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

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.

Determining Burst Firing Time Distributions from Multiple Spike Trains

Recent experimental findings have shown the presence of robust and cell-type-specific intraburst firing patterns in bursting neurons. We address the problem of characterizing these patterns under the assumption that the bursts exhibit well-defined firing time distributions. We propose a method for estimating these distributions based on a burst alignment algorithm that minimizes the overlap among the firing time distributions of the different spikes within the burst. This method provides a good approximation to the burst's intrinsic temporal structure as a set of firing time distributions. In addition, the method allows labeling the spikes in any particular burst, establishing a correspondence between each spike and the distribution that best explains it, and identifying missing spikes. Our results on both simulated and experimental data from the lobster stomatogastric ganglion show that the proposed method provides a reliable characterization of the intraburst firing patterns and avoids the errors derived from missing spikes. This method can also be applied to nonbursting neurons as a general tool for the study and the interpretation of firing time distributions as part of a temporal neural code.

Neural signatures: multiple coding in spiking-bursting cells

Recent experiments have revealed the existence of neural signatures in the activity of individual cells of the pyloric central pattern generator (CPG) of crustacean. The neural signatures consist of cell-specific spike timings in the bursting activity of the neurons. The role of these intraburst neural fingerprints is still unclear. It has been reported previously that some muscles can reflect small changes in the spike timings of the neurons that innervate them. However, it is unclear to what extent neural signatures contribute to the command message that the muscles receive from the motoneurons. It is also unknown whether the signatures have any functional meaning for the neurons that belong to the same CPG or to other interconnected CPGs. In this paper, we use realistic neural models to study the ability of single cells and small circuits to recognize individual neural signatures. We show that model cells and circuits can respond distinctly to the incoming neural fingerprints in addition to the properties of the slow depolarizing waves. Our results suggest that neural signatures can be a general mechanism of spiking-bursting cells to implement multicoding.

What is signal and what is noise in the brain?

The response of a cortical neuron to a stimulus can show a very large variability when repeatedly stimulated by exactly the same stimulus. This has been quantified in terms of inter-spike-interval (ISI) statistics by several researchers (e.g., [Softky, W., Koch, C., 1993. The highly irregular firing of cortical cells is inconsistent with temporal integration of random EPSPs. J. Neurosci. 13(1), 334-350.]). The common view is that this variability reflects noisy information processing based on redundant representation in large neuron populations. This view has been challenged by the idea that the apparent noise inherent in brain activity that is not strictly related or temporally coupled to the experiment could be functionally significant. In this work we examine the ISI statistics and discuss these views in a recently published model of interacting cortical areas [Knoblauch, A., Palm, G., 2002. Scene segmentation by spike synchronization in reciprocally connected visual areas. I. Local effects of cortical feedback. Biol. Cybernet. 87(3), 151-167.]. From the results of further single neuron simulations we can isolate temporally modulated synaptic input as a main contributor for high ISI variability in our model and possibly in real neurons. In contrast to alternative mechanisms, our model suggests a function of the temporal modulations for short-term binding and segmentation of figures from background. Moreover, we show that temporally modulated inputs lead to ISI statistics which fit better to the neurophysiological data than alternative mechanisms.

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 modulation of the interspike interval signatures of bursting pyloric neurons

The pyloric network of the lobster stomatogastric nervous system is one of the best described assemblies of oscillatory neurons producing bursts of action potentials. While the temporal patterns of bursts have been investigated in detail, those of spikes have received less attention. Here we analyze the intraburst firing patterns of pyloric neurons and the synaptic interactions shaping their dynamics in millisecond time scales not performed before. We find that different pyloric neurons express characteristic, cell-specific firing patterns in their bursts. Nonlinear analysis of the interspike intervals (ISIs) reveals distinctive temporal structures ('interspike interval signatures'), which are found to depend on the synaptic connectivity of the network. We compare ISI patterns of the pyloric dilator (PD), lateral pyloric (LP), and ventricular dilator (VD) neurons in 1) normal conditions, 2) after blocking glutamatergic synaptic connections, and 3) in various functional configurations of the three neurons. Manipulation of the synaptic connectivity results in characteristic changes in the ISI signatures of the postsynaptic neurons. The intraburst firing pattern of the PD neuron is regularized by the inhibitory synaptic connection from the LP neuron as revealed in current-clamp experiments and also as reconstructed with a dynamic clamp. On the other hand, mutual inhibition between the LP and VD neurons tend to produce more irregular bursts with increased spike jitter. The results show that synaptic interactions fine-tune the output of pyloric neurons. The present data also suggest a way of processing of synaptic information: bursting neurons are capable of encoding incoming signals by altering the fine structure of their intraburst spike patterns.

