Cell Type-Specific Properties of Subicular GABAergic Currents Shape Hippocampal Output Firing Mode

GABAergic function of the subiculum is central to the regulation of hippocampal output activity. Subicular neuronal networks are indeed under potent control by local inhibition. However, information about the properties of GABAergic currents generated by neurons of this parahippocampal area in normal tissue is still missing. Here, we describe GABA_A receptor (GABA_AR)-mediated phasic and tonic currents generated by principal cells (PCs) and interneurons (INs) of the rat subiculum. We show that in spite of similar synaptic current densities, INs generate spontaneous IPSCs (sIPSCs) that occur less frequently and exhibit smaller charge transfer, thus receiving less synaptic total current than PCs. Further distinction of PCs between intrinsically bursting (IB) and regular-spiking (RS) neurons suggested that sIPSCs generated by the two PC sub-types are likely to be similar. PCs and INs are also controlled by a similar tonic inhibition. However, whereas a comparable tonic current density is found in RS cells and INs, IB neurons are constrained by a greater inhibitory tone. Finally, pharmacological blockade of GABA_AR did not promote functional switch of RS neurons to IB mode, but influenced the bursting propensity of IB cells and released fast spiking activity in INs. Our findings reveal differences in GABAergic currents between PCs and INs as well as within PC sub-types. We propose that GABAergic inhibition may shape hippocampal output activity by providing cell type-specific fine-tuning of subicular excitatory and inhibitory drives.     

Oscillations in Large-Scale Cortical Networks: Map-Based Model

We develop a new computationally efficient approach for the analysis of complex large-scale neurobiological networks. Its key element is the use of a new phenomenological model of a neuron capable of replicating important spike pattern characteristics and designed in the form of a system of difference equations (a map). We developed a set of map-based models that replicate spiking activity of cortical fast spiking, regular spiking and intrinsically bursting neurons. Interconnected with synaptic currents these model neurons demonstrated responses very similar to those found with Hodgkin-Huxley models and in experiments. We illustrate the efficacy of this approach in simulations of one- and two-dimensional cortical network models consisting of regular spiking neurons and fast spiking interneurons to model sleep and activated states of the thalamocortical system. Our study suggests that map-based models can be widely used for large-scale simulations and that such models are especially useful for tasks where the modeling of specific firing patterns of different cell classes is important.     

Minimal Hodgkin–Huxley type models for different classes of cortical and thalamic neurons

We review here the development of Hodgkin–Huxley (HH) type models of cerebral cortex and thalamic neurons for network simulations. The intrinsic electrophysiological properties of cortical neurons were analyzed from several preparations, and we selected the four most prominent electrophysiological classes of neurons. These four classes are “fast spiking” “regular spiking” “intrinsically bursting” and “low-threshold spike” cells. For each class, we fit “minimal” HH type models to experimental data. The models contain the minimal set of voltage-dependent currents to account for the data. To obtain models as generic as possible, we used data from different preparations in vivo and in vitro, such as rat somatosensory cortex and thalamus, guinea-pig visual and frontal cortex, ferret visual cortex, cat visual cortex and cat association cortex. For two cell classes, we used automatic fitting procedures applied to several cells, which revealed substantial cell-to-cell variability within each class. The selection of such cellular models constitutes a necessary step towards building network simulations of the thalamocortical system with realistic cellular dynamical properties.     

Feedback enhances feedforward figure-ground segmentation by changing firing mode

In the visual cortex, feedback projections are conjectured to be crucial in figure-ground segregation. However, the precise function of feedback herein is unclear. Here we tested a hypothetical model of reentrant feedback. We used a previous developed 2-layered feedforward spiking network that is able to segregate figure from ground and included feedback connections. Our computer model data show that without feedback, neurons respond with regular low-frequency (～9 Hz) bursting to a figure-ground stimulus. After including feedback the firing pattern changed into a regular (tonic) spiking pattern. In this state, we found an extra enhancement of figure responses and a further suppression of background responses resulting in a stronger figure-ground signal. Such push-pull effect was confirmed by comparing the figure-ground responses with the responses to a homogenous texture. We propose that feedback controls figure-ground segregation by influencing the neural firing patterns of feedforward projecting neurons.     

