Determining Ultrasound Parameters for Bursting Polymer Microbubbles for Future Use in Spinal Cord Injury

OBJECTIVE

We believe our poly(lactic acid) (PLA) microbubbles are well suited for therapeutic delivery to spinal cord injury (SCI) using ultrasound-triggered bursting. We investigated the feasibility of clinical ultrasound bursting in situ, the optimal bursting parameters in vitro and the loading and release of a model bio-active DNA.

METHODS

Microbubbles were tested using clinical ultrasound in a rat cadaver SCI model. Burst pressure thresholds were determined using the change in enhancement after ultrasound exposure. Resonance frequency, acoustic enhancement, sizing and morphology were evaluated by comparing two microbubble porogens, ammonium carbonate and ammonium carbamate. Oligonucleotides were loaded into the shell and released using the found optimized ultrasound bursting parameters.

RESULTS

In situ imaging and bursting were successful. In vitro bursting thresholds using frequencies 1, 2.25 and 5 MHz were identified between peak negative pressures 0.2 and 0.5 MPa, believed to be safe for spinal cord. The pressure threshold decreased with decreasing frequencies. PLA bursting was optimized near the resonance frequency of 2.5 to 3.0 MHz using 2.25 MHz and not at lower frequencies. PLA microbubbles, initially with a mean size of approximately 2 µm, remained in one piece, collapsed to between 0.5 and 1 µm and did not fragment. Significantly more oligonucleotide was released after ultrasound bursting of loaded microbubbles. Microbubble-sized debris was detected when using ammonium carbamate, leading to inaccurate microbubble concentration measurements.

CONCLUSION

PLA microbubbles made with ammonium carbonate and burst at appropriate parameters have the potential to safely improve intrathecal therapeutic delivery to SCI using targeted ultrasound.

Perspective: Operate on lumbar synovial cysts and avoid ineffective percutaneous techniques

Background

Lumbar synovial cysts (LSC), best diagnosed on MR studies, may cause symptoms/signs ranging from unilateral radiculopathy to cauda equina compressive syndromes. Attempts at percutaneous treatment of LSC typically fail. Rather, greater safety/efficacy is associated with direct surgical resection with/without fusion.

Methods

Treatment of LSC with percutaneous techniques, including cyst aspiration/perforation, injection (i.e., with/without steroids, saline/other), dilatation, and/or disruption/bursting, classically fail. This is because LSCs' tough, thickened, and adherent fibrous capsules cause extensive thecal sac/nerve root compression, and contain minimal central "fluid" (i.e., "crank-case" and non-aspirable). Multiple percutaneous attempts at decompression, therefore, typically cause several needle puncture sites risking dural tears (DT)/cerebrospinal fluid (CSF) leaks, direct root injuries, failure to decompress the thecal sac/nerve roots, infections, hematomas, and over the longer-term, adhesive arachnoiditis.

Results

Alternatively, many studies document the success of direct or even partial resection of LSC (i.e., partial removal with marked cyst/dural adhesions with shrinking down the remnant of capsular tissue). Surgical decompressions of LSC, ranging from focal laminotomies to laminectomies, may or may not warrant additional fusions.

Conclusions

Symptomatic LSC are best managed with direct or even partial operative resection/decompression with/without fusion. The use of varying percutaneous techniques classically fails, and increases multiple perioperative risks.

Nonlinear mechanism for the enhanced bursting activities induced by fast inhibitory autapse and reduced activities by fast excitatory autapse

The paradoxical phenomena that excitatory modulation does not enhance but reduces or inhibitory modulation not suppresses but promotes neural firing activities have attracted increasing attention. In the present study, paradoxical phenomena induced by both fast excitatory and inhibitory autapses in a "Fold/Big Homoclinic" bursting are simulated, and the corresponding nonlinear and biophysical mechanisms are presented. Firstly, the enhanced conductance of excitatory autapse induces the number of spikes per burst and firing rate reduced, while the enhanced inhibitory autapse cause both indicators increased. Secondly, with fast-slow variable dissection, the burst of bursting is identified to locate between a fold bifurcation and a big saddle-homoclinic orbit bifurcation of the fast subsystem. Enhanced excitatory or inhibitory autapses cannot induce changes of both bifurcation points, i.e., burst width. However, width of slow variable between two successive spikes within a burst becomes wider for the excitatory autapse and narrower for the inhibitory autapse, resulting in the less and more spikes per burst, respectively. Last, the autaptic current of fast autapse mainly plays a role during the peak of action potential, differing from the slow autaptic current with exponential decay, which can play roles following the peak of action potential. The fast excitatory autaptic current enhances the amplitude of the action potential and reduces the repolarization of the action potential to lengthen the interspike interval (ISI) of the spiking of the fast subsystem, resulting in the wide width of slow variable between successive spikes. The fast inhibitory autaptic current reduces the amplitude of action potential and ISI of spiking, resulting in narrow width of slow variable. The novel example of the paradoxical responses for both fast modulations and nonlinear mechanism extend the contents of neurodynamics, which presents potential functions of the fast autapse.

Synergising single-cell resolution and 4sU labelling boosts inference of transcriptional bursting

Despite the recent rise of RNA-seq datasets combining single-cell (sc) resolution with 4-thiouridine (4sU) labelling, analytical methods exploiting their power to dissect transcriptional bursting are lacking. Here, we present a mathematical model and Bayesian inference implementation to facilitate genome-wide joint parameter estimation and confidence quantification (R package: burstMCMC). We demonstrate that, unlike conventional scRNA-seq, 4sU scRNA-seq resolves temporal parameters and furthermore boosts inference of dimensionless parameters via a synergy between single-cell resolution and 4sU labelling. We apply our method to published 4sU scRNA-seq data and linked with ChIP-seq data, we uncover previously obscured associations between different parameters and histone modifications.

CaV3.1 T-type calcium channels are important for spatial memory processing in the dorsal subiculum

The dorsal subiculum (dSub) is one of the key structures responsible for the formation of hippocampal memory traces but the contribution of individual ionic currents to its cognitive function is not well studied. Although we recently reported that low-voltage-activated T-type calcium channels (T-channels) are crucial for the burst firing pattern regulation in the dSub pyramidal neurons, their potential role in learning and memory remains unclear. Here we used in vivo local field potential recordings and miniscope calcium imaging in freely behaving mice coupled with pharmacological and genetic tools to address this gap in knowledge. We show that the CaV3.1 isoform of T-channels is critically involved in controlling neuronal activity in the dSub in vivo. Altering neuronal excitability by inhibiting T-channel activity markedly affects calcium dynamics, synaptic plasticity, neuronal oscillations and phase-amplitude coupling in the dSub, thereby disrupting spatial learning. These results provide an important causative link between the CaV3.1 channels, burst firing of dSub neurons and memory formation, thus further supporting the notion that changes in neuronal excitability regulate memory processing. We posit that subicular CaV3.1 T-channels could be a promising novel drug target for cognitive disorders.

