Phase sensitivity and entrainment in a modeled bursting neuron

Contribution of Afferent Feedback to Adaptive Hindlimb Walking in Cats: A Neuromusculoskeletal Modeling Study

Mammalian locomotion is generated by central pattern generators (CPGs) in the spinal cord, which produce alternating flexor and extensor activities controlling the locomotor movements of each limb. Afferent feedback signals from the limbs are integrated by the CPGs to provide adaptive control of locomotion. Responses of CPG-generated neural activity to afferent feedback stimulation have been previously studied during fictive locomotion in immobilized cats. Yet, locomotion in awake, behaving animals involves dynamic interactions between central neuronal circuits, afferent feedback, musculoskeletal system, and environment. To study these complex interactions, we developed a model simulating interactions between a half-center CPG and the musculoskeletal system of a cat hindlimb. Then, we analyzed the role of afferent feedback in the locomotor adaptation from a dynamic viewpoint using the methods of dynamical systems theory and nullcline analysis. Our model reproduced limb movements during regular cat walking as well as adaptive changes of these movements when the foot steps into a hole. The model generates important insights into the mechanism for adaptive locomotion resulting from dynamic interactions between the CPG-based neural circuits, the musculoskeletal system, and the environment.

Phase-Dependent Response to Afferent Stimulation During Fictive Locomotion: A Computational Modeling Study

Central pattern generators (CPGs) in the spinal cord generate rhythmic neural activity and control locomotion in vertebrates. These CPGs operate under the control of sensory feedback that affects the generated locomotor pattern and adapt it to the animal's biomechanics and environment. Studies of the effects of afferent stimulation on fictive locomotion in immobilized cats have shown that brief stimulation of peripheral nerves can reset the ongoing locomotor rhythm. Depending on the phase of stimulation and the stimulated nerve, the applied stimulation can either shorten or prolong the current locomotor phase and the locomotor cycle. Here, we used a mathematical model of a half-center CPG to investigate the phase-dependent effects of brief stimulation applied to CPG on the CPG-generated locomotor oscillations. The CPG in the model consisted of two half-centers mutually inhibiting each other. The rhythmic activity in each half-center was based on a slowly inactivating, persistent sodium current. Brief stimulation was applied to CPG half-centers in different phases of the locomotor cycle to produce phase-dependent changes in CPG activity. The model reproduced several results from experiments on the effect of afferent stimulation of fictive locomotion in cats. The mechanisms of locomotor rhythm resetting under different conditions were analyzed using dynamic systems theory methods.

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.

Maladaptive neural synchrony in tinnitus: origin and restoration

Tinnitus is the conscious perception of sound heard in the absence of physical sound sources external or internal to the body, reflected in aberrant neural synchrony of spontaneous or resting-state brain activity. Neural synchrony is generated by the nearly simultaneous firing of individual neurons, of the synchronization of membrane-potential changes in local neural groups as reflected in the local field potentials, resulting in the presence of oscillatory brain waves in the EEG. Noise-induced hearing loss, often resulting in tinnitus, causes a reorganization of the tonotopic map in auditory cortex and increased spontaneous firing rates and neural synchrony. Spontaneous brain rhythms rely on neural synchrony. Abnormal neural synchrony in tinnitus appears to be confined to specific frequency bands of brain rhythms. Increases in delta-band activity are generated by deafferented/deprived neuronal networks resulting from hearing loss. Coordinated reset (CR) stimulation was developed in order to specifically counteract such abnormal neuronal synchrony by desynchronization. The goal of acoustic CR neuromodulation is to desynchronize tinnitus-related abnormal delta-band oscillations. CR neuromodulation does not require permanent stimulus delivery in order to achieve long-lasting desynchronization or even a full-blown anti-kindling but may have cumulative effects, i.e., the effect of different CR epochs separated by pauses may accumulate. Unlike other approaches, acoustic CR neuromodulation does not intend to reduce tinnitus-related neuronal activity by employing lateral inhibition. The potential efficacy of acoustic CR modulation was shown in a clinical proof of concept trial, where effects achieved in 12 weeks of treatment delivered 4–6 h/day persisted through a preplanned 4-week therapy pause and showed sustained long-term effects after 10 months of therapy, leading to 75% responders.

