Ion concentration dynamics as a mechanism for neuronal bursting

We describe a simple conductance-based model neuron that includes intra- and extracellular ion concentration dynamics and show that this model exhibits periodic bursting. The bursting arises as the fast-spiking behavior of the neuron is modulated by the slow oscillatory behavior in the ion concentration variables and vice versa. By separating these time scales and studying the bifurcation structure of the neuron, we catalog several qualitatively different bursting profiles that are strikingly similar to those seen in experimental preparations. Our work suggests that ion concentration dynamics may play an important role in modulating neuronal excitability in real biological systems.     

A comparative analysis of multi-conductance neuronal models <Emphasis Type="Italic">in silico</Emphasis>

We demonstrate that a previously presented flexible silicon–neuron architecture can implement three disparate conductance-based neuron models with both fast and slow dynamics. By exploiting the real-time nature of this physical implementation, we mapped the model dynamics across a large region of parameter space. We also found that two of these dynamically different models represent points in a contiguous bursting space that spans between the two models. By systematically varying the model parameters, we also found that multiple, diverse trajectories in parameter space connected the two canonical bursting points. In addition, we found that the combination of parameter values keeps the neuron in the bursting region. These findings demonstrate the usefulness of the silicon–neuron architecture as a neural-modeling tool and illustrate its versatility as a platform for a multi-behavioral neuron that resembles its living analog.     

A Hamilton-Jacobi-Bellman approach for termination of seizure-like bursting

We use Hamilton-Jacobi-Bellman methods to find minimum-time and energy-optimal control strategies to terminate seizure-like bursting behavior in a conductance-based neural model. Averaging is used to eliminate fast variables from the model, and a target set is defined through bifurcation analysis of the slow variables of the model. This method is illustrated for a single neuron model and for a network model to illustrate its efficacy in terminating bursting once it begins. This work represents a numerical proof-of-concept that a new class of control strategies can be employed to mitigate bursting, and could ultimately be adapted to treat medically intractible epilepsy in patient-specific models.     

Synchronous and asynchronous bursting states: role of intrinsic neural dynamics

Brain signals such as local field potentials often display gamma-band oscillations (30-70 Hz) in a variety of cognitive tasks. These oscillatory activities possibly reflect synchronization of cell assemblies that are engaged in a cognitive function. A type of pyramidal neurons, i.e., chattering neurons, show fast rhythmic bursting (FRB) in the gamma frequency range, and may play an active role in generating the gamma-band oscillations in the cerebral cortex. Our previous phase response analyses have revealed that the synchronization between the coupled bursting neurons significantly depends on the bursting mode that is defined as the number of spikes in each burst. Namely, a network of neurons bursting through a Ca(2+)-dependent mechanism exhibited sharp transitions between synchronous and asynchronous firing states when the neurons exchanged the bursting mode between singlet, doublet and so on. However, whether a broad class of bursting neuron models commonly show such a network behavior remains unclear. Here, we analyze the mechanism underlying this network behavior using a mathematically tractable neuron model. Then we extend our results to a multi-compartment version of the NaP current-based neuron model and prove a similar tight relationship between the bursting mode changes and the network state changes in this model. Thus, the synchronization behavior couples tightly to the bursting mode in a wide class of networks of bursting neurons.     

An Algorithmic Method for Reducing Conductance-based Neuron Models

Although conductance-based neural models provide a realistic depiction of neuronal activity, their complexity often limits effective implementation and analysis. Neuronal model reduction methods provide a means to reduce model complexity while retaining the original model’s realism and relevance. Such methods, however, typically include ad hoc components that require that the modeler already be intimately familiar with the dynamics of the original model. We present an automated, algorithmic method for reducing conductance-based neuron models using the method of equivalent potentials (Kelper et al., Biol Cybern 66(5):381–387, 1992) Our results demonstrate that this algorithm is able to reduce the complexity of the original model with minimal performance loss, and requires minimal prior knowledge of the model’s dynamics. Furthermore, by utilizing a cost function based on the contribution of each state variable to the total conductance of the model, the performance of the algorithm can be significantly improved.     

Extracting non-linear integrate-and-fire models from experimental data using dynamic I–V curves

The dynamic I–V curve method was recently introduced for the efficient experimental generation of reduced neuron models. The method extracts the response properties of a neuron while it is subject to a naturalistic stimulus that mimics in vivo-like fluctuating synaptic drive. The resulting history-dependent, transmembrane current is then projected onto a one-dimensional current–voltage relation that provides the basis for a tractable non-linear integrate-and-fire model. An attractive feature of the method is that it can be used in spike-triggered mode to quantify the distinct patterns of post-spike refractoriness seen in different classes of cortical neuron. The method is first illustrated using a conductance-based model and is then applied experimentally to generate reduced models of cortical layer-5 pyramidal cells and interneurons, in injected-current and injected- conductance protocols. The resulting low-dimensional neuron models—of the refractory exponential integrate-and-fire type—provide highly accurate predictions for spike-times. The method therefore provides a useful tool for the construction of tractable models and rapid experimental classification of cortical neurons.     

