Dominant ionic mechanisms explored in spiking and bursting using local low-dimensional reductions of a biophysically realistic model neuron

The large number of variables involved in many biophysical models can conceal potentially simple dynamical mechanisms governing the properties of its solutions and the transitions between them as parameters are varied. To address this issue, we extend a novel model reduction method, based on “scales of dominance,” to multi-compartment models. We use this method to systematically reduce the dimension of a two-compartment conductance-based model of a crustacean pyloric dilator (PD) neuron that exhibits distinct modes of oscillation—tonic spiking, intermediate bursting and strong bursting. We divide trajectories into intervals dominated by a smaller number of variables, resulting in a locally reduced hybrid model whose dimension varies between two and six in different temporal regimes. The reduced model exhibits the same modes of oscillation as the 16 dimensional model over a comparable parameter range, and requires fewer ad hoc simplifications than a more traditional reduction to a single, globally valid model. The hybrid model highlights low-dimensional organizing structure in the dynamics of the PD neuron, and the dependence of its oscillations on parameters such as the maximal conductances of calcium currents. Our technique could be used to build hybrid low-dimensional models from any large multi-compartment conductance-based model in order to analyze the interactions between different modes of activity.     

Mechanisms of gamma oscillations in the hippocampus of the behaving rat

Gamma frequency oscillations (30-100 Hz) have been suggested to underlie various cognitive and motor functions. Here, we examine the generation of gamma oscillation currents in the hippocampus, using two-dimensional, 96-site silicon probes. Two gamma generators were identified, one in the dentate gyrus and another in the CA3-CA1 regions. The coupling strength between the two oscillators varied during both theta and nontheta states. Both pyramidal cells and interneurons were phase-locked to gamma waves. Anatomical connectivity, rather than physical distance, determined the coupling strength of the oscillating neurons. CA3 pyramidal neurons discharged CA3 and CA1 interneurons at latencies indicative of monosynaptic connections. Intrahippocampal gamma oscillation emerges in the CA3 recurrent system, which entrains the CA1 region via its interneurons.     

Molecular dynamics simulation of fluid properties by the streamwise oscillation of the solid wall

In this paper, the hydrodynamics of streamwise wall oscillations on a Couette flow are studied using the molecular dynamics method. Firstly, based on the two-dimensional Couette flow model which is made up of copper wall and argon fluid, the characters of the fluid near the streamwise oscillation wall are simulated under the condition of varied oscillating parameters. By scrupulous data processing, some significative results such as the velocity distribution, the density distribution, the potential energy curves of the flow field and the frictional force of the wall are obtained. Secondly, the mechanism how the wall oscillation brings about change to the frictional drag at liquid–solid interface is investigated. And the results indicate that the frictional drag can be reduced significantly by applying appropriate streamwise oscillation to the solid wall. The drag reduction rate mainly depends on the oscillation parameters. In addition, the decrease in the fluid’s density near the wall is another important reason behind the frictional drag reduction.     

Regulation of photosynthesis: Analysis of a model for sensitivity of delayed luminescence oscillation and the CO<Subscript>2</Subscript> fixation rate to variation of the model parameters

The oscillation behavior of delayed luminescence was addressed using the earlier proposed mathematical model [Karavaev and Kukushkin, Biofizika 38, 958 (1993)]. The oscillation frequency and damping factor were calculated by Lyapunov analysis. In calculations, each of the model parameters was varied in a broad range. The results suggest that, besides the oscillation mode observed experimentally, there may be other oscillations that are more rapidly damped. Analysis of how variation in the model parameters affects the CO₂ fixation rate has shown that CO₂ assimilation differently depends on the light absorbed by photosystems I and II.     

Homeostasis of the auto-oscillation period in a model of a cellular circadian clock

A mathematical model was numerically investigated which describes the autooscillatory temporal organization of futile cycles of the carbohydrate branch of energy metabolism. Using an optimization method we found a region in the parametric space of the model in which the circadian oscillation period was practically constant, although the major eight parameters varied in a wide range. Such homeostasis of the period is due to synergistic effects of the four feed-back mechanisms regulating activities of the key enzymes. The result obtained supports the metabolic theory of the circadian cell clock.     

