Two types of independent bursting mechanisms in inspiratory neurons: an integrative model

The network of coupled neurons in the pre-Bötzinger complex (pBC) of the medulla generates a bursting rhythm, which underlies the inspiratory phase of respiration. In some of these neurons, bursting persists even when synaptic coupling in the network is blocked and respiratory rhythmic discharge stops. Bursting in inspiratory neurons has been extensively studied, and two classes of bursting neurons have been identified, with bursting mechanism depends on either persistent sodium current or changes in intracellular Ca²⁺, respectively. Motivated by experimental evidence from these intrinsically bursting neurons, we present a two-compartment mathematical model of an isolated pBC neuron with two independent bursting mechanisms. Bursting in the somatic compartment is modeled via inactivation of a persistent sodium current, whereas bursting in the dendritic compartment relies on Ca²⁺ oscillations, which are determined by the neuromodulatory tone. The model explains a number of conflicting experimental results and is able to generate a robust bursting rhythm, over a large range of parameters, with a frequency adjusted by neuromodulators.     

Endogenous nature of spontaneous bursting in hippocampal pyramidal neurons

The normal spontaneous bursting behavior of hippocampal pyramidal neurons was investigated. Bursting frequency was found to be membrane potential dependent, the frequency increasing with maintained depolarization and decreasing upon hyperpolarization. Short depolarizing-current pulses would trigger bursts which outlasted the stimulus, and bursting continued when synaptic transmission had been blocked. The spontaneous bursts of these neurons, in contrast to bursts induced by convulsive agents, appear to exhibit the classical behavior of endogenous bursts as observed in invertebrate neurons. The endogenous bursts in hippocampal neurons may result, also, from an interplay of intrinsic membrane currents.     

Neural rate equations for bursting dynamics derived from conductance-based equations

A method of obtaining rate equations from conductance-based equations is developed and applied to fast-spiking and bursting neocortical neurons. It involves splitting systems of conductance-based equations into fast and slow subsystems, and averaging the effects of fast terms that drive the slowly varying quantities by showing that their average is closely proportional to the firing rate. The dependence of the firing rate on the injected current is then approximated in the analysis. The resulting behavior of the slow variables is then substituted back into the fast equations, with the further approximation of replacing the fast voltages in these terms by effective values. For bursting neurons the method yields two coupled limit-cycle oscillators: a self-exciting oscillator for the slow variables that commences limit-cycle oscillations at a critical current and modulates a fast spike-generating oscillator, thereby leading to slowly modulated bursts with a group of spikes in each burst. The dynamics of these coupled oscillators are then verified against those of the conductance-based equations. Finally, it is shown how to place the results in a form suitable for use in mean-field equations for neural population dynamics.     

Firing responses of bursting neurons with delayed feedback

Thalamic neurons, which play important roles in the genesis of rhythmic activities of the brain, show various bursting behaviors, particularly modulated by complex thalamocortical feedback via cortical neurons. As a first step to explore this complex neural system and focus on the effects of the feedback on the bursting behavior, a simple loop structure delayed in time and scaled by a coupling strength is added to a recent mean-field model of bursting neurons. Depending on the coupling strength and delay time, the modeled neurons show two distinct response patterns: one entrained to the unperturbed bursting frequency of the neurons and one entrained to the resonant frequency of the loop structure. Transitions between these two patterns are explored in the model’s parameter space via extensive numerical simulations. It is found that at a fixed loop delay, there is a critical coupling strength at which the dominant response frequency switches from the unperturbed bursting frequency to the loop-induced one. Furthermore, alternating occurrence of these two response frequencies is observed when the delay varies at fixed coupling strength. The results demonstrate that bursting is coupled with feedback to yield new dynamics, which will provide insights into such effects in more complex neural systems.     

From Plateau to Pseudo-Plateau Bursting: Making the Transition

Bursting electrical activity is ubiquitous in excitable cells such as neurons and many endocrine cells. The technique of fast/slow analysis, which takes advantage of time scale differences, is typically used to analyze the dynamics of bursting in mathematical models. Two classes of bursting oscillations that have been identified with this technique, plateau and pseudo-plateau bursting, are often observed in neurons and endocrine cells, respectively. These two types of bursting have very different properties and likely serve different functions. This latter point is supported by the divergent expression of the bursting patterns into different cell types, and raises the question of whether it is even possible for a model for one type of cell to produce bursting of the type seen in the other type without large changes to the model. Using fast/slow analysis, we show here that this is possible, and we provide a procedure for achieving this transition. This suggests that the design principles for bursting in endocrine cells are just quantitative variations of those for bursting in neurons.     

