A basic bifurcation structure from bursting to spiking of injured nerve fibers in a two-dimensional parameter space

Learning-based sliding mode synchronization for fractional-order Hindmarsh-Rose neuronal models with deterministic learning

Introduction

In recent years, extensive research has been conducted on the synchronous behavior of neural networks. It is found that the synchronization ability of neurons is related to the performance of signal reception and transmission between neurons, which in turn affects the function of the organism. However, most of the existing synchronization methods are faced with two difficulties, one is the structural parameter dependency, which limits the promotion and application of synchronous methods in practical problems. The other is the limited adaptability, that is, even when faced with the same control tasks, for most of the existing control methods, the control parameters still need to be retrained. To this end, the present study investigates the synchronization problem of the fractional-order HindmarshRose (FOHR) neuronal models in unknown dynamic environment.

Methods

Inspired by the human experience of knowledge acquiring, memorizing, and application, a learning-based sliding mode control algorithm is proposed by using the deterministic learning (DL) mechanism. Firstly, the unknown dynamics of the FOHR system under unknown dynamic environment is locally accurately identified and stored in the form of constant weight neural networks through deterministic learning without dependency of the system parameters. Then, based on the identified and stored system dynamics, the model-based and relearning-based sliding mode controller are designed for similar as well as new synchronization tasks, respectively.

Results

The synchronization process can be started quickly by recalling the empirical dynamics of neurons. Therefore, fast synchronization effect is achieved by reducing the online computing time. In addition, because of the convergence of the identification and synchronization process, the control experience can be constantly replenished and stored for reutilization, so as to improve the synchronization speed and accuracy continuously.

Discussion

The thought of this article will also bring inspiration to the related research in other fields.

Three-dimensional memristive Morris-Lecar model with magnetic induction effects and its FPGA implementation

To characterize the magnetic induction flow induced by neuron membrane potential, a three-dimensional (3D) memristive Morris-Lecar (ML) neuron model is proposed in this paper. It is achieved using a memristor induction current to replace the slow modulation current in the existing 3D ML neuron model with fast-slow structure. The magnetic induction effects on firing activities are explained by the spiking/bursting firings with period-adding bifurcation and periodic/chaotic spiking-bursting patterns, and the bifurcation mechanisms of the bursting patterns are elaborated using the fast-slow analysis method to create two bifurcation sets. In particular, the 3D memristive ML model can also exhibit the homogeneous coexisting bursting patterns when switching the memristor initial states, which are effectively illustrated by the theoretical analysis and numerical simulations. Finally, a digitally FPGA-based hardware platform is developed for the 3D memristive ML model and the experimentally measured results well verify the numerical ones.

Bifurcations in the firing of neuronal population caused by a small difference in pulse parameters during sustained stimulations in rat hippocampus in vivo

OBJECTIVE

The bifurcation of neuronal firing is one of important nonlinear phenomena in the nervous system and is characterized by a significant change in the rate or temporal pattern of neuronal firing on responding to a small disturbance from external inputs. Previous studies have reported firing bifurcations for individual neurons, not for a population of neurons. We hypothesized that the integrated firing of a neuronal population could also show a bifurcation behavior that should be important in certain situations such as deep brain stimulations. The hypothesis was verified by experiments of rat hippocampus in vivo.

METHODS

Stimulation sequences of paired-pulses with two different inter-pulse-intervals (IPIs) or with two different pulse intensities were applied on the alveus of hippocampal CA1 region in anaesthetized rats. The amplitude and area of antidromic population spike (APS) were used as indices to evaluate the differences in the responses of neuronal population to the different pulses in stimulations.

RESULTS

During sustained paired-pulse stimulations with a high mean pulse frequency such as ~130 Hz, a small difference of only a few percent in the two IPIs or in the two intensities was able to generate a sequence of evoked APSs with a substantial bifurcation in their amplitudes and areas.

CONCLUSION

Small differences in the excitatory inputs can cause nonlinearly enlarged differences in the induced firing of neuronal populations.

SIGNIFICANCE

The novel dynamics and bifurcation of neuronal responses to electrical stimulations provide important clues for developing new paradigms to extend neural stimulations to treat more diseases.