A spiking computational model for striatal cholinergic interneurons

Cholinergic interneurons in the striatum, also known as tonically active interneurons or TANs, are thought to have a strong effect on corticostriatal plasticity and on striatal activity and outputs, which in turn play a critical role in modulating downstream basal ganglia activity and movement. Striatal TANs can exhibit a variety of firing patterns and responses to synaptic inputs; furthermore, they have been found to display various surges and pauses in activity associated with sensory cues and reward delivery in learning as well as with motor tic production. To help explain the factors that contribute to TAN activity patterns and to provide a resource for future studies, we present a novel conductance-based computational model of a striatal TAN. We show that this model produces the various characteristic firing patterns observed in recordings of TANs. With a single baseline tuning associated with tonic firing, the model also captures a wide range of TAN behaviors found in previous experiments involving a variety of manipulations. In addition to demonstrating these results, we explain how various ionic currents in the model contribute to them. Finally, we use this model to explore the contributions of the acetylcholine released by TANs to the production of surges and pauses in TAN activity in response to strong excitatory inputs. These results provide predictions for future experimental testing that may help with efforts to advance our understanding of the role of TANs in reinforcement learning and in motor disorders such as Tourette's syndrome.

Machine learning provides insight into models of heterogeneous electrical activity in human beta-cells

Understanding how heterogeneous cellular responses emerge from cell-to-cell variations in expression and function of subcellular components is of general interest. Here, we focus on human insulin-secreting beta-cells, which are believed to constitute a population in which heterogeneity is of physiological importance. We exploit recent single-cell electrophysiological data that allow biologically realistic population modeling of human beta-cells that accounts for cellular heterogeneity and correlation between ion channel parameters. To investigate how ion channels influence the dynamics of our updated mathematical model of human pancreatic beta-cells, we explore several machine learning techniques to determine which model parameters are important for determining the qualitative patterns of electrical activity of the model cells. As expected, K^＋ channels promote absence of activity, but once a cell is active, they increase the likelihood of having action potential firing. HERG channels were of great importance for determining cell behavior in most of the investigated scenarios. Fast bursting is influenced by the time scales of ion channel activation and, interestingly, by the type of Ca^2＋ channels coupled to BK channels in BK-CaV complexes. Slow, metabolically driven oscillations are promoted mostly by K(ATP) channels. In summary, combining population modeling with machine learning analysis provides insight into the model and generates new hypotheses to be investigated both experimentally, via simulations and through mathematical analysis.

The role of astrocytes in place cell formation: A computational modeling study

Place cells develop spatially-tuned receptive fields during the early stages of novel environment exploration. The generative mechanism underlying these spatially-selective responses remains largely elusive, but has been associated with theta rhythmicity. An important factor implicating the transformation of silent cells to place cells is a spatially-uniform depolarization that is mediated by a persistent sodium current. This neuronal current is modulated by extracellular calcium concentration, which, in turn, is actively controlled by astrocytes. However, there is no established relationship between the neuronal depolarization and astrocytic activity. To consider this link, we designed a bioplausible computational model of a neuronal-astrocytic network, where astrocytes induced the transient emergence of place fields in silent cells, and accelerated the plasticity-induced consolidation of place cells. Interestingly, theta oscillations emerged naturally at the network level, resulting from the astrocytic modulation of subcellular neuronal properties. Our results suggest that astrocytes participate in spatial mapping and exploration, and further highlight the computational roles of these cells in the brain.

Exact mean-field models for spiking neural networks with adaptation

Networks of spiking neurons with adaption have been shown to be able to reproduce a wide range of neural activities, including the emergent population bursting and spike synchrony that underpin brain disorders and normal function. Exact mean-field models derived from spiking neural networks are extremely valuable, as such models can be used to determine how individual neurons and the network they reside within interact to produce macroscopic network behaviours. In the paper, we derive and analyze a set of exact mean-field equations for the neural network with spike frequency adaptation. Specifically, our model is a network of Izhikevich neurons, where each neuron is modeled by a two dimensional system consisting of a quadratic integrate and fire equation plus an equation which implements spike frequency adaptation. Previous work deriving a mean-field model for this type of network, relied on the assumption of sufficiently slow dynamics of the adaptation variable. However, this approximation did not succeed in establishing an exact correspondence between the macroscopic description and the realistic neural network, especially when the adaptation time constant was not large. The challenge lies in how to achieve a closed set of mean-field equations with the inclusion of the mean-field dynamics of the adaptation variable. We address this problem by using a Lorentzian ansatz combined with the moment closure approach to arrive at a mean-field system in the thermodynamic limit. The resulting macroscopic description is capable of qualitatively and quantitatively describing the collective dynamics of the neural network, including transition between states where the individual neurons exhibit asynchronous tonic firing and synchronous bursting. We extend the approach to a network of two populations of neurons and discuss the accuracy and efficacy of our mean-field approximations by examining all assumptions that are imposed during the derivation. Numerical bifurcation analysis of our mean-field models reveals bifurcations not previously observed in the models, including a novel mechanism for emergence of bursting in the network. We anticipate our results will provide a tractable and reliable tool to investigate the underlying mechanism of brain function and dysfunction from the perspective of computational neuroscience.

Canard solutions in neural mass models: consequences on critical regimes

Mathematical models at multiple temporal and spatial scales can unveil the fundamental mechanisms of critical transitions in brain activities. Neural mass models (NMMs) consider the average temporal dynamics of interconnected neuronal subpopulations without explicitly representing the underlying cellular activity. The mesoscopic level offered by the neural mass formulation has been used to model electroencephalographic (EEG) recordings and to investigate various cerebral mechanisms, such as the generation of physiological and pathological brain activities. In this work, we consider a NMM widely accepted in the context of epilepsy, which includes four interacting neuronal subpopulations with different synaptic kinetics. Due to the resulting three-time-scale structure, the model yields complex oscillations of relaxation and bursting types. By applying the principles of geometric singular perturbation theory, we unveil the existence of the canard solutions and detail how they organize the complex oscillations and excitability properties of the model. In particular, we show that boundaries between pathological epileptic discharges and physiological background activity are determined by the canard solutions. Finally we report the existence of canard-mediated small-amplitude frequency-specific oscillations in simulated local field potentials for decreased inhibition conditions. Interestingly, such oscillations are actually observed in intracerebral EEG signals recorded in epileptic patients during pre-ictal periods, close to seizure onsets.