Transition between two excitabilities in mesencephalic V neurons

Neurons can make different responses to identical inputs. According to the emerging frequency of repetitive firing, neurons are classified into two types: type 1 and type 2 excitability. Though in mathematical simulations, minor modifications of parameters describing ionic currents can result in transitions between these two excitabilities, empirical evidence to support these theoretical possibilities is scarce. Here we report a joint theoretical and experimental study to test the hypothesis that changes in parameters describing ionic currents cause predictable transitions between the two excitabilities in mesencephalic V (Mes V) neurons. We developed a simple mathematical model of Mes V neurons. Using bifurcation analysis and model simulation, we then predicted that changes in conductance of two low-threshold currents would result in transitions between type 1 and type 2. Finally, by applying specific channel blockers, we observed the transition between two excitabilities forecast by the mathematical model.     

A new regime for highly robust gamma oscillation with co-exist of accurate and weak synchronization in excitatory–inhibitory networks

A great number of biological experiments show that gamma oscillation occurs in many brain areas after the presentation of stimulus. The neural systems in these brain areas are highly heterogeneous. Specifically, the neurons and synapses in these neural systems are diversified; the external inputs and parameters of these neurons and synapses are heterogeneous. How the gamma oscillation generated in such highly heterogeneous networks remains a challenging problem. Aiming at this problem, a highly heterogeneous complex network model that takes account of many aspects of real neural circuits was constructed. The network model consists of excitatory neurons and fast spiking interneurons, has three types of synapses (GABA_A, AMPA, and NMDA), and has highly heterogeneous external drive currents. We found a new regime for robust gamma oscillation, i.e. the oscillation in inhibitory neurons is rather accurate but the oscillation in excitatory neurons is weak, in such highly heterogeneous neural networks. We also found that the mechanism of the oscillation is a mixture of interneuron gamma (ING) and pyramidal-interneuron gamma (PING). We explained the mixture ING and PING mechanism in a consistent-way by a compound post-synaptic current, which has a slowly rising-excitatory stage and a sharp decreasing-inhibitory stage.     

Heartbeat control in the medicinal leech: A model system for understanding the origin,coordination, and modulation of rhythmic motor patterns

We have analyzed in detail the neuronal network that generates heartbeat in the leech. Reciprocally inhibitory pairs of heart interneurons form oscillators that pace the heartbeat rhythm. Other heart interneurons coordinate these oscillators. These coordinating interneurons, along with the oscillator interneurons, form an eight-cell timing oscillator network for heartbeat. Still other interneurons, along with the oscillator interneurons, inhibit heart motor neurons, sculpting their activity into rhythmic bursts. Critical switch interneurons interface between the oscillator interneurons and the other premotor interneurons to produce two alternating coordination states of the motor neurons. The periods of the oscillator interneurons are modulated by endogenous RFamide neuropeptides. We have explored the ionic currents and graded and spike-mediated synaptic transmission that promote oscillation in the oscillator interneurons and have incorporated these data into a conductance-based computer model. This model has been of considerable predictive value and has led to new insights into how reciprocally inhibitory neurons produce oscillation. We are now in a strong position to expand this model upward, to encompass the entire heartbeat network, horizontally, to elucidate the mechanisms of FMRFamide modulation, and downward, to incorporate cellular morphology. By studying the mechanisms of motor pattern formation in the leech, using modeling studies in conjunction with parallel physiological experiments, we can contribute to a deeper understanding of how rhythmic motor acts are generated, coordinated, modulated, and reconfigured at the level of networks, cells, ionic currents, and synapses. © 1995 John Wiley & Sons, Inc.     