Classification of bursting patterns: A tale of two ducks

Bursting is one of the fundamental rhythms that excitable cells can generate either in response to incoming stimuli or intrinsically. It has been a topic of intense research in computational biology for several decades. The classification of bursting oscillations in excitable systems has been the subject of active research since the early 1980s and is still ongoing. As a by-product, it establishes analytical and numerical foundations for studying complex temporal behaviors in multiple timescale models of cellular activity. In this review, we first present the seminal works of Rinzel and Izhikevich in classifying bursting patterns of excitable systems. We recall a complementary mathematical classification approach by Bertram and colleagues, and then by Golubitsky and colleagues, which, together with the Rinzel-Izhikevich proposals, provide the state-of-the-art foundations to these classifications. Beyond classical approaches, we review a recent bursting example that falls outside the previous classification systems. Generalizing this example leads us to propose an extended classification, which requires the analysis of both fast and slow subsystems of an underlying slow-fast model and allows the dissection of a larger class of bursters. Namely, we provide a general framework for bursting systems with both subthreshold and superthreshold oscillations. A new class of bursters with at least 2 slow variables is then added, which we denote folded-node bursters, to convey the idea that the bursts are initiated or annihilated via a folded-node singularity. Key to this mechanism are so-called canard or duck orbits, organizing the underpinning excitability structure. We describe the 2 main families of folded-node bursters, depending upon the phase (active/spiking or silent/nonspiking) of the bursting cycle during which folded-node dynamics occurs. We classify both families and give examples of minimal systems displaying these novel bursting patterns. Finally, we provide a biophysical example by reinterpreting a generic conductance-based episodic burster as a folded-node burster, showing that the associated framework can explain its subthreshold oscillations over a larger parameter region than the fast subsystem approach.     

Reliable circuits from irregular neurons: A dynamical approach to understanding central pattern generators

Central pattern generating neurons from the lobster stomatogastric ganglion were analyzed using new nonlinear methods. The LP neuron was found to have only four or five degrees of freedom in the isolated condition and displayed chaotic behavior. We show that this chaotic behavior could be regularized by periodic pulses of negative current injected into the neuron or by coupling it to another neuron via inhibitory connections. We used both a modified Hindmarsh-Rose model to simulate the neurons behavior phenomenologically and a more realistic conductance-based model so that the modeling could be linked to the experimental observations. Both models were able to capture the dynamics of the neuron behavior better than previous models. We used the Hindmarsh-Rose model as the basis for building electronic neurons which could then be integrated into the biological circuitry. Such neurons were able to rescue patterns which had been disabled by removing key biological neurons from the circuit.     

Conductance-based refractory density model of primary visual cortex

A layered continual population model of primary visual cortex has been constructed, which reproduces a set of experimental data, including postsynaptic responses of single neurons on extracellular electric stimulation and spatially distributed activity patterns in response to visual stimulation. In the model, synaptically interacting excitatory and inhibitory neuronal populations are described by a conductance-based refractory density approach. Populations of two-compartment excitatory and inhibitory neurons in cortical layers 2/3 and 4 are distributed in the 2-d cortical space and connected by AMPA, NMDA and GABA type synapses. The external connections are pinwheel-like, according to the orientation of a stimulus. Intracortical connections are isotropic local and patchy between neurons with similar orientations. The model proposes better temporal resolution and more detailed elaboration than conventional mean-field models. In comparison to large network simulations, it excludes a posteriori statistical data manipulation and provides better computational efficiency and minimal parametrization.     

Hodgkin–Huxley type modelling and parameter estimation of GnRH neurons

In this paper a simple one compartment Hodgkin–Huxley type electrophysiological model of GnRH neurons is presented, that is able to reasonably reproduce the most important qualitative features of the firing pattern, such as baseline potential, depolarization amplitudes, sub-baseline hyperpolarization phenomenon and average firing frequency in response to excitatory current. In addition, the same model provides an acceptable numerical fit of voltage clamp (VC) measurement results. The parameters of the model have been estimated using averaged VC traces, and characteristic values of measured current clamp traces originating from GnRH neurons in hypothalamic slices. The resulting parameter values show a good agreement with literature data in most of the cases. Applying parametric changes, which lead to the increase of baseline potential and enhance cell excitability, the model becomes capable of bursting. The effects of various parameters to burst length have been analyzed by simulation.     