Noise-induced burst and spike synchronizations in an inhibitory small-world network of subthreshold bursting neurons

We are interested in noise-induced firings of subthreshold neurons which may be used for encoding environmental stimuli. Noise-induced population synchronization was previously studied only for the case of global coupling, unlike the case of subthreshold spiking neurons. Hence, we investigate the effect of complex network architecture on noise-induced synchronization in an inhibitory population of subthreshold bursting Hindmarsh–Rose neurons. For modeling complex synaptic connectivity, we consider the Watts–Strogatz small-world network which interpolates between regular lattice and random network via rewiring, and investigate the effect of small-world connectivity on emergence of noise-induced population synchronization. Thus, noise-induced burst synchronization (synchrony on the slow bursting time scale) and spike synchronization (synchrony on the fast spike time scale) are found to appear in a synchronized region of the J–D plane (J: synaptic inhibition strength and D: noise intensity). As the rewiring probability p is decreased from 1 (random network) to 0 (regular lattice), the region of spike synchronization shrinks rapidly in the J–D plane, while the region of the burst synchronization decreases slowly. We separate the slow bursting and the fast spiking time scales via frequency filtering, and characterize the noise-induced burst and spike synchronizations by employing realistic order parameters and statistical-mechanical measures introduced in our recent work. Thus, the bursting and spiking thresholds for the burst and spike synchronization transitions are determined in terms of the bursting and spiking order parameters, respectively. Furthermore, we also measure the degrees of burst and spike synchronizations in terms of the statistical-mechanical bursting and spiking measures, respectively.     

Generating oscillatory bursts from a network of regular spiking neurons without inhibition

Avian nucleus isthmi pars parvocellularis (Ipc) neurons are reciprocally connected with the layer 10 (L10) neurons in the optic tectum and respond with oscillatory bursts to visual stimulation. Our in vitro experiments show that both neuron types respond with regular spiking to somatic current injection and that the feedforward and feedback synaptic connections are excitatory, but of different strength and time course. To elucidate mechanisms of oscillatory bursting in this network of regularly spiking neurons, we investigated an experimentally constrained model of coupled leaky integrate-and-fire neurons with spike-rate adaptation. The model reproduces the observed Ipc oscillatory bursting in response to simulated visual stimulation. A scan through the model parameter volume reveals that Ipc oscillatory burst generation can be caused by strong and brief feedforward synaptic conductance changes. The mechanism is sensitive to the parameter values of spike-rate adaptation. In conclusion, we show that a network of regular-spiking neurons with feedforward excitation and spike-rate adaptation can generate oscillatory bursting in response to a constant input.     

Independent Components of Neural Activity Carry Information on Individual Populations

Local field potential (LFP), the low-frequency part of the potential recorded extracellularly in the brain, reflects neural activity at the population level. The interpretation of LFP is complicated because it can mix activity from remote cells, on the order of millimeters from the electrode. To understand better the relation between the recordings and the local activity of cells we used a large-scale network thalamocortical model to compute simultaneous LFP, transmembrane currents, and spiking activity. We used this model to study the information contained in independent components obtained from the reconstructed Current Source Density (CSD), which smooths transmembrane currents, decomposed further with Independent Component Analysis (ICA). We found that the three most robust components matched well the activity of two dominating cell populations: superior pyramidal cells in layer 2/3 (rhythmic spiking) and tufted pyramids from layer 5 (intrinsically bursting). The pyramidal population from layer 2/3 could not be well described as a product of spatial profile and temporal activation, but by a sum of two such products which we recovered in two of the ICA components in our analysis, which correspond to the two first principal components of PCA decomposition of layer 2/3 population activity. At low noise one more cell population could be discerned but it is unlikely that it could be recovered in experiment given typical noise ranges.     

