Hysteresis phenomena between periodic and stationary solutions in a model of pacemaker and nonpacemaker coupled cardiac cells

We were interested in investigating the behaviour of a cardiac electrophysiological model including coupled pacemaker (PM) and nonpacemaker (NPM) cells. To this aim, a modified version of the model of Van Capelle and Durrer was used. First, few discrete values were assigned to coupling resistance (CR) and respective cell sizes and numerical simulations versus time showed three possible kinds of response pattern: sustained rhythmic activity, subthreshold oscillations, and complete inhibition. Then, after setting a fixed value to PM cell size, we undertake a thorough study of the system by using bifurcation-continuation techniques and CR was chosen as the continuation parameter. On the maximum action potential — CR plane representation, we could describe five behavioural zones: complete inhibition, coexistence of complete inhibition and NPM large oscillations, NPM large oscillations, coexistence of NPM large oscillations and subthreshold oscillations, subthreshold oscillations. Within the zones of qualitatively different coexisting solutions, a detailed exploration clearly demonstrated the presence of hysteresis cycles. Indeed, the status of the system depended on its immediate previous story within narrow ranges of CR values. Such a coexistence of stable solutions for identical values of CR may suggest an explanation of the intermittant activity elicited from abnormal ectopic foci observed in certain ventricular rhythm disturbances. In addition, a Hopf bifurcation point, from which emerged stationary and periodic solutions, was followed on the PM cell size — CR plane and from this representation we could deduce that the smaller the PM cell, the higher the CR must be for the PM cell to escape from the NPM cell inhibition.     

Dynamics of a physiologically structured population in a time-varying environment

Physiologically structured population models have become a valuable tool to model the dynamics of populations. In a stationary environment such models can exhibit equilibrium solutions as well as periodic solutions. However, for many organisms the environment is not stationary, but varies more or less regularly. In order to understand the interaction between an external environmental forcing and the internal dynamics in a population, we examine the response of a physiologically structured population model to a periodic variation in the food resource. We explore the addition of forcing in two cases: (A) where the population dynamics is in equilibrium in a stationary environment, and (B) where the population dynamics exhibits a periodic solution in a stationary environment. When forcing is applied in case A, the solutions are mainly periodic. In case B the forcing signal interacts with the oscillations of the unforced system, and both periodic and irregular (quasi-periodic or chaotic) solutions occur. In both cases the periodic solutions include one and multiple period cycles, and each cycle can have several reproduction pulses.     

A conceptual model of the dynamics of the Syamozero lake community

We have developed a conceptual mathematical model of the dynamics of a spatially heterogeneous population system, the prototype of which is the Syamozero lake fish community. Based on the analysis of solutions of the model, we show that interrelations between prey and predator populations in two neighboring habitats (pelagic and offshore zones) can lead to both undamped oscillations and stationary values of the population size. The population density was found to be close to the values oblained in the course of long-term observations of the biota of the Syamozero lake. Besides, we showed that the transition to the stationary states can be accompanied by long-term (dozens and hundreds of years) damped oscillations of the prey and predator population size. In natural waters, long-term transition periods can prevent the attainment of stationary regimes of fish community functioning.     

Control Analysis for Autonomously Oscillating Biochemical Networks

It has hitherto not been possible to analyze the control of oscillatory dynamic cellular processes in other than qualitative ways. The control coefficients, used in metabolic control analyses of steady states, cannot be applied directly to dynamic systems. We here illustrate a way out of this limitation that uses Fourier transforms to convert the time domain into the stationary frequency domain, and then analyses the control of limit cycle oscillations. In addition to the already known summation theorems for frequency and amplitude, we reveal summation theorems that apply to the control of average value, waveform, and phase differences of the oscillations. The approach is made fully operational in an analysis of yeast glycolytic oscillations. It follows an experimental approach, sampling from the model output and using discrete Fourier transforms of this data set. It quantifies the control of various aspects of the oscillations by the external glucose concentration and by various internal molecular processes. We show that the control of various oscillatory properties is distributed over the system enzymes in ways that differ among those properties. The models that are described in this paper can be accessed on http://jjj.biochem.sun.ac.za.     

