Simulation of the bursting activity of neuron R15 in Aplysia: role of ionic currents, calcium balance, and modulatory transmitters

Pairing cellular and synaptic dynamics into building blocks of rhythmic neural circuits. A tutorial

In this study we focus on two subnetworks common in the circuitry of swim central pattern generators (CPGs) in the sea slugs, Melibe leonina and Dendronotus iris and show that they are independently capable of stably producing emergent network bursting. This observation raises the question of whether the coordination of redundant bursting mechanisms plays a role in the generation of rhythm and its regulation in the given swim CPGs. To address this question, we investigate two pairwise rhythm-generating networks and examine the properties of their fundamental components: cellular and synaptic, which are crucial for proper network assembly and its stable function. We perform a slow-fast decomposition analysis of cellular dynamics and highlight its significant bifurcations occurring in isolated and coupled neurons. A novel model for slow synapses with high filtering efficiency and temporal delay is also introduced and examined. Our findings demonstrate the existence of two modes of oscillation in bicellular rhythm-generating networks with network hysteresis: i) a half-center oscillator and ii) an excitatory-inhibitory pair. These 2-cell networks offer potential as common building blocks combined in modular organization of larger neural circuits preserving robust network hysteresis.

Voltage- and Calcium-Gated Membrane Currents Tune the Plateau Potential Properties of Multiple Neuron Types

Many neurons exhibit regular firing that is limited to the duration and intensity of depolarizing stimuli. However, some neurons exhibit all-or-nothing plateau potentials that, once elicited, can lead to prolonged activity that is independent of stimulus intensity or duration. To better understand this diversity of information processing, we compared the voltage-gated and Ca2+-gated currents of three identified neurons from hermaphroditic Aplysia californica Two of these neurons, B51 and B64, generated plateau potentials and a third neuron, B8, exhibited regular firing and was incapable of generating a plateau potential. With the exception of the Ca2+-gated potassium current (I KCa), all three neuron types expressed a similar array of outward and inward currents, but with distinct voltage-dependent properties for each neuron type. Inhibiting voltage-gated Ca2+ channels with Ni+ prolonged the plateau potential, indicating I KCa is important for plateau potential termination. In contrast, inhibiting persistent Na+ (I NaP) blocked plateau potentials, empirically and in simulations. Surprisingly, the properties and level of expression of I NaP were similar in all three neurons, indicating that the presence of I NaP does not distinguish between regular-firing neurons and neurons capable of generating plateau potentials. Rather, the key distinguishing factor is the relationship between I NaP and outward currents such as the delayed outward current (I D), and I KCa We then demonstrated a technique for predicting complex physiological properties such as plateau duration, plateau amplitude, and action potential duration as a function of parameter values, by fitting a curve in parameter space and projecting the curve beyond the tested values.SIGNIFICANCE STATEMENT Plateau potentials are intrinsic properties of neurons that are important for information processing in a wide variety of nervous systems. We examined three identified neurons in Aplysia californica with different propensities to generate a plateau potential. No single conductance was found to distinguish plateau generating neurons. Instead, plateau generation depended on the ratio between persistent Na+ current (I NaP), which favored plateaus, and outward currents such as I KCa, which facilitated plateau termination. Computational models revealed a relationship between the individual currents that predicted the features of simulated plateau potentials. These results provide a more solid understanding of the conductances that mediate plateau generation.

Activity-Dependent Fluctuations in Interstitial [K+]: Investigations Using Ion-Sensitive Microelectrodes

In the course of action potential firing, all axons and neurons release K⁺ from the intra- cellular compartment into the interstitial space to counteract the depolarizing effect of Na⁺ influx, which restores the resting membrane potential. This efflux of K⁺ from axons results in K⁺ accumulation in the interstitial space, causing depolarization of the K⁺ reversal potential (E_K), which can prevent subsequent action potentials. To ensure optimal neuronal function, the K⁺ is buffered by astrocytes, an energy-dependent process, which acts as a sink for interstitial K⁺, absorbing it at regions of high concentration and distributing it through the syncytium for release in distant regions. Pathological processes in which energy production is compromised, such as anoxia, ischemia, epilepsy and spreading depression, can lead to excessive interstitial K⁺ accumulation, disrupting sensitive trans-membrane ion gradients and attenuating neuronal activity. The changes that occur in interstitial [K⁺] resulting from both physiological and pathological processes can be monitored accurately in real time using K⁺-sensitive microelectrodes, an invaluable tool in electrophysiological studies.

Invertebrate neurons as a simple model to study the hyperexcitable state of epileptic disorders in single cells, monosynaptic connections, and polysynaptic circuits

Epilepsy is a neurological disorder characterized by a hyperexcitable state in neurons from different brain regions. Much is unknown about epilepsy and seizures development, depicting a growing field of research. Animal models have provided important clues about the underlying mechanisms of seizure-generating neuronal circuits. Mammalian complexity still makes it difficult to define some principles of nervous system function, and non-mammalian models have played pivotal roles depending on the research question at hand. Mollusks and the Helix land snail have been used to study epileptic-like behavior in neurons. Neurons from these organisms confer advantages as single-cell identification, isolation, and culture, either as single cells or as physiological relevant monosynaptic or polysynaptic circuits, together with amenability to different protocols and treatments. This review's purpose consists in presenting relevant papers in order to gain a better understanding of Helix neurons, their characteristics, uses, and capabilities for studying the fundamental mechanisms of epileptic disorders and their treatment, to facilitate their more expansive use in epilepsy research.

Organelle calcium-derived voltage oscillations in pacemaker neurons drive the motor program for food-seeking behavior in Aplysia

The expression of motivated behaviors depends on both external and internally arising neural stimuli, yet the intrinsic releasing mechanisms for such variably occurring behaviors remain elusive. In isolated nervous system preparations of Aplysia, we have found that irregularly expressed cycles of motor output underlying food-seeking behavior arise from regular membrane potential oscillations of varying magnitude in an identified pair of interneurons (B63) in the bilateral buccal ganglia. This rhythmic signal, which is specific to the B63 cells, is generated by organelle-derived intracellular calcium fluxes that activate voltage-independent plasma membrane channels. The resulting voltage oscillation spreads throughout a subset of gap junction-coupled buccal network neurons and by triggering plateau potential-mediated bursts in B63, can initiate motor output driving food-seeking action. Thus, an atypical neuronal pacemaker mechanism, based on rhythmic intracellular calcium store release and intercellular propagation, can act as an autonomous intrinsic releaser for the occurrence of a motivated behavior.