Control of abnormal synchronization in neurological disorders

In the nervous system, synchronization processes play an important role, e.g., in the context of information processing and motor control. However, pathological, excessive synchronization may strongly impair brain function and is a hallmark of several neurological disorders. This focused review addresses the question of how an abnormal neuronal synchronization can specifically be counteracted by invasive and non-invasive brain stimulation as, for instance, by deep brain stimulation for the treatment of Parkinson’s disease, or by acoustic stimulation for the treatment of tinnitus. On the example of coordinated reset (CR) neuromodulation, we illustrate how insights into the dynamics of complex systems contribute to successful model-based approaches, which use methods from synergetics, non-linear dynamics, and statistical physics, for the development of novel therapies for normalization of brain function and synaptic connectivity. Based on the intrinsic multistability of the neuronal populations induced by spike timing-dependent plasticity (STDP), CR neuromodulation utilizes the mutual interdependence between synaptic connectivity and dynamics of the neuronal networks in order to restore more physiological patterns of connectivity via desynchronization of neuronal activity. The very goal is to shift the neuronal population by stimulation from an abnormally coupled and synchronized state to a desynchronized regime with normalized synaptic connectivity, which significantly outlasts the stimulation cessation, so that long-lasting therapeutic effects can be achieved.

Desynchronizing electrical and sensory coordinated reset neuromodulation

Coordinated reset (CR) stimulation is a desynchronizing stimulation technique based on timely coordinated phase resets of sub-populations of a synchronized neuronal ensemble. It has initially been computationally developed for electrical deep brain stimulation (DBS), to enable an effective desynchronization and unlearning of pathological synchrony and connectivity (anti-kindling). Here we computationally show for ensembles of spiking and bursting model neurons interacting via excitatory and inhibitory adaptive synapses that a phase reset of neuronal populations as well as a desynchronization and an anti-kindling can robustly be achieved by direct electrical stimulation or indirect (synaptically-mediated) excitatory and inhibitory stimulation. Our findings are relevant for DBS as well as for sensory stimulation in neurological disorders characterized by pathological neuronal synchrony. Based on the obtained results, we may expect that the local effects in the vicinity of a depth electrode (realized by direct stimulation of the neurons' somata or stimulation of axon terminals) and the non-local CR effects (realized by stimulation of excitatory or inhibitory efferent fibers) of deep brain CR neuromodulation may be similar or even identical. Furthermore, our results indicate that an effective desynchronization and anti-kindling can even be achieved by non-invasive, sensory CR neuromodulation. We discuss the concept of sensory CR neuromodulation in the context of neurological disorders.

Responses of a bursting pacemaker to excitation reveal spatial segregation between bursting and spiking mechanisms

Central pattern generators (CPGs) frequently include bursting neurons that serve as pacemakers for rhythm generation. Phase resetting curves (PRCs) can provide insight into mechanisms underlying phase locking in such circuits. PRCs were constructed for a pacemaker bursting complex in the pyloric circuit in the stomatogastric ganglion of the lobster and crab. This complex is comprised of the Anterior Burster (AB) neuron and two Pyloric Dilator (PD) neurons that are all electrically coupled. Artificial excitatory synaptic conductance pulses of different strengths and durations were injected into one of the AB or PD somata using the Dynamic Clamp. Previously, we characterized the inhibitory PRCs by assuming a single slow process that enabled synaptic inputs to trigger switches between an up state in which spiking occurs and a down state in which it does not. Excitation produced five different PRC shapes, which could not be explained with such a simple model. A separate dendritic compartment was required to separate the mechanism that generates the up and down phases of the bursting envelope (1) from synaptic inputs applied at the soma, (2) from axonal spike generation and (3) from a slow process with a slower time scale than burst generation. This study reveals that due to the nonlinear properties and compartmentalization of ionic channels, the response to excitation is more complex than inhibition.

Phase-response curves and synchronized neural networks

We review the principal assumptions underlying the application of phase-response curves (PRCs) to synchronization in neuronal networks. The PRC measures how much a given synaptic input perturbs spike timing in a neural oscillator. Among other applications, PRCs make explicit predictions about whether a given network of interconnected neurons will synchronize, as is often observed in cortical structures. Regarding the assumptions of the PRC theory, we conclude: (i) The assumption of noise-tolerant cellular oscillations at or near the network frequency holds in some but not all cases. (ii) Reduced models for PRC-based analysis can be formally related to more realistic models. (iii) Spike-rate adaptation limits PRC-based analysis but does not invalidate it. (iv) The dependence of PRCs on synaptic location emphasizes the importance of improving methods of synaptic stimulation. (v) New methods can distinguish between oscillations that derive from mutual connections and those arising from common drive. (vi) It is helpful to assume linear summation of effects of synaptic inputs; experiments with trains of inputs call this assumption into question. (vii) Relatively subtle changes in network structure can invalidate PRC-based predictions. (viii) Heterogeneity in the preferred frequencies of component neurons does not invalidate PRC analysis, but can annihilate synchronous activity.