Anti-control of periodic firing in HR model in the aspects of position, amplitude and frequency

This paper proposes a novel controller to control position, amplitude and frequency of periodic firing activity in Hindmarsh-Rose model based on Hopf bifurcation theory which is composed of linear control gain and nonlinear control gain. First, we select the activation of the fast ion channel as control parameter. Based on explicit criterion of Hopf bifurcation, a series of conditions are obtained to derive the linear gains of controller responsible for control of the location where the periodic firing activity occurs. Then, based on the control parameter, a series of conditions are obtained to derive the nonlinear gains of controller responsible for controlling the amplitude and frequency of periodic firing activity by using center manifold and normal form. Finally, the numerical experiments show that our controller can make the periodic firing activity occur at designed value and control the amplitude and frequency of periodic firing activity by adjusting nonlinear control gain of controller.

Numerical Bifurcation Analysis of Pacemaker Dynamics in a Model of Smooth Muscle Cells

Evidence from experimental studies shows that oscillations due to electro-mechanical coupling can be generated spontaneously in smooth muscle cells. Such cellular dynamics are known as pacemaker dynamics. In this article, we address pacemaker dynamics associated with the interaction of [Formula: see text] and [Formula: see text] fluxes in the cell membrane of a smooth muscle cell. First we reduce a pacemaker model to a two-dimensional system equivalent to the reduced Morris-Lecar model and then perform a detailed numerical bifurcation analysis of the reduced model. Existing bifurcation analyses of the Morris-Lecar model concentrate on external applied current, whereas we focus on parameters that model the response of the cell to changes in transmural pressure. We reveal a transition between Type I and Type II excitabilities with no external current required. We also compute a two-parameter bifurcation diagram and show how the transition is explained by the bifurcation structure.

Coherence resonance for neuronal bursting with spike undershoot

Although the bursting patterns with spike undershoot are involved with the achievement of physiological or cognitive functions of brain with synaptic noise, noise induced-coherence resonance (CR) from resting state or subthreshold oscillations instead of bursting has been widely identified to play positive roles in information process. Instead, in the present paper, CR characterized by the increase firstly and then decease of peak value of power spectrum of spike trains is evoked from a bursting pattern with spike undershoot, which means that the minimal membrane potential within burst is lower than that of the subthreshold oscillations between bursts, while CR cannot be evoked from the bursting pattern without spike undershoot. With bifurcations and fast-slow variable dissection method, the bursting patterns with and without spike undershoot are classified into "Sub-Hopf/Fold" bursting and "Fold/Homoclinic" bursting, respectively. For the bursting with spike undershoot, the trajectory of the subthreshold oscillations is very close to that of the spikes within burst. Therefore, noise can induce more spikes from the subthreshold oscillations and modulate the bursting regularity, which leads to the appearance of CR. For the bursting pattern without spike undershoot, the trajectory of the quiescent state is not close to that of the spikes within burst, and noise cannot induce spikes from the quiescent state between bursts, which is cause for non-CR. The result provides a novel case of CR phenomenon and extends the scopes of CR concept, presents that noise can enhance rather than suppress information of the bursting patterns with spike undershoot, which are helpful for understanding the dynamics and the potential physiological or cognitive functions of the nerve fiber or brain neurons with such bursting patterns.

Effect of potassium channel noise on nerve discharge based on the Chay model

BACKGROUND:

The nervous system senses and transmits information through the firing behavior of neurons, and this process is affected by various noises. However, in the previous study of the influence of noise on nerve discharge, the channel of some noise effects is not clear, and the difference from other noises was not examined.

OBJECTIVE:

To construct ion channel noise which is more biologically significant, and to clarify the basic characteristics of the random firing rhythm of neurons generated by different types of noise acting on ion channels.

Method:

Based on the dynamics of the ion channel, we constructed ion channel noise. We simulated the nerve discharge based on the Chay model of potassium ion channel noise, and used the nonlinear time series analysis method to measure the certainty and randomness of nerve discharge.

RESULTS:

In the Chay model with potassium ion noise, the chaotic rhythm defined by the original model could be effectively unified with the random rhythm simulated by the previous random Chay model into a periodic bifurcation process.

CONCLUSION:

This method clarified the influence of ion channel noise on nerve discharge, better understood the randomness of nerve discharge and provided a more reasonable explanation for the mechanism of nerve discharge.