Pattern selection in coupled neurons under high-low frequency electric field

Biological neurons exposed to an external electric field can induce polarization and charge fluctuation. Indeed, the exchange of calcium, sodium and potassium ions across the cell membrane can induce an electric field as a result of a time-varying electromagnetic field set up. This field could further modulate the cell electrical activity by inducing multiple firing modes. Based on the physical law of electric field, an improved model which includes an additive electrical field variable is constructed for a network of neurons to study wave propagation and mode transition by exploring the longtime dynamics of slightly perturbed plane waves in the network. Wave pattern and mode transition dependence on the different parameters of external electric field are discussed. It is found that the plane wave propagating in the coupled system breaks down to localized structures under the activation of modulational instability. The network under high-low external electric field supports bursting synchronization. This could be a fruitful avenue to discern the occurrence of paroxysmal epilepsy.

Analysis of temperature-dependent abnormal bursting patterns of neurons in Aplysia

Temperature affects the firing pattern and electrical activity of neurons in animals, eliciting diverse responses depending on neuronal cell type. However, the mechanisms underlying such diverse responses are not well understood. In the present study, we performed in vitro recording of abdominal ganglia cells of Aplysia juliana, and analyzed their burst firing patterns. We identified atypical bursting patterns dependent on temperature that were totally different from classical bursting patterns observed in R15 neurons of A. juliana. We classified these abnormal bursting patterns into type 1 and type 2; type 1 abnormal single bursts are composed of two kinds of spikes with a long interspike interval (ISI) followed by short ISI regular firing, while type 2 abnormal single bursts are composed of complex multiplets. To investigate the mechanism underlying the temperature dependence of abnormal bursting, we employed simulations using a modified Plant model and determined that the temperature dependence of type 2 abnormal bursting is related to temperature-dependent scaling factors and activation or inactivation of potassium or sodium channels.

Instant activation of TRP channels by NH4 + promotes neuronal bursting and glutamate spikes in CA1 neurons

Ammonia ( NH 4 + ) is a by-product of cell metabolism and may elicit subcellular effects with specific physiological responses. Chronic effects have been implicated in several neurological diseases and attributed to persistent elevation in blood ammonia levels transferred to the brain. In previous studies the activities of neurons and astrocytes have been examined at ammonia concentrations an order of magnitude higher than measured in the blood. The effects developed within several minutes. Here we focused upon acute responses of neurons to ammonia and whether they may occur at much lower doses. To this end, we combined patch-clamp in CA1 neurons with glutamate imaging in hippocampal slices. Particular attention was paid to the Rett syndrome that has been originally attributed to hyperammonemia. We compared the responses in the wild-type (WT) and model Rett mice (MECP2-null, RTT) to ammonia doses from 0.3 mM on. In both preparations NH 4 + promptly depolarized neurons and increased the ambient glutamate. The bursting activity emerged in WT and it was augmented in RTT. Searching for subcellular mechanisms we examined possible modulation of ion channels by ammonia. We did not find any changes in HCN- and Ca²⁺ currents, which substantially contribute to the bursting activity. The non-selective cation channels were markedly potentiated by ammonia. ASIC channels had a major contribution to the augmentation of neuronal activity by ammonia. Interestingly, their general blocker amiloride (100 μM) moderately excited CA1 cells akin to NH 4 + . In its presence subsequent ammonia effects were markedly compromised. Blockade of TRPC1 channels partially occluded NH 4 + effects. ASIC and TRPC1 blockers decreased the amplitude of excitatory postsynaptic currents (EPSC) and neuronal bursts, congruent with a postsynaptic location of the channels. Inhibition of TRPV1 channels potentiated the responses to NH 4 + . EPSC amplitudes did not change, but the frequency decreased, indicating presynaptic effects. All extracellular NH 4 + actions were observed at concentrations as low as 0.3 mM and the neurons reacted immediately after ammonia arrived the slice. We propose that a brief augmentation of neuronal activity by NH 4 + may occur either spontaneously during arousal or induced by inhalation of smelling salts.

Mixture distributions in a stochastic gene expression model with delayed feedback: a WKB approximation approach

Noise in gene expression can be substantively affected by the presence of production delay. Here we consider a mathematical model with bursty production of protein, a one-step production delay (the passage of which activates the protein), and feedback in the frequency of bursts. We specifically focus on examining the steady-state behaviour of the model in the slow-activation (i.e. large-delay) regime. Using a formal asymptotic approach, we derive an autonomous ordinary differential equation for the inactive protein that applies in the slow-activation regime. If the differential equation is monostable, the steady-state distribution of the inactive (active) protein is approximated by a single Gaussian (Poisson) mode located at the globally stable fixed point of the differential equation. If the differential equation is bistable (due to cooperative positive feedback), the steady-state distribution of the inactive (active) protein is approximated by a mixture of Gaussian (Poisson) modes located at the stable fixed points; the weights of the modes are determined from a WKB approximation to the stationary distribution. The asymptotic results are compared to numerical solutions of the chemical master equation.

Coherence resonance for neuronal bursting with spike undershoot

Although the bursting patterns with spike undershoot are involved with the achievement of physiological or cognitive functions of brain with synaptic noise, noise induced-coherence resonance (CR) from resting state or subthreshold oscillations instead of bursting has been widely identified to play positive roles in information process. Instead, in the present paper, CR characterized by the increase firstly and then decease of peak value of power spectrum of spike trains is evoked from a bursting pattern with spike undershoot, which means that the minimal membrane potential within burst is lower than that of the subthreshold oscillations between bursts, while CR cannot be evoked from the bursting pattern without spike undershoot. With bifurcations and fast-slow variable dissection method, the bursting patterns with and without spike undershoot are classified into "Sub-Hopf/Fold" bursting and "Fold/Homoclinic" bursting, respectively. For the bursting with spike undershoot, the trajectory of the subthreshold oscillations is very close to that of the spikes within burst. Therefore, noise can induce more spikes from the subthreshold oscillations and modulate the bursting regularity, which leads to the appearance of CR. For the bursting pattern without spike undershoot, the trajectory of the quiescent state is not close to that of the spikes within burst, and noise cannot induce spikes from the quiescent state between bursts, which is cause for non-CR. The result provides a novel case of CR phenomenon and extends the scopes of CR concept, presents that noise can enhance rather than suppress information of the bursting patterns with spike undershoot, which are helpful for understanding the dynamics and the potential physiological or cognitive functions of the nerve fiber or brain neurons with such bursting patterns.

