High Prevalence of Multistability of Rest States and Bursting in a Database of a Model Neuron

Flexibility in neuronal circuits has its roots in the dynamical richness of their neurons. Depending on their membrane properties single neurons can produce a plethora of activity regimes including silence, spiking and bursting. What is less appreciated is that these regimes can coexist with each other so that a transient stimulus can cause persistent change in the activity of a given neuron. Such multistability of the neuronal dynamics has been shown in a variety of neurons under different modulatory conditions. It can play either a functional role or present a substrate for dynamical diseases. We considered a database of an isolated leech heart interneuron model that can display silent, tonic spiking and bursting regimes. We analyzed only the cases of endogenous bursters producing functional half-center oscillators (HCOs). Using a one parameter (the leak conductance ()) bifurcation analysis, we extended the database to include silent regimes (stationary states) and systematically classified cases for the coexistence of silent and bursting regimes. We showed that different cases could exhibit two stable depolarized stationary states and two hyperpolarized stationary states in addition to various spiking and bursting regimes. We analyzed all cases of endogenous bursters and found that 18% of the cases were multistable, exhibiting coexistences of stationary states and bursting. Moreover, 91% of the cases exhibited multistability in some range of . We also explored HCOs built of multistable neuron cases with coexisting stationary states and a bursting regime. In 96% of cases analyzed, the HCOs resumed normal alternating bursting after one of the neurons was reset to a stationary state, proving themselves robust against this perturbation.     

Spiking resonances in models with the same slow resonant and fast amplifying currents but different subthreshold dynamic properties

The generation of spiking resonances in neurons (preferred spiking responses to oscillatory inputs) requires the interplay of the intrinsic ionic currents that operate at the subthreshold voltage level and the spiking mechanisms. Combinations of the same types of ionic currents in different parameter regimes may give rise to different types of nonlinearities in the voltage equation (e.g., parabolic- and cubic-like), generating subthreshold (membrane potential) oscillations patterns with different properties. These nonlinearities are not apparent in the model equations, but can be uncovered by plotting the voltage nullclines in the phase-plane diagram. We investigate the spiking resonant properties of conductance-based models that are biophysically equivalent at the subthreshold level (same ionic currents), but dynamically different (parabolic- and cubic-like voltage nullclines). As a case study we consider a model having a persistent sodium and a hyperpolarization-activated (h-) currents, which exhibits subthreshold resonance in the theta frequency band. We unfold the concept of spiking resonance into evoked and output spiking resonance. The former focuses on the input frequencies that are able to generate spikes, while the latter focuses on the output spiking frequencies regardless of the input frequency that generated these spikes. A cell can exhibit one or both types of resonances. We also measure spiking phasonance, which is an extension of subthreshold phasonance (zero-phase-shift response to oscillatory inputs) to the spiking regime. The subthreshold resonant properties of both types of models are communicated to the spiking regime for low enough input amplitudes as the voltage response for the subthreshold resonant frequency band raises above threshold. For higher input amplitudes evoked spiking resonance is no longer present in these models, but output spiking resonance is present primarily in the parabolic-like model due to a cycle skipping mechanism (involving mixed-mode oscillations), while the cubic-like model shows a better 1:1 entrainment. We use dynamical systems tools to explain the underlying mechanisms and the mechanistic differences between the resonance types. Our results demonstrate that the effective time scales that operate at the subthreshold regime to generate intrinsic subthreshold oscillations, mixed-mode oscillations and subthreshold resonance do not necessarily determine the existence of a preferred spiking response to oscillatory inputs in the same frequency band. The results discussed in this paper highlight both the complexity of the suprathreshold responses to oscillatory inputs in neurons having resonant and amplifying currents with different time scales and the fact that the identity of the participating ionic currents is not enough to predict the resulting patterns, but additional dynamic information, captured by the geometric properties of the phase-space diagram, is needed.     

