Spontaneous spiking in an autaptic Hodgkin-Huxley setup

Dynamical response of Autaptic Izhikevich Neuron disturbed by Gaussian white noise

Using the improved memristive Izhikevich neuron model, the effects of autaptic connection as well as electromagnetic induction are studied on the dynamical behavior of neuronal spiking. Using bifurcation analysis for membrane potentials, the effects of autaptic and electromagnetic parameters on the mode transition in electrical activities of the neuron model are investigated. Furthermore, white Gaussian noise is considered in the neuron model, to evaluate the effect of electromagnetic disturbance on the firing pattern of the neuron using the coefficient of variation. The bifurcation diagram versus autaptic conductance and time delay has been extensively studied. The results show that the effects of autaptic connection as well as electromagnetic induction on the spiking behavior of neurons can be well demonstrated by using the Izhikevich model. The electrical activities of the Izhikevich neuron model become more complex when the effects of autaptic connection and electromagnetic induction are considered in the neuron model. Using the Izhikevich neuron model, the high variety of spiking/bursting patterns is represented in the bifurcation diagram of inter-spike interval versus autaptic or electromagnetic parameters. Noise can have distinct effects on the spiking activity of the neuron, for the subthreshold input current, increasing the intensity of the electromagnetic noise increases the regularity of the neuron spiking, but for the suprathreshold input current, the regularity of spiking decreases with noise.

Transport and nonequilibrium phase transitions in polygonal urn models

We study the deterministic dynamics of N point particles moving at a constant speed in a 2D table made of two polygonal urns connected by an active rectangular channel, which applies a feedback control on the particles, inverting the horizontal component of their velocities when their number in the channel exceeds a fixed threshold. Such a bounce-back mechanism is non-dissipative: it preserves volumes in phase space. An additional passive channel closes the billiard table forming a circuit in which a stationary current may flow. Under specific constraints on the geometry and on the initial conditions, the large N limit allows nonequilibrium phase transitions between homogeneous and inhomogeneous phases. The role of ergodicity in making a probabilistic theory applicable is discussed for both rational and irrational urns. The theoretical predictions are compared with the numerical simulation results. Connections with the dynamics of feedback-controlled biological systems are highlighted.

Impacts of autapse on chaotic resonance in single neurons and small-world neuronal networks

Chaotic resonance (CR) is a new phenomenon induced by an intermediate level of chaotic signal intensity in neuronal systems. In the current study, we investigated the effects of autapse on the CR phenomenon in single neurons and small-world (SW) neuronal networks. In single neurons, we assume that the neuron has only one autapse modelled as electrical, excitatory chemical and inhibitory chemical synapse, respectively. Then, we analysed the effects of each one on the CR, separately. Obtained results revealed that, regardless of its type, autapse significantly increases the chaotic resonance of the appropriate autaptic parameter's values. It is also observed that, at the optimal chaotic current intensity, the multiple CR emerges depending on autaptic time delay for all the autapse types when the autaptic delay time or its integer multiples match the half period or period of the weak signal. In SW networks, we investigated the effects of chaotic activity on the prorogation of pacemaker activity, where pacemaker neurons have different kinds of autapse as considered in single neuron cases. Obtained results revealed that excitatory and electrical autapses prominently increase the prorogation of pacemaker activity, whereas inhibitory autapse reduces or does not change it. Also, the best propagation was obtained when the autapse was excitatory. This article is part of the theme issue 'Vibrational and stochastic resonance in driven nonlinear systems (part 2)'.

Contribution of AMPA and NMDA receptors in the spontaneous firing patterns of single neurons in autaptic culture

Single neurons in an autaptic culture exhibit various types of firing pattern with different firing durations and rhythms. However, a neuron with autapses has often been modeled as an oscillator providing a monotonic firing pattern with a constant periodicity because of the lack of a mathematical model. In the work described in this study, we use computational simulation and whole-cell patch-clamp recording to elucidate and model the mechanism by which such neurons generate various firing pattens. In the computational simulation, three types of spontaneous firing pattern, i.e., short, long-lasting, and periodic burst firing patterns are realized by changing the combination ratio of N-methyl-d-aspartate (NMDA) to α-amino-3-hydroxyl-5-methyl-4-isoxazole-propionate (AMPA) conductance. These three types of firing patterns are also observed in the experiments where neurons are cultured in isolation on micropatterned substrates. Using the AMPA and NMDA current models, we discuss that, in principle, autapses can regulate rhythmicity and information selection in neuronal networks.