A Hodgkin-Huxley model exhibiting bursting oscillations

We investigate bursting behaviour generated in an electrophysiological model of pituitary corticotrophs. The active and silent phases of this mode of bursting are generated by moving between two stable oscillatory solutions. The bursting is indirectly driven by slow modulation of the endoplasmic reticulum Ca²⁺ concentration. The model exhibits different modes of bursting, and we investigate mode transitions and similar modes of bursting in other Hodgkin-Huxley models. Bifurcation analysis and the use of null-surfaces facilitate a geometric interpretation of the model bursting modes and action potential generation, respectively.     

Morphologically accurate reduced order modeling of spiking neurons

Accurately simulating neurons with realistic morphological structure and synaptic inputs requires the solution of large systems of nonlinear ordinary differential equations. We apply model reduction techniques to recover the complete nonlinear voltage dynamics of a neuron using a system of much lower dimension. Using a proper orthogonal decomposition, we build a reduced-order system from salient snapshots of the full system output, thus reducing the number of state variables. A discrete empirical interpolation method is then used to reduce the complexity of the nonlinear term to be proportional to the number of reduced variables. Together these two techniques allow for up to two orders of magnitude dimension reduction without sacrificing the spatially-distributed input structure, with an associated order of magnitude speed-up in simulation time. We demonstrate that both nonlinear spiking behavior and subthreshold response of realistic cells are accurately captured by these low-dimensional models.     

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.     

A database of computational models of a half-center oscillator for analyzing how neuronal parameters influence network activity

A half-center oscillator (HCO) is a common circuit building block of central pattern generator networks that produce rhythmic motor patterns in animals. Here we constructed an efficient relational database table with the resulting characteristics of the Hill et al.’s (J Comput Neurosci 10:281–302, 2001) HCO simple conductance-based model. The model consists of two reciprocally inhibitory neurons and replicates the electrical activity of the oscillator interneurons of the leech heartbeat central pattern generator under a variety of experimental conditions. Our long-range goal is to understand how this basic circuit building block produces functional activity under a variety of parameter regimes and how different parameter regimes influence stability and modulatability. By using the latest developments in computer technology, we simulated and stored large amounts of data (on the order of terabytes). We systematically explored the parameter space of the HCO and corresponding isolated neuron models using a brute-force approach. We varied a set of selected parameters (maximal conductance of intrinsic and synaptic currents) in all combinations, resulting in about 10 million simulations. We classified these HCO and isolated neuron model simulations by their activity characteristics into identifiable groups and quantified their prevalence. By querying the database, we compared the activity characteristics of the identified groups of our simulated HCO models with those of our simulated isolated neuron models and found that regularly bursting neurons compose only a small minority of functional HCO models; the vast majority was composed of spiking neurons.     

Firing pattern of bursting neurons under sinusoidal drive in mean-field modeling

Bursting has been observed in many sensory neurons, and is thought to be important in neural signaling, sleep, and some disorders of the brain. Bursting neurons have been studied via various types of conductance-based models at the single-neuron level. Important features of bursting have been reproduced by this type of model, but it is not certain how well the behavior of populations of bursting neurons can be represented solely by that of individual neurons. To study bursting neurons at the population level, a conductance-based model is incorporated into a mean-field model to yield a mean-field bursting model. The responses of the model to sinusoidal inputs are studied, showing that neurons with various different initial states are capable of phase-locked or intermittent firing, depending on their baseline voltage. Furthermore, depending on this voltage, the bursting frequency either slaves to the original unperturbed bursting frequency or approaches a steady value when the external driving frequency increases. Finally, use of white noise perturbations shows that the bursting frequency of the neurons remains the same even under a more general external stimulus.     

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.     

Routes to chaos in a model of a bursting neuron.

Chaotic regimens have been observed experimentally in neurons as well as in deterministic neuronal models. The R15 bursting cell in the abdominal ganglion of Aplysia has been the subject of extensive mathematical modeling. Previously, the model of Plant and Kim has been shown to exhibit both bursting and beating modes of electrical activity. In this report, we demonstrate (a) that a chaotic regime exists between the bursting and beating modes of the model, and (b) that the model approaches chaos from both modes by a period doubling cascade. The bifurcation parameter employed is the external stimulus current. In addition to the period doubling observed in the model-generated trajectories, a period three "window" was observed, power spectra that demonstrate the approaches to chaos were generated, and the Lyaponov exponents and the fractal dimension of the chaotic attractors were calculated. Chaotic regimes have been observed in several similar models, which suggests that they are a general characteristic of cells that exhibit both bursting and beating modes.     

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.     