Firing synchronization and temporal order in noisy neuronal networks

Noise-induced complete synchronization and frequency synchronization in coupled spiking and bursting neurons are studied firstly. The effects of noise and coupling are discussed. It is found that bursting neurons are easier to achieve firing synchronization than spiking ones, which means that bursting activities are more important for information transfer in neuronal networks. Secondly, the effects of noise on firing synchronization in a noisy map neuronal network are presented. Noise-induced synchronization and temporal order are investigated by means of the firing rate function and the order index. Firing synchronization and temporal order of excitatory neurons can be greatly enhanced by subthreshold stimuli with resonance frequency. Finally, it is concluded that random perturbations play an important role in firing activities and temporal order in neuronal networks.     

Neuronal oscillator robustness to multiple global perturbations

Neuronal activity depends on ion channels and biophysical processes that are strongly and differentially sensitive to physical variables such as temperature and pH. Nonetheless, neuronal oscillators can be surprisingly resilient to perturbations in these variables. We study a three-neuron pacemaker ensemble that drives the pyloric rhythm of the crab, Cancer borealis. These crabs routinely experience a number of global perturbations, including changes in temperature and pH. Although pyloric oscillations are robust to such changes, for sufficiently large deviations the rhythm reversibly breaks down. As temperature increases beyond a tipping point, oscillators transition to silence. Acidic pH deviations also show tipping points, with a reliable transition first to tonic spiking, then to silence. Surprisingly, robustness to perturbations in pH only moderately affects temperature robustness. Consistent with high animal-to-animal variability in biophysical circuit parameters, tipping points in temperature and pH vary across animals. However, the ordering and discrete classes of transitions at critical points are conserved. This implies that qualitative oscillator dynamics are preserved across animals despite high quantitative parameter variability. A universal model of bursting dynamics predicts the existence of these transition types and the order in which they occur.     

Exploring neuronal bistability at the depolarization block

Many neurons display bistability-coexistence of two firing modes such as bursting and tonic spiking or tonic spiking and silence. Bistability has been proposed to endow neurons with richer forms of information processing in general and to be involved in short-term memory in particular by allowing a brief signal to elicit long-lasting changes in firing. In this paper, we focus on bistability that allows for a choice between tonic spiking and depolarization block in a wide range of the depolarization levels. We consider the spike-producing currents in two neurons, models of which differ by the parameter values. Our dopaminergic neuron model displays bistability in a wide range of applied currents at the depolarization block. The Hodgkin-Huxley model of the squid giant axon shows no bistability. We varied parameter values for the model to analyze transitions between the two parameter sets. We show that bistability primarily characterizes the inactivation of the Na(+) current. Our study suggests a connection between the amount of the Na(+) window current and the length of the bistability range. For the dopaminergic neuron we hypothesize that bistability can be linked to a prolonged action of antipsychotic drugs.     

State-Dependent Effects of Na Channel Noise on Neuronal Burst Generation

We explore the effects of stochastic sodium (Na) channel activation on the variability and dynamics of spiking and bursting in a model neuron. The complete model segregates Hodgin-Huxley-type currents into two compartments, and undergoes applied current-dependent bifurcations between regimes of periodic bursting, chaotic bursting, and tonic spiking. Noise is added to simulate variable, finite sizes of the population of Na channels in the fast spiking compartment.During tonic firing, Na channel noise causes variability in interspike intervals (ISIs). The variance, as well as the sensitivity to noise, depend on the model's biophysical complexity. They are smallest in an isolated spiking compartment; increase significantly upon coupling to a passive compartment; and increase again when the second compartment also includes slow-acting currents. In this full model, sufficient noise can convert tonic firing into bursting.During bursting, the actions of Na channel noise are state-dependent. The higher the noise level, the greater the jitter in spike timing within bursts. The noise makes the burst durations of periodic regimes variable, while decreasing burst length duration and variance in a chaotic regime. Na channel noise blurs the sharp transitions of spike time and burst length seen at the bifurcations of the noise-free model. Close to such a bifurcation, the burst behaviors of previously periodic and chaotic regimes become essentially indistinguishable.We discuss biophysical mechanisms, dynamical interpretations and physiological implications. We suggest that noise associated with finite populations of Na channels could evoke very different effects on the intrinsic variability of spiking and bursting discharges, depending on a biological neuron's complexity and applied current-dependent state. We find that simulated channel noise in the model neuron qualitatively replicates the observed variability in burst length and interburst interval in an isolated biological bursting neuron.     