Sigmoidal synaptic learning produces mutual stabilization in chaotic FitzHugh-Nagumo model

This paper investigates the interaction between two coupled neurons at the terminal end of a long chain of neurons. Specifically, we examine a bidirectional, two-cell FitzHugh-Nagumo neural model capable of exhibiting chaotic dynamics. Analysis of this model shows how mutual stabilization of the chaotic dynamics can occur through sigmoidal synaptic learning. Initially, this paper begins with a bifurcation analysis of an adapted version of a previously studied FitzHugh-Nagumo model that indicates regions of periodic and chaotic behaviors. Through allowing the synaptic properties to change dynamically via neural learning, it is shown how the system can evolve from chaotic to stable periodic behavior. The driving factor between this transition is representative of a stimulus coming down a long neural pathway. The result that two chaotic neurons can mutually stabilize via a synaptic learning implies that this may be a mechanism whereby neurons can transition from a disordered, chaotic state to a stable, ordered periodic state that persists. This approach shows that even at the simplest level of two terminal neurons, chaotic behavior can become stable, sustained periodic behavior. This is achieved without the need for a large network of neurons.

Autaptic Cultures: Methods and Applications

Neurons typically form daisy chains of synaptic connections with other neurons, but they can also form synapses with themselves. Although such self-synapses, or autapses, are comparatively rare in vivo, they are surprisingly common in dissociated neuronal cultures. At first glance, autapses in culture seem like a mere curiosity. However, by providing a simple model system in which a single recording electrode gives simultaneous access to the pre- and postsynaptic compartments, autaptic cultures have proven to be invaluable in facilitating important and elegant experiments in the area of synaptic neuroscience. Here, I provide detailed protocols for preparing and recording from autaptic cultures (also called micro-island or microdot cultures). Variations on the basic procedure are presented, as well as practical tips for optimizing the outcomes. I also illustrate the utility of autaptic cultures by reviewing the types of experiments that have used them over the past three decades. These examples serve to highlight the power and elegance of this simple model system, and will hopefully inspire new experiments for the interrogation of synaptic function.

Different dynamical behaviors induced by slow excitatory feedback for type II and III excitabilities

Neuronal excitability is classified as type I, II, or III, according to the responses of electronic activities, which play different roles. In the present paper, the effect of an excitatory autapse on type III excitability is investigated and compared to type II excitability in the Morris-Lecar model, based on Hopf bifurcation and characteristics of the nullcline. The autaptic current of a fast-decay autapse produces periodic stimulations, and that of a slow-decay autapse highly resembles sustained stimulations. Thus, both fast- and slow-decay autapses can induce a resting state for type II excitability that changes to repetitive firing. However, for type III excitability, a fast-decay autapse can induce a resting state to change to repetitive firing, while a slow-decay autapse can induce a resting state to change to a resting state following a transient spike instead of repetitive spiking, which shows the abnormal phenomenon that a stronger excitatory effect of a slow-decay autapse just induces weaker responses. Our results uncover a novel paradoxical phenomenon of the excitatory effect, and we present potential functions of fast- and slow-decay autapses that are helpful for the alteration and maintenance of type III excitability in the real nervous system related to neuropathic pain or sound localization.

Autapses enhance bursting and coincidence detection in neocortical pyramidal cells

Autapses are synaptic contacts of a neuron’s axon onto its own dendrite and soma. In the neocortex, self-inhibiting autapses in GABAergic interneurons are abundant in number and play critical roles in regulating spike precision and network activity. Here we examine whether the principal glutamatergic pyramidal cells (PCs) also form functional autapses. In patch-clamp recording from both rodent and human PCs, we isolated autaptic responses and found that these occur predominantly in layer-5 PCs projecting to subcortical regions, with very few in those projecting to contralateral prefrontal cortex and layer 2/3 PCs. Moreover, PC autapses persist during development into adulthood. Surprisingly, they produce giant postsynaptic responses (∼5 fold greater than recurrent PC-PC synapses) that are exclusively mediated by AMPA receptors. Upon activation, autapses enhance burst firing, neuronal responsiveness and coincidence detection of synaptic inputs. These findings indicate that PC autapses are functional and represent an important circuit element in the neocortex.

While autapses are synapses made by a neuron onto itself, its functional significance in pyramidal cells are not clear. Here, the authors show that in the mammalian neocortex, autapses of pyramidal cells can enhance burst firing and coincidence detection from other inputs.