Intrinsic and network rhythmogenesis in a reduced traub model for CA3 neurons

We have developed a two-compartment, eight-variable model of a CA3 pyramidal cell as a reduction of a complex 19-compartment cable model [Traub et al, 1991]. Our reduced model segregates the fast currents for sodium spiking into a proximal, soma-like, compartment and the slower calcium and calcium-mediated currents into a dendrite-like compartment. In each model periodic bursting gives way to repetitive soma spiking as somatic injected current increases. Steady dendritic stimulation can produce periodic bursting of significantly higher frequency (8–20 Hz) than can steady somatic input (<8 Hz). Bursting in our model occurs only for an intermediate range of electronic coupling conductance. It depends on the segregation of channel types and on the coupling current that flows back-and-forth between compartments. When the soma and dendrite are tightly coupled electrically, our model reduces to a single compartment and does not burst. Network simulations with our model using excitatory AMPA and NMDA synapses (without inhibition) give results similar to those obtained with the complex cable model [Traub et al, 1991; Traub et al, 1992]. Brief stimulation of a single cell in a resting network produces multiple synchronized population bursts, with fast AMPA synapses providing the dominant synchronizing mechanism. The number of bursts increases with the level of maximal NMDA conductance. For high enough maximal NMDA conductance synchronized bursting repeats indefinitely. We find that two factors can cause the cells to desynchronize when AMPA synapses are blocked: heterogeneity of properties amongst cells and intrinsically chaotic burst dynamics. But even when cells are identical, they may synchronize only approximately rather than exactly. Since our model has a limited number of parameters and variables, we have studied its cellular and network dynamics computationally with relative ease and over wide parameter ranges. Thereby, we identify some qualitative features that parallel or are distinguished from those of other neuronal systems; e.g., we discuss how bursting here differs from that in some classical models.     

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.     

Firing patterns in the adaptive exponential integrate-and-fire model

For simulations of large spiking neuron networks, an accurate, simple and versatile single-neuron modeling framework is required. Here we explore the versatility of a simple two-equation model: the adaptive exponential integrate-and-fire neuron. We show that this model generates multiple firing patterns depending on the choice of parameter values, and present a phase diagram describing the transition from one firing type to another. We give an analytical criterion to distinguish between continuous adaption, initial bursting, regular bursting and two types of tonic spiking. Also, we report that the deterministic model is capable of producing irregular spiking when stimulated with constant current, indicating low-dimensional chaos. Lastly, the simple model is fitted to real experiments of cortical neurons under step current stimulation. The results provide support for the suitability of simple models such as the adaptive exponential integrate-and-fire neuron for large network simulations.     

An integrate-and-fire model for synchronized bursting in a network of cultured cortical neurons

It has been suggested that spontaneous synchronous neuronal activity is an essential step in the formation of functional networks in the central nervous system. The key features of this type of activity consist of bursts of action potentials with associated spikes of elevated cytoplasmic calcium. These features are also observed in networks of rat cortical neurons that have been formed in culture. Experimental studies of these cultured networks have led to several hypotheses for the mechanisms underlying the observed synchronized oscillations. In this paper, bursting integrate-and-fire type mathematical models for regular spiking (RS) and intrinsic bursting (IB) neurons are introduced and incorporated through a small-world connection scheme into a two-dimensional excitatory network similar to those in the cultured network. This computer model exhibits spontaneous synchronous activity through mechanisms similar to those hypothesized for the cultured experimental networks. Traces of the membrane potential and cytoplasmic calcium from the model closely match those obtained from experiments. We also consider the impact on network behavior of the IB neurons, the geometry and the small world connection scheme. Action Editor: David Golomb     

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.     

A Unified Approach to Linking Experimental,Statistical and Computational Analysis of Spike Train Data

A fundamental issue in neuroscience is how to identify the multiple biophysical mechanisms through which neurons generate observed patterns of spiking activity. In previous work, we proposed a method for linking observed patterns of spiking activity to specific biophysical mechanisms based on a state space modeling framework and a sequential Monte Carlo, or particle filter, estimation algorithm. We have shown, in simulation, that this approach is able to identify a space of simple biophysical models that were consistent with observed spiking data (and included the model that generated the data), but have yet to demonstrate the application of the method to identify realistic currents from real spike train data. Here, we apply the particle filter to spiking data recorded from rat layer V cortical neurons, and correctly identify the dynamics of an slow, intrinsic current. The underlying intrinsic current is successfully identified in four distinct neurons, even though the cells exhibit two distinct classes of spiking activity: regular spiking and bursting. This approach – linking statistical, computational, and experimental neuroscience – provides an effective technique to constrain detailed biophysical models to specific mechanisms consistent with observed spike train data.     

