Huaguang Gu

Integer multiple spiking in neuronal pacemakers without external periodic stimulation

Abstract An integer multiple spiking, whose interspike intervals (ISIs) were mostly located at the integer multiples of a basic ISI, was observed in the experiment on a neuronal pacemaker without external periodic stimulation. It was observed between the period 1 spiking and the subthreshold oscillation when extra-cellular calcium concentration ([Ca ++ ] o ) was changed as a control parameter. A Hopf bifurcation point across which the period 1 spiking changed into the subthreshold oscillation directly was observed in the corresponding parameter region in the deterministic Chay model. In the stochastic Chay model, if the parameter v c (corresponding to [Ca ++ ] o in the experiment) was changed across the Hopf bifurcation point, the integer multiple spiking, appeared between the period 1 spiking and the subthreshold oscillation, was simulated. It exhibited similar characters to those in the experiment and was verified to be caused by autonomous stochastic resonance (ASR). The result revealed the sequence of firing patterns near the Hopf bifurcation point and indicated that ASR was probably the cause of the generation of the integer multiple spiking in the experiment.

Experimental observation of the stochastic bursting caused by coherence resonance in a neural pacemaker

The inter-burst intervals (IBIs) of the stochastic bursting caused by the effect of coherence resonance (CR) in Hindmarsh-Rose model exhibited multiple mode and were approximately integer multiples of a basic IBI. A similar bursting pattern was found in the experiment on an experimental neural pacemaker perfused with solution whose extracellular calcium concentration ([Ca2+]o) was lower than the normal level in which the firing pattern was period 1 spiking. The results verified the existence of the stochastic bursting and implied its CR mechanism. In addition, the distinction between the integer multiple bursting and the integer multiple spiking, another classical firing pattern caused by CR, was identified. The integer multiple bursting was generated in the parameter region where [Ca2+]o was lower than normal.

A SERIES OF BIFURCATION SCENARIOS IN THE FIRING PATTERN TRANSITIONS IN AN EXPERIMENTAL NEURAL PACEMAKER

Various bifurcation scenarios from period-1 bursting to period-1 spiking via a complex procedure were simulated in previous theoretical studies on neuronal models. The results revealed a general principle of neuronal firing pattern transitions. In this letter, three types of bifurcation scenarios with respect to extracellar calcium concentration ([Ca++]o) were discovered in experiments on neural pacemakers. Such a series of bifurcation scenarios implied complex structure of bifurcations in the firing pattern transitions of neurons. In the two-dimensional parameter space of Chay model, three classical kinds of bifurcation scenarios with respect to the bifurcation parameter vc (the reverse potential of calcium concentration) were simulated. By the variation of the conditional parameter, λn, the relationship among the three bifurcation scenarios was revealed. The results not only verified the existence of different bifurcation scenarios in real neuronal system, but also indicated that the differences among t...

IDENTIFYING DISTINCT STOCHASTIC DYNAMICS FROM CHAOS: A STUDY ON MULTIMODAL NEURAL FIRING PATTERNS

Some chaotic and a series of stochastic neural firings are multimodal. Stochastic multimodal firing patterns are of special importance because they indicate a possible utility of noise. A number of previous studies confused the dynamics of chaotic and stochastic multimodal firing patterns. The confusion resulted partly from inappropriate interpretations of estimations of nonlinear time series measures. With deliberately chosen examples the present paper introduces strategies and methods of identification of stochastic firing patterns from chaotic ones. Aided by theoretical simulation we show that the stochastic multimodal firing patterns result from the effects of noise on neuronal systems near to a bifurcation between two simpler attractors, such as a point attractor and a limit cycle attractor or two limit cycle attractors. In contrast, the multimodal chaotic firing trains are generated by the dynamics of a specific strange attractor. Three systems were carefully chosen to elucidate these two mechanisms. An experimental neural pacemaker model and the Chay mathematical model were used to show the stochastic dynamics, while the deterministic Wang model was used to show the deterministic dynamics. The usage and interpretation of nonlinear time series measures were systematically tested by applying them to firing trains generated by the three systems. We successfully identified the distinct differences between stochastic and chaotic multimodal firing patterns and showed the dynamics underlying two categories of stochastic firing patterns. The first category results from the effects of noise on the neuronal system near a Hopf bifurcation. The second category results from the effects of noise on the period-adding bifurcation between two limit cycles. Although direct application of nonlinear measures to interspike interval series of these firing trains misleadingly implies chaotic properties, definition of eigen events based on more appropriate judgments of the underlying dynamics leads to accurate identifications of the stochastic properties.