Autapse-induced multiple stochastic resonances in a modular neuronal network

This study investigates the nontrivial effects of autapse on stochastic resonance in a modular neuronal network subjected to bounded noise. The resonance effect of autapse is detected by imposing a self-feedback loop with autaptic strength and autaptic time delay to each constituent neuron. Numerical simulations have demonstrated that bounded noise with the proper level of amplitude can induce stochastic resonance; moreover, the noise induced resonance dynamics can be significantly shaped by the autapse. In detail, for a specific range of autaptic strength, multiple stochastic resonances can be induced when the autaptic time delays are appropriately adjusted. These appropriately adjusted delays are detected to nearly approach integer multiples of the period of the external weak signal when the autaptic strength is very near zero; otherwise, they do not match the period of the external weak signal when the autaptic strength is slightly greater than zero. Surprisingly, in both cases, the differences between arbitrary two adjacent adjusted autaptic delays are always approximately equal to the period of the weak signal. The phenomenon of autaptic delay induced multiple stochastic resonances is further confirmed to be robust against the period of the external weak signal and the intramodule probability of subnetwork. These findings could have important implications for weak signal detection and information propagation in realistic neural systems.

Transitions between classes of neuronal excitability and bifurcations induced by autapse

Neuronal excitabilities behave as the basic and important dynamics related to the transitions between firing and resting states, and are characterized by distinct bifurcation types and spiking frequency responses. Switches between class I and II excitabilities induced by modulations outside the neuron (for example, modulation to M-type potassium current) have been one of the most concerning issues in both electrophysiology and nonlinear dynamics. In the present paper, we identified switches between 2 classes of excitability and firing frequency responses when an autapse, which widely exists in real nervous systems and plays important roles via self-feedback, is introduced into the Morris-Lecar (ML) model neuron. The transition from class I to class II excitability and from class II to class I spiking frequency responses were respectively induced by the inhibitory and excitatory autapse, which are characterized by changes of bifurcations, frequency responses, steady-state current-potential curves, and nullclines. Furthermore, we identified codimension-1 and -2 bifurcations and the characteristics of the current-potential curve that determine the transitions. Our results presented a comprehensive relationship between 2 classes of neuronal excitability/spiking characterized by different types of bifurcations, along with a novel possible function of autapse or self-feedback control on modulating neuronal excitability.

Spike-timing-dependent plasticity enhanced synchronization transitions induced by autapses in adaptive Newman-Watts neuronal networks

In this paper, we numerically study the effect of spike-timing-dependent plasticity (STDP) on synchronization transitions induced by autaptic activity in adaptive Newman-Watts Hodgkin-Huxley neuron networks. It is found that synchronization transitions induced by autaptic delay vary with the adjusting rate A_p of STDP and become strongest at a certain A_p value, and the A_p value increases when network randomness or network size increases. It is also found that the synchronization transitions induced by autaptic delay become strongest at a certain network randomness and network size, and the values increase and related synchronization transitions are enhanced when A_p increases. These results show that there is optimal STDP that can enhance the synchronization transitions induced by autaptic delay in the adaptive neuronal networks. These findings provide a new insight into the roles of STDP and autapses for the information transmission in neural systems.

The Space-Clamped Hodgkin-Huxley System with Random Synaptic Input: Inhibition of Spiking by Weak Noise and Analysis with Moment Equations

We consider a classical space-clamped Hodgkin-Huxley model neuron stimulated by synaptic excitation and inhibition with conductances represented by Ornstein-Uhlenbeck processes. Using numerical solutions of the stochastic model system obtained by an Euler method, it is found that with excitation only, there is a critical value of the steady-state excitatory conductance for repetitive spiking without noise, and for values of the conductance near the critical value, small noise has a powerfully inhibitory effect. For a given level of inhibition, there is also a critical value of the steady-state excitatory conductance for repetitive firing, and it is demonstrated that noise in either the excitatory or inhibitory processes or both can powerfully inhibit spiking. Furthermore, near the critical value, inverse stochastic resonance was observed when noise was present only in the inhibitory input process. The system of deterministic differential equations for the approximate first- and second-order moments of the model is derived. They are solved using Runge-Kutta methods, and the solutions are compared with the results obtained by simulation for various sets of parameters, including some with conductances obtained by experiment on pyramidal cells of rat prefrontal cortex. The mean and variance obtained from simulation are in good agreement when there is spiking induced by strong stimulation and relatively small noise or when the voltage is fluctuating at subthreshold levels. In the occasional spike mode sometimes exhibited by spinal motoneurons and cortical pyramidal cells, the assumptions underlying the moment equation approach are not satisfied. The simulation results show that noisy synaptic input of either an excitatory or inhibitory character or both may lead to the suppression of firing in neurons operating near a critical point and this has possible implications for cortical networks. Although suppression of firing is corroborated for the system of moment equations, there seem to be substantial differences between the dynamical properties of the original system of stochastic differential equations and the much larger system of moment equations.