Density dependent neurodynamics

The dynamics of a neural network depends on density parameters at (at least) two different levels: the subcellular density of ion channels in single neurons, and the density of cells and synapses at a network level. For the Frankenhaeuser-Huxley (FH) neural model, the density of sodium (Na) and potassium (K) channels determines the behaviour of a single neuron when exposed to an external stimulus. The features of the onset of single neuron oscillations vary qualitatively among different regions in the channel density plane. At a network level, the density of neurons is reflected in the global connectivity. We study the relation between the two density levels in a network of oscillatory FH neurons, by qualitatively distinguishing between three regions, where the mean network activity is (1) spiking, (2) oscillating with enveloped frequencies, and (3) bursting, respectively. We demonstrate that the global activity can be shifted between regions by changing either the density of ion channels at the subcellular level, or the connectivity at the network level, suggesting that different underlying mechanisms can explain similar global phenomena. Finally, we model a possible effect of anaesthesia by blocking specific inhibitory ion channels.     

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.     

Frequency-domain order parameters for the burst and spike synchronization transitions of bursting neurons

We are interested in characterization of synchronization transitions of bursting neurons in the frequency domain. Instantaneous population firing rate (IPFR) R(t), which is directly obtained from the raster plot of neural spikes, is often used as a realistic collective quantity describing population activities in both the computational and the experimental neuroscience. For the case of spiking neurons, a realistic time-domain order parameter, based on R(t), was introduced in our recent work to characterize the spike synchronization transition. Unlike the case of spiking neurons, the IPFR R(t) of bursting neurons exhibits population behaviors with both the slow bursting and the fast spiking timescales. For our aim, we decompose the IPFR R(t) into the instantaneous population bursting rate R_b(t) (describing the bursting behavior) and the instantaneous population spike rate R_s(t) (describing the spiking behavior) via frequency filtering, and extend the realistic order parameter to the case of bursting neurons. Thus, we develop the frequency-domain bursting and spiking order parameters which are just the bursting and spiking “coherence factors” β_b and β_s of the bursting and spiking peaks in the power spectral densities of R_b and R_s (i.e., “signal to noise” ratio of the spectral peak height and its relative width). Through calculation of β_b and β_s, we obtain the bursting and spiking thresholds beyond which the burst and spike synchronizations break up, respectively. Consequently, it is shown in explicit examples that the frequency-domain bursting and spiking order parameters may be usefully used for characterization of the bursting and the spiking transitions, respectively.     

The Sodium-Potassium Pump Controls the Intrinsic Firing of the Cerebellar Purkinje Neuron

In vitro, cerebellar Purkinje cells can intrinsically fire action potentials in a repeating trimodal or bimodal pattern. The trimodal pattern consists of tonic spiking, bursting, and quiescence. The bimodal pattern consists of tonic spiking and quiescence. It is unclear how these firing patterns are generated and what determines which firing pattern is selected. We have constructed a realistic biophysical Purkinje cell model that can replicate these patterns. In this model, Na⁺/K⁺ pump activity sets the Purkinje cell''s operating mode. From rat cerebellar slices we present Purkinje whole cell recordings in the presence of ouabain, which irreversibly blocks the Na⁺/K⁺ pump. The model can replicate these recordings. We propose that Na⁺/K⁺ pump activity controls the intrinsic firing mode of cerbellar Purkinje cells.     

Dynamics of in-phase and anti-phase bursting in the coupled pre-Bötzinger complex cells

Activity of neurons in the pre-Bötzinger complex within the mammalian brain stem has an important role in the generation of respiratory rhythms. Previous experimental results have shown that the dynamics of sodium and calcium within each cell may be responsible for various bursting mechanisms. In this paper, we study the bursting dynamics of the two-coupled pre-Bötzinger complex neurons. Using a combination of fast-slow decomposition and two-parameter bifurcation analysis, we explore the possible forms of dynamics that the model network can produce as well the transitions of in-phase and anti-phase bursting respectively.