Transients in the inhibitory driving of neurons and their postsynaptic consequences

Global dynamics of a stochastic neuronal oscillator

Nonlinear oscillators have been used to model neurons that fire periodically in the absence of input. These oscillators, which are called neuronal oscillators, share some common response structures with other biological oscillations such as cardiac cells. In this study, we analyze the dependence of the global dynamics of an impulse-driven stochastic neuronal oscillator on the relaxation rate to the limit cycle, the strength of the intrinsic noise, and the impulsive input parameters. To do this, we use a Markov operator that both reflects the density evolution of the oscillator and is an extension of the phase transition curve, which describes the phase shift due to a single isolated impulse. Previously, we derived the Markov operator for the finite relaxation rate that describes the dynamics of the entire phase plane. Here, we construct a Markov operator for the infinite relaxation rate that describes the stochastic dynamics restricted to the limit cycle. In both cases, the response of the stochastic neuronal oscillator to time-varying impulses is described by a product of Markov operators. Furthermore, we calculate the number of spikes between two consecutive impulses to relate the dynamics of the oscillator to the number of spikes per unit time and the interspike interval density. Specifically, we analyze the dynamics of the number of spikes per unit time based on the properties of the Markov operators. Each Markov operator can be decomposed into stationary and transient components based on the properties of the eigenvalues and eigenfunctions. This allows us to evaluate the difference in the number of spikes per unit time between the stationary and transient responses of the oscillator, which we show to be based on the dependence of the oscillator on past activity. Our analysis shows how the duration of the past neuronal activity depends on the relaxation rate, the noise strength, and the impulsive input parameters.

Stochastic phase transition operator

In this study a Markov operator is introduced that represents the density evolution of an impulse-driven stochastic biological oscillator. The operator's stochastic kernel is constructed using the asymptotic expansion of stochastic processes instead of solving the Fokker-Planck equation. The Markov operator is shown to successfully approximate the density evolution of the biological oscillator considered. The response of the oscillator to both periodic and time-varying impulses can be analyzed using the operator's transient and stationary properties. Furthermore, an unreported stochastic dynamic bifurcation for the biological oscillator is obtained by using the eigenvalues of the product of the Markov operators.

Influence of the temporal distribution of electric pulses on transcallosal single unit responses

We examined how differently timed stimuli to one auditory cortex affect the spike trains they drive in the controlateral homotopic field of anesthetized rats. Bipolar electrical stimulations consisted of trains of pulses (100 micro s, <500 micro A) at rates of 25, 50 or 125 pulses/s and with different stimulus patterns (i.e., dispersions, sequences), called "pacemaker", "accelerando" or "decelerando". Trains lasted for 342 ms and were separated by 4 s. When trains were evaluated over times comparable to the stimulus duration changes characteristically involved an initial slowing followed by recovery and several discharges both stimulus- and neuron-dependent. When evaluated by cross-correlations between cortical cell pairs, the changes extended far beyond the stimulus end. Results suggest that interhemispheric projections, by way of their averages and patterns, play key, long duration roles in the spike-dependent properties of cortical synapses (e.g., potentiation, depression) and thus of cortical circuit operations.

Pooled spike trains of correlated presynaptic inputs as realizations of cluster point processes

The pooled spike trains of correlated presynaptic terminals acting synchronously upon a single neuron are realizations of cluster point processes: the notions of spikes synchronizing in bursts and of points bunching in clusters are conceptually identical. The primary processes constituent specifies the timing of the cluster series; subsidiary processes and poolings specify burst structure and tightness. This representation and the Poisson process representation of independent terminals complete the formal approach to pooled trains. The notion's usefulness was illustrated by expressing physiological questions in terms of those constituents, each possessing a clear biological embodiment; constituents provided the control variables in simulations using leaky integrate-and-fire postsynaptic neurons excited by multiple weak terminals. Regular or irregular primary processes and bursts series determined low or high postsynaptic dispersions. When convergent set synchrony increased, its postsynaptic consequences approached those of single powerful synapses; concomitantly, output spike trains approached periodic, quasiperiodic, or aperiodic behaviors. The sequence in which terminals fired within bursts affected the predictee and predictor roles of presynaptic and postsynaptic spikes; when inhibition was added, EPSP and IPSP delays and order were influential (summation was noncommutative). Outputs to different correlations were heterogeneous; heterogeneity was accentuated by conditioning by variables such as DC biases.