A Deterministic Model Predicts the Properties of Stochastic Calcium Oscillations in Airway Smooth Muscle Cells

The inositol trisphosphate receptor () is one of the most important cellular components responsible for oscillations in the cytoplasmic calcium concentration. Over the past decade, two major questions about the have arisen. Firstly, how best should the be modeled? In other words, what fundamental properties of the allow it to perform its function, and what are their quantitative properties? Secondly, although calcium oscillations are caused by the stochastic opening and closing of small numbers of , is it possible for a deterministic model to be a reliable predictor of calcium behavior? Here, we answer these two questions, using airway smooth muscle cells (ASMC) as a specific example. Firstly, we show that periodic calcium waves in ASMC, as well as the statistics of calcium puffs in other cell types, can be quantitatively reproduced by a two-state model of the , and thus the behavior of the is essentially determined by its modal structure. The structure within each mode is irrelevant for function. Secondly, we show that, although calcium waves in ASMC are generated by a stochastic mechanism, stochasticity is not essential for a qualitative prediction of how oscillation frequency depends on model parameters, and thus deterministic models demonstrate the same level of predictive capability as do stochastic models. We conclude that, firstly, calcium dynamics can be accurately modeled using simplified models, and, secondly, to obtain qualitative predictions of how oscillation frequency depends on parameters it is sufficient to use a deterministic model.     

Single generation cycles and delayed feedback cycles are not separate phenomena

We study a simple model for generation cycles, which are oscillations with a period of one or a few generation times of the species. The model is formulated in terms of a single delay-differential equation for the population density of an adult stage, with recruitment to the adult stage depending on the intensity of competition during the juvenile phase. This model is a simplified version of a group of models proposed by Gurney and Nisbet, who were the first to distinguish between single-generation cycles and delayed-feedback cycles. According to these authors, the two oscillation types are caused by different mechanisms and have periods in different intervals, which are one to two generation times for single-generation cycles and two to four generation times for delayed-feedback cycles. By abolishing the strict coupling between the maturation time and the time delay between competition and its effect on the population dynamics, we find that single-generation cycles and delayed-feedback cycles occur in the same model version, with a gradual transition between the two as the model parameters are varied over a sufficiently large range. Furthermore, cycle periods are not bounded to lie within single octaves. This implies that a clear distinction between different types of generation cycles is not possible. Cycles of all periods and even chaos can be generated by varying the parameters that determine the time during which individuals from different cohorts compete with each other. This suggests that life-cycle features in the juvenile stage and during the transition to the adult stage are important determinants of the dynamics of density limited populations.     

Sodium Dynamics Underlying Burst Firing and Putative Mechanisms for the Regulation of the Firing Pattern in Midbrain Dopamine Neurons: A Computational Approach

A physiologically based multicompartmental computational model of a midbrain dopamine (DA) neuron, calibrated using data from the literature, was developed and used to test the hypothesis that sodium dynamics drive the generation of a slow oscillation postulated to underlie NMDA-evoked bursting activity in a slice preparation. The full compartmental model was reduced to three compartments and ultimately to two variables, while retaining the biophysical interpretation of all parameters. A phase-plane analysis then suggested two mechanisms for the regulation of the firing pattern: (1) bursting activity is favored by manipulations that enhance the region of negative slope in the whole-cell IV curve and inhibited by those manipulations, such as increasing linear currents, that tend to dampen this region and (2) assuming a region of negative slope is present in the IV curve, the bias of the system can be altered, either enabling or disabling bursting. The model provides a coherent framework for interpreting the effects of glutamate, aspartate, NMDA, and GABA agonists and antagonists under current-clamp conditions, as well as the effects of NMDA and barium under voltage-clamp conditions.     

Exploring neuronal bistability at the depolarization block

Many neurons display bistability-coexistence of two firing modes such as bursting and tonic spiking or tonic spiking and silence. Bistability has been proposed to endow neurons with richer forms of information processing in general and to be involved in short-term memory in particular by allowing a brief signal to elicit long-lasting changes in firing. In this paper, we focus on bistability that allows for a choice between tonic spiking and depolarization block in a wide range of the depolarization levels. We consider the spike-producing currents in two neurons, models of which differ by the parameter values. Our dopaminergic neuron model displays bistability in a wide range of applied currents at the depolarization block. The Hodgkin-Huxley model of the squid giant axon shows no bistability. We varied parameter values for the model to analyze transitions between the two parameter sets. We show that bistability primarily characterizes the inactivation of the Na(+) current. Our study suggests a connection between the amount of the Na(+) window current and the length of the bistability range. For the dopaminergic neuron we hypothesize that bistability can be linked to a prolonged action of antipsychotic drugs.     