Unpacking Transient Event Dynamics in Electrophysiological Power Spectra

Electrophysiological recordings of neuronal activity show spontaneous and task-dependent changes in their frequency-domain power spectra. These changes are conventionally interpreted as modulations in the amplitude of underlying oscillations. However, this overlooks the possibility of underlying transient spectral ‘bursts’ or events whose dynamics can map to changes in trial-average spectral power in numerous ways. Under this emerging perspective, a key challenge is to perform burst detection, i.e. to characterise single-trial transient spectral events, in a principled manner. Here, we describe how transient spectral events can be operationalised and estimated using Hidden Markov Models (HMMs). The HMM overcomes a number of the limitations of the standard amplitude-thresholding approach to burst detection; in that it is able to concurrently detect different types of bursts, each with distinct spectral content, without the need to predefine frequency bands of interest, and does so with less dependence on a priori threshold specification. We describe how the HMM can be used for burst detection and illustrate its benefits on simulated data. Finally, we apply this method to empirical data to detect multiple burst types in a task-MEG dataset, and illustrate how we can compute burst metrics, such as the task-evoked timecourse of burst duration.

Temperature effects on neuronal firing rates and tonic-to-bursting transitions

Temperature fluctuations can affect neurological processes at a variety of levels, with the overall output that higher temperatures in general increase neuronal activity. While variations in firing rates can happen with the neuronal system maintaining its homeostatic firing pattern of tonic firing, or bursting, changes in firing rates can also be associated with transitions between the two patterns of firing. Our computer simulations suggest a possible mechanism directly related to the shortening of the duration of the action potential for higher firing rates with temperature increase. Increased temperatures also shorten the period doubling cascade and chaos transition between tonic and burting regimes.

A Computational model of the mammalian external tufted cell

We introduce a novel detailed conductance-based model of the bursting activity in external tufted (ET) cells of the olfactory bulb. We investigate the mechanisms underlying their bursting, and make experimentally-testable predictions. The ionic currents included in the model are specific to ET cells, and their kinetic and other parameters are based on experimental recordings. We validate the model by showing that its bursting characteristics under various conditions (e.g. blocking various currents) are consistent with experimental observations. Further, we identify the bifurcation structure and dynamics that explain bursting behavior. This analysis allows us to make predictions of the response of the cell to current pulses at different burst phases. We find that depolarizing (but not hyperpolarizing) inputs received during the interburst interval can advance burst timing, creating the substrate for synchronization by excitatory connections. It has been hypothesized that such synchronization among the ET cells within one glomerulus might help coordinate the glomerular output. Next we investigate model parameter sensitivity and identify parameters that play the most prominent role in controlling each burst characteristic, such as the burst frequency and duration. Finally, the response of the cell to periodic inputs is examined, reflecting the sniffing-modulated input that these cell receive in vivo. We find that individual cells can be better entrained by inputs with higher, rather than lower, frequencies than the intrinsic bursting frequency of the cell. Nevertheless, a heterogeneous population of ET cells (as may be found in a glomerulus) is able to produce reliable periodic population responses even at lower input frequencies.

Energy expenditure computation of a single bursting neuron

Brief bursts of high-frequency spikes are a common firing pattern of neurons. The cellular mechanisms of bursting and its biological significance remain a matter of debate. Focusing on the energy aspect, this paper proposes a neural energy calculation method based on the Chay model of bursting. The flow of ions across the membrane of the bursting neuron with or without current stimulation and its power which contributes to the change of the transmembrane electrical potential energy are analyzed here in detail. We find that during the depolarization of spikes in bursting this power becomes negative, which was also discovered in previous research with another energy model. We also find that the neuron’s energy consumption during bursting is minimal. Especially in the spontaneous state without stimulation, the total energy consumption (2.152 × 10⁻⁷ J) during 30 s of bursting is very similar to the biological energy consumption (2.468 × 10⁻⁷ J) during the generation of a single action potential, as shown in Wang et al. (Neural Plast 2017, 2017a). Our results suggest that this property of low energy consumption could simply be the consequence of the biophysics of generating bursts, which is consistent with the principle of energy minimization. Our results also imply that neural energy plays a critical role in neural coding, which opens a new avenue for research of a central challenge facing neuroscience today.

Control of clustered action potential firing in a mathematical model of entorhinal cortex stellate cells

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An SDE model of entorhinal cortex (EC) stellate cells is proposed.

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Experimentally observed action potential clustering is investigated in the model.

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Clusters are generated by subcritical-Hopf/homoclinic type bursting.

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Potential mechanisms underlying changes in EC dynamics in dementia are presented.

The entorhinal cortex is a crucial component of our memory and spatial navigation systems and is one of the first areas to be affected in dementias featuring tau pathology, such as Alzheimer’s disease and frontotemporal dementia. Electrophysiological recordings from principle cells of medial entorhinal cortex (layer II stellate cells, mEC-SCs) demonstrate a number of key identifying properties including subthreshold oscillations in the theta (4–12 Hz) range and clustered action potential firing. These single cell properties are correlated with network activity such as grid firing and coupling between theta and gamma rhythms, suggesting they are important for spatial memory. As such, experimental models of dementia have revealed disruption of organised dorsoventral gradients in clustered action potential firing.

To better understand the mechanisms underpinning these different dynamics, we study a conductance based model of mEC-SCs. We demonstrate that the model, driven by extrinsic noise, can capture quantitative differences in clustered action potential firing patterns recorded from experimental models of tau pathology and healthy animals. The differential equation formulation of our model allows us to perform numerical bifurcation analyses in order to uncover the dynamic mechanisms underlying these patterns. We show that clustered dynamics can be understood as subcritical Hopf/homoclinic bursting in a fast-slow system where the slow sub-system is governed by activation of the persistent sodium current and inactivation of the slow A-type potassium current. In the full system, we demonstrate that clustered firing arises via flip bifurcations as conductance parameters are varied. Our model analyses confirm the experimentally suggested hypothesis that the breakdown of clustered dynamics in disease occurs via increases in AHP conductance.

Transitions between bursting modes in the integrated oscillator model for pancreatic β-cells

Insulin-secreting β-cells of pancreatic islets of Langerhans produce bursts of electrical impulses, resulting in intracellular Ca²⁺ oscillations and pulsatile insulin secretion. The mechanism for this bursting activity has been the focus of mathematical modeling for more than three decades, and as new data are acquired old models are modified and new models are developed. Comprehensive models must now account for the various modes of bursting observed in islet β-cells, which include fast bursting, slow bursting, and compound bursting. One such model is the Integrated Oscillator Model (IOM), in which β-cell electrical activity, intracellular Ca²⁺, and glucose metabolism interact via numerous feedforward and feedback pathways. These interactions can produce metabolic oscillations with a sawtooth time course or a pulsatile time course, reflecting very different oscillation mechanisms. In this report, we determine conditions favorable to one form of oscillations or the other, and examine the transitions between modes of bursting and the relationship of the transitions to the patterns of metabolic oscillations. Importantly, this work clarifies what can be expected in experimental measurements of β-cell oscillatory activity, and suggests pathways through which oscillations of one type can be converted to oscillations of another type.