Mechanisms of firing patterns in fast-spiking cortical interneurons

Cortical fast-spiking (FS) interneurons display highly variable electrophysiological properties. Their spike responses to step currents occur almost immediately following the step onset or after a substantial delay, during which subthreshold oscillations are frequently observed. Their firing patterns include high-frequency tonic firing and rhythmic or irregular bursting (stuttering). What is the origin of this variability? In the present paper, we hypothesize that it emerges naturally if one assumes a continuous distribution of properties in a small set of active channels. To test this hypothesis, we construct a minimal, single-compartment conductance-based model of FS cells that includes transient Na(+), delayed-rectifier K(+), and slowly inactivating d-type K(+) conductances. The model is analyzed using nonlinear dynamical system theory. For small Na(+) window current, the neuron exhibits high-frequency tonic firing. At current threshold, the spike response is almost instantaneous for small d-current conductance, gd, and it is delayed for larger gd. As gd further increases, the neuron stutters. Noise substantially reduces the delay duration and induces subthreshold oscillations. In contrast, when the Na(+) window current is large, the neuron always fires tonically. Near threshold, the firing rates are low, and the delay to firing is only weakly sensitive to noise; subthreshold oscillations are not observed. We propose that the variability in the response of cortical FS neurons is a consequence of heterogeneities in their gd and in the strength of their Na(+) window current. We predict the existence of two types of firing patterns in FS neurons, differing in the sensitivity of the delay duration to noise, in the minimal firing rate of the tonic discharge, and in the existence of subthreshold oscillations. We report experimental results from intracellular recordings supporting this prediction.     

First return maps for the dynamics of synaptically coupled conditional bursters

The pre-Bötzinger complex (preBötc) in the mammalian brainstem has an important role in generating respiratory rhythms. An influential differential equation model for the activity of individual neurons in the preBötc yields transitions from quiescence to bursting to tonic spiking as a parameter is varied. Further, past work has established that bursting dynamics can arise from a pair of tonic model cells coupled with synaptic excitation. In this paper, we analytically derive one- and two-dimensional maps from the differential equations for a self-coupled neuron and a two-neuron network, respectively. Using a combination of analysis and simulations of these maps, we explore the possible forms of dynamics that the model networks can produce as well as which transitions between dynamic regimes are mathematically possible.     

Theta oscillations by synaptic excitation in a neocortical circuit model

Neocortical theta-band oscillatory activity is associated with cognitive tasks involving learning and memory. This oscillatory activity is proposed to originate from the synchronization of interconnected layer V intrinsic bursting (IB) neurons by recurrent excitation. To test this hypothesis, a sparsely connected spiking circuit model based on empirical data was simulated using Hodgkin-Huxley-type bursting neurons and use-dependent depressing synaptic connections. In response to a heterogeneous tonic current stimulus, the model generated coherent and robust oscillatory activity throughout the theta-band (4-12 Hz). These oscillations were not, however, self-sustaining without a driving current, and not dependent on N-methyl-D-aspartate receptor synaptic currents. At realistic connection strengths, synaptic depression was necessary to avoid instability and expanded the basin of attraction for theta oscillations by controlling the gain of recurrent excitation. These results support the hypothesis that IB neuron networks can generate robust and coherent theta-band oscillations in neocortex.     

Impulse pattern in bi-directionally coupled model neurons of different dynamics

The effects of bi-directional gap junction coupling of two model neurons with subthreshold oscillations have been examined when the individual neurons are operating at different dynamical states either in the tonic or bursting firing mode. Our simulations indicate that intermediate coupling strengths mostly lead to highly variable, often chaotic impulse patterns whereas transition to completely synchronized activity at high coupling strengths is generally going along with transitions to regular limit cycle activity. The synchronized activity pattern, however, can be completely different from the original pattern of the uncoupled neurons.     