Low frequency electromagnetic fields and the Belousov-Zhabotinsky reaction

Low frequency magnetic fields can influence biochemical reactions and consequently physiological rhythms and oscillations. To test this for a model reaction we used the chemical Belousov-Zhabotinsky (BZ) reaction, which is one of the simplest chemical oscillators. The oscillation frequency of the reaction was tracked optically by the absorption of blue light. Field treatment was carried out at room temperature in the middle of two Helmholtz coils. After starting the reaction, for 5 min the oscillations were monitored as control measurement, then during the next 10 min monitoring was with a magnetic field switched on, followed by a period of 5 min with the field switched off. A variety of exposure conditions have been tested: the frequency was varied between 5 and 1000 Hz, the field strength was varied up to 2.7 mT, different pulse shapes were used, the influence of the exposure temperature was tested, and the influence of the optimum exposure conditions (static magnetic field and the frequency of the dynamic field) as predicted by the ion parametric resonance (IPR) model has been measured. In conclusion, in no case any statistical significant influence of the magnetic treatment on the oscillation frequency of the BZ reaction could be detected (P > .05, t-test).     

One-dimensional computer analysis of oscillatory flow in rigid tubes.

The dynamic characteristics of catheter-transducer systems using rigid tubes with compliance lumped in the transducer and oscillatory flow of fluid in rigid tubes were analyzed. A digital computer model based on one dimensional laminar oscillatory flow was developed and verified by exact solution of the Navier-Stokes Equation. Experimental results indicated that the damping ratio and resistance is much higher at higher frequencies of oscillation than predicted by the one dimensional model. An empirical correction factor was developed and incorporated into the computer model to correct the model to the experimental data. Amplitude of oscillation was found to have no effect on damping ratio so it was concluded that the increased damping ratio and resistance at higher frequencies was not due to turbulence but to two dimensional flow effects. Graphs and equations were developed to calculate damping ratio and undamped natural frequency of a catheter-transducer system from system parameters. Graphs and equations were also developed to calculate resistance and inertance for oscillatory flow in rigid tubes from system parameters and frequency of oscillation.     

Origins and suppression of oscillations in a computational model of Parkinson’s disease

Efficacy of deep brain stimulation (DBS) for motor signs of Parkinson’s disease (PD) depends in part on post-operative programming of stimulus parameters. There is a need for a systematic approach to tuning parameters based on patient physiology. We used a physiologically realistic computational model of the basal ganglia network to investigate the emergence of a 34 Hz oscillation in the PD state and its optimal suppression with DBS. Discrete time transfer functions were fit to post-stimulus time histograms (PSTHs) collected in open-loop, by simulating the pharmacological block of synaptic connections, to describe the behavior of the basal ganglia nuclei. These functions were then connected to create a mean-field model of the closed-loop system, which was analyzed to determine the origin of the emergent 34 Hz pathological oscillation. This analysis determined that the oscillation could emerge from the coupling between the globus pallidus external (GPe) and subthalamic nucleus (STN). When coupled, the two resonate with each other in the PD state but not in the healthy state. By characterizing how this oscillation is affected by subthreshold DBS pulses, we hypothesize that it is possible to predict stimulus frequencies capable of suppressing this oscillation. To characterize the response to the stimulus, we developed a new method for estimating phase response curves (PRCs) from population data. Using the population PRC we were able to predict frequencies that enhance and suppress the 34 Hz pathological oscillation. This provides a systematic approach to tuning DBS frequencies and could enable closed-loop tuning of stimulation parameters.     

Stochastic resonance induced by fluctuation in liquid membrane oscillator without input signals

We investigated numerically the dynamic behavior of the oil/water liquid membrane, which is a promising model for excitable bio-membrane. When we use noise to modulate the parameters in simulation, noise-induced coherent oscillation is observed. With the increment of the noise intensity, the coherence of noise-induced oscillation can go through a maximum, which indicating the occurrence of stochastic resonance (SR) without input signals. We compared the SR effects under the condition that noise is added to different control parameters. When noise was added to both of the parameters, a complicated SR-like phenomemon was observed. The interaction of coherent SRs induced by two independent noises is discussed. The possibly constructive role of noise in some sensory cells is discussed also.     