Phase resetting and phase locking in hybrid circuits of one model and one biological neuron

To determine why elements of central pattern generators phase lock in a particular pattern under some conditions but not others, we tested a theoretical pattern prediction method. The method is based on the tabulated open loop pulsatile interactions of bursting neurons on a cycle-by-cycle basis and was tested in closed loop hybrid circuits composed of one bursting biological neuron and one bursting model neuron coupled using the dynamic clamp. A total of 164 hybrid networks were formed by varying the synaptic conductances. The prediction of 1:1 phase locking agreed qualitatively with the experimental observations, except in three hybrid circuits in which 1:1 locking was predicted but not observed. Correct predictions sometimes required consideration of the second order phase resetting, which measures the change in the timing of the second burst after the perturbation. The method was robust to offsets between the initiation of bursting in the presynaptic neuron and the activation of the synaptic coupling with the postsynaptic neuron. The quantitative accuracy of the predictions fell within the variability (10%) in the experimentally observed intrinsic period and phase resetting curve (PRC), despite changes in the burst duration of the neurons between open and closed loop conditions.     

Reduction of a model for an Onchidium pacemaker neuron

The eight-variable model for the giant neuron localized in the esophageal ganglia of the marine pulmonate mollusk Onchidium verruculatum is reduced to four-and-three-dimensional systems by regrouping variables with similar time scales. These reduced models replicate the complex behavior including beating, periodic bursting and aperiodic bursting displayed by the original full model when the parameter I _ext representing the intensity of the constant DC current stimulation is varied across a wide range. The complex behavior of the full model arises from the interaction of fast and slow dynamics, and depends on the time scale C _s of the slow dynamics. The four-variable reduced model is constructed independently from the parameter C _s so that it reproduces the two-dimensional bifurcation structure of the full model for the two parameters I _ext and C _s . The three-variable reduced model is derived for a specific value of C _s . The parameters of this model are tuned so that its one-parameter bifurcation diagram for I _ext closely matches that of the full model. Correspondence between bifurcation structures ensures that both reduced models reproduce the various discharge patterns of the full model. Similarity between the full and reduced models is also confirmed by comparing mean firing frequencies and membrane potential waveforms in various regimes. The reduction exposes the factors essential for reproducing the dynamics of the full model; indeed, it shows that the eight variables representing the membrane potential and seven gating variables of six ionic currents in the full model account, in fact, for three basic processes responsible for excitability, post-discharge refractoriness and slow membrane modulation. Received: 7 May 1996 / Accepted 4 December 1997     

On dependency properties of the ISIs generated by a two-compartmental neuronal model

One-dimensional leaky integrate and fire neuronal models describe interspike intervals (ISIs) of a neuron as a renewal process and disregarding the neuron geometry. Many multi-compartment models account for the geometrical features of the neuron but are too complex for their mathematical tractability. Leaky integrate and fire two-compartment models seem a good compromise between mathematical tractability and an improved realism. They indeed allow to relax the renewal hypothesis, typical of one-dimensional models, without introducing too strong mathematical difficulties. Here, we pursue the analysis of the two-compartment model studied by Lansky and Rodriguez (Phys D 132:267–286, 1999), aiming of introducing some specific mathematical results used together with simulation techniques. With the aid of these methods, we investigate dependency properties of ISIs for different values of the model parameters. We show that an increase of the input increases the strength of the dependence between successive ISIs.     

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.     

Shaping bursting by electrical coupling and noise

Gap-junctional coupling is an important way of communication between neurons and other excitable cells. Strong electrical coupling synchronizes activity across cell ensembles. Surprisingly, in the presence of noise synchronous oscillations generated by an electrically coupled network may differ qualitatively from the oscillations produced by uncoupled individual cells forming the network. A prominent example of such behavior is the synchronized bursting in islets of Langerhans formed by pancreatic β-cells, which in isolation are known to exhibit irregular spiking (Sherman and Rinzel, Biophys J 54:411–425, 1988; Sherman and Rinzel, Biophys J 59:547–559, 1991). At the heart of this intriguing phenomenon lies denoising, a remarkable ability of electrical coupling to diminish the effects of noise acting on individual cells. In this paper, building on an earlier analysis of denoising in networks of integrate-and-fire neurons (Medvedev, Neural Comput 21 (11):3057–3078, 2009) and our recent study of spontaneous activity in a closely related model of the Locus Coeruleus network (Medvedev and Zhuravytska, The geometry of spontaneous spiking in neuronal networks, submitted, 2012), we derive quantitative estimates characterizing denoising in electrically coupled networks of conductance-based models of square wave bursting cells. Our analysis reveals the interplay of the intrinsic properties of the individual cells and network topology and their respective contributions to this important effect. In particular, we show that networks on graphs with large algebraic connectivity (Fiedler, Czech Math J 23(98):298–305, 1973) or small total effective resistance (Bollobas, Modern graph theory, Graduate Texts in Mathematics, vol. 184, Springer, New York, 1998) are better equipped for implementing denoising. As a by-product of the analysis of denoising, we analytically estimate the rate with which trajectories converge to the synchronization subspace and the stability of the latter to random perturbations. These estimates reveal the role of the network topology in synchronization. The analysis is complemented by numerical simulations of electrically coupled conductance-based networks. Taken together, these results explain the mechanisms underlying synchronization and denoising in an important class of biological models.