Autapse-induced multiple coherence resonance in single neurons and neuronal networks

We study the effects of electrical and chemical autapse on the temporal coherence or firing regularity of single stochastic Hodgkin-Huxley neurons and scale-free neuronal networks. Also, we study the effects of chemical autapse on the occurrence of spatial synchronization in scale-free neuronal networks. Irrespective of the type of autapse, we observe autaptic time delay induced multiple coherence resonance for appropriately tuned autaptic conductance levels in single neurons. More precisely, we show that in the presence of an electrical autapse, there is an optimal intensity of channel noise inducing the multiple coherence resonance, whereas in the presence of chemical autapse the occurrence of multiple coherence resonance is less sensitive to the channel noise intensity. At the network level, we find autaptic time delay induced multiple coherence resonance and synchronization transitions, occurring at approximately the same delay lengths. We show that these two phenomena can arise only at a specific range of the coupling strength, and that they can be observed independently of the average degree of the network.

Accurate Langevin approaches to simulate Markovian channel dynamics

The stochasticity of ion-channels dynamic is significant for physiological processes on neuronal cell membranes. Microscopic simulations of the ion-channel gating with Markov chains can be considered to be an accurate standard. However, such Markovian simulations are computationally demanding for membrane areas of physiologically relevant sizes, which makes the noise-approximating or Langevin equation methods advantageous in many cases. In this review, we discuss the Langevin-like approaches, including the channel-based and simplified subunit-based stochastic differential equations proposed by Fox and Lu, and the effective Langevin approaches in which colored noise is added to deterministic differential equations. In the framework of Fox and Lu's classical models, several variants of numerical algorithms, which have been recently developed to improve accuracy as well as efficiency, are also discussed. Through the comparison of different simulation algorithms of ion-channel noise with the standard Markovian simulation, we aim to reveal the extent to which the existing Langevin-like methods approximate results using Markovian methods. Open questions for future studies are also discussed.

Autaptic activity-induced synchronization transitions in Newman-Watts network of Hodgkin-Huxley neurons

In this paper, we numerically study the effect of autapse on the synchronization of Newman-Watts small-world Hodgkin-Huxley neuron network. It is found that the neurons exhibit synchronization transitions as autaptic self-feedback delay is varied, and the phenomenon becomes strongest when autaptic self-feedback strength is optimal. This phenomenon also changes with the change of coupling strength and network randomness and become strongest when they are optimal. There are similar synchronization transitions for electrical and chemical autapse, but the synchronization transitions for chemical autapse occur more frequently and are stronger than those for electrical synapse. The underlying mechanisms are briefly discussed in quality. These results show that autaptic activity plays a subtle role in the synchronization of the neuronal network. These findings may find potential implications of autapse for the information processing and transmission in neural systems.

Effect of autaptic activity on the response of a Hodgkin-Huxley neuron

An autapse is a special synapse that connects a neuron to itself. In this study, we investigated the effect of an autapse on the responses of a Hodgkin-Huxley neuron to different forms of external stimuli. When the neuron was subjected to a DC stimulus, the firing frequencies and the interspike interval distributions of the output spike trains showed periodic behaviors as the autaptic delay time increased. When the input was a synaptic pulse-like train with random interspike intervals, we observed low-pass and band-pass filtering behaviors. Moreover, the region over which the output ISIs are distributed and the mean firing frequency display periodic behaviors with increasing autaptic delay time. When specific autaptic parameters were chosen, most of the input ISIs could be filtered, and the response spike trains were nearly regular, even with a highly random input. The background mechanism of these observed dynamics has been analyzed based on the phase response curve method. We also found that the information entropy of the output spike train could be modified by the autapse. These results also suggest that the autapse can serve as a regulator of information response in the nervous system.

Autapse-induced spiral wave in network of neurons under noise

Autapse plays an important role in regulating the electric activity of neuron by feedbacking time-delayed current on the membrane of neuron. Autapses are considered in a local area of regular network of neurons to investigate the development of spatiotemporal pattern, and emergence of spiral wave is observed while it fails to grow up and occupy the network completely. It is found that spiral wave can be induced to occupy more area in the network under optimized noise on the network with periodical or no-flux boundary condition being used. The developed spiral wave with self-sustained property can regulate the collective behaviors of neurons as a pacemaker. To detect the collective behaviors, a statistical factor of synchronization is calculated to investigate the emergence of ordered state in the network. The network keeps ordered state when self-sustained spiral wave is formed under noise and autapse in local area of network, and it independent of the selection of periodical or no-flux boundary condition. The developed stable spiral wave could be helpful for memory due to the distinct self-sustained property.