Sexual dimorphism in the complexity of cardiac pacemaker activity

This study explored the effects of gender and aging on the complexity of cardiac pacemaker activity. Electrocardiogram signals were studied in normal women (n = 240) and men (n = 240) ranging in age from 40 to 79 yr. Nonlinear analysis of short-term resting R-R intervals was performed using the correlation dimension (CD), approximate entropy (ApEn), and largest Lyapunov exponent (LLE). Evidence of nonlinear structure was obtained by the surrogate data test. CD, ApEn, and LLE were negatively correlated with age. Despite similar means and SDs of the R-R intervals, women had a significantly higher CD, ApEn, and LLE compared with men in the age strata of 40-44 and 45-49 yr. CD and ApEn were strongly (r > 0.71) correlated with low- and high-frequency components. We conclude that the resting cardiac pacemaker activity of women is more complex than that of men in middle age, and the gender-related difference diminishes after the age of 50 yr. The higher complexity implies a more comprehensive neural modulation.

Possible neural coding with interevent intervals of synchronous firing

Neural networks composed of excitable neurons with noise generate rich nonlinear dynamics with spatiotemporal structures of neuronal spikes. Among various spatiotemporal patterns of spikes, synchronous firing has been studied most extensively both with physiological experimentation and with theoretical analysis. In this paper, we consider nonlinear neurodynamics in terms of synchronous firing and possibility of neural coding with such synchronous firing, which may be used in the "noisy brain." In particular, reconstruction of a chaotic attractor modeling a dynamical environment is explored with interevent intervals of synchronous firing from the perspective of nonlinear time series analysis and stochastic resonance.

Analytical characterization of spontaneous firing in networks of developing rat cultured cortical neurons

We have used a multiunit electrode array in extracellular recording to investigate changes in the firing patterns in networks of developing rat cortical neurons. The spontaneous activity of continual asynchronous firing or the alternation of asynchronous spikes and synchronous bursts changed over time so that activity in the later stages consisted exclusively of synchronized bursts. The spontaneous coordinated activity in bursts produced a variability in interburst interval (IBI) sequences that is referred to as "form." The stochastic and nonlinear dynamical analysis of IBI sequences revealed that these sequences reflected a largely random process and that the form for relatively immature neurons was largely oscillatory while the form for the more mature neurons was Poisson-like. The observed IBI sequences thus showed changes in form associated with both the intrinsic properties of the developing cells and the neural response to correlated synaptic inputs due to interaction between the developing neural circuits.

Nonlinear behavior of sinusoidally forced pyloric pacemaker neurons

Periodic current forcing was used to investigate the intrinsic dynamics of a small group of electrically coupled neurons in the pyloric central pattern generator (CPG) of the lobster. This group contains three neurons, namely the two pyloric dilator (PD) motoneurons and the anterior burster (AB) interneuron. Intracellular current injection, using sinusoidal waveforms of varying amplitude and frequency, was applied in three configurations of the pacemaker neurons: 1) the complete pacemaker group, 2) the two PDs without the AB, and 3) the AB neuron isolated from the PDs. Depending on the frequency and amplitude of the injected current, the intact pacemaker group exhibited a wide variety of nonlinear behaviors, including synchronization to the forcing, quasiperiodicity, and complex dynamics. In contrast, a single, broad 1:1 entrainment zone characterized the response of the PD neurons when isolated from the main pacemaker neuron AB. The isolated AB responded to periodic forcing in a manner similar to the complete pacemaker group, but with wider zones of synchronization. We have built an analog electronic circuit as an implementation of a modified Hindmarsh-Rose model for simulating the membrane potential activity of pyloric neurons. We subjected this electronic model neuron to the same periodic forcing as used in the biological experiments. This four-dimensional electronic model neuron reproduced the autonomous oscillatory firing patterns of biological pyloric pacemaker neurons, and it expressed the same stationary nonlinear responses to periodic forcing as its biological counterparts. This adds to our confidence in the model. These results strongly support the idea that the intact pyloric pacemaker group acts as a uniform low-dimensional deterministic nonlinear oscillator, and the regular pyloric oscillation is the outcome of cooperative behavior of strongly coupled neurons, having different dynamical and biophysical properties when isolated.