Mechanisms of Generation of Pacemaker Activity by Identified Helix Neurons

We studied the mechanisms of generation of pacemaker activity in identified neurons of Helix pomatia. For this purpose, we isolated the PPa2 and PPa7 neurons generating spontaneous rhythmic monomodal activity and PPa1 neuron with bursting activity. It was demonstrated that isolated PPa2 and PPa7 cells produce endogenous rhythmic activity that was not considerably modified by external application of 1 mM CdCl₂. Sometimes, only low-amplitude dendritic action potentials (AP) were observed instead of generation of full-amplitude somatic AP. In contrast, isolation of the PPa1 neuron eliminated its bursting activity, but subsequent application of oxytocin on this neuron recovered such activity. This finding shows that the bursting activity of the PPa1 neuron is of an exogenous nature. Application of 1 mM CdCl₂ suppressed this bursting activity, but when Cd²⁺ was applied against the background of superfusion of the neuron with Ringer solution containing a bursting activity-initiating neuropeptide obtained from the molluscan CNS, this blocker was incapable of suppressing the bursting activity. A blocker of the hyperpolarization-activated current (I _h , H current), Cs⁺ (10 mM) exerted no noticeable effect on the activity of the studied neurons. Our findings allow us to conclude that the pacemaker activity is initiated within the dendritic tree of a cell and is then electrotonically spread to the soma, where full-amplitude AP are generated. It seems probable that Ca²⁺ ions and H current are not directly involved in generation of the pacemaker activity in the studied snail neurons.     

The significance of action potential bursting in the brain reward circuit

The brain reward circuit consists of specialized cortical and subcortical structural components that code for various cognitive aspects of goal-directed behavior. These components include the prefrontal cortex (PFC), amygdala (AMY), nucleus accumbens (Nac), subiculum (SUB) of the hippocampal formation, and the dopamine (DA) neurons in the ventral tegmental area (VTA). Both serial and parallel processing in the different components of the circuit code the various aspects of reward-related behavior. Individual neurons within each component have developed specialized intrinsic membrane properties that have led them to be typically defined as either single spiking or high frequency burst-firing neurons. However, a strict definition based on the output mode may not be appropriate. Under the right conditions, neurons can switch between bursting and single-spiking modes, therefore providing a conditional output state. The preferred mode of each individual neuron depends on a combination of different plastic neuronal properties such as, dendritic architecture, neuromodulation, intracellular calcium (Ca(++)) buffering, excitatory and inhibitory synaptic strength, and the spatial distribution and density of voltage and ligand-gated channels. It is likely that, in vivo, most neurons in the circuit, despite variations in intrinsic membrane properties, are conditional output neurons equipped with the versatility of switching between output modes under appropriate conditions. Bursting mode may be used to boost the gain of neural signaling of important or novel events by enhancing transmitter release and enhancing dendritic depolarization, thereby increasing synaptic potentiation. Conversely, single spiking mode may be used to dampen neuronal signaling and may be associated with habituation to unimportant events. Mode switching may provide flexibility to the circuit allowing different sets of neurons to conditionally code for the various aspects of reward-related memory and behavior.     

Firing patterns in a conductance-based neuron model: bifurcation, phase diagram, and chaos

Responding to various stimuli, some neurons either remain resting or can fire several distinct patterns of action potentials, such as spiking, bursting, subthreshold oscillations, and chaotic firing. In particular, Wilson’s conductance-based neocortical neuron model, derived from the Hodgkin–Huxley model, is explored to understand underlying mechanisms of the firing patterns. Phase diagrams describing boundaries between the domains of different firing patterns are obtained via extensive numerical computations. The boundaries are further studied by standard instability analyses, which demonstrates that the chaotic neural firing could develop via period-doubling and/or period- adding cascades. Sequences of the firing patterns often observed in many neural experiments are also discussed in the phase diagram framework developed. Our results lay the groundwork for wider use of the model, especially for incorporating it into neural field modeling of the brain.     