Neuronal oscillator robustness to multiple global perturbations

Neuronal activity depends on ion channels and biophysical processes that are strongly and differentially sensitive to physical variables such as temperature and pH. Nonetheless, neuronal oscillators can be surprisingly resilient to perturbations in these variables. We study a three-neuron pacemaker ensemble that drives the pyloric rhythm of the crab, Cancer borealis. These crabs routinely experience a number of global perturbations, including changes in temperature and pH. Although pyloric oscillations are robust to such changes, for sufficiently large deviations the rhythm reversibly breaks down. As temperature increases beyond a tipping point, oscillators transition to silence. Acidic pH deviations also show tipping points, with a reliable transition first to tonic spiking, then to silence. Surprisingly, robustness to perturbations in pH only moderately affects temperature robustness. Consistent with high animal-to-animal variability in biophysical circuit parameters, tipping points in temperature and pH vary across animals. However, the ordering and discrete classes of transitions at critical points are conserved. This implies that qualitative oscillator dynamics are preserved across animals despite high quantitative parameter variability. A universal model of bursting dynamics predicts the existence of these transition types and the order in which they occur.

Learning dominant physical processes with data-driven balance models

Throughout the history of science, physics-based modeling has relied on judiciously approximating observed dynamics as a balance between a few dominant processes. However, this traditional approach is mathematically cumbersome and only applies in asymptotic regimes where there is a strict separation of scales in the physics. Here, we automate and generalize this approach to non-asymptotic regimes by introducing the idea of an equation space, in which different local balances appear as distinct subspace clusters. Unsupervised learning can then automatically identify regions where groups of terms may be neglected. We show that our data-driven balance models successfully delineate dominant balance physics in a much richer class of systems. In particular, this approach uncovers key mechanistic models in turbulence, combustion, nonlinear optics, geophysical fluids, and neuroscience.

Where to look and how to look: Combining global sensitivity analysis with fast/slow analysis to study multi-timescale oscillations

Parameterized systems of nonlinear ordinary differential equations, the type of system that is often used in mathematical models for biological systems, can be of sufficient complexity that it can take years to appreciate the full range of behaviors that can be produced. Global sensitivity analysis is one tool that has been developed for determining which parameters have the largest impact on the behavior of the model. Thus, it provides the user with a tool to know where to look in parameter space for important changes in behavior. However, it says nothing about the underlying mechanism mediating a change in behavior. For this, other tools exist. If the system dynamics occur over multiple highly-separated time scales then one useful analysis tool is fast/slow geometric analysis, also known as geometric singular perturbation analysis. This is based on bifurcation analysis of a fast or slow subsystem, and can shed light on the influence that a parameter has on structures of either subsystem, and thus on the system dynamics. Hence, once one knows where to look in parameter space for interesting behavior, this technique describes how to look at the system to extract information about how parameter changes influence the behavior of the system. In this study, we combine the two techniques in the analysis of bursting behavior in a model of insulin-secreting pancreatic β-cells, with the goal of determining the key parameters setting the period of the bursting oscillations, and understanding why they are so influential. This can be viewed as a case study for combining mathematical techniques to build on the strengths of each and thereby achieve a better understanding of what most influences the range of model behaviors and how this influence is brought about.

A unified model for two modes of bursting in GnRH neurons

Gonadotropin-releasing hormone (GnRH) neurons exhibit at least two intrinsic modes of action potential burst firing, referred to as parabolic and irregular bursting. Parabolic bursting is characterized by a slow wave in membrane potential that can underlie periodic clusters of action potentials with increased interspike interval at the beginning and at the end of each cluster. Irregular bursting is characterized by clusters of action potentials that are separated by varying durations of interburst intervals and a relatively stable baseline potential. Based on recent studies of isolated ionic currents, a stochastic Hodgkin-Huxley (HH)-like model for the GnRH neuron is developed to reproduce each mode of burst firing with an appropriate set of conductances. Model outcomes for bursting are in agreement with the experimental recordings in terms of interburst interval, interspike interval, active phase duration, and other quantitative properties specific to each mode of bursting. The model also shows similar outcomes in membrane potential to those seen experimentally when tetrodotoxin (TTX) is used to block action potentials during bursting, and when estradiol transitions cells exhibiting slow oscillations to irregular bursting mode in vitro. Based on the parameter values used to reproduce each mode of bursting, the model suggests that GnRH neurons can switch between the two through changes in the maximum conductance of certain ionic currents, notably the slow inward Ca(2+) current I s, and the Ca(2+) -activated K(+) current I KCa. Bifurcation analysis of the model shows that both modes of bursting are similar from a dynamical systems perspective despite differences in burst characteristics.

Temperature-dependent bursting pattern analysis by modified Plant model

Many electrophysiological properties of neuron including firing rates and rhythmical oscillation change in response to a temperature variation, but the mechanism underlying these correlations remains unverified. In this study, we analyzed various action potential (AP) parameters of bursting pacemaker neurons in the abdominal ganglion of Aplysia juliana to examine whether or not bursting patterns are altered in response to temperature change. Here we found that the inter-burst interval, burst duration, and number of spike during burst decreased as temperature increased. On the other hand, the numbers of bursts per minute and numbers of spikes per minute increased and then decreased, but interspike interval during burst firstly decreased and then increased. We also tested the reproducibility of temperature-dependent changes in bursting patterns and AP parameters. Finally we performed computational simulations of these phenomena by using a modified Plant model composed of equations with temperature-dependent scaling factors to mathematically clarify the temperature-dependent changes of bursting patterns in burst-firing neurons. Taken together, we found that the modified Plant model could trace the ionic mechanism underlying the temperature-dependent change in bursting pattern from experiments with bursting pacemaker neurons in the abdominal ganglia of Aplysia juliana.

Decreased response to acetylcholine during aging of aplysia neuron R15

How aging affects the communication between neurons is poorly understood. To address this question, we have studied the electrophysiological properties of identified neuron R15 of the marine mollusk Aplysia californica. R15 is a bursting neuron in the abdominal ganglia of the central nervous system and is implicated in reproduction, water balance, and heart function. Exposure to acetylcholine (ACh) causes an increase in R15 burst firing. Whole-cell recordings of R15 in the intact ganglia dissected from mature and old Aplysia showed specific changes in burst firing and properties of action potentials induced by ACh. We found that while there were no significant changes in resting membrane potential and latency in response to ACh, the burst number and burst duration is altered during aging. The action potential waveform analysis showed that unlike mature neurons, the duration of depolarization and the repolarization amplitude and duration did not change in old neurons in response to ACh. Furthermore, single neuron quantitative analysis of acetylcholine receptors (AChRs) suggested alteration of expression of specific AChRs in R15 neurons during aging. These results suggest a defect in cholinergic transmission during aging of the R15 neuron.