Multiple bifurcations and coexistence in an inertial two-neuron system with multiple delays

In this paper, we construct an inertial two-neuron system with multiple delays, which is described by three first-order delayed differential equations. The neural system presents dynamical coexistence with equilibria, periodic orbits, and even quasi-periodic behavior by employing multiple types of bifurcations. To this end, the pitchfork bifurcation of trivial equilibrium is analyzed firstly by using center manifold reduction and normal form method. The system presents different sequences of supercritical and subcritical pitchfork bifurcations. Further, the nontrivial equilibrium bifurcated from trivial equilibrium presents a secondary pitchfork bifurcation. The system exhibits stable coexistence of multiple equilibria. Using the pitchfork bifurcation curves, we divide the parameter plane into different regions, corresponding to different number of equilibria. To obtain the effect of time delays on system dynamical behaviors, we analyze equilibrium stability employing characteristic equation of the system. By the Hopf bifurcation, the system illustrates a periodic orbit near the trivial equilibrium. We give the stability regions in the delayed plane to illustrate stability switching. The neural system is illustrated to have Hopf-Hopf bifurcation points. The coexistence with two periodic orbits is presented near these bifurcation points. Finally, we present some mixed dynamical coexistence. The system has a stable coexistence with periodic orbit and equilibrium near the pitchfork-Hopf bifurcation point. Moreover, multiple frequencies of the system induce the presentation of quasi-periodic behavior. The system presents stable coexistence with two periodic orbits and one quasi-periodic behavior.

Temperature effect on memristive ion channels

Neuron shows distinct dependence of electrical activities on membrane patch temperature, and the mode transition of electrical activity is induced by the patch temperature through modulating the opening and closing rates of ion channels. In this paper, inspired by the physical effect of memristor, the potassium and sodium ion channels embedded in the membrane patch are updated by using memristor-based voltage gate variables, and an external stimulus is applied to detect the variety of mode selection in electrical activities under different patch temperatures. It is found that each ion channel can be regarded as a physical memristor, and the shape of pinched hysteresis loop of memristor is dependent on both input voltage and patch temperature. The pinched hysteresis loops of two ion-channel memristors are dramatically enlarged by increasing patch temperature, and the hysteresis lobe areas are monotonously reduced with the increasing of excitation frequency if the frequency of external stimulus exceeds certain threshold. However, for the memristive potassium channel, the AREA1 corresponding to the threshold frequency is increased with the increasing of patch temperature. The amplitude of conductance for two ion-channel memristors depends on the variation of patch temperature. The results of this paper might provide insights to modulate the neural activities in appropriate temperature condition completely, and involvement of external stimulus enhance the effect of patch temperature.

Energy expenditure computation of a single bursting neuron

Brief bursts of high-frequency spikes are a common firing pattern of neurons. The cellular mechanisms of bursting and its biological significance remain a matter of debate. Focusing on the energy aspect, this paper proposes a neural energy calculation method based on the Chay model of bursting. The flow of ions across the membrane of the bursting neuron with or without current stimulation and its power which contributes to the change of the transmembrane electrical potential energy are analyzed here in detail. We find that during the depolarization of spikes in bursting this power becomes negative, which was also discovered in previous research with another energy model. We also find that the neuron’s energy consumption during bursting is minimal. Especially in the spontaneous state without stimulation, the total energy consumption (2.152 × 10⁻⁷ J) during 30 s of bursting is very similar to the biological energy consumption (2.468 × 10⁻⁷ J) during the generation of a single action potential, as shown in Wang et al. (Neural Plast 2017, 2017a). Our results suggest that this property of low energy consumption could simply be the consequence of the biophysics of generating bursts, which is consistent with the principle of energy minimization. Our results also imply that neural energy plays a critical role in neural coding, which opens a new avenue for research of a central challenge facing neuroscience today.

Computational study on neuronal activities arising in the pre-Bötzinger complex

Experimental investigations have shown that the pre-Bötzinger complex (pre-BötC) within the mammalian brainstem generates the inspiratory phase of respiratory rhythm. Based on a single-compartment model of a pre-BötC inspiratory neuron, we, in this paper, use semi-analytical, numerical as well as fast-slow dynamical methods to investigate the effects of sodium conductance ([Formula: see text]) and potassium conductance ([Formula: see text]) on the firing activities of pre-BötC and try to reveal the dynamical mechanisms behind them. We show how [Formula: see text] and [Formula: see text] affect the bifurcations of the fast-subsystem and how the the firing patterns of pre-BötC transit according to the bifurcations.