the eighth nerve of cats maintained at two spatial positions both with and without an added click perturbation. The cells were demonstrated to present directional sensitivity, multivaluedness and wandering of mean rate trajectories. To the wandering were fitted appropriate stability boundaries corresponding to recent theories of finite time stability. Results supported the hypothesis that physiological stimuli result in local deformations of a flexible trampoline-like macula.

Notes: Abstract The relation between a maintained spatial orientation and the corresponding fully adapted discharge rate was multivalued in all the afferents tonically sensitive to maintained spatial orientation observed in isolated utricles of Rhinobates productus. The spread of rate values was of the order of changes produced by natural tilts. The occurrence of multivaluedness in isolated receptors indicated that peripheral issues are sufficient. Two factors contributed: firstly, the side from which the orientation had been reached (i.e. “hysteresis”): higher adapted rates occurred when the preceding orientation was characterized by lower rates and when the corresponding transition caused acceleration; secondly, “spontaneous” rate variations, some of which resembled markedly, and interacted with, the effects of tilts. It was not possible to identify the basic mechanisms underlying these factors. The multivaluedness in the coding of maintained position, because of its constancy and magnitude, cannot be ignored. It, as well as the sensitivity to fast transients, must be taken into account in utricular models, in evaluations of information transmission, and in psychophysical explorations.

Notes: Zusammenfassung Mit Hilfe eines Digitalrechners wurden die Eingangs-und Ausgangsbeziehungen auf synaptischer Ebene untersucht und dargestellt. Diese Untersuchung erstreckt sich auf erregende Synapsen und analysiert die Veränderungen postsynaptischer Aktionspotentiale, die auftreten, 1. wenn die Anzahl der präsynaptischen axonischen Endigungen ansteigt, während andererseits die Amplitude des EPSP abnimmt; 2. wenn sich die statistische Struktur oder „Form” der Spike-Kette in jeder präsynaptischen Faser verändert; und/oder 3. wenn die Beziehungen zwischen den präsynaptischen Fasern von völliger Unabhängigkeit bis zu starker Abhängigkeit variiert werden. 1Unabhängige präsynaptische Endigungen. Mit zunehmender Anzahl präsynaptischer Endigungen bei gleichzeitiger proportionaler Abnahme des EPSP (Input Form konstant) treten folgende Veränderungen auf: a) das durchschnittliche Intervall zwischen den postsynaptischen Spikes nimmt etwas zu; b) die mittlere statistische Abweichung (standard deviation) nimmt erheblich ab; c) die Form des Histogramms wird eng umschrieben; und d) das Autokorrelogramm nimmt „periodischen” Charakter an. Wenn andererseits die präsynaptische Form verändert wird (während die Anzahl der Endigungen sowie die Größe des EPSP konstant bleibt), hängt der am postsynaptischen Ausgang registrierte Effekt von der gegebenen Anzahl der Endigungen und von der Größe des EPSP ab. Ist die Anzahl der Endigungen gering und das EPSP groß, dann variiert der Ausgang mit der präsynaptischen Form. Der postsynaptische Variationskoeffizient steht dann in linearer Abhängigkeit vom präsynaptischen Variationskoeffizienten, wobei die Steigung der Geraden mit zunehmendem Eingang abnimmt. Sind die Endigungen zahlreich und die Größen der EPSPs gering, dann übt die präsynaptische Form keinen Einfluß mehr aus, und das von der postsynaptischen Zelle erzeugte Ausgangsprodukt wird unabhängig von der detaillierten Struktur des Eingangs. Für eine jegliche Kombination von unabhängigen und schwachen Eingangsformen stellt sich das Ausgangsprodukt in Form einer sehr regelmäßig gestalteten und durch gleichmäßige Abstände gekennzeichneten Spike-Kette dar (diese Folgerung gilt nur für die Fälle, in denen die präsynaptischen Endigungen sich nicht äußerst langsam entladen). Diese Resultate können mathematisch an Hand eines einfachen Membranmodells abgeleitet werden (s. Appendix). Wenn das EPSP in Größe ansteigt, alle anderen Variablen jedoch gleich bleiben, dann nimmt das postsynaptische Intervall fortwährend ab. Dieser Abfall ist entweder gleichmäßig (präsynaptische Form: „Poisson”) oder stufenweise (präsynaptische Form: „pacemaker”). Die postsynaptischen Aktionspotentiale werden durch eine vergleichsweise kleine Anzahl von hemmenden Endigungen wirkungsvoll blockiert. 2Abhängige präsynaptische Endigungen. Wenn sich der Grad der Abhängigkeit zwischen den präsynaptischen Endigungen in physiologischen Grenzen bewegt, dann kann die Aktivität der postsynaptischen Zelle als eine Funktion der statistischen Form der Eingangskanäle angegeben werden, und das sogar, wenn die letzteren zahlreich und schwach sind. Dieser Fall tritt dann ein, wenn die Abhängigkeit zwischen den präsynaptischen Endigungen nur einen Teil aller Endigungen oder nur die Endigungen innerhalb getrennter und unabhängiger Gruppen betrifft. Um die im Nervensystem stattfindenden Übertragungen zu verstehen, ist es notwendig, diejenigen präsynaptischen Statistiken zu idendifizieren, die entsprechende postsynaptische Entladungen beeinflussen. Wenn präsynaptische Endigungen große PSPs hervorrufen, dann ist ihr Einfluß dominierend und wird entsprechend der präzisen statistischen Form ausgeübt. Wenn die Endigungen kleine PSPs hervorrufen, dann hängt ihr Einfluß weitgehend von dem Grad der Abhängigkeit voneinander ab. Wenn die präsynaptischen Endigungen unkorreliert sind, dann vermitteln ihre durchschnittlichen, präsynaptischen Entladungsgeschwindigkeiten eine gleichmäßige Regulierung des postsynaptischen Membranpotentials und der postsynaptischen Entladungsgeschwindigkeiten. Sind andererseits die präsynaptischen Endigungen korreliert, dann nehmen sie eine dominierende Funktion ein, und die Beziehungen zwischen präsynaptischer Regulierung und postsynaptischer Entladung können präzise definiert werden. Somit stellt sich der Grad der Abhängigkeit zwischen den präsynaptischen Endigungen als eine funktionell bedeutende Variable dar.