A multivariate population density model of the dLGN/PGN relay

Using a population density approach we study the dynamics of two interacting collections of integrate-and-fire-or-burst (IFB) neurons representing thalamocortical (TC) cells from the dorsal lateral geniculate nucleus (dLGN) and thalamic reticular (RE) cells from the perigeniculate nucleus (PGN). Each population of neurons is described by a multivariate probability density function that satisfies a conservation equation with appropriately defined probability fluxes and boundary conditions. The state variables of each neuron are the membrane potential and the inactivation gating variable of the low-threshold Ca²⁺ current I_T. The synaptic coupling of the populations and external excitatory drive are modeled by instantaneous jumps in the membrane potential of postsynaptic neurons. The population density model is validated by comparing its response to time-varying retinal input to Monte Carlo simulations of the corresponding IFB network composed of 100 to 1000 cells per population. In the absence of retinal input, the population density model exhibits rhythmic bursting similar to the 7 to 14 Hz oscillations associated with slow wave sleep that require feedback inhibition from RE to TC cells. When the TC and RE cell potassium leakage conductances are adjusted to represent cholingergic neuromodulation and arousal of the network, rhythmic bursting of the probability density model may either persists or be eliminated depending on the number of excitatory (TC to RE) or inhibitory (RE to TC) connections made by each presynaptic cell. When the probability density model is stimulated with constant retinal input (10–100 spikes/sec), a wide range of responses are observed depending on cellular parameters and network connectivity. These include asynchronous burst and tonic spikes, sleep spindle-like rhythmic bursting, and oscillations in population firing rate that are distinguishable from sleep spindles due to their amplitude, frequency, or the presence of tonic spikes. In this context of dLGN/PGN network modeling, we find the population density approach using 2,500 mesh points and resolving membrane voltage to 0.7 mV is over 30 times more efficient than 1000-cell Monte Carlo simulations. Action Editor: David Golomb     

Slow variable dominance and phase resetting in phantom bursting

Bursting oscillations are common in neurons and endocrine cells. One type of bursting model with two slow variables has been called ‘phantom bursting’ since the burst period is a blend of the time constants of the slow variables. A phantom bursting model can produce bursting with a wide range of periods: fast (short period), medium, and slow (long period). We describe a measure, which we call the ‘dominance factor’, of the relative contributions of the two slow variables to the bursting produced by a simple phantom bursting model. Using this tool, we demonstrate how the control of different phases of the burst can be shifted from one slow variable to another by changing a model parameter. We then show that the dominance curves obtained as a parameter is varied can be useful in making predictions about the resetting properties of the model cells. Finally, we demonstrate two mechanisms by which phase-independent resetting of a burst can be achieved, as has been shown to occur in the electrical activity of pancreatic islets.     

Monoamine release by neurons of a primitive nervous system: an amperometric study

We measured monoamine release from dissociated neurons of the sea pansy Renilla koellikeri, a representative of the most evolutionarily ancient animals with nervous systems, by real-time monitoring of exocytosis using the amperometric method with carbon-fiber microelectrodes. Depolarization-induced, as well as spontaneously active, neurons exhibited calcium-dependent exocytotic events at both the soma and the terminal bulb of neuritic processes. All spontaneously active neurons exhibited a bursting activity pattern in which amplitudes of exocytotic events appeared to be distributed in a quantal-like fashion. Fast Fourier transform analysis of bursting activity in 20 such neurons revealed burst harmonics with a major frequency of 8 Hz and a dominant rate of 95 Hz for individual exocytotic events within bursts. The results suggest that exocytotic transmitter release is as ancient as neurons and that endogenously bursting neurons in the sea pansy are as complex as those of higher animals. In addition, the observation that both soma and neuritic terminals of the same neuron can release transmitter suggests that local release sites in these cnidarian neurons are not critical for nerve net function.     