Autaptic self-inhibition of cortical GABAergic neurons: synaptic narcissism or useful introspection?

Fast synaptic inhibition sculpts all forms of cortical activity by means of a specialized connectivity pattern between highly heterogeneous inhibitory interneurons and principal excitatory cells. Importantly, inhibitory neurons connect also to each other extensively, following a detailed blueprint, and, indeed, specific forms of disinhibition affect important behavioral functions. Here we discuss a peculiar form of cortical disinhibition: the massive autaptic self-inhibition of parvalbumin-(PV) positive basket cells. Despite being described long ago, autaptic inhibition onto PV basket cells is rarely included in cortical circuit diagrams, perhaps because of its still elusive function. We propose here a potential dual role of autaptic feedback inhibition in temporally coordinating PV basket cells during cortical network activity.

Control of bursting behavior in neurons by autaptic modulation

Firing properties of biological neurons have long been recognized to be determined by extrinsic synaptic afferents that neurons receive and intrinsic ionic mechanisms that neurons possess, however, previous researches have also demonstrated that firing behavior of single neurons can be modulated by the neurons themselves, realized by the autapses. Thus in this investigation, we argued that how autaptic modulations shape the bursting behavior of biological neurons. We considered the issue from the following two aspects: autaptic-excitation and -inhibition. Our results suggested that for autaptic-excitation, under the condition of relatively weak stimulus, regular bursting was more incline to occur when the autaptic strength was weak, while regular spiking was more likely to appear when the autaptic strength was strong. However, larger stimulus would diminish the portion of bursting, but increase the portion of spiking. For autaptic-inhibition, under relatively weak stimulus, a wide range of regular bursting emerges when the autaptic strength was small, but when stronger stimulus were applied, the range of regular bursting shrinked into a small region. Meanwhile, we observed that synaptic delays have no obvious effects in the case of autaptic-excitation, while a subtle effect of synaptic delays was observed in the case of autaptic-inhibition. These results showed that bursting behavior of neurons could be controlled and modulated by the autaptic mechanisms that biological neurons intrinsically possess, and the final results may further promote the understanding in the generation of various neuronal firing patterns.

In-phase and anti-phase synchronization in noisy Hodgkin-Huxley neurons

We numerically investigate the influence of intrinsic channel noise on the dynamical response of delay-coupling in neuronal systems. The stochastic dynamics of the spiking is modeled within a stochastic modification of the standard Hodgkin-Huxley model wherein the delay-coupling accounts for the finite propagation time of an action potential along the neuronal axon. We quantify this delay-coupling of the Pyragas-type in terms of the difference between corresponding presynaptic and postsynaptic membrane potentials. For an elementary neuronal network consisting of two coupled neurons we detect characteristic stochastic synchronization patterns which exhibit multiple phase-flip bifurcations: The phase-flip bifurcations occur in form of alternate transitions from an in-phase spiking activity towards an anti-phase spiking activity. Interestingly, these phase-flips remain robust for strong channel noise and in turn cause a striking stabilization of the spiking frequency.

Effects of time delay on the stochastic resonance in small-world neuronal networks

The effects of time delay on stochastic resonance in small-world neuronal networks are investigated. Without delay, an intermediate intensity of additive noise is able to optimize the temporal response of the neural system to the subthreshold periodic signal imposed on all neurons constituting the network. The time delay in the coupling process can either enhance or destroy stochastic resonance of neuronal activity in the small-world network. In particular, appropriately tuned delays can induce multiple stochastic resonances, which appear intermittently at integer multiples of the oscillation period of weak external forcing. It is found that the delay-induced multiple stochastic resonances are most efficient when the forcing frequency is close to the global-resonance frequency of each individual neuron. Furthermore, the impact of time delay on stochastic resonance is largely independent of the small-world topology, except for resonance peaks. Considering that information transmission delays are inevitable in intra- and inter-neuronal communication, the presented results could have important implications for the weak signal detection and information propagation in neural systems.

Collective modes of coupled phase oscillators with delayed coupling

We study the effects of delayed coupling on timing and pattern formation in spatially extended systems of dynamic oscillators. Starting from a discrete lattice of coupled oscillators, we derive a generic continuum theory for collective modes of long wavelengths. We use this approach to study spatial phase profiles of cellular oscillators in the segmentation clock, a dynamic patterning system of vertebrate embryos. Collective wave patterns result from the interplay of coupling delays and moving boundary conditions. We show that the phase profiles of collective modes depend on coupling delays.