Age-associated bidirectional modulation of gene expression in single identified R15 neuron of Aplysia

Background

Despite the advances in our understanding of aging-associated behavioral decline, relatively little is known about how aging affects neural circuits that regulate specific behaviors, particularly the expression of genes in specific neural circuits during aging. We have addressed this by exploring a peptidergic neuron R15, an identified neuron of the marine snail Aplysia californica. R15 is implicated in reproduction and osmoregulation and responds to neurotransmitters such as acetylcholine, serotonin and glutamate and is characterized by its action potential bursts.

Results

We examined changes in gene expression in R15 neurons during aging by microarray analyses of RNAs from two different age groups, mature and old animals. Specifically we find that 1083 ESTs are differentially regulated in mature and old R15 neurons. Bioinformatics analyses of these genes have identified specific biological pathways that are up or downregulated in mature and old neurons. Comparison with human signaling networks using pathway analyses have identified three major networks [(1) cell signaling, cell morphology, and skeletal muscular system development (2) cell death and survival, cellular function maintenance and embryonic development and (3) neurological diseases, developmental and hereditary disorders] altered in old R15 neurons. Furthermore, qPCR analysis of single R15 neurons to quantify expression levels of candidate regulators involved in transcription (CREB1) and translation (S6K) showed that aging is associated with a decrease in expression of these regulators, and similar analysis in three other neurons (L7, L11 and R2) showed that gene expression change during aging could be bidirectional.

Conclusions

We find that aging is associated with bidirectional changes in gene expression. Detailed bioinformatics analyses and human homolog searches have identified specific biological processes and human-relevant signaling pathways in R15 that are affected during aging. Evaluation of gene expression changes in different neurons suggests specific transcriptomic signature of single neurons during aging.

The effects of temperature on the stability of a neuronal oscillator

The crab Cancer borealis undergoes large daily fluctuations in environmental temperature (8-24°C) and must maintain appropriate neural function in the face of this perturbation. In the pyloric circuit of the crab stomatogastric ganglion, we pharmacologically isolated the pacemaker kernel (the AB and PD neurons) and characterized its behavior in response to temperature ramps from 7°C to 31°C. For moderate temperatures, the pacemaker displayed a frequency-temperature curve statistically indistinguishable from that of the intact circuit, and like the intact circuit maintained a constant duty cycle. At high temperatures (above 23°C), a variety of different behaviors were seen: in some preparations the pacemaker increased in frequency, in some it slowed, and in many preparations the pacemaker stopped oscillating ("crashed"). Furthermore, these crashes seemed to fall into two qualitatively different classes. Additionally, the animal-to-animal variability in frequency increased at high temperatures. We used a series of Morris-Lecar mathematical models to gain insight into these phenomena. The biophysical components of the final model have temperature sensitivities similar to those found in nature, and can crash via two qualitatively different mechanisms that resemble those observed experimentally. The crash type is determined by the precise parameters of the model at the reference temperature, 11°C, which could explain why some preparations seem to crash in one way and some in another. Furthermore, even models with very similar behavior at the reference temperature diverge greatly at high temperatures, resembling the experimental observations.

Improved stimulus representation by short interspike intervals in primary auditory cortex

We analyzed the receptive field information conveyed by interspike intervals (ISIs) in the auditory cortex. In the visual system, different ISIs may both code for different visual features and convey differing amounts of stimulus information. To determine their potential role in auditory signal processing, we obtained extracellular recordings in the primary auditory cortex (AI) of the cat while presenting a dynamic moving ripple stimulus and then used the responses to construct spectrotemporal receptive fields (STRFs). For each neuron, we constructed three STRFs, one for short-ISI events (ISI < 15 ms); one for isolated, long-ISI events (ISI > 15 ms); and one including all events. To characterize stimulus encoding, we calculated the feature selectivity and event information for each of the STRFs. Short-ISI spikes were more feature selective and conveyed information more efficiently. The different ISI regimens of AI neurons did not represent different stimulus features, but short-ISI spike events did contribute over-proportionately to the full spike train STRF information. Thus short-ISIs constitute a robust representation of auditory features, and they are particularly effective at driving postsynaptic activity. This suggests that short-ISI events are especially suited to provide noise immunity and high-fidelity information transmission in AI.

Mechanism, dynamics, and biological existence of multistability in a large class of bursting neurons

Multistability, the coexistence of multiple attractors in a dynamical system, is explored in bursting nerve cells. A modeling study is performed to show that a large class of bursting systems, as defined by a shared topology when represented as dynamical systems, is inherently suited to support multistability. We derive the bifurcation structure and parametric trends leading to multistability in these systems. Evidence for the existence of multirhythmic behavior in neurons of the aquatic mollusc Aplysia californica that is consistent with our proposed mechanism is presented. Although these experimental results are preliminary, they indicate that single neurons may be capable of dynamically storing information for longer time scales than typically attributed to nonsynaptic mechanisms.

Capturing the bursting dynamics of a two-cell inhibitory network using a one-dimensional map

Out-of-phase bursting is a functionally important behavior displayed by central pattern generators and other neural circuits. Understanding this complex activity requires the knowledge of the interplay between the intrinsic cell properties and the properties of synaptic coupling between the cells. Here we describe a simple method that allows us to investigate the existence and stability of anti-phase bursting solutions in a network of two spiking neurons, each possessing a T-type calcium current and coupled by reciprocal inhibition. We derive a one-dimensional map which fully characterizes the genesis and regulation of anti-phase bursting arising from the interaction of the T-current properties with the properties of synaptic inhibition. This map is the burst length return map formed as the composition of two distinct one-dimensional maps that are each regulated by a different set of model parameters. Although each map is constructed using the properties of a single isolated model neuron, the composition of the two maps accurately captures the behavior of the full network. We analyze the parameter sensitivity of these maps to determine the influence of both the intrinsic cell properties and the synaptic properties on the burst length, and to find the conditions under which multistability of several bursting solutions is achieved. Although the derivation of the map relies on a number of simplifying assumptions, we discuss how the principle features of this dimensional reduction method could be extended to more realistic model networks.

Generation of Very Slow Neuronal Rhythms and Chaos Near the Hopf Bifurcation in Single Neuron Models

We have presented a new generation mechanism of slow spiking or repetitive discharges with extraordinarily long inter-spike intervals using the modified Hodgkin-Huxley equations (Doi and Kumagai, 2001). This generation process of slow firing is completely different from that of the well-known potassium A-current in that the steady-state current-voltage relation of the neuronal model is monotonic rather than the N-shaped one of the A-current. In this paper, we extend the previous results and show that the very slow spiking generically appears in both the three-dimensional Hodgkin-Huxley equations and the three dimensional Bonhoeffer-van der Pol (or FitzHugh-Nagumo) equations. The generation of repetitive discharges or the destabilization of the unique equilibrium point (resting potential) is a simple Hopf bifurcation. We also show that the generation of slow spiking does not depend on the stability of the Hopf bifurcation: supercritical or subcritical. The dynamics of slow spiking is investigated in detail and we demonstrate that the phenomenology of slow spiking can be categorized into two types according to the type of the corresponding bifurcation of a fast subsystem: Hopf or saddle-node bifurcation.