Bursting synchronization dynamics of pancreatic β-cells with electrical and chemical coupling

Based on bifurcation analysis, the synchronization behaviors of two identical pancreatic β-cells connected by electrical and chemical coupling are investigated, respectively. Various firing patterns are produced in coupled cells when a single cell exhibits tonic spiking or square-wave bursting individually, irrespectively of what the cells are connected by electrical or chemical coupling. On the one hand, cells can burst synchronously for both weak electrical and chemical coupling when an isolated cell exhibits tonic spiking itself. In particular, for electrically coupled cells, under the variation of the coupling strength there exist complex transition processes of synchronous firing patterns such as “fold/limit cycle” type of bursting, then anti-phase continuous spiking, followed by the “fold/torus” type of bursting, and finally in-phase tonic spiking. On the other hand, it is shown that when the individual cell exhibits square-wave bursting, suitable coupling strength can make the electrically coupled system generate “fold/Hopf” bursting via “fold/fold” hysteresis loop; whereas, the chemically coupled cells generate “fold/subHopf” bursting. Especially, chemically coupled bursters can exhibit inverse period-adding bursting sequence. Fast–slow dynamics analysis is applied to explore the generation mechanism of these bursting oscillations. The above analysis of bursting types and the transition may provide us with better insight into understanding the role of coupling in the dynamic behaviors of pancreatic β-cells.     

Brief Subthreshold Events Can Act as Hebbian Signals for Long-Term Plasticity

Background

Action potentials are thought to be determinant for the induction of long-term synaptic plasticity, the cellular basis of learning and memory. However, neuronal activity does not lead systematically to an action potential but also, in many cases, to synaptic depolarizing subthreshold events. This is particularly exemplified in corticostriatal information processing. Indeed, the striatum integrates information from the whole cerebral cortex and, due to the membrane properties of striatal medium spiny neurons, cortical inputs do not systematically trigger an action potential but a wide range of subthreshold postsynaptic depolarizations. Accordingly, we have addressed the following question: does a brief subthreshold event act as a Hebbian signal and induce long-term synaptic efficacy changes?

Methodology/Principal Findings

Here, using perforated patch-clamp recordings on rat brain corticostriatal slices, we demonstrate, that brief (30 ms) subthreshold depolarizing events in quasi-coincidence with presynaptic activity can act as Hebbian signals and are sufficient to induce long-term synaptic plasticity at corticostriatal synapses. This “subthreshold-depolarization dependent plasticity” (SDDP) induces strong, significant and bidirectional long-term synaptic efficacy changes at a very high occurrence (81%) for time intervals between pre- and postsynaptic stimulations (Δt) of −110<Δt<+110 ms. Such subthreshold depolarizations are able to induce robust long-term depression (cannabinoid type-1 receptor-activation dependent) as well as long-term potentiation (NMDA receptor-activation dependent).

Conclusion/Significance

Our data show the existence of a robust, reliable and timing-dependent bidirectional long-term plasticity induced by brief subthreshold events paired with presynaptic activity. The existence of a subthreshold-depolarization dependent plasticity extends considerably, beyond the action potential, the neuron''s capabilities to express long-term synaptic efficacy changes.     

Long-Term Plasticity Is Proportional to Theta-Activity

Background

Theta rhythm in the hippocampal formation is a main feature of exploratory behaviour and is believed to enable the encoding of new spatial information and the modification of synaptic weights. Cyclic changes of dentate gyrus excitability during theta rhythm are related to its function, but whether theta epochs per se are able to alter network properties of dentate gyrus for long time-periods is still poorly understood.

Methodology/Principal Findings

We used low-frequency stimulation protocols that amplify the power of endogenous theta oscillations, in order to estimate the plasticity effect of endogenous theta oscillations on a population level. We found that stimulation-induced augmentation of the theta rhythm is linked to a subsequent increase of neuronal excitability and decrease of the synaptic response. This EPSP-to-Spike uncoupling is related to an increased postsynaptic spiking on the positive phases of theta frequency oscillations. Parallel increase of the field EPSP slope and the population spike occurs only after concurrent pre- and postsynaptic activation. Furthermore, we observed that long-term potentiation (>24 h) occurs in the dentate gyrus of freely behaving adult rats after phasic activity of entorhinal afferents in the theta-frequency range. This plasticity is proportional to the field bursting activity of granule cells during the stimulation, and may comprise a key step in spatial information transfer. Long-term potentiation of the synaptic component occurs only when the afferent stimulus precedes the evoked population burst, and is input-specific.