Oxygen, pH value, and carbon source induced changes of the mode of oscillation in synchronous continuous culture of Saccharomyces cerevisiae

Oscillations of measured process parameters occur in continuous cultures of Saccharomyces cerevisiae owing to a partial synchronization of budding. Intentional changes of the oxygen concentration, pH value, and carbon source cause effects on the period length similar to those known from variations of the dilution rate. The generation times of parent and daughter cells frequently differ in synchronous culture. To analyze the oscillation the term mode IJ of oscillation is used, which is defined as the ratio IJ of the generation times of parent and daughter cells. When the dissolved oxygen concentration was reduced to zero, the mode of oscillation changed within two periods from mode 12 to mode 11, caused by a decrease of the generation time of daughter cells and an increase of that of the parent cells. When the pH value was slowly reduced from 5.0 to 3.9, a change from mode 112 to mode 13 was observed. Mode 13, representing one parent and three daughter cell populations (the start of budding of each of the three being delayed by one period), denotes an elongated generation time of the daughter cells compared to mode 112, marked by one parent and two different daughter cell classes. When the carbon source galactose was replaced by glucose a mode change from mode 12 to mode 11 was observed. This alteration of the mode was found to be dependent on the status of the cell cycle at the time when the carbon source is changed. The population distribution in batch cultures with glucose or galactose as a substrate was analysed by dyeing the DNA and counting the bud scars. Galactose provoked higher growth rates for the older cells. According to the model for stationary synchronous growth parameters like DO, pH value or the type of carbon source can be varied within a certain range without effecting the period length. If the variation imposes a certain stress, the culture switches to a new mode. These kinds of parameters therefore provide selective measures to influence the period lengths and the modes of oscillation.     

A qualitative mathematical model of the ontogeny of a circadian rhythm in crayfish

Based on experimental work on the ontogeny of the electroretinogram circadian rhythm in crayfish, we present a mathematical model simulating changes in both frequency and amplitude of the electroretinogram oscillation during several developmental stages until shortly before the adult age. Simultaneously, we propose a hypothetical oscillation in the hormonal release whose frequency is imposed on the electroretinogram oscillation. The model consists of two coupled nonlinear oscillators in which a dynamical response is obtained mainly through an Andronov-Hopf bifurcation. Through the construction of the model, a biological hypothesis about the essential elements underlying the ERG circadian rhythm and their interrelations is formulated and discussed.     

Modeling a Neural Oscillator that Paces Heartbeat in the Medicinal Leech

SYNOPSIS. Heartbeat in the medicinal leech is paced by a neuraloscillator comprising two elemental oscillators whose activityis coordinated intersegmental coordinating fibers. The elementaloscillators each consist of a bilateral pair of heart interneuronslinked by reciprocal inhibitory synapses. The activity cycleof each elemental oscillator consists of alternating burstsof action potentials (plateau/burst phase) and periods inhibition(inactive phase). Oscillation ensues in the reciprocally inhibitorypairs because each neuron is able to escape from the inhibitionits contralateral partner and thus move on to the plateau/burstphase. We have identified and described membrane currents thatcontribute to oscillation and studied graded synaptic transmissionbetween the neurons, using discontinuous current clamp and switchingsingle electrode voltage clamp techniques. A hyperpolarization-activatedinward current, Ih, plays a major role in escape from inhibition,and Ca²⁺ currents produce plateau potentials that support burstformation and mediate graded synaptic transmission. To consolidate our knowledge and guide future research, we haveconstructed a first generation computer model of a neural oscillatorbased on reciprocal inhibition, using Hodgkin-Huxley equationsand a synaptic transfer model, derived from our biophysicalstudies, with Nodus software (De Schutter, 1989). This modelhas confirmed an important role for I_h in sustaining oscillationand has implicated a similarly important role for outward currents(particularly I_A), which remain to be studied. Neural oscillatorsbased on reciprocal inhibition appear to be ubiquitous, andour studies, biophysical and computational, provide insightsinto how they may operate.