Fast-Slow Bursters in the Unfolding of a High Codimension Singularity and the Ultra-slow Transitions of Classes

Bursting is a phenomenon found in a variety of physical and biological systems. For example, in neuroscience, bursting is believed to play a key role in the way information is transferred in the nervous system. In this work, we propose a model that, appropriately tuned, can display several types of bursting behaviors. The model contains two subsystems acting at different time scales. For the fast subsystem we use the planar unfolding of a high codimension singularity. In its bifurcation diagram, we locate paths that underlie the right sequence of bifurcations necessary for bursting. The slow subsystem steers the fast one back and forth along these paths leading to bursting behavior. The model is able to produce almost all the classes of bursting predicted for systems with a planar fast subsystem. Transitions between classes can be obtained through an ultra-slow modulation of the model's parameters. A detailed exploration of the parameter space allows predicting possible transitions. This provides a single framework to understand the coexistence of diverse bursting patterns in physical and biological systems or in models.

Timescales and Mechanisms of Sigh-Like Bursting and Spiking in Models of Rhythmic Respiratory Neurons

Neural networks generate a variety of rhythmic activity patterns, often involving different timescales. One example arises in the respiratory network in the pre-Bötzinger complex of the mammalian brainstem, which can generate the eupneic rhythm associated with normal respiration as well as recurrent low-frequency, large-amplitude bursts associated with sighing. Two competing hypotheses have been proposed to explain sigh generation: the recruitment of a neuronal population distinct from the eupneic rhythm-generating subpopulation or the reconfiguration of activity within a single population. Here, we consider two recent computational models, one of which represents each of the hypotheses. We use methods of dynamical systems theory, such as fast-slow decomposition, averaging, and bifurcation analysis, to understand the multiple-timescale mechanisms underlying sigh generation in each model. In the course of our analysis, we discover that a third timescale is required to generate sighs in both models. Furthermore, we identify the similarities of the underlying mechanisms in the two models and the aspects in which they differ.

Calcium Oscillation Frequency-Sensitive Gene Regulation and Homeostatic Compensation in Pancreatic β-Cells

Pancreatic islet [Formula: see text]-cells are electrically excitable cells that secrete insulin in an oscillatory fashion when the blood glucose concentration is at a stimulatory level. Insulin oscillations are the result of cytosolic [Formula: see text] oscillations that accompany bursting electrical activity of [Formula: see text]-cells and are physiologically important. ATP-sensitive [Formula: see text] channels (K(ATP) channels) play the key role in setting the overall activity of the cell and in driving bursting, by coupling cell metabolism to the membrane potential. In humans, when there is a defect in K(ATP) channel function, [Formula: see text]-cells fail to respond appropriately to changes in the blood glucose level, and electrical and [Formula: see text] oscillations are lost. However, mice compensate for K(ATP) channel defects in islet [Formula: see text]-cells by employing alternative mechanisms to maintain electrical and [Formula: see text] oscillations. In a recent study, we showed that in mice islets in which K(ATP) channels are genetically knocked out another [Formula: see text] current, provided by inward-rectifying [Formula: see text] channels, is increased. With mathematical modeling, we demonstrated that a sufficient upregulation in these channels can account for the paradoxical electrical bursting and [Formula: see text] oscillations observed in these [Formula: see text]-cells. However, the question of determining the correct level of upregulation that is necessary for this compensation remained unanswered, and this question motivates the current study. [Formula: see text] is a well-known regulator of gene expression, and several examples have been shown of genes that are sensitive to the frequency of the [Formula: see text] signal. In this mathematical modeling study, we demonstrate that a [Formula: see text] oscillation frequency-sensitive gene transcription network can adjust the gene expression level of a compensating [Formula: see text] channel so as to rescue electrical bursting and [Formula: see text] oscillations in a model [Formula: see text]-cell in which the key K(ATP) current is removed. This is done without the prescription of a target [Formula: see text] level, but evolves naturally as a consequence of the feedback between the [Formula: see text]-dependent enzymes and the cell's electrical activity. More generally, the study indicates how [Formula: see text] can provide the link between gene expression and cellular electrical activity that promotes wild-type behavior in a cell following gene knockout.

Computational study on neuronal activities arising in the pre-Bötzinger complex

Experimental investigations have shown that the pre-Bötzinger complex (pre-BötC) within the mammalian brainstem generates the inspiratory phase of respiratory rhythm. Based on a single-compartment model of a pre-BötC inspiratory neuron, we, in this paper, use semi-analytical, numerical as well as fast-slow dynamical methods to investigate the effects of sodium conductance ([Formula: see text]) and potassium conductance ([Formula: see text]) on the firing activities of pre-BötC and try to reveal the dynamical mechanisms behind them. We show how [Formula: see text] and [Formula: see text] affect the bifurcations of the fast-subsystem and how the the firing patterns of pre-BötC transit according to the bifurcations.

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.

A basic bifurcation structure from bursting to spiking of injured nerve fibers in a two-dimensional parameter space

Two different bifurcation scenarios of firing patterns with decreasing extracellular calcium concentrations were observed in identical sciatic nerve fibers of a chronic constriction injury (CCI) model when the extracellular 4-aminopyridine concentrations were fixed at two different levels. Both processes proceeded from period-1 bursting to period-1 spiking via complex or simple processes. Multiple typical experimental examples manifested dynamics closely matching those simulated in a recently proposed 4-dimensional model to describe the nonlinear dynamics of the CCI model, which included most cases of the bifurcation scenarios. As the extracellular 4-aminopyridine concentrations is increased, the structure of the bifurcation scenario becomes more complex. The results provide a basic framework for identifying the relationships between different neural firing patterns and different bifurcation scenarios and for revealing the complex nonlinear dynamics of neural firing patterns. The potential roles of the basic bifurcation structures in identifying the information process mechanism are discussed.

Propagation and synchronization of reverberatory bursts in developing cultured networks

Developing networks of neural systems can exhibit spontaneous, synchronous activities called neural bursts, which can be important in the organization of functional neural circuits. Before the network matures, the activity level of a burst can reverberate in repeated rise-and-falls in periods of hundreds of milliseconds following an initial wave-like propagation of spiking activity, while the burst itself lasts for seconds. To investigate the spatiotemporal structure of the reverberatory bursts, we culture dissociated, rat cortical neurons on a high-density multi-electrode array to record the dynamics of neural activity over the growth and maturation of the network. We find the synchrony of the spiking significantly reduced following the initial wave and the activities become broadly distributed spatially. The synchrony recovers as the system reverberates until the end of the burst. Using a propagation model we infer the spreading speed of the spiking activity, which increases as the culture ages. We perform computer simulations of the system using a physiological model of spiking networks in two spatial dimensions and find the parameters that reproduce the observed resynchronization of spiking in the bursts. An analysis of the simulated dynamics suggests that the depletion of synaptic resources causes the resynchronization. The spatial propagation dynamics of the simulations match well with observations over the course of a burst and point to an interplay of the synaptic efficacy and the noisy neural self-activation in producing the morphology of the bursts.