Dynamics of learning in cultured neuronal networks with antagonists of glutamate receptors

Cognitive dysfunction may result from abnormality of ionotropic glutamate receptors. Although various forms of synaptic plasticity in learning that rely on altering of glutamate receptors have been considered, the evidence is insufficient from an informatics view. Dynamics could reflect neuroinformatics encoding, including temporal pattern encoding, spatial pattern encoding, and energy distribution. Discovering informatics encoding is fundamental and crucial to understanding the working principle of the neural system. In this article, we analyzed the dynamic characteristics of response activities during learning training in cultured hippocampal networks under normal and abnormal conditions of ionotropic glutamate receptors, respectively. The rate, which is one of the temporal configurations, was decreased markedly by inhibition of alpha-amino-3-hydroxy-5-methylisoxazole-4-proprionic acid (AMPA) receptors. Moreover, the energy distribution in different characteristic frequencies was changed markedly by inhibition of AMPA receptors. Spatial configurations, including regularization, correlation, and synchrony, were changed significantly by inhibition of N-methyl-d-aspartate receptors. These results suggest that temporal pattern encoding and energy distribution of response activities in cultured hippocampal neuronal networks during learning training are modulated by AMPA receptors, whereas spatial pattern encoding of response activities is modulated by N-methyl-d-aspartate receptors.

Simulation-Based Training In Electrophysiology By iCELL

A simulation-based training resource was developed for electrophysiology. The resource focuses on computational modeling of cellular bioelectric activity and is called the interactive cell modeling tool (http://ssd1.bme.memphis.edu/icell). iCell can be used as a simulation-based teaching and learning tool for electrophysiology. The site consists of JAVA models of various cardiac cells and neurons, and provides simulation data of their bioelectric activities at single cell level. Each JAVA-based model is menu-driven and presents options to change model parameters or conditions, run and view simulation results. iCell has been used as a teaching and learning tool for seven graduate courses at the Joint Biomedical Engineering Program of University of Memphis and University of Tennessee. Scientists from the fields of biosciences, engineering, life sciences and medicine in 17 different countries have also tested and utilized iCell as a simulation-based teaching, learning and collaboration environment.

Interactive cell modeling web-resource, iCell, as a simulation-based teaching and learning tool to supplement electrophysiology education

An interactive cell modeling web site, iCell (http://ssd1.bme.memphis.edu/icell/), that integrates research and education, was developed to present and to disseminate JAVA-coded models of cellular activities, and to supplement physiology education. iCell can be used to supplement the text-book material as a simulation-based teaching and learning tool. Specifically, iCell allows the students to supplement their learning experiences of the text-book cellular physiology material by running simulations in an interactive environment. The site consists of JAVA-coded models of various cardiac cells and neurons, and provides simulation data of their bioelectric transport activities at cellular level. Each JAVA-coded model allows the user to go through menu options to change model parameters, run and view simulation results. The site also has a glossary section for the scientific terms. iCell has been used as a teaching and learning tool for seven graduate courses at the Joint Biomedical Engineering Program of University of Memphis and University of Tennessee. This modeling tool was also used as a collaboration site among our physiology colleagues interested in simulations of cell membrane activities. Scientists from the fields of biosciences, engineering, life sciences and medical sciences in 17 countries have tested and utilized iCell as a simulation-based teaching, learning and collaboration environment. iCell provides us with an interactive, platform-independent, and user-friendly teaching and learning resource, and also a collaboration environment for electrophysiology to be shared over the Internet. The usage of simulations for teaching and learning will continue advancing simulation-based engineering and sciences for research and development.

Beyond Two-Cell Networks: Experimental Measurement of Neuronal Responses to Multiple Synaptic Inputs

Oscillations of large populations of neurons are thought to be important in the normal functioning of the brain. We have used phase response curve (PRC) methods to characterize the dynamics of single neurons and predict population dynamics. Our past experimental work was limited to special circumstances (e.g., 2-cell networks of periodically firing neurons). Here, we explore the feasibility of extending our methods to predict the synchronization properties of stellate cells (SCs) in the rat entorhinal cortex under broader conditions. In particular, we test the hypothesis that PRCs in SCs scale linearly with changes in synaptic amplitude, and measure how well responses to Poisson process-driven inputs can be predicted in terms of PRCs. Although we see nonlinear responses to excitatory and inhibitory inputs, we find that models based on weak coupling account for scaling and Poisson process-driven inputs reasonably accurately.