Hybrid integrate-and-fire model of a bursting neuron

We present a reduction of a Hodgkin-Huxley (HH)--style bursting model to a hybridized integrate-and-fire (IF) formalism based on a thorough bifurcation analysis of the neuron's dynamics. The model incorporates HH--style equations to evolve the subthreshold currents and includes IF mechanisms to characterize spike events and mediate interactions between the subthreshold and spiking currents. The hybrid IF model successfully reproduces the dynamic behavior and temporal characteristics of the full model over a wide range of activity, including bursting and tonic firing. Comparisons of timed computer simulations of the reduced model and the original model for both single neurons and moderately sized networks (n < or = 500) show that this model offers improvement in computational speed over the HH--style bursting model.

Modeling interactions between electrical activity and second-messenger cascades in Aplysia neuron R15

The biophysical properties of neuron R15 in Aplysia endow it with the ability to express multiple modes of oscillatory electrical activity, such as beating and bursting. Previous modeling studies examined the ways in which membrane conductances contribute to the electrical activity of R15 and the ways in which extrinsic modulatory inputs alter the membrane conductances by biochemical cascades and influence the electrical activity. The goals of the present study were to examine the ways in which electrical activity influences the biochemical cascades and what dynamical properties emerge from the ongoing interactions between electrical activity and these cascades. The model proposed by Butera et al. in 1995 was extended to include equations for the binding of Ca(2+) to calmodulin (CaM) and the actions of Ca(2+)/CaM on both adenylyl cyclase and phosphodiesterase. Simulations indicated that levels of cAMP oscillated during bursting and that these oscillations were approximately antiphasic to the oscillations of Ca(2+). In the presence of cAMP oscillations, brief perturbations could switch the electrical activity between bursting and beating (bistability). Compared with a constant-cAMP model, oscillations of cAMP substantially expanded the range of bistability. Moreover, the integrated electrical/biochemical model simulated some early experimental results such as activity-dependent inactivation of the anomalous rectifier. The results of the present study suggest that the endogenous activity of R15 depends, in part, on interactions between electrical activity and biochemical cascades.

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.

Impairment of a parabolic bursting rhythm by the ectopic expression of a small conductance Ca2+-activated K+ channel in Aplysia neuron R15

The electrical properties of neurons are produced by the coordinated activity of ion channels. K+ channels play a key role in shaping action potentials and in determining neural firing patterns. Small conductance Ca2+-activated K+ (SK(Ca)) channels are involved in modulating the slow component of afterhyperpolarization (AHP). Here we examine whether rat type 2 SK(Ca) (rSK2) channels can affect the shape of the action potential and the neural firing pattern, by overexpressing rat SK2 channels in Aplysia neuron R15. Our results show that rSK2 overexpression decreased the intra-burst frequency and changed the regular bursting activity of neurons to an irregular bursting or beating pattern in R15. Furthermore, the overexpression of rSK2 channels increased AHP and reduced the duration of the action potential. Thus, our results suggest that ectopic SK(Ca) channels play an important role in regulating the firing pattern and the shape of the action potential.

Is there chaos in the brain? II. Experimental evidence and related models

The search for chaotic patterns has occupied numerous investigators in neuroscience, as in many other fields of science. Their results and main conclusions are reviewed in the light of the most recent criteria that need to be satisfied since the first descriptions of the surrogate strategy. The methods used in each of these studies have almost invariably combined the analysis of experimental data with simulations using formal models, often based on modified Huxley and Hodgkin equations and/or of the Hindmarsh and Rose models of bursting neurons. Due to technical limitations, the results of these simulations have prevailed over experimental ones in studies on the nonlinear properties of large cortical networks and higher brain functions. Yet, and although a convincing proof of chaos (as defined mathematically) has only been obtained at the level of axons, of single and coupled cells, convergent results can be interpreted as compatible with the notion that signals in the brain are distributed according to chaotic patterns at all levels of its various forms of hierarchy. This chronological account of the main landmarks of nonlinear neurosciences follows an earlier publication [Faure, Korn, C. R. Acad. Sci. Paris, Ser. III 324 (2001) 773-793] that was focused on the basic concepts of nonlinear dynamics and methods of investigations which allow chaotic processes to be distinguished from stochastic ones and on the rationale for envisioning their control using external perturbations. Here we present the data and main arguments that support the existence of chaos at all levels from the simplest to the most complex forms of organization of the nervous system. We first provide a short mathematical description of the models of excitable cells and of the different modes of firing of bursting neurons (Section 1). The deterministic behavior reported in giant axons (principally squid), in pacemaker cells, in isolated or in paired neurons of Invertebrates acting as coupled oscillators is then described (Section 2). We also consider chaotic processes exhibited by coupled Vertebrate neurons and of several components of Central Pattern Generators (Section 3). It is then shown that as indicated by studies of synaptic noise, deterministic patterns of firing in presynaptic interneurons are reliably transmitted, to their postsynaptic targets, via probabilistic synapses (Section 4). This raises the more general issue of chaos as a possible neuronal code and of the emerging concept of stochastic resonance Considerations on cortical dynamics and of EEGs are divided in two parts. The first concerns the early attempts by several pioneer authors to demonstrate chaos in experimental material such as the olfactory system or in human recordings during various forms of epilepsies, and the belief in 'dynamical diseases' (Section 5). The second part explores the more recent period during which surrogate-testing, definition of unstable periodic orbits and period-doubling bifurcations have been used to establish more firmly the nonlinear features of retinal and cortical activities and to define predictors of epileptic seizures (Section 6). Finally studies of multidimensional systems have founded radical hypothesis on the role of neuronal attractors in information processing, perception and memory and two elaborate models of the internal states of the brain (i.e. 'winnerless competition' and 'chaotic itinerancy'). Their modifications during cognitive functions are given special attention due to their functional and adaptive capabilities (Section 7) and despite the difficulties that still exist in the practical use of topological profiles in a state space to identify the physical underlying correlates. The reality of 'neurochaos' and its relations with information theory are discussed in the conclusion (Section 8) where are also emphasized the similarities between the theory of chaos and that of dynamical systems. Both theories strongly challenge computationalism and suggest that new models are needed to describe how the external world is represented in the brain.