Modeling of synchronization behavior of bursting neurons at nonlinearly coupled dynamical networks

Synchronization behaviors of bursting neurons coupled through electrical and dynamic chemical synapses are investigated. The Izhikevich model is used with random and small world network of bursting neurons. Various currents which consist of diffusive electrical and time-delayed dynamic chemical synapses are used in the simulations to investigate the influences of synaptic currents and couplings on synchronization behavior of bursting neurons. The effects of parameters, such as time delay, inhibitory synaptic strengths, and decay time on synchronization behavior are investigated. It is observed that in random networks with no delay, bursting synchrony is established with the electrical synapse alone, single spiking synchrony is observed with hybrid coupling. In small world network with no delay, periodic bursting behavior with multiple spikes is observed when only chemical and only electrical synapse exist. Single-spike and multiple-spike bursting are established with hybrid couplings. A decrease in the synchronization measure is observed with zero time delay, as the decay time is increased in random network. For synaptic delays which are above active phase period, synchronization measure increases with an increase in synaptic strength and time delay in small world network. However, in random network, it increases with only an increase in synaptic strength.

Bursting dynamics remarkably improve the performance of neural networks on liquid computing

Burst firings are functionally important behaviors displayed by neural circuits, which plays a primary role in reliable transmission of electrical signals for neuronal communication. However, with respect to the computational capability of neural networks, most of relevant studies are based on the spiking dynamics of individual neurons, while burst firing is seldom considered. In this paper, we carry out a comprehensive study to compare the performance of spiking and bursting dynamics on the capability of liquid computing, which is an effective approach for intelligent computation of neural networks. The results show that neural networks with bursting dynamic have much better computational performance than those with spiking dynamics, especially for complex computational tasks. Further analysis demonstrate that the fast firing pattern of bursting dynamics can obviously enhance the efficiency of synaptic integration from pre-neurons both temporally and spatially. This indicates that bursting dynamic can significantly enhance the complexity of network activity, implying its high efficiency in information processing.

Multiple timescale mixed bursting dynamics in a respiratory neuron model

Experimental results in rodent medullary slices containing the pre-Bötzinger complex (pre-BötC) have identified multiple bursting mechanisms based on persistent sodium current (I _NaP) and intracellular Ca²⁺. The classic two-timescale approach to the analysis of pre-BötC bursting treats the inactivation of I _NaP, the calcium concentration, as well as the Ca²⁺-dependent inactivation of IP ₃ as slow variables and considers other evolving quantities as fast variables. Based on its time course, however, it appears that a novel mixed bursting (MB) solution, observed both in recordings and in model pre-BötC neurons, involves at least three timescales. In this work, we consider a single-compartment model of a pre-BötC inspiratory neuron that can exhibit both I _NaP and Ca²⁺ oscillations and has the ability to produce MB solutions. We use methods of dynamical systems theory, such as phase plane analysis, fast-slow decomposition, and bifurcation analysis, to better understand the mechanisms underlying the MB solution pattern. Rather surprisingly, we discover that a third timescale is not actually required to generate mixed bursting solutions. Through our analysis of timescales, we also elucidate how the pre-BötC neuron model can be tuned to improve the robustness of the MB solution.

Multi-timescale systems and fast-slow analysis

Mathematical models of biological systems often have components that vary on different timescales. This multi-timescale character can lead to problems when doing computer simulations, which can require a great deal of computer time so that the components that change on the fastest time scale can be resolved. Mathematical analysis of these multi-timescale systems can be greatly simplified by partitioning them into subsystems that evolve on different time scales. The subsystems are then analyzed semi-independently, using a technique called fast-slow analysis. In this review we describe the fast-slow analysis technique and apply it to relaxation oscillations, neuronal bursting oscillations, canard oscillations, and mixed-mode oscillations. Although these examples all involve neural systems, the technique can and has been applied to other biological, chemical, and physical systems. It is a powerful analysis method that will become even more useful in the future as new experimental techniques push forward the complexity of biological models.

BK Channels and the Control of the Pituitary

The pituitary gland provides the important link between the nervous system and the endocrine system and regulates a diverse range of physiological functions. The pituitary is connected to the hypothalamus by the pituitary stalk and is comprised primarily of two lobes. The anterior lobe consists of five hormone-secreting cell types which are electrically excitable and display single-spike action potentials as well as complex bursting patterns. Bursting is of particular interest as it raises intracellular calcium to a greater extent than spiking and is believed to underlie secretagogue-induced hormone secretion. BK channels have been identified as a key regulator of bursting in anterior pituitary cells. Experimental data and mathematical modeling have demonstrated that BK activation during the upstroke of an action potential results in a prolonged depolarization and an increase in intracellular calcium. In contrast, the posterior lobe is primarily composed of axonal projections of magnocellular neurosecretory cells which extend from the supraoptic and paraventricular nuclei of the hypothalamus. In these neuroendocrine cells, BK channel activation results in a decrease in excitability and hormone secretion. The opposite effect of BK channels in the anterior and posterior pituitary highlights the diverse role of BK channels in regulating the activity of excitable cells. Further studies of pituitary cell excitability and the specific role of BK channels would lead to a greater understanding of how pituitary cell excitability is regulated by both hypothalamic secretagogues and negative feedback loops, and could ultimately lead to novel treatments to pituitary-related disorders.

A biophysical model of supercoiling dependent transcription predicts a structural aspect to gene regulation

Background

Transcription in Escherichia coli generates positive supercoiling in the DNA, which is relieved by the enzymatic activity of gyrase. Recently published experimental evidence suggests that transcription initiation and elongation are inhibited by the buildup of positive supercoiling. It has therefore been proposed that intermittent binding of gyrase plays a role in transcriptional bursting. Considering that transcription is one of the most fundamental cellular processes, it is desirable to be able to account for the buildup and release of positive supercoiling in models of transcription.

Results

Here we present a detailed biophysical model of gene expression that incorporates the effects of supercoiling due to transcription. By directly linking the amount of positive supercoiling to the rate of transcription, the model predicts that highly transcribed genes’ mRNA distributions should substantially deviate from Poisson distributions, with enhanced density at low mRNA copy numbers. Additionally, the model predicts a high degree of correlation between expression levels of genes inside the same supercoiling domain.

Conclusions

Our model, incorporating the supercoiling state of the gene, makes specific predictions that differ from previous models of gene expression. Genes in the same supercoiling domain influence the expression level of neighboring genes. Such structurally dependent regulation predicts correlations between genes in the same supercoiling domain. The topology of the chromosome therefore creates a higher level of gene regulation, which has broad implications for understanding the evolution and organization of bacterial genomes.