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.

Synchronization in hybrid neuronal networks of the hippocampal formation

Understanding the mechanistic bases of neuronal synchronization is a current challenge in quantitative neuroscience. We studied this problem in two putative cellular pacemakers of the mammalian hippocampal theta rhythm: glutamatergic stellate cells (SCs) of the medial entorhinal cortex and GABAergic oriens-lacunosum-molecular (O-LM) interneurons of hippocampal region CA1. We used two experimental methods. First, we measured changes in spike timing induced by artificial synaptic inputs applied to individual neurons. We then measured responses of free-running hybrid neuronal networks, consisting of biological neurons coupled (via dynamic clamp) to biological or virtual counterparts. Results from the single-cell experiments predicted network behaviors well and are compatible with previous model-based predictions of how specific membrane mechanisms give rise to empirically measured synchronization behavior. Both cell types phase lock stably when connected via homogeneous excitatory-excitatory (E-E) or inhibitory-inhibitory (I-I) connections. Phase-locked firing is consistently synchronous for either cell type with E-E connections and nearly anti-synchronous with I-I connections. With heterogeneous connections (e.g., excitatory-inhibitory, as might be expected if members of a given population had heterogeneous connections involving intermediate interneurons), networks often settled into phase locking that was either stable or unstable, depending on the order of firing of the two cells in the hybrid network. Our results imply that excitatory SCs, but not inhibitory O-LM interneurons, are capable of synchronizing in phase via monosynaptic mutual connections of the biologically appropriate polarity. Results are largely independent of synaptic strength and synaptic kinetics, implying that our conclusions are robust and largely unaffected by synaptic plasticity.

Alternative to hand-tuning conductance-based models: construction and analysis of databases of model neurons

Conventionally, the parameters of neuronal models are hand-tuned using trial-and-error searches to produce a desired behavior. Here, we present an alternative approach. We have generated a database of about 1.7 million single-compartment model neurons by independently varying 8 maximal membrane conductances based on measurements from lobster stomatogastric neurons. We classified the spontaneous electrical activity of each model neuron and its responsiveness to inputs during runtime with an adaptive algorithm and saved a reduced version of each neuron's activity pattern. Our analysis of the distribution of different activity types (silent, spiking, bursting, irregular) in the 8-dimensional conductance space indicates that the coarse grid of conductance values we chose is sufficient to capture the salient features of the distribution. The database can be searched for different combinations of neuron properties such as activity type, spike or burst frequency, resting potential, frequency-current relation, and phase-response curve. We demonstrate how the database can be screened for models that reproduce the behavior of a specific biological neuron and show that the contents of the database can give insight into the way a neuron's membrane conductances determine its activity pattern and response properties. Similar databases can be constructed to explore parameter spaces in multicompartmental models or small networks, or to examine the effects of changes in the voltage dependence of currents. In all cases, database searches can provide insight into how neuronal and network properties depend on the values of the parameters in the models.

Computational model of the ventricular action potential in adult spontaneously hypertensive rats

INTRODUCTION

Cardiac hypertrophy has substantial clinical significance because many hypertrophic cells have markedly prolonged repolarization behavior, which may lead to increased risk for cardiac arrhythmias. Spontaneously hypertensive rat (SHR) is one model of hypertension that is studied extensively and is considered to be the best laboratory model of human hypertension. We extended our previously published model of the rat ventricular myocyte to simulate the effects of hypertrophy in SHR.

METHODS AND RESULTS

In SHR it has been shown that the membrane capacitance is increased, the density of transient outward K+ current is decreased, the sarcoplasmic reticulum Ca2+ ATPase activity is reduced, and the cell volumes are increased compared to those of the normal rat. We introduced these changes into our previous model of the rat ventricular myocyte and simulated the ventricular action potential of SHR. Our results demonstrated increased action potential duration (APD) and increased peak systolic value of the intracellular calcium transient in SHR. Simulations with reduced extracellular K+ concentration ([K+]o) have shown that there is increased APD shortening in SHR compared to that of the normal rat.