A nonlinear dynamic approach reveals a long-term stroke effect on cerebral blood flow regulation at multiple time scales

Cerebral autoregulation (CA) is an important vascular control mechanism responsible for relatively stable cerebral blood flow despite changes of systemic blood pressure (BP). Impaired CA may leave brain tissue unprotected against potentially harmful effects of BP fluctuations. It is generally accepted that CA is less effective or even inactive at frequencies >∼0.1 Hz. Without any physiological foundation, this concept is based on studies that quantified the coupling between BP and cerebral blood flow velocity (BFV) using transfer function analysis. This traditional analysis assumes stationary oscillations with constant amplitude and period, and may be unreliable or even invalid for analysis of nonstationary BP and BFV signals. In this study we propose a novel computational tool for CA assessment that is based on nonlinear dynamic theory without the assumption of stationary signals. Using this method, we studied BP and BFV recordings collected from 39 patients with chronic ischemic infarctions and 40 age-matched non-stroke subjects during baseline resting conditions. The active CA function in non-stroke subjects was associated with an advanced phase in BFV oscillations compared to BP oscillations at frequencies from ∼0.02 to 0.38 Hz. The phase shift was reduced in stroke patients even at > = 6 months after stroke, and the reduction was consistent at all tested frequencies and in both stroke and non-stroke hemispheres. These results provide strong evidence that CA may be active in a much wider frequency region than previously believed and that the altered multiscale CA in different vascular territories following stroke may have important clinical implications for post-stroke recovery. Moreover, the stroke effects on multiscale cerebral blood flow regulation could not be detected by transfer function analysis, suggesting that nonlinear approaches without the assumption of stationarity are more sensitive for the assessment of the coupling of nonstationary physiological signals.     

Metastable spiking networks in the replica-mean-field limit

Characterizing metastable neural dynamics in finite-size spiking networks remains a daunting challenge. We propose to address this challenge in the recently introduced replica-mean-field (RMF) limit. In this limit, networks are made of infinitely many replicas of the finite network of interest, but with randomized interactions across replicas. Such randomization renders certain excitatory networks fully tractable at the cost of neglecting activity correlations, but with explicit dependence on the finite size of the neural constituents. However, metastable dynamics typically unfold in networks with mixed inhibition and excitation. Here, we extend the RMF computational framework to point-process-based neural network models with exponential stochastic intensities, allowing for mixed excitation and inhibition. Within this setting, we show that metastable finite-size networks admit multistable RMF limits, which are fully characterized by stationary firing rates. Technically, these stationary rates are determined as the solutions of a set of delayed differential equations under certain regularity conditions that any physical solutions shall satisfy. We solve this original problem by combining the resolvent formalism and singular-perturbation theory. Importantly, we find that these rates specify probabilistic pseudo-equilibria which accurately capture the neural variability observed in the original finite-size network. We also discuss the emergence of metastability as a stochastic bifurcation, which can be interpreted as a static phase transition in the RMF limits. In turn, we expect to leverage the static picture of RMF limits to infer purely dynamical features of metastable finite-size networks, such as the transition rates between pseudo-equilibria.     

Modeling different regimes of bioelectrical activity in the brain in normal subjects and in "increased readiness" in a network of neuron-like elements

Desynchronous (low voltage fast activity), synchronous (high voltage slow waves) as well as convulsive brain activities were stimulated by a computer model of neuronal population. Network excitatory and inhibitory elements possessed fundamental dynamic properties of real neurones. Being independent both of the excitability of elements and of external influence efficacy, synchronous (desynchronous) network activity resulted from the increase (decrease) of the average power of "neuronal" interconnections which imitated mutual and recurrent excitation and inhibition. The inhibition efficacy being reduced as compared with excitation, synchronization of elements became intensified. As a consequence, the rhythmic activity amplitude increased and the appearance of self-sustained oscillations simulating convulsive activity was facilitated. The probable mechanism of EEG activation by virtue of the reduction of mutual and recurrent excitation and inhibition efficacy as well as the significance of inhibitory mechanism deficiency for epileptogenesis are discussed.     

Self-Excitation of Low-Frequency Oscillations in the Plasma Ring Formed by an ECR Discharge in a Narrow Coaxial Cavity

The conditions for the excitation of low-frequency oscillations in a plasma ring formed by an ECR discharge in a narrow coaxial cavity filled with argon were studied experimentally. The domain of the discharge parameters where these oscillations are stable is determined. It is supposed that the oscillations recorded are excited due to the appearance of an electrostatic wave propagating in the azimuthal direction.     

Exploring Age-Related Changes in Dynamical Non-Stationarity in Electroencephalographic Signals during Early Adolescence

Dynamics of brain signals such as electroencephalogram (EEG) can be characterized as a sequence of quasi-stable patterns. Such patterns in the brain signals can be associated with coordinated neural oscillations, which can be modeled by non-linear systems. Further, these patterns can be quantified through dynamical non-stationarity based on detection of qualitative changes in the state of the systems underlying the observed brain signals. This study explored age-related changes in dynamical non-stationarity of the brain signals recorded at rest, longitudinally with 128-channel EEG during early adolescence (10 to 13 years of age, 56 participants). Dynamical non-stationarity was analyzed based on segmentation of the time series with subsequent grouping of the segments into clusters with similar dynamics. Age-related changes in dynamical non-stationarity were described in terms of the number of stationary states and the duration of the stationary segments. We found that the EEG signal became more non-stationary with age. Specifically, the number of states increased whereas the mean duration of the stationary segment decreased with age. These two effects had global and parieto-occipital distribution, respectively, with the later effect being most dominant in the alpha (around 10 Hz) frequency band.     