Integer multiple spiking in the stochastic Chay model and its dynamical generation mechanism

Abstract Integer multiple spiking induced by Gaussian white noise in the stochastic Chay system may occur near a subcritical Hopf bifurcation of the corresponding deterministic system without external periodic forcing. The deterministic Chay system near the subcritical Hopf bifurcation can exhibit a damped subthreshold oscillation of membrane potential, and can easily fire a spike in response to some kinds of weak pulses. Therefore, on account of the characteristic of Gaussian white noise, it can be shown that the deterministic Chay system supplied with certain weak pulses can fire integer multiple spiking similar to that in the corresponding stochastic system. Furthermore, on the light of the characteristic of the unstable limit cycle “before” a subcritical Hopf bifurcation in the deterministic Chay model, it can be explained that integer multiple spiking only appears in a certain range of the parameter VK.

Different classifications of UPOs in the parametrically different chaotic ISI series of a neural pacemaker

Various interspike intervals (ISIs) in a series of spontaneous discharges of experimental neural pacemakers were identified as chaotic and as lying between stable period 2 and stable period 3. The method of So et al. for detection of unstable periodic orbits (UPOs) was applied to analyze two chaotic ISI series generated under the action of [Ca2+]o in different concentrations. In the chaotic ISI series near the stable period 2, an unstable period 2 orbit was detected. In the chaotic ISI series near the stable period 3, both an unstable period 2 orbit and an unstable period 3 orbit were detected. The location of the unstable period 2 orbit was close to that of the stable period 2 in the return map, while the location of the unstable period 3 orbit was close to that of the stable period 3. The results not only revealed the structures of various chaotic ISI series, but also indicated that the classification of UPOs could reflect the experimental control parameters at which the chaotic ISI series were generated. The previously discovered period adding bifurcation in the ISI series generated by experimental neural pacemakers [12] was further described in terms of the evolution of the period orbits.

COHERENCE RESONANCE–INDUCED STOCHASTIC NEURAL FIRING AT A SADDLE-NODE BIFURCATION

Coherence resonance at a saddle-node bifurcation point and the corresponding stochastic firing patterns are simulated in a theoretical neuronal model. The characteristics of noise-induced neural firing pattern, such as exponential decay in histogram of interspike interval (ISI) series, independence and stochasticity within ISI series are identified. Firing pattern similar to the simulated results was discovered in biological experiment on a neural pacemaker. The difference between this firing and integer multiple firing generated at a Hopf bifurcation point is also given. The results not only revealed the stochastic dynamics near a saddle-node bifurcation, but also gave practical approaches to identify the saddle-node bifurcation and to distinguish it from the Hopf bifurcation in neuronal system. In addition, many previously observed firing patterns can be attribute to stochastic firing pattern near such a saddle-node bifurcation.

Noise Induced Multi-Mode Neural Firings in a Period Adding Bifurcation Scenario

A series of new firing patterns lying between two kinds of periodic bursting patterns and randomly alternating on the two kinds of periodic bursts, were discovered in the experiment on neural pacemaker and induced by noise in Chay model near each bifurcation point in a period adding bifurcation scenario without chaos, exhibiting multi-mode characteristics that was the signature of autonomous stochastic resonance in neuronal firing. Two endogenous characteristics near each bifurcation point that made noise induce the new firing patterns were identified. One was that there existed critical phase between the trajectories of the two kinds of bursting patterns. The other was that the periods of the two kinds of bursting patterns near each bifurcation point were small.

Experimental Discovery of a Complex Bifurcation Scenario from Period 1 Bursting to Period 1 Spiking

A complex bifurcation scenario from period 1 bursting to period 1 spiking containing period increment bifurcation scenario in bursting patterns, simulated in theoretical model in previous studies, awaited experimental demonstration. In this Letter, the bifurcation scenario was discovered in an experimental neural pacemaker. The characteristics of the bifurcation scenario, the period increment bifurcation scenario and the transition from bursting to spiking pattern were also identified. The results demonstrate the reality of the theoretical prediction of firing pattern transition regularity.

Neural Information Encoding Based on a Bifurcation Machinery

Neural information encoding mechanism based on a bifurcation machinery was proposed based on the results on two experimental models and theoretical models. The framework structures of bifurcation scenarios of neuronal firing pattern were acquired in the experimental neural pacemaker. The dynamics of dynamic neural firing pattern and the neural information encoding of firing patterns were elucidated in the experiment on rabbit depressor nerve, based on the bifurcation structure of firing patterns, and the mean level and temporal procedure of blood pressure.