Synaptic transmission of chaotic spike trains between primary afferent fiber and spinal dorsal horn neuron in the rat

Primary sensory neurons can generate irregular burst firings in which the existence of significant deterministic behaviors of chaotic dynamics has been proved with nonlinear time series analysis. But how well the deterministic characteristics and neural information of presynaptic chaotic spike trains were transmitted into postsynaptic spike trains is still an open question. Here we investigated the synaptic transmission of chaotic spike trains between primary Adelta afferent fiber and spinal dorsal horn neuron. Two kinds of basic stimulus unit, brief burst and single pulse, were employed by us to comprise chaotic stimulus trains. For time series analysis, we defined "events" as the longest sequences of spikes with all interspike intervals less than or equal to a certain threshold and extracted the interevent intervals (IEIs) from spike trains. Return map analysis of the IEI series showed that the main temporal structure of chaotic input trains could be detected in postsynaptic output trains, especially under brief-burst stimulation. Using correlation dimension and nonlinear prediction methods, we found that synaptic transmission could influence the nonlinear characteristics of chaotic trains, such as fractal dimension and short-term predictability, with greater influence made under single-pulse stimulation. By calculating the mutual information between input and output trains, we found the information carried by presynaptic spike trains could not be completely transmitted at primary afferent synapses, and that brief bursts could more reliably transmit the information carried by chaotic input trains across synapses. These results indicate that although unreliability exists during synaptic transmission, the main deterministic characteristics of chaotic burst trains can be transmitted across primary afferent synapses. Moreover, brief bursts that come from the periphery can more reliably transmit neural information between primary afferent fibers and spinal dorsal horn neurons.

An analysis of the reliability phenomenon in the FitzHugh-Nagumo model

The reliability of single neurons on realistic stimuli has been experimentally confirmed in a wide variety of animal preparations. We present a theoretical study of the reliability phenomenon in the FitzHugh-Nagumo model on white Gaussian stimulation. The analysis of the model's dynamics is performed in three regimes-the excitable, bistable, and oscillatory ones. We use tools from the random dynamical systems theory, such as the pullbacks and the estimation of the Lyapunov exponents and rotation number. The results show that for most stimulus intensities, trajectories converge to a single stochastic equilibrium point, and the leading Lyapunov exponent is negative. Consequently, in these regimes the discharge times are reliable in the sense that repeated presentation of the same aperiodic input segment evokes similar firing times after some transient time. Surprisingly, for a certain range of stimulus intensities, unreliable firing is observed due to the onset of stochastic chaos, as indicated by the estimated positive leading Lyapunov exponents. For this range of stimulus intensities, stochastic chaos occurs in the bistable regime and also expands in adjacent parts of the excitable and oscillating regimes. The obtained results are valuable in the explanation of experimental observations concerning the reliability of neurons stimulated with broad-band Gaussian inputs. They reveal two distinct neuronal response types. In the regime where the first Lyapunov has negative values, such inputs eventually lead neurons to reliable firing, and this suggests that any observed variance of firing times in reliability experiments is mainly due to internal noise. In the regime with positive Lyapunov exponents, the source of unreliable firing is stochastic chaos, a novel phenomenon in the reliability literature, whose origin and function need further investigation.

Rate coding in a chain of pulse-coupled oscillators

The input-output relation of a chain of pulse-coupled oscillators is examined. The oscillators capture the essential aspect of the dynamics of pacemaker neurons. Inputs consist of pacemaker, and noisy trains impinging upon the first unit in the chain. The response of the chain is defined as the spike train emitted by the last unit. We observe two important phenomena in the response of the chain for a given input train, whether pacemaker or noisy. First, the mean output rate of the chain is equal to the mean input rate in the range of input rate in which one input pulse corresponds to one output spike without phase locking (1:1 alternation). Second, for the same input range, the output interspike intervals tend to the average of the input interpulse intervals in a long chain of oscillators. This behavior contrasts with the fact that the response of a single unit depends on both input rate and pattern. We show that the response of the chain is reproduced by the phase transition curve which represents the phase shift due to a single isolated pulse stimulus. This analysis reveals that the averaging of the output interspike intervals is due to the geometrical aspect of the phase transition curve. This geometrical aspect causes the dependence of the response of a single unit on input pattern.

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.

Analysis of the response of a pacemaker neuron model to transient inputs

The response of a pacemaker neuron model to a train of transient inhibitory impulsive perturbations is examined. The model reproduces the heterogeneous discharge forms and abrupt switchings displayed by the crayfish slowly adapting stretch receptor organ (Segundo et al., 1994, Neuroscience 62(2), pp. 459-480). The non-monotonous aspect of the instantaneous firing rate of the model reflects the fact that in some regimes input and output rates are both increasing, despite the inhibitory effect of the former. We determine how such paradoxical acceleration takes place by analyzing the response of the model using its phase transition curve. We show that paradoxical acceleration results from the fact that the phase transition curve exhibits a large slowly increasing, almost linear section similar to that of living preparations.

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.

XNBC: a simulation tool. Application to the study of neural coding using hybrid networks

XNBC is a software package for simulating biological neural networks. Two neuron models are available, a leaky integrator model and an ion-conductance model. Inputs to the simulated neurons can be provided by experimental data stored in files, allowing the creation of 'hybrid' networks. Graphic tools are used to describe the modeled neurons as well as the network. Neuron and network parameters can be modified during the simulation, to mimic electrical stimulations and drugs action. The temporal evolution of the network and of selected neurons can be visualized. A point process, frequency or dynamic analysis of the simulator output can be performed. The successive stages of the creation of a hybrid network are explained.

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)

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)