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.

Modeling stability in neuron and network function: the role of activity in homeostasis

Individual neurons display characteristic firing patterns determined by the number and kind of ion channels in their membranes. We describe experimental and computational studies that suggest that neurons use activity sensors to regulate the number and kind of ion channels and receptors in their membrane to maintain a stable pattern of activity and to compensate for ongoing processes of degradation, synthesis and insertion of ion channels and receptors. We show that similar neuronal and network outputs can be produced by a number of different combinations of ion channels and synapse strengths. This suggests that individual neurons of the same class may each have found an acceptable solution to a genetically determined pattern of activity, and that networks of neurons in different animals may produce similar output patterns by somewhat variable underlying mechanisms.

Cellular, synaptic and network effects of neuromodulation

All network dynamics emerge from the complex interaction between the intrinsic membrane properties of network neurons and their synaptic connections. Nervous systems contain numerous amines and neuropeptides that function to both modulate the strength of synaptic connections and the intrinsic properties of network neurons. Consequently network dynamics can be tuned and configured in different ways, as a function of the actions of neuromodulators. General principles of the organization of modulatory systems in nervous systems include: (a) many neurons and networks are multiply modulated, (b) there is extensive convergence and divergence in modulator action, and (c) some modulators may be released extrinsically to the modulated circuit, while others may be released by some of the circuit neurons themselves, and act intrinsically. Some of the computational consequences of these features of modulator action are discussed.

Mechanisms underlying fictive feeding in aplysia: coupling between a large neuron with plateau potentials activity and a spiking neuron

The buccal ganglia of Aplysia contain a central pattern generator (CPG) that organizes the rhythmic movements of the radula and buccal mass during feeding. Many of the cellular and synaptic elements of this CPG have been identified and characterized. However, the roles that specific cellular and synaptic properties play in generating patterns of activity are not well understood. To examine these issues, the present study developed computational models of a portion of this CPG and used simulations to investigate processes underlying the initiation of patterned activity. Simulations were done with the SNNAP software package. The simulated network contained two neurons, B31/B32 and B63. The development of the model was guided and constrained by the available current-clamp data that describe the properties of these two protraction-phase interneurons B31/B32 and B63, which are coupled via electrical and chemical synapses. Several configurations of the model were examined. In one configuration, a fast excitatory postsynaptic potential (EPSP) from B63 to B31/B32 was implemented in combination with an endogenous plateau-like potential in B31/B32. In a second configuration, the excitatory synaptic connection from B63 to B31/B32 produced both fast and slow EPSPs in B31/B32 and the plateau-like potential was removed from B31/B32. Simulations indicated that the former configuration (i.e., electrical and fast chemical coupling in combination with a plateau-like potential) gave rise to a circuit that was robust to changes in parameter values and stochastic fluctuations, that closely mimicked empirical observations, and that was extremely sensitive to inputs controlling the onset of a burst. The coupling between the two simulated neurons served to amplify exogenous depolarizations via a positive feedback loop and the subthreshold activation of the plateau-like potential. Once a burst was initiated, the circuit produced the program in an all-or-none fashion. The slow kinetics of the simulated plateau-like potential played important roles in both initiating and maintaining the burst activity. Thus the present study identified cellular and network properties that contribute to the ability of the simulated network to integrate information over an extended period before a decision is made to initiate a burst of activity and suggests that similar mechanisms may operate in the buccal ganglia in initiating feeding movements.

Global structure, robustness, and modulation of neuronal models

The electrical characteristics of many neurons are remarkably robust in the face of changing internal and external conditions. At the same time, neurons can be highly sensitive to neuromodulators. We find correlates of this dual robustness and sensitivity in a global analysis of the structure of a conductance-based model neuron. We vary the maximal conductance parameters of the model neuron and, for each set of parameters tested, characterize the activity pattern generated by the cell as silent, tonically firing, or bursting. Within the parameter space of the five maximal conductances of the model, we find directions, representing concerted changes in multiple conductances, along which the basic pattern of neural activity does not change. In other directions, relatively small concurrent changes in a few conductances can induce transitions between these activity patterns. The global structure of the conductance-space maps implies that neuromodulators that alter a sensitive set of conductances will have powerful, and possibly state-dependent, effects. Other modulators that may have no direct impact on the activity of the neuron may nevertheless change the effects of such direct modulators via this state dependence. Some of the results and predictions arising from the model studies are replicated and verified in recordings of stomatogastric ganglion neurons using the dynamic clamp.

Nonlinear behavior of sinusoidally forced pyloric pacemaker neurons

Periodic current forcing was used to investigate the intrinsic dynamics of a small group of electrically coupled neurons in the pyloric central pattern generator (CPG) of the lobster. This group contains three neurons, namely the two pyloric dilator (PD) motoneurons and the anterior burster (AB) interneuron. Intracellular current injection, using sinusoidal waveforms of varying amplitude and frequency, was applied in three configurations of the pacemaker neurons: 1) the complete pacemaker group, 2) the two PDs without the AB, and 3) the AB neuron isolated from the PDs. Depending on the frequency and amplitude of the injected current, the intact pacemaker group exhibited a wide variety of nonlinear behaviors, including synchronization to the forcing, quasiperiodicity, and complex dynamics. In contrast, a single, broad 1:1 entrainment zone characterized the response of the PD neurons when isolated from the main pacemaker neuron AB. The isolated AB responded to periodic forcing in a manner similar to the complete pacemaker group, but with wider zones of synchronization. We have built an analog electronic circuit as an implementation of a modified Hindmarsh-Rose model for simulating the membrane potential activity of pyloric neurons. We subjected this electronic model neuron to the same periodic forcing as used in the biological experiments. This four-dimensional electronic model neuron reproduced the autonomous oscillatory firing patterns of biological pyloric pacemaker neurons, and it expressed the same stationary nonlinear responses to periodic forcing as its biological counterparts. This adds to our confidence in the model. These results strongly support the idea that the intact pyloric pacemaker group acts as a uniform low-dimensional deterministic nonlinear oscillator, and the regular pyloric oscillation is the outcome of cooperative behavior of strongly coupled neurons, having different dynamical and biophysical properties when isolated.