CONCLUSIONS

Our computational model qualitatively simulated the electrophysiologic changes observed in SHR and provided the plausible mechanistic linkage between the prolonged APD and increased inotropy. Our model results also demonstrated the electrophysiologic changes observed with reduced [K+]o in SHR, a finding that is clinically significant in hypertensive patients with left ventricular hypertrophy undergoing diuretic treatment.

Dynamics from a time series: can we extract the phase resetting curve from a time series?

Recordings of the membrane potential from a bursting neuron were used to reconstruct the phase curve for that neuron for a limited set of perturbations. These perturbations were inhibitory synaptic conductance pulses able to shift the membrane potential below the most hyperpolarized level attained in the free running mode. The extraction of the phase resetting curve from such a one-dimensional time series requires reconstruction of the periodic activity in the form of a limit cycle attractor. Resetting was found to have two components. In the first component, if the pulse was applied during a burst, the burst was truncated, and the time until the next burst was shortened in a manner predicted by movement normal to the limit cycle. By movement normal to the limit cycle, we mean a switch between two well-defined solution branches of a relaxation-like oscillator in a hysteretic manner enabled by the existence of a singular dominant slow process (variable). In the second component, the onset of the burst was delayed until the end of the hyperpolarizing pulse. Thus, for the pulse amplitudes we studied, resetting was independent of amplitude but increased linearly with pulse duration. The predicted and the experimental phase resetting curves for a pyloric dilator neuron show satisfactory agreement. The method was applied to only one pulse per cycle, but our results suggest it could easily be generalized to accommodate multiple inputs.

The functional consequences of changes in the strength and duration of synaptic inputs to oscillatory neurons

We studied the effect of synaptic inputs of different amplitude and duration on neural oscillators by simulating synaptic conductance pulses in a bursting conductance-based pacemaker model and by injecting artificial synaptic conductance pulses into pyloric pacemaker neurons of the lobster stomatogastric ganglion using the dynamic clamp. In the model and the biological neuron, the change in burst period caused by inhibitory and excitatory inputs of increasing strength saturated, such that synaptic inputs above a certain strength all had the same effect on the firing pattern of the oscillatory neuron. In contrast, increasing the duration of the synaptic conductance pulses always led to changes in the burst period, indicating that neural oscillators are sensitive to changes in the duration of synaptic input but are not sensitive to changes in the strength of synaptic inputs above a certain conductance. This saturation of the response to progressively stronger synaptic inputs occurs not only in bursting neurons but also in tonically spiking neurons. We identified inward currents at hyperpolarized potentials as the cause of the saturation in the model neuron. Our findings imply that activity-dependent or modulator-induced changes in synaptic strength are not necessarily accompanied by changes in the functional impact of a synapse on the timing of postsynaptic spikes or bursts.

The functional consequences of changes in the strength and duration of synaptic inputs to oscillatory neurons

We studied the effect of synaptic inputs of different amplitude and duration on neural oscillators by simulating synaptic conductance pulses in a bursting conductance-based pacemaker model and by injecting artificial synaptic conductance pulses into pyloric pacemaker neurons of the lobster stomatogastric ganglion using the dynamic clamp. In the model and the biological neuron, the change in burst period caused by inhibitory and excitatory inputs of increasing strength saturated, such that synaptic inputs above a certain strength all had the same effect on the firing pattern of the oscillatory neuron. In contrast, increasing the duration of the synaptic conductance pulses always led to changes in the burst period, indicating that neural oscillators are sensitive to changes in the duration of synaptic input but are not sensitive to changes in the strength of synaptic inputs above a certain conductance. This saturation of the response to progressively stronger synaptic inputs occurs not only in bursting neurons but also in tonically spiking neurons. We identified inward currents at hyperpolarized potentials as the cause of the saturation in the model neuron. Our findings imply that activity-dependent or modulator-induced changes in synaptic strength are not necessarily accompanied by changes in the functional impact of a synapse on the timing of postsynaptic spikes or bursts.

A mathematical model of the electrophysiological alterations in rat ventricular myocytes in type-I diabetes