Notes: Abstract A mathematical model designed originally for pacemaker neurons and post synaptic potentials represents acceptably respiratory driving by a pump through the Hering-Breuer reflex (Vibert et al. 1981). Discrepancies with the respiratory embodiment arose firstly in the zig-zag plots of average driving and driven intervals where the boundaries between positively and negatively-sloped segments differed, and where a hysteresis-like dependence on the order of the observations occurred. Secondly, they arose from the fact that estimates of the slope A and intercept B of the linear dependence of the delay on the arrival phase of the driver differed when based on the model's formulae but using different aspects of the same data. The question thus arose as to which model shortcomings cause discrepancies with this particular embodiment. This communication presents computer simulations that explore the consequences of modifying, or imposing variabilities, upon several parameters. A reflects how the driver's (i.e. the inflation's) consequences increase as it arrives later in the driven (i.e. phrenic) cycle: when it departs from around 0.7, (as required by the model) and approaches 1. The positively sloped segments broaden at the expense of the negatively-sloped ones which disappear when A reaches 1. The intercept B reflects the effect of the driver when it arrives just after the driven event: its increases shift a relatively unchanged plot up and to the right. When the driving period I exhibits variability or the driver one N exhibits variability or trends, as happens often with the respiratory embodiment, the model-predicted behavior remains acceptably natural-like: in particular, that with a jittery driven period mimics well observations on cats with high CO2 or superficially anesthetized. For each driving period, the driven intervals pertained to a limited number of classes (e.g. to one, two or three) which occurred in an invariant sequence.