Membrane Potential Oscillations Produced by Excitatory Amino Acids in Isolated Spinal Neurons of the Lamprey Lampetra fluviatilis

Membrane potential (MP) oscillations produced by excitatory amino acids (EAA) have been studied in branching neurons isolated by an enzymatic-mechanical method from the lamprey spinal cord. It was shown that (1) all studied EAA (glutamate, kainate, NMDA, aspartate, and quisqualate) evoke an ion current and a short-term reversible depolarization in studied cells; (2) EAA added to perfusion solution may produce MP oscillations, with kinetic parameters and duration of the oscillation depending on the amino acid used (the most effective are kainate and NMDA, the least effective, quisqualate); (3) oscillations can be irregular (of the type of a synaptic noise or of a long-term plateau of depolarization with action potentials—AP) or regular, with frequency of 0.5–1.5 Hz. Amplitude of both oscillation types depends on MP level, frequency is more steady for each cell and less depends on MP. In 68 out of 128 studied cells, oscillations could be evoked, which indicates that a significant part of lamprey spinal neurons have intrinsic capability for MP oscillations and probably pacemaker properties. The functional role of oscillations can be different. They can take cells out from the profound inhibition state, synchronize activity of rhythm generation neurons and/or be the base for trigger signals (AP firing) sent by locomotor neuronal circuits to trunk muscles.     

Dynamics of Sparsely Connected Networks of Excitatory and Inhibitory Spiking Neurons

The dynamics of networks of sparsely connected excitatory and inhibitory integrate-and-fire neurons are studied analytically. The analysis reveals a rich repertoire of states, including synchronous states in which neurons fire regularly; asynchronous states with stationary global activity and very irregular individual cell activity; and states in which the global activity oscillates but individual cells fire irregularly, typically at rates lower than the global oscillation frequency. The network can switch between these states, provided the external frequency, or the balance between excitation and inhibition, is varied. Two types of network oscillations are observed. In the fast oscillatory state, the network frequency is almost fully controlled by the synaptic time scale. In the slow oscillatory state, the network frequency depends mostly on the membrane time constant. Finite size effects in the asynchronous state are also discussed.     

Responses of nervous elements of the visual cortex of the chipmunk Eutamias sibiricus to moving and stationary stimuli]

The receptive field organization of cortical units has been studied in experiments with testing by moving and stationary light spots. The size of the receptive fields varied from 3 degrees to 10 degrees. Receptive fields which were tested by a stationary light spot exhibited various types of organization. Some of the neurons produced extensive excitatory on- and off-responses to stimulation by a light spot. Neuronal excitation evoked by light decreased if the stimulus was near the field boundary. Some of the neurons produced either on- or off-responses in any point of the receptive field. A small part of neurons had receptive fields with on- and off-reactions in the center, and either on- or off-responses at the peripheral zones. Most of the neurons exhibited specialization with respect to high-speed motion.     

Revisiting excitation pattern design for magnetic resonance imaging through optimisation of the signal contrast efficiency

The design of excitation signals for Magnetic Resonance Imaging (MRI) is cast as an optimal control problem. Here, we demonstrate that signals other than pulse excitations, which are ubiquitous in MRI, can provide adequate excitation, thus challenging the optimality and ubiquity of pulsed signals. A class of on-resonance piecewise continuous amplitude modulated signals is introduced. It is shown that despite the bilinear nature of the Bloch equations, the spins system response is largely analytically tractable for this class of signals, using Galerkin approximation methods. To challenge the optimality of the pulse excitation, an appropriate cost criterion, the Signal Contrast Efficiency (SCE), is developed. It is to be optimised subject to dynamics expressed by the Bloch equations. To solve the problem the Bloch equation is transferred to the excitation dependent rotating frame of reference. The numerical solutions to the problem for different tissue types show that for a short period of time, pulse excitations provide the maximum signal contrast. However, the problem should be solved for longer periods of time which may result in a different answer than a pulse. For this purpose, the approximate analytic solution which is derived based on averaging the Bloch equation in the excitation dependent rotating frame of reference will be used to find the optimal excitation pattern. The solution to the optimisation problem is potentially useful for all forms of MRI including structural and functional imaging. The objective of this paper is to show that while classically transient response of pulses have been monitored so far, the optimal excitation pattern may be the steady state response of a non-pulse excitation.     