Spiking patterns and their functional implications in the antennal lobe of the tobacco hornworm Manduca sexta

Bursting as well as tonic firing patterns have been described in various sensory systems. In the olfactory system, spontaneous bursts have been observed in neurons distributed across several synaptic levels, from the periphery, to the olfactory bulb (OB) and to the olfactory cortex. Several in vitro studies indicate that spontaneous firing patterns may be viewed as "fingerprints" of different types of neurons that exhibit distinct functions in the OB. It is still not known, however, if and how neuronal burstiness is correlated with the coding of natural olfactory stimuli. We thus conducted an in vivo study to probe this question in the OB equivalent structure of insects, the antennal lobe (AL) of the tobacco hornworm Manduca sexta. We found that in the moth's AL, both projection (output) neurons (PNs) and local interneurons (LNs) are spontaneously active, but PNs tend to produce spike bursts while LNs fire more regularly. In addition, we found that the burstiness of PNs is correlated with the strength of their responses to odor stimulation--the more bursting the stronger their responses to odors. Moreover, the burstiness of PNs was also positively correlated with the spontaneous firing rate of these neurons, and pharmacological reduction of bursting resulted in a decrease of the neurons' responsiveness. These results suggest that neuronal burstiness reflects a physiological state of these neurons that is directly linked to their response characteristics.     

Bursting excitable cell models by a slow Ca2+ current

Bursting in excitable cells is a phenomenon that has attracted the interest of many electrophysiologists and non-linear dynamicists. In this paper, we present two models that give rise to bursting in action potentials. The membrane of the first model contains a voltage-activated Ca2+ channel that inactivates very slowly upon depolarization and a delayed K+ channel that is activated by voltage. This model consists of three dynamic variables--the gating variable of K+ channel (n), inactivation gating variable of the Ca2+ channel (f), and membrane potential (V). The membrane of the second model contains a voltage-activated Na+ channel that inactivates rather fast upon depolarization. This model contains altogether five dynamic variables--the Na+ inactivation gating variable (h) and Ca2+ activation variable (d), in addition to the three dynamic variables in the first model. With the first model, we show how various interesting bursting patterns may arise from such a simple three dynamic variable model. We also demonstrate that a slowly inactivating voltage-dependent Ca2+ channel may play the key role in the genesis of bursting. With the second model, we show how the participation of a quickly inactivating fast inward current may lead to a neuronal type of bursting, multi-peaked oscillations, and chaos, as the rates of the gating variables change.     

A parameter-space search algorithm tested on a Hodgkin–Huxley model

We demonstrate a parameter-space search algorithm using a computational model of a single-compartment neuron with conductance-based Hodgkin-Huxley dynamics. To classify bursting (the desired behavior), we use a simple cost function whose inputs are derived from the frequency content of the neural output. Our method involves the repeated use of a stochastic gradient descent-type algorithm to locate parameter values that allow the neural model to produce bursting within a specified tolerance. We demonstrate good results, including those showing that the utility of our algorithm improves as the pre-defined allowable parameter ranges increase and that the initial approach to our method is computationally efficient.     

Electrical Resonance in the θ Frequency Range in Olfactory Amygdala Neurons

The cortical amygdala receives direct olfactory inputs and is thought to participate in processing and learning of biologically relevant olfactory cues. As for other brain structures implicated in learning, the principal neurons of the anterior cortical nucleus (ACo) exhibit intrinsic subthreshold membrane potential oscillations in the θ-frequency range. Here we show that nearly 50% of ACo layer II neurons also display electrical resonance, consisting of selective responsiveness to stimuli of a preferential frequency (2–6 Hz). Their impedance profile resembles an electrical band-pass filter with a peak at the preferred frequency, in contrast to the low-pass filter properties of other neurons. Most ACo resonant neurons displayed frequency preference along the whole subthreshold voltage range. We used pharmacological tools to identify the voltage-dependent conductances implicated in resonance. A hyperpolarization-activated cationic current depending on HCN channels underlies resonance at resting and hyperpolarized potentials; notably, this current also participates in resonance at depolarized subthreshold voltages. KV7/KCNQ K⁺ channels also contribute to resonant behavior at depolarized potentials, but not in all resonant cells. Moreover, resonance was strongly attenuated after blockade of voltage-dependent persistent Na⁺ channels, suggesting an amplifying role. Remarkably, resonant neurons presented a higher firing probability for stimuli of the preferred frequency. To fully understand the mechanisms underlying resonance in these neurons, we developed a comprehensive conductance-based model including the aforementioned and leak conductances, as well as Hodgkin and Huxley-type channels. The model reproduces the resonant impedance profile and our pharmacological results, allowing a quantitative evaluation of the contribution of each conductance to resonance. It also replicates selective spiking at the resonant frequency and allows a prediction of the temperature-dependent shift in resonance frequency. Our results provide a complete characterization of the resonant behavior of olfactory amygdala neurons and shed light on a putative mechanism for network activity coordination in the intact brain.