Birhythmicity, trirhythmicity and chaos in bursting calcium oscillations

We have analyzed various types of complex calcium oscillations. The oscillations are explained with a model based on calcium-induced calcium release (CICR). In addition to the endoplasmic reticulum as the main intracellular Ca2+ store, mitochondrial and cytosolic Ca2+ binding proteins are also taken into account. This model was previously proposed for the study of the physiological role of mitochondria and the cytosolic proteins in gene rating complex Ca2+ oscillations [1]. Here, we investigated the occurrence of different types of Ca2+ oscillations obtained by the model, i.e. simple oscillations, bursting, and chaos. In a bifurcation diagram, we have shown that all these various modes of oscillatory behavior are obtained by a change of only one model parameter, which corresponds to the physiological variability of an agonist. Bursting oscillations were studied in more detail because they express birhythmicity, trirhythmicity and chaotic behavior. Two different routes to chaos are observed in the model: in addition to the usual period doubling cascade, we also show intermittency. For the characterization of the chaotic behavior, we made use of return maps and Lyapunov exponents. The potential biological role of chaos in intracellular signaling is discussed.

Studies of chemoreceptor perception in mollusks

The formation and operation of the peripheral nervous system can be observed and studied in mollusks using as an example the primitive chemoreceptor organ, the osphradium, which is connected to the visceral arch of the CNS and analyzes the physicochemical properties of water in the mantle cavity. Nerve recordings showed that the osphradium is sensitive to excess osmotic pressure, sodium chloride, and amino acids. In addition, the osphradium responds to the quality of the water in which the animal is living. The osphradium of the pond snail retains its ancient multisensory function, uniting the perception of various chemical and physical stimuli. Patch clamp recordings at fixed potential or current were used to study membrane currents in identified ganglion and receptor cells, associated with increases in the concentrations of Na+ and L-aspartate in the solution bathing the osphradium. The influx current appears to be a sodium and/or calcium current, and is not blocked by tetraethylammonium, while the efflux current is a potassium current, as has been shown for the taste cells of vertebrates.

Minimal model for intracellular calcium oscillations and electrical bursting in melanotrope cells of Xenopus laevis

A minimal model is presented to explain changes in frequency, shape, and amplitude of Ca2+ oscillations in the neuroendocrine melanotrope cell of Xenopus Laevis. It describes the cell as a plasma membrane oscillator with influx of extracellular Ca2+ via voltage-gated Ca2+ channels in the plasma membrane. The Ca2+ oscillations in the Xenopus melanotrope show specific features that cannot be explained by previous models for electrically bursting cells using one set of parameters. The model assumes a KCa-channel with slow Ca2+-dependent gating kinetics that initiates and terminates the bursts. The slow kinetics of this channel cause an activation of the Kca-channel with a phase shift relative to the intracellular Ca2+ concentration. The phase shift, together with the presence of a Na+ channel that has a lower threshold than the Ca2+ channel, generate the characteristic features of the Ca2+ oscillations in the Xenopus melanotrope cell.

Simulation of axonal excitability using a Spreadsheet template created in Microsoft Excel

The objective of this present study was to implement an established simulation protocol (A.M. Brown, A methodology for simulating biological systems using Microsoft Excel, Comp. Methods Prog. Biomed. 58 (1999) 181-90) to model axonal excitability. The simulation protocol involves the use of in-cell formulas directly typed into a spreadsheet and does not require any programming skills or use of the macro language. Once the initial spreadsheet template has been set up the simulations described in this paper can be executed with a few simple keystrokes. The model axon contained voltage-gated ion channels that were modeled using Hodgkin Huxley style kinetics. The basic properties of axonal excitability modeled were: (1) threshold of action potential firing, demonstrating that not only are the stimulus amplitude and duration critical in the generation of an action potential, but also the resting membrane potential; (2) refractoriness, the phenomenon of reduced excitability immediately following an action potential. The difference between the absolute refractory period, when no amount of stimulus will elicit an action potential, and relative refractory period, when an action potential may be generated by applying increased stimulus, was demonstrated with regard to the underlying state of the Na(+) and K(+) channels; (3) temporal summation, a process by which two sub-threshold stimuli can unite to elicit an action potential was shown to be due to conductance changes outlasting the first stimulus and summing with the second stimulus-induced conductance changes to drive the membrane potential past threshold; (4) anode break excitation, where membrane hyperpolarization was shown to produce an action potential by removing Na(+) channel inactivation that is present at resting membrane potential. The simulations described in this paper provide insights into mechanisms of axonal excitation that can be carried out by following an easily understood protocol.

A phase model of temperature-dependent mammalian cold receptors

We present a tractable stochastic phase model of the temperature sensitivity of a mammalian cold receptor. Using simple linear dependencies of the amplitude, frequency, and bias on temperature, the model reproduces the experimentally observed transitions between bursting, beating, and stochastically phase-locked firing patterns. We analyze the model in the deterministic limit and predict, using a Strutt map, the number of spikes per burst for a given temperature. The inclusion of noise produces a variable number of spikes per burst and also extends the dynamic range of the neuron, both of which are analyzed in terms of the Strutt map. Our analysis can be readily applied to other receptors that display various bursting patterns following temperature changes.

Role of the axodendritic tree in the functioning of helix bursting neurons: generation of pacemaker activity and propagation of action potentials along the axon

The purpose of this study was to determine the role of the axodendritic tree in the generation of bursting pacemaker activity in the identified Helix RPa1 neuron, which is homologous to the Aplysia R15 cell, and propagation of action potentials along the axons. In doing so, I used recording of RPa1 neuron electrical activity after cutting off the right or left parietovisceral connections, the two-electrode voltage-clamp technique, registration of electrical activity of visceral nerve containing RPa1 axon branches, isolation of the RPa1 neuron and puff application of oxytocin on it. Cutting of the right (but not left) parietovisceral connection in all cases (more than 15 preparations) evoked complete disappearance of bursting pacemaker activity in the RPa1 neuron and hyperpolarization of its membrane potential up to -65 to -67 mV. Such silent state of the RPa1 neuron was maintained after its complete isolation from the ganglion. The described cutting did not result in a change of bursting activity of the pacemaker neuron V7 located in the visceral ganglion, although isolation of the V7 neuron also eliminated its own activity. Puff application of oxytocin (10 microM in a micropipette) on to the RPa1 neuron both after cutting the right parietovisceral connection or isolation of the neuron from the ganglion resulted in all cases (more than 10 cells) in transient depolarization with development of beating, oscillatory or bursting activity. Voltage clamping of RPa1 soma in the intact ganglion at a level close to zero membrane current sometimes, and, as a rule, at a more depolarized level, revealed bursting-like oscillations of membrane current, reflecting electrical bursting activity in the unclamped remote region of a neuron, most likely in the dendritic tree. Voltage clamping of RPa1 soma possessing bursting activity reveals bursting-like oscillation of membrane current and prevents propagation of corresponding axon action potentials in the visceral nerve. Controversially, clamping of RPa1 soma possessing beating activity exhibits a beating-like oscillation of membrane current and does not prevent propagation of corresponding action potentials in the nerve. Within the framework of the developed hypothesis that persistent bursting pacemaker activity of the RPa1 neuron is due to a constant activation of its peptidergic synaptic inputs [Kononenko N. I. (1993) Comp. Biochem. Physiol. 106A, 135-147], the experimental results were interpreted in the manner that these synapses and, correspondingly, the locus of electrical bursting activity generation, are localized on the dendritic tree of the RPa1 neuron mainly or possibly exclusively in the visceral ganglion. It is hypothesized that bursting and beating neuronal activities are due to functioning of different loci of the dendritic tree, regarding their electrical relations with axon branches.