Periodic and traveling wave solutions to Volterra-Lotka equations with diffusion

Analytic and numerical solutions to two coupled nonlinear diffusion equations are studied. They are the modified equations of Volterra and Lotka for the spatially stratified predatorprey population model. In a bounded domain with the reflecting boundary, equilibrium, stability, and transition to time-periodic solutions are analyzed. For a wide class of initial states, the solutions to the initial boundary-value problem evolve into their corresponding stable, space-homogeneous, periodic oscillations. In an unbounded domain, a family of traveling wave solutions is found for certain exponential, initial distributions in the limit as the diffusion coefficientv ₁ of the prey tends to zero. In the presence of both diffusions, the results of a numerical simulation to an initial-value problem showed the rapid formation of the Pursuit-Evasion Waves whose speed of propagation and amplitudes increase with the diffusion coefficientv ₁. Presented at the 1974 SIAM Fall Meeting.     

Deep modulation of a high-current relativistic electron beam in a system of coupled cavities

The excitation of inphase (0-type) and antiphase (π-type) electromagnetic oscillations by a relativistic electron beam in a system of identical coupled cavities is considered. It is shown that, in the case of excitation of antiphase oscillations, instability develops in a shorter system of cavities than it does when inphase oscillations are excited. In the nonlinear stage of the excitation of antiphase oscillations in a system of coupled cavities, a virtual cathode forms that breaks the initially uniform relativistic electron beam into a periodic sequence of spatially separated short bunches.     

The synaptic mechanism of direction selectivity in distal processes of starburst amacrine cells

Patch-clamp recordings revealed that distal processes of starburst amacrine cells (SACs) received largely excitatory synaptic input from the receptive field center and nearly purely inhibitory inputs from the surround during both stationary and moving light stimulations. The direct surround inhibition was mediated mainly by reciprocal GABA(A) synapses between opposing SACs, which provided leading and prolonged inhibition during centripetal stimulus motion. Simultaneous Ca(2+) imaging and current-clamp recording during apparent-motion stimulation further demonstrated the contributions of both centrifugal excitation and GABA(A/C)-receptor-mediated centripetal inhibition to the direction-selective Ca(2+) responses in SAC distal processes. Thus, by placing GABA release sites in electrotonically semi-isolated distal processes and endowing these sites with reciprocal GABA(A) synapses, SACs use a radial-symmetric center-surround receptive field structure to build a polar-asymmetric circuitry. This circuitry may integrate at least three levels of interactions--center excitation, surround inhibition, and reciprocal inhibitions that amplify the center--surround antagonism-to generate robust direction selectivity in the distal processes.     

Neurotrophin action on a rapid timescale

Mechanisms underlying the fast action of neurotrophins include intracellular Ca(2+) signaling, neuronal excitation, augmentation of synaptic excitation by modulation of N-methyl-d-aspartate receptor activity and control of synaptic inhibition through the regulation of the K(+)-Cl(-) cotransporter KCC2. The fastest action of brain-derived neurotrophic factor and neurotrophin-4/5 occurs within milliseconds, and involves activation of TrkB and the opening of the Na(+) channel Na(v)1.9. Through these rapid actions, neurotrophins shape neuronal activity, modulate synaptic transmission and produce instructive signals for the induction of long-term changes in the efficacy of synaptic transmission.     

On the Possibility of Excitation of Oscillations in a Schumann Resonator on Mars

Plasma Physics Reports - The possibility of excitation of oscillations in a Schumann resonator in the atmosphere of Mars is discussed. On Earth, the main source of energy in the resonator cavity...     

Sustained oscillations generated by mutually inhibiting neurons with adaptation

Autonomic oscillatory activities exist in almost every living thing and most of them are produced by rhythmic activities of the corresponding neural systems (locomotion, respiration, heart beat, etc.). This paper mathematically discusses sustained oscillations generated by mutual inhibition of the neurons which are represented by a continuous-variable model with a kind of fatigue or adaptation effect. If the neural network has no stable stationary state for constant input stimuli, it will generate and sustain some oscillation for any initial state and for any disturbance. Some sufficient conditions for that are given to three types of neural networks: lateral inhibition networks of linearly arrayed neurons, symmetric inhibition networks and cyclic inhibition networks. The result suggests that the adaptation of the neurons plays a very important role for the appearance of the oscillations. Some computer simulations of rhythic activities are also presented for cyclic inhibition networks consisting of a few neurons.     

A conceptual model of the dynamics of the Syamozero Lake community

We present a conceptual mathematical model of the dynamics of a spatially heterogeneous population system whose prototype is the fish community of Lake Syamozero. Analysis of the solutions of this model is used to demonstrate that interactions between the predator and prey populations in two neighboring biotopes (the pelagic and coastal zones) may result in either undamped oscillations or steady states of the population sizes. The model population densities are of the same order of magnitude as the values obtained in long-term observations of the Syamozero biota. It is also demonstrated that the transition to steady states may be accompanied by long-term (dozens or hundreds of years) damped oscillations of the prey and predator population densities. Under natural conditions, long transitional periods may prevent fish communities from reaching stationary modes.