Our mathematical model of the rat ventricular myocyte (Pandit et al., 2001) was utilized to explore the ionic mechanism(s) that underlie the altered electrophysiological characteristics associated with the short-term model of streptozotocin-induced, type-I diabetes. The simulations show that the observed reductions in the Ca(2+)-independent transient outward K(+) current (I(t)) and the steady-state outward K(+) current (I(ss)), along with slowed inactivation of the L-type Ca(2+) current (I(CaL)), can result in the prolongation of the action potential duration, a well-known experimental finding. In addition, the model demonstrates that the slowed reactivation kinetics of I(t) in diabetic myocytes can account for the more pronounced rate-dependent action potential duration prolongation in diabetes, and that a decrease in the electrogenic Na(+)-K(+) pump current (I(NaK)) results in a small depolarization in the resting membrane potential (V(rest)). This depolarization reduces the availability of the Na(+) channels (I(Na)), thereby resulting in a slower upstroke (dV/dt(max)) of the diabetic action potential. Additional simulations suggest that a reduction in the magnitude of I(CaL), in combination with impaired sarcoplasmic reticulum uptake can lead to a decreased sarcoplasmic reticulum Ca(2+) load. These factors contribute to characteristic abnormal [Ca(2+)](i) homeostasis (reduced peak systolic value and rate of decay) in myocytes from diabetic animals. In combination, these simulation results provide novel information and integrative insights concerning plausible ionic mechanisms for the observed changes in cardiac repolarization and excitation-contraction coupling in rat ventricular myocytes in the setting of streptozotocin-induced, type-I diabetes.

A mathematical model of action potential heterogeneity in adult rat left ventricular myocytes

Mathematical models were developed to reconstruct the action potentials (AP) recorded in epicardial and endocardial myocytes isolated from the adult rat left ventricle. The main goal was to obtain additional insight into the ionic mechanisms responsible for the transmural AP heterogeneity. The simulation results support the hypothesis that the smaller density and the slower reactivation kinetics of the Ca(2+)-independent transient outward K(+) current (I(t)) in the endocardial myocytes can account for the longer action potential duration (APD), and more prominent rate dependence in that cell type. The larger density of the Na(+) current (I(Na)) in the endocardial myocytes results in a faster upstroke (dV/dt(max)). This, in addition to the smaller magnitude of I(t), is responsible for the larger peak overshoot of the simulated endocardial AP. The prolonged APD in the endocardial cell also leads to an enhanced amplitude of the sustained K(+) current (I(ss)), and a larger influx of Ca(2+) ions via the L-type Ca(2+) current (I(CaL)). The latter results in an increased sarcoplasmic reticulum (SR) load, which is mainly responsible for the higher peak systolic value of the Ca(2+) transient [Ca(2+)](i), and the resultant increase in the Na(+)-Ca(2+) exchanger (I(NaCa)) activity, associated with the simulated endocardial AP. In combination, these calculations provide novel, quantitative insights into the repolarization process and its naturally occurring transmural variations in the rat left ventricle.

Models of respiratory rhythm generation in the pre-Bötzinger complex. I. Bursting pacemaker neurons

A network of oscillatory bursting neurons with excitatory coupling is hypothesized to define the primary kernel for respiratory rhythm generation in the pre-Bötzinger complex (pre-BötC) in mammals. Two minimal models of these neurons are proposed. In model 1, bursting arises via fast activation and slow inactivation of a persistent Na+ current INaP-h. In model 2, bursting arises via a fast-activating persistent Na+ current INaP and slow activation of a K+ current IKS. In both models, action potentials are generated via fast Na+ and K+ currents. The two models have few differences in parameters to facilitate a rigorous comparison of the two different burst-generating mechanisms. Both models are consistent with many of the dynamic features of electrophysiological recordings from pre-BötC oscillatory bursting neurons in vitro, including voltage-dependent activity modes (silence, bursting, and beating), a voltage-dependent burst frequency that can vary from 0.05 to >1 Hz, and a decaying spike frequency during bursting. These results are robust and persist across a wide range of parameter values for both models. However, the dynamics of model 1 are more consistent with experimental data in that the burst duration decreases as the baseline membrane potential is depolarized and the model has a relatively flat membrane potential trajectory during the interburst interval. We propose several experimental tests to demonstrate the validity of either model and to differentiate between the two mechanisms.

Evidence for a novel bursting mechanism in rodent trigeminal neurons

We investigated bursting behavior in rodent trigeminal neurons. The essential mechanisms operating in the biological systems were determined based on testable predictions of mathematical models. Bursting activity in trigeminal motoneurons is consistent with a traditional mechanism employing a region of negative slope resistance in the steady-state current-voltage relationship (Smith, T. G. 1975. Nature. 253:450-452). However, the bursting dynamics of trigeminal interneurons is inconsistent with the traditional mechanisms, and is far more effectively explained by a new model of bursting that exploits the unique stability properties associated with spike threshold (Baer, S. M., T. Erneux, and J. Rinzel. 1989. SIAM J. Appl. Math. 49:55-71).