Computational model of the serotonergic modulation of sensory neurons in Aplysia

Serotonergic modulation of the sensory neurons that mediate the gill- and tail-withdrawal reflexes of Aplysia is a useful model system for studies of neuronal plasticity that contributes to learning and memory. The effects of serotonin (5-HT) are mediated, in part, via two protein kinases (protein kinase A, PKA, and protein kinase C, PKC), which in turn, modulate at least four membrane currents, including a S ("serotonin-sensitive") K(+) current (I(K, S)), a steeply voltage-dependent K(+) current (I(K-V)), a slow component of the Ca(2+)-activated K(+) current (I(K,Ca-S)), and a L-type Ca(2+) current (I(Ca-L)). The present study investigated how the modulation of these currents altered the spike duration and excitability of sensory neurons and examined the relative contributions of PKA- and PKC-mediated effects to the actions of 5-HT. A Hodgkin-Huxley type model was developed that described the ionic conductances in the somata of sensory neurons. The descriptions of these currents and their modulation were based largely on voltage-clamp data from sensory neurons. Simulations were preformed with the program SNNAP (Simulator for Neural Networks and Action Potentials). The model was sufficient to replicate empirical data that describes the membrane currents, action potential waveform and excitability as well as their modulation by application of 5-HT, increased levels of adenosine cyclic monophosphate or application of active phorbol esters. In the model, modulation of I(K-V) by PKC played a dominate role in 5-HT-induced spike broadening, whereas the concurrent modulation of I(K,S) and I(K,Ca-S) by PKA primarily accounted for 5-HT-induced increases in excitability. Finally, simulations indicated that a PKC-induced increase in excitability resulted from decreases of I(K,S) and I(K,Ca-S), which was likely the indirect result of cross-talk between the PKC and PKA systems. The results provide several predictions that warrant additional experimental investigation and illustrate the importance of considering indirect as well as direct effects of modulatory agents on the modulation of membrane currents.

Calcium dynamics underlying pacemaker-like and burst firing oscillations in midbrain dopaminergic neurons: a computational study

A mathematical model of midbrain dopamine neurons has been developed to understand the mechanisms underlying two types of calcium-dependent firing patterns that these cells exhibit in vitro. The first is the regular, pacemaker-like firing exhibited in a slice preparation, and the second is a burst firing pattern sometimes exhibited in the presence of apamin. Because both types of oscillations are blocked by nifedipine, we have focused on the slow calcium dynamics underlying these firing modes. The underlying oscillations in membrane potential are best observed when action potentials are blocked by the application of TTX. This converts the regular single-spike firing mode to a slow oscillatory potential (SOP) and apamin-induced bursting to a slow square-wave oscillation. We hypothesize that the SOP results from the interplay between the L-type calcium current (I(Ca,L)) and the apamin-sensitive calcium-activated potassium current (I(K,Ca,SK)). We further hypothesize that the square-wave oscillation results from the alternating voltage activation and calcium inactivation of I(Ca,L). Our model consists of two components: a Hodgkin-Huxley-type membrane model and a fluid compartment model. A material balance on Ca(2+) is provided in the cytosolic fluid compartment, whereas calcium concentration is considered constant in the extracellular compartment. Model parameters were determined using both voltage-clamp and calcium-imaging data from the literature. In addition to modeling the SOP and square-wave oscillations in dopaminergic neurons, the model provides reasonable mimicry of the experimentally observed response of SOPs to TEA application and elongation of the plateau duration of the square-wave oscillations in response to calcium chelation.

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.

Outward currents influencing bursting dynamics in guinea pig trigeminal motoneurons

To initiate and maintain bursts (and plateau potentials) in the presence of serotonin, guinea pig trigeminal motoneurons utilize L-type Ca2+ and persistent Na+ inward currents. However, the intrinsic currents that contribute to burst termination and determine the duration of the interburst interval are unknown. Therefore we investigated the roles of outward currents, whose slow activation is coupled to cytosolic cation (Ca2+ and Na+) accumulation. First we examined a Ca2+-dependent K+ current (IK-Ca) with apamin and Ba2+-substituted, low-Ca2+ solution. Blockade of IK-Ca lengthened burst duration and cycle time but did not abolish bursting. Next we studied the Na+/K+-ATPase pump current (Ip) with cardiac glycosides. In the presence of apamin or low-Ca2+/Ba2+ solution, blocking Ip (with ouabain or strophanthidin) decreased both burst duration and cycle time and ultimately transformed bursting into tonic spiking. We conclude that IK-Ca and Ip contribute to burst termination in trigeminal motoneurons. These currents influence temporal bursting properties such as burst duration and cycle time and may help determine the phasic activity of motoneurons during rhythmic oral-motor behaviors.

A methodology for simulating biological systems using Microsoft Excel

The objective of this present study was to develop a simple, easily understood methodology for solving biologically based models using a Microsoft Excel spreadsheet. The method involves the use of in-cell formulas in which Rows and Columns of new data are generated from data typed into the spreadsheet, but does not require any programming skills or use of the macro language. The approach involves entering the key parameter values into the spreadsheet and conducting the simulation by solving a set of equations based on these parameter values. The examples used in this paper are firstly, a simple voltage clamp simulation in which initial parameter values are used to calculate a system in steady state. The second example is a current clamp simulation where steady state is not reached and the solution of the equations for each time increment is used as the input for the next time increment in the simulation. The calculations are based on the Hodgkin Huxley mathematical equations that describe the voltage dependence of ion channel behavior. The problems and flexibility of the method are briefly discussed. The methodology developed in this present study should help novice modelers to create simple simulations without the need to learn a programming language or purchase expensive software.