Electronic supplementary material

The online version of this article (doi:10.1186/s13628-016-0027-0) contains supplementary material, which is available to authorized users.

Phase-locking of bursting neuronal firing to dominant LFP frequency components

Neuronal firing in the hippocampal formation relative to the phase of local field potentials (LFP) has a key role in memory processing and spatial navigation. Firing can be in either tonic or burst mode. Although bursting neurons are common in the hippocampal formation, the characteristics of their locking to LFP phase are not completely understood. We investigated phase-locking properties of bursting neurons using simulations generated by a dual compartmental model of a pyramidal neuron adapted to match the bursting activity in the subiculum of a rat. The model was driven with stochastic input signals containing a power spectral profile consistent with physiologically relevant frequencies observed in LFP. The single spikes and spike bursts fired by the model were locked to a preferred phase of the predominant frequency band where there was a peak in the power of the driving signal. Moreover, the preferred phase of locking shifted with increasing burst size, providing evidence that LFP phase can be encoded by burst size. We also provide initial support for the model results by analysing example data of spontaneous LFP and spiking activity recorded from the subiculum of a single urethane-anaesthetised rat. Subicular neurons fired single spikes, two-spike bursts and larger bursts that locked to a preferred phase of either dominant slow oscillations or theta rhythms within the LFP, according to the model prediction. Both power-modulated phase-locking and gradual shift in the preferred phase of locking as a function of burst size suggest that neurons can use bursts to encode timing information contained in LFP phase into a spike-count code.

Dynamical analysis of periodic bursting in piece-wise linear planar neuron model

A piece-wise linear planar neuron model, namely, two-dimensional McKean model with periodic drive is investigated in this paper. Periodical bursting phenomenon can be observed in the numerical simulations. By assuming the formal solutions associated with different intervals of this non-autonomous system and introducing the generalized Jacobian matrix at the non-smooth boundaries, the bifurcation mechanism for the bursting solution induced by the slowly varying periodic drive is presented. It is shown that, the discontinuous Hopf bifurcation occurring at the non-smooth boundaries, i.e., the bifurcation taking place at the thresholds of the stimulation, leads the alternation between the rest state and spiking state. That is, different oscillation modes of this non-autonomous system convert periodically due to the non-smoothness of the vector field and the slow variation of the periodic drive as well.

Modelling the vascular response to sympathetic postganglionic nerve activity

This paper explores the influence of burst properties of the sympathetic nervous system on arterial contractility. Specifically, a mathematical model is constructed of the pathway from action potential generation in a sympathetic postganglionic neurone to contraction of an arterial smooth muscle cell. The differential equation model is a synthesis of models of the individual physiological processes, and is shown to be consistent with physiological data.

The model is found to be unresponsive to tonic (regular) stimulation at typical frequencies recorded in sympathetic efferents. However, when stimulated at the same average frequency, but with repetitive respiratory-modulated burst patterns, it produces marked contractions. Moreover, the contractile force produced is found to be highly dependent on the number of spikes in each burst. In particular, when the model is driven by preganglionic spike trains recorded from wild-type and spontaneously hypertensive rats (which have increased spiking during each burst) the contractile force was found to be 10-fold greater in the hypertensive case. An explanation is provided in terms of the summative increased release of noradrenaline. Furthermore, the results suggest the marked effect that hypertensive spike trains had on smooth muscle cell tone can provide a significant contribution to the pathology of hypertension.

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We model the sympathetic-driven contraction of a vascular smooth muscle cell.

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The cell is unresponsive to tonic stimulation at typical sympathetic frequencies.

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We quantify the force produced by the cell in response to sympathetic bursting.

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The response of the cell is strongly dependent on burst amplitude and duration.

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Recordings from hypertensive animals produce significant contractile forces.

Role of small conductance Ca²⁺-activated K⁺ channels in controlling CA1 pyramidal cell excitability

Small-conductance Ca(2+)-activated K(+) (SK or K(Ca)2) channels are widely expressed in the CNS. In several types of neurons, these channels were shown to become activated during repetitive firing, causing early spike frequency adaptation. In CA1 pyramidal cells, SK channels in dendritic spines were shown to regulate synaptic transmission. However, the presence of functional SK channels in the somata and their role in controlling the intrinsic firing of these neurons has been controversial. Using whole-cell voltage-clamp and current-clamp recordings in acute hippocampal slices and focal applications of irreversible and reversible SK channel blockers, we provide evidence that functional SK channels are expressed in the somata and proximal dendrites of adult rat CA1 pyramidal cells. Although these channels can generate a medium duration afterhyperpolarizing current, they play only an auxiliary role in controlling the intrinsic excitability of these neurons, secondary to the low voltage-activating, noninactivating K(V)7/M channels. As long as K(V)7/M channels are operative, activation of SK channels during repetitive firing does not notably affect the spike output of CA1 pyramidal cells. However, when K(V)7/M channel activity is compromised, SK channel activation significantly and uniquely reduces spike output of these neurons. Therefore, proximal SK channels provide a "second line of defense" against intrinsic hyperexcitability, which may play a role in multiple conditions in which K(V)7/M channels activity is compromised, such as hyposmolarity.

Decreased bursting and novel object-specific cell firing in the hippocampus after mild traumatic brain injury

OBJECTIVE

mild traumatic brain injury (mTBI) can produce lasting memory deficits even in the absence of cell loss. We investigated changes in hippocampal firing patterns during exploration and during a novel object recognition (NOR) task.

METHODS

six male Sprague-Dawley rats were subjected to mTBI via fluid percussion injury and were compared with sham-operated rats. Microelectrodes were implanted into CA1 and CA3 and multiple units were recorded from the pyramidal cell layer. Spontaneous "burst" characteristics were analyzed and temporal firing patterns were correlated with object encounters to establish object-specific firing patterns.

RESULTS

mTBI was associated with significantly less hippocampal bursting (p<0.05) with a trend toward longer bursts and lower interburst spike frequency. mTBI was also associated with no preference for a novel object at 12h (p<0.05). During the NOR task, a subset of pyramidal cells were identified which consistently demonstrated a transiently increased firing rate upon encounter of a specific object ("object-specific" cell). Across both groups, there was a significant (p<0.05) correlation between preference for object novelty and the difference between the total number of novel object-specific cells and familiar object-specific cells. The proportion of object-specific cells that responded to the unexpected (novel) object compared to those responding to the familiar object was significantly smaller in rats that had been exposed to mTBI (p<0.05).

CONCLUSION

memory deficits after mTBI are associated with decreased intrinsic burst activity and impaired context-specific firing patterns in the hippocampus during object exploration.

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.