Conclusions/Significance

Our data confirm the role of the dentate gyrus in filtering information to the subsequent network during the activated state of the hippocampus.     

Cell Type-Specific Separation of Subicular Principal Neurons during Network Activities

The hippocampal output structure, the subiculum, expresses two major memory relevant network rhythms, sharp wave ripple and gamma frequency oscillations. To this date, it remains unclear how the two distinct types of subicular principal cells, intrinsically bursting and regular spiking neurons, participate in these two network rhythms. Using concomitant local field potential and intracellular recordings in an in vitro mouse model that allows the investigation of both network rhythms, we found a cell type-specific segregation of principal neurons into participating intrinsically bursting and non-participating regular spiking cells. However, if regular spiking cells were kept at a more depolarized level, they did participate in a specific manner, suggesting a potential bimodal working model dependent on the level of excitation. Furthermore, intrinsically bursting and regular spiking cells exhibited divergent intrinsic membrane and synaptic properties in the active network. Thus, our results suggest a cell-type-specific segregation of principal cells into two separate groups during network activities, supporting the idea of two parallel streams of information processing within the subiculum.     

Coexistence of Tonic Spiking Oscillations in a Leech Neuron Model

The leech neuron model studied here has a remarkable dynamical plasticity. It exhibits a wide range of activities including various types of tonic spiking and bursting. In this study we apply methods of the qualitative theory of dynamical systems and the bifurcation theory to analyze the dynamics of the leech neuron model with emphasis on tonic spiking regimes. We show that the model can demonstrate bi-stability, such that two modes of tonic spiking coexist. Under a certain parameter regime, both tonic spiking modes are represented by the periodic attractors. As a bifurcation parameter is varied, one of the attractors becomes chaotic through a cascade of period-doubling bifurcations, while the other remains periodic. Thus, the system can demonstrate co-existence of a periodic tonic spiking with either periodic or chaotic tonic spiking. Pontryagins averaging technique is used to locate the periodic orbits in the phase space.     

Neurons controlling Aplysia feeding inhibit themselves by continuous NO production

Background

Neural activity can be affected by nitric oxide (NO) produced by spiking neurons. Can neural activity also be affected by NO produced in neurons in the absence of spiking?

Methodology/Principal Findings

Applying an NO scavenger to quiescent Aplysia buccal ganglia initiated fictive feeding, indicating that NO production at rest inhibits feeding. The inhibition is in part via effects on neurons B31/B32, neurons initiating food consumption. Applying NO scavengers or nitric oxide synthase (NOS) blockers to B31/B32 neurons cultured in isolation caused inactive neurons to depolarize and fire, indicating that B31/B32 produce NO tonically without action potentials, and tonic NO production contributes to the B31/B32 resting potentials. Guanylyl cyclase blockers also caused depolarization and firing, indicating that the cGMP second messenger cascade, presumably activated by the tonic presence of NO, contributes to the B31/B32 resting potential. Blocking NO while voltage-clamping revealed an inward leak current, indicating that NO prevents this current from depolarizing the neuron. Blocking nitrergic transmission had no effect on a number of other cultured, isolated neurons. However, treatment with NO blockers did excite cerebral ganglion neuron C-PR, a command-like neuron initiating food-finding behavior, both in situ, and when the neuron was cultured in isolation, indicating that this neuron also inhibits itself by producing NO at rest.

Conclusion/Significance

Self-inhibitory, tonic NO production is a novel mechanism for the modulation of neural activity. Localization of this mechanism to critical neurons in different ganglia controlling different aspects of a behavior provides a mechanism by which a humeral signal affecting background NO production, such as the NO precursor L-arginine, could control multiple aspects of the behavior.     