Pharmacological tests of the mechanism of the periodic rhythm caused by veratramine in the sinoatrial node of the guinea pig

1. We investigated the effects of several drugs and extracellular ions on the periodic sinoatrial node rhythm caused by high concentrations of veratramine (>2 microM) in isolated guinea pig sinus atria. 2. During the active phase of this rhythm, pacemaker activity appeared to be due to transient afterdepolarizations resembling the delayed afterdepolarizations attributed to Ca++-induced Ca++ release in cardiac tissue. 3. Ryanodine (200-2200 nM) did not decrease the transient afterdepolarizations, and instead increased the heart rate during the active phase, prolonged the active phase, and sometimes caused conversion to regular rhythm. 4. Dichlorobenzamyl (10-110 microM), a blocker of electrogenic Na+-Ca++ exchange, did not slow or stop beating during periodic rhythm, but rather increased average heart rate and, at a higher concentration, caused conversion to regular rhythm. 5. Ouabain (0.1 microM), an inhibitor of the sodium pump and electrogenic Na+-K+ exchange, had little effect on veratramine periodic rhythm, but at higher concentrations it caused increased average heart rate and conversion to regular rhythm. 6. The chronotropic effect of Ca++ was normally weakly positive; however, in the presence of veratramine, and before the appearance of periodic rhythm, the chronotropic effect of Ca++ was weakly negative, and was associated with destabilization of the heart rate, leading to frequency oscillations or periodic rhythm. 7. Veratramine changed the chronotropic effect of K+ from weakly negative to moderately positive. 8. When half the Na+ or Cl- in the bathing medium was replaced by an impermeant ion, in the absence of veratramine the average heart rate was slightly decreased, whereas, in the presence of veratramine and periodic rhythm the average rate was increased, although the increase was not statistically significant in the case of low Na+. 9. These observations indicate that Ca++-induced Ca++ release, Na+-Ca++ exchange, and probably electrogenic Na+-K+ exchange play no important role in generation of periodic rhythm. The increased K+ dependence suggests an altered pacemaker mechanism.

Bursting calcium rotors in cultured cardiac myocyte monolayers

Rotating waves (rotors) of cellular activity were observed in nonconfluent cultures of embryonic chick heart cells by using a macroscopic imaging system that detected fluorescence from intracellular Ca2+. Unlike previous observations of rotors or spiral waves in other systems, the rotors did not persist but exhibited a repetitive pattern of spontaneous onset and offset leading to a bursting rhythm. Similar dynamics were observed in a cellular automaton model of excitable media that incorporates spontaneous initiation of activity, and a decrease of excitability as a consequence of rapid activity (fatigue). These results provide a mechanism for bursting dynamics in normal and pathological biological processes.

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).

Multirhythmic bursting

A complex modeled bursting neuron [C. C. Canavier, J. W. Clark, and J. H. Byrne, J. Neurophysiol. 66, 2107-2124 (1991)] has been shown to possess seven coexisting limit cycle solutions at a given parameter set [Canavier et al., J. Neurophysiol 69, 2252-2259 (1993); 72, 872-882 (1994)]. These solutions are unique in that the limit cycles are concentric in the space of the slow variables. We examine the origin of these solutions using a minimal 4-variable bursting cell model. Poincare maps are constructed using a saddle-node bifurcation of a fast subsystem such as our Poincare section. This bifurcation defines a threshold between the active and silent phases of the burst cycle in the space of the slow variables. The maps identify parameter spaces with single limit cycles, multiple limit cycles, and two types of chaotic bursting. To investigate the dynamical features which underlie the unique shape of the maps, the maps are further decomposed into two submaps which describe the solution trajectories during the active and silent phases of a single burst. From these findings we postulate several necessary criteria for a bursting model to possess multiple stable concentric limit cycles. These criteria are demonstrated in a generalized 3-variable model. Finally, using a less direct numerical procedure, similar return maps are calculated for the original complex model [C. C. Canavier, J. W. Clark, and J. H. Byrne, J. Neurophysiol. 66, 2107-2124 (1991)], with the resulting mappings appearing qualitatively similar to those of our 4-variable model. These multistable concentric bursting solutions cannot occur in a bursting model with one slow variable. This type of multistability arises when a bursting system has two or more slow variables and is viewed as an essentially second-order system which receives discrete perturbations in a state-dependent manner. (c) 1998 American Institute of Physics.

Dynamics and kinematics of simple neural systems

The dynamics of simple neural systems is of interest to both biologists and physicists. One of the possible roles of such systems is the production of rhythmic patterns, and their alterations (modification of behavior, processing of sensory information, adaptation, control). In this paper, the neural systems are considered as a subject of modeling by the dynamical systems approach. In particular, we analyze how a stable, ordinary behavior of a small neural system can be described by simple finite automata models, and how more complicated dynamical systems modeling can be used. The approach is illustrated by biological and numerical examples: experiments with and numerical simulations of the stomatogastric central pattern generators network of the California spiny lobster. (c) 1996 American Institute of Physics.

Dissection and reduction of a modeled bursting neuron

An 11-variable Hodgkin-Huxley type model of a bursting neuron was investigated using numerical bifurcation analysis and computer simulations. The results were applied to develop a reduced model of the underlying subthreshold oscillations (slow-wave) in membrane potential. Two different low-order models were developed: one 3-variable model, which mimicked the slow-wave of the full model in the absence of action potentials and a second 4-variable model, which included expressions accounting for the perturbational effects of action potentials on the slow-wave. The 4-variable model predicted more accurately the activity mode (bursting, beating, or silence) in response to application of extrinsic stimulus current or modulatory agents. The 4-variable model also possessed a phase-response curve that was very similar to that of the original 11-variable model. The results suggest that low-order models of bursting cells that do not consider the effects of action potentials may erroneously predict modes of activity and transient responses of the full model on which the reductions are based. These results also show that it is possible to develop low-order models that retain many of the characteristics of the activity of the higher-order system.

Modeling slowly bursting neurons via calcium store and voltage-independent calcium current

Recent experiments indicate that the calcium store (e.g., endoplasmic reticulum) is involved in electrical bursting and [Ca2+]i oscillation in bursting neuronal cells. In this paper, we formulate a mathematical model for bursting neurons, which includes Ca2+ in the intracellular Ca2+ stores and a voltage-independent calcium channel (VICC). This VICC is activated by a depletion of Ca2+ concentration in the store, [Ca2+]cs. In this model, [Ca2+]cs oscillates slowly, and this slow dynamic in turn gives rise to electrical bursting. The newly formulated model thus is radically different from existing models of bursting excitable cells, whose mechanism owes its origin to the ion channels in the plasma membrane and the [Ca2+]i dynamics. In addition, this model is capable of providing answers to some puzzling phenomena, which the previous models could not (e.g., why cAMP, glucagon, and caffeine have ability to change the burst periodicity). Using mag-fura-2 fluorescent dyes, it would be interesting to verify the prediction of the model that (1) [Ca2+]cs oscillates in bursting neurons such as Aplysia neuron and (2) the neurotransmitters and hormones that affect the adenylate cyclase pathway can influence this oscillation.