Implications of cellular models of dopamine neurons for schizophrenia

Midbrain dopamine neurons are pacemakers in vitro, but in vivo they fire less regularly and occasionally in bursts that can lead to a temporary cessation in firing produced by depolarization block. The therapeutic efficacy of antipsychotic drugs used to treat the positive symptoms of schizophrenia has been attributed to their ability to induce depolarization block within a subpopulation of dopamine neurons. We summarize the results of experiments characterizing the physiological mechanisms underlying the ability of these neurons to enter depolarization block in vitro, and our computational simulations of those experiments. We suggest that the inactivation of voltage-dependent Na(+) channels, and, in particular, the slower component of this inactivation, is critical in controlling entry into depolarization block. In addition, an ether-a-go-related gene (ERG) K(+) current also appears to be involved by delaying entry into and speeding recovery from depolarization block. Since many antipsychotic drugs share the ability to block this current, ERG channels may contribute to the therapeutic effects of these drugs.

Modeling cancelation of periodic inputs with burst-STDP and feedback

Prediction and cancelation of redundant information is an important feature that many neural systems must display in order to efficiently code external signals. We develop an analytic framework for such cancelation in sensory neurons produced by a cerebellar-like structure in wave-type electric fish. Our biologically plausible mechanism is motivated by experimental evidence of cancelation of periodic input arising from the proximity of conspecifics as well as tail motion. This mechanism involves elements present in a wide range of systems: (1) stimulus-driven feedback to the neurons acting as detectors, (2) a large variety of temporal delays in the pathways transmitting such feedback, responsible for producing frequency channels, and (3) burst-induced long-term plasticity. The bursting arises from back-propagating action potentials. Bursting events drive the input frequency-dependent learning rule, which in turn affects the feedback input and thus the burst rate. We show how the mean firing rate and the rate of production of 2- and 4-spike bursts (the main learning events) can be estimated analytically for a leaky integrate-and-fire model driven by (slow) sinusoidal, back-propagating and feedback inputs as well as rectified filtered noise. The effect of bursts on the average synaptic strength is also derived. Our results shed light on why bursts rather than single spikes can drive learning in such networks "online", i.e. in the absence of a correlative discharge. Phase locked spiking in frequency specific channels together with a frequency-dependent STDP window size regulate burst probability and duration self-consistently to implement cancelation.

Qualitative validation of the reduction from two reciprocally coupled neurons to one self-coupled neuron in a respiratory network model

The pre-Bötzinger complex of the mammalian brainstem is a heterogeneous neuronal network, and individual neurons within the network have varying strengths of the persistent sodium and calcium-activated nonspecific cationic currents. Individually, these currents have been the focus of modeling efforts. Previously, Dunmyre et al. (J Comput Neurosci 1-24, 2011) proposed a model and studied the interactions of these currents within one self-coupled neuron. In this work, I consider two identical, reciprocally coupled model neurons and validate the reduction to the self-coupled case. I find that all of the dynamics of the two model neuron network and the regions of parameter space where these distinct dynamics are found are qualitatively preserved in the reduction to the self-coupled case.

A Computational Model of the Temperature-dependent Changes in Firing Patterns in Aplysia Neurons

We performed experiments using Aplysia neurons to identify the mechanism underlying the changes in the firing patterns in response to temperature changes. When the temperature was gradually increased from 11℃ to 31℃ the firing patterns changed sequentially from the silent state to beating, doublets, beating-chaos, bursting-chaos, square-wave bursting, and bursting-oscillation patterns. When the temperature was decreased over the same temperature range, these sequential changes in the firing patterns reappeared in reverse order. To simulate this entire range of spiking patterns we modified nonlinear differential equations that Chay and Lee made using temperature-dependent scaling factors. To refine the equations, we also analyzed the spike pattern changes in the presence of potassium channel blockers. Based on the solutions of these equations and potassium channel blocker experiments, we found that, as temperature increases, the maximum value of the potassium channel relaxation time constant, τ(n)(t) increases, but the maximum value of the probabilities of openings for activation of the potassium channels, n(t) decreases. Accordingly, the voltage-dependent potassium current is likely to play a leading role in the temperature-dependent changes in the firing patterns in Aplysia neurons.

Chronic Cellular Hyperexcitability in Elderly Epileptic Rats with Spontaneous Seizures Induced by Kainic Acid Status Epilepticus while Young Adults

Emerging data indicate that age-related brain changes alter seizure susceptibility, seizure-associated neurodegeneration, and responsiveness to AEDs. The present study assessed long-term animal survival in the Kainic Acid (KA) model along with in-vivo spontaneous seizure frequency, cellular hyperexcitability in CA1 in-vitro and in-vivo in subiculum, and responsiveness of in-vitro CA1 hyperexcitability to topiramate. Sprague-Dawley male rats were given KA to induce convulsive status epilepticus (KA-SE) at 2-3 months of age. The one-month mortality after KA-SE was 27%. One-month survivor rats had 37% sudden unexplained late mortality after KA-SE as compared to none in saline controls during their second year of life. In-vivo seizure frequency was examined prior to terminal experiments. The diurnal average seizure frequency in the KA-SE group at age 2 years was 1.06 ± 0.24 seizures/hour while no seizures were observed in the saline age-matched controls (p<0.001). In-vitro recordings of CA1 pyramidal neurons revealed that depolarizing current injection from -60 mV evoked an increased number of action potentials in the aged KA-SE group compared to controls (p<0.002). Topiramate exhibited dose-dependent inhibition of action potential firing evoked by current injections into CA1 pyramidal neurons of KA-SE rats. In subiculum, KA-SE rats had frequent interictal spikes associated with high frequency oscillations while only rare spontaneous EPSPs were recorded in saline controls. Our experiments revealed that the hippocampal formation of aged epileptic rats shares features of hyperexcitability previously described in young adult epileptic rats using the KA model.

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.

The role of spiking and bursting pacemakers in the neuronal control of breathing

Breathing is controlled by a distributed network involving areas in the neocortex, cerebellum, pons, medulla, spinal cord, and various other subcortical regions. However, only one area seems to be essential and sufficient for generating the respiratory rhythm: the preBötzinger complex (preBötC). Lesioning this area abolishes breathing and following isolation in a brain slice the preBötC continues to generate different forms of respiratory activities. The use of slice preparations led to a thorough understanding of the cellular mechanisms that underlie the generation of inspiratory activity within this network. Two types of inward currents, the persistent sodium current (I(NaP)) and the calcium-activated non-specific cation current (I(CAN)), play important roles in respiratory rhythm generation. These currents give rise to autonomous pacemaker activity within respiratory neurons, leading to the generation of intrinsic spiking and bursting activity. These membrane properties amplify as well as activate synaptic mechanisms that are critical for the initiation and maintenance of inspiratory activity. In this review, we describe the dynamic interplay between synaptic and intrinsic membrane properties in the generation of the respiratory rhythm and we relate these mechanisms to rhythm generating networks involved in other behaviors.