Population oscillations in a discrete model of neural networks of the brain

A discrete model of biological neural networks is used to find out how synchronized firing of neurons emerges in a randomly connected neural population. The objective is to understand the mechanisms underlying brain waves and to find and characterize conditions which support spontaneous switching from disordered to rhythmic population activity as in case of an epileptic seizure. The model is kept as simple as possible to achieve on one hand a fast performance of computer simulations of networks with up to 10,000 neurons and to keep on the other hand an overview of parameter dependences. Dynamics of the model can be classified into different regimes: random fluctuations, rhythmic oscillations and silence. When the ratio of the inhibitory/excitatory connectivity is raised the system crosses from the fluctuating regime through the rhythmic oscillating region to the silence regime. Close to the boundary between the fluctuating and the oscillating regimes the network shows spontaneous bursting of high amplitude rhythmic oscillations, which is characteristic of epileptiform behavior. The simulation results are in agreement with recent theories saying that focal epilepsy after injury of the brain could result from axonal sprouting of GABAergic neurons in the injured region.     

Oscillations, resonances and noise: basis of flexible neuronal pattern generation

Modulation of neuronal impulse pattern is examined by means of a simplified Hodgkin-Huxley type computer model which refers to experimental recordings of cold receptor discharges. This model essentially consists of two potentially oscillating subsystems: a spike generator and a subthreshold oscillator. With addition of noise the model successfully mimics the major types of experimentally recorded impulse patterns and thereby elucidate different resonance behaviors. (1) There is a range of rhythmic spiking or bursting where the spike generator is strongly coupled to the subthreshold oscillator. (2) There is a pacemaker activity of more complex interactions where the spike generator has overtaken part of the control. (3) There is a situation where the two subsystems are decoupled and only resonate with the help of noise.     

A model of a thalamic neuron

We modify our recent three equilibrium-point model of neuronal bursting by a means of a small deformation of the nullclines in the x-y phase plane to give a model that can have as many as five equilibrium points. In this model the middle stable equilibrium point (e.p.) is separated from the outer stable and unstable e.ps by two saddle points. If the system is started at rest at the middle stable e.p. it has the following complex properties: A short suprathreshold current pulse switches the model from a silent state to a bursting state, or to give a single burst, depending on the choice of parameters. A subthreshold depolarizing current step gives a passive response at rest, but if the model is either constantly hyperpolarized or constantly depolarized, then the same current step gives different active responses. At a hyperpolarized level this consists of a burst response that shows refractoriness. At a depolarized level it consists of tonic firing with a linear frequency--current relationship. Hyperpolarization from rest is followed by post-inhibitory rebound. The model responds in a unique and characteristic way to an applied current ramp. These properties are very similar to those that have been recently recorded intracellularly from neurons in the mammalian thalamus. In the x-y phase plane our models of the repetitively firing neuron, the bursting neuron and the thalamic neuron form a progression of models in which the y nullcline in the subthreshold region is deformed once to give the burst neuron model, and a second time to give the thalamic neuron model. Each deformation can be interpreted as corresponding to the inclusion of a slow inward current in the model. As these currents are included so the associated firing properties increase in complexity.     

Noise-induced impulse pattern modifications at different dynamical period-one situations in a computer model of temperature encoding

We used a minimal Hodgkin-Huxley type model of cold receptor discharges to examine how noise interferes with the non-linear dynamics of the ionic mechanisms of neuronal stimulus encoding. The model is based on the assumption that spike-generation depends on subthreshold oscillations. With physiologically plausible temperature scaling, it passes through different impulse patterns which, with addition of noise, are in excellent agreement with real experimental data. The interval distributions of purely deterministic simulations, however, exhibit considerable differences compared to the noisy simulations especially at the bifurcations of deterministically period-one discharges. We, therefore, analyzed the effects of noise in different situations of deterministically regular period-one discharges: (1) at high-temperatures near the transition to subthreshold oscillations and to burst discharges, and (2) at low-temperatures close to and more far away from the bifurcations to chaotic dynamics. The data suggest that addition of noise can considerably extend the dynamical behavior of the system with coexistence of different dynamical situations at deterministically fixed parameter constellations. Apart from well-described coexistence of spike-generating and subthreshold oscillations also mixtures of tonic and bursting patterns can be seen and even transitions to unstable period-one orbits seem to appear. The data indicate that cooperative effects between low- and high-dimensional dynamics have to be considered as qualitatively important factors in neuronal encoding.