Coarse-grained reduction and analysis of a network model of cortical response: I. Drifting grating stimuli

We present a reduction of a large-scale network model of visual cortex developed by McLaughlin, Shapley, Shelley, and Wielaard. The reduction is from many integrate-and-fire neurons to a spatially coarse-grained system for firing rates of neuronal subpopulations. It accounts explicitly for spatially varying architecture, ordered cortical maps (such as orientation preference) that vary regularly across the cortical layer, and disordered cortical maps (such as spatial phase preference or stochastic input conductances) that may vary widely from cortical neuron to cortical neuron. The result of the reduction is a set of nonlinear spatiotemporal integral equations for phase-averaged firing rates of neuronal subpopulations across the model cortex, derived asymptotically from the full model without the addition of any extra phenomological constants. This reduced system is used to study the response of the model to drifting grating stimuli—where it is shown to be useful for numerical investigations that reproduce, at far less computational cost, the salient features of the point-neuron network and for analytical investigations that unveil cortical mechanisms behind the responses observed in the simulations of the large-scale computational model. For example, the reduced equations clearly show (1) phase averaging as the source of the time-invariance of cortico-cortical conductances, (2) the mechanisms in the model for higher firing rates and better orientation selectivity of simple cells which are near pinwheel centers, (3) the effects of the length-scales of cortico-cortical coupling, and (4) the role of noise in improving the contrast invariance of orientation selectivity.     

Stochastically Gating Ion Channels Enable Patterned Spike Firing through Activity-Dependent Modulation of Spike Probability

The transformation of synaptic input into patterns of spike output is a fundamental operation that is determined by the particular complement of ion channels that a neuron expresses. Although it is well established that individual ion channel proteins make stochastic transitions between conducting and non-conducting states, most models of synaptic integration are deterministic, and relatively little is known about the functional consequences of interactions between stochastically gating ion channels. Here, we show that a model of stellate neurons from layer II of the medial entorhinal cortex implemented with either stochastic or deterministically gating ion channels can reproduce the resting membrane properties of stellate neurons, but only the stochastic version of the model can fully account for perithreshold membrane potential fluctuations and clustered patterns of spike output that are recorded from stellate neurons during depolarized states. We demonstrate that the stochastic model implements an example of a general mechanism for patterning of neuronal output through activity-dependent changes in the probability of spike firing. Unlike deterministic mechanisms that generate spike patterns through slow changes in the state of model parameters, this general stochastic mechanism does not require retention of information beyond the duration of a single spike and its associated afterhyperpolarization. Instead, clustered patterns of spikes emerge in the stochastic model of stellate neurons as a result of a transient increase in firing probability driven by activation of HCN channels during recovery from the spike afterhyperpolarization. Using this model, we infer conditions in which stochastic ion channel gating may influence firing patterns in vivo and predict consequences of modifications of HCN channel function for in vivo firing patterns.     

The potassium A-current,low firing rates and rebound excitation in Hodgkin-Huxley models

It is widely believed, following the work of Connor and Stevens (1971,J. Physiol. Lond. 214, 31–53) that the ability to fire action potentials over a wide frequency range, especially down to very low rates, is due to the transient, potassium A-current (I _A). Using a reduction of the classical Hodgkin-Huxley model, we study the effects ofI _A on steady firing rate, especially in the near-threshold regime for the onset of firing. A minimum firing rate of zero corresponds to a homoclinic bifurcation of periodic solutions at a critical level of stimulating current. It requires that the membrane's steady-state current-voltage relation be N-shaped rather than monotonic. For experimentally based genericI _A parameters, the model does not fire at arbitrarily low rates, although it can for the more atypicalI _A parameters given by Connor and Stevens for the crab axon. When theI _A inactivation rate is slow, we find that the transient potassium current can mediate more complex firing patterns, such as periodic bursting in some parameter regimes. The number of spikes per burst increases asg _A decreases and as inactivation rate decreases. We also study howI _A affects properties of transient voltage responses, such as threshold and firing latency for anodal break excitation. We provide mathematical explanations for several of these dynamic behaviors using bifurcation theory and averaging methods.     

Solvent Effects on the Shape of a Tetramer

Abstract

The solvent effect on the shape of a tetramer with increasing temperature is analyzed. For this purpose models of an isolated chain and a chain immersed in a solvent have been simulated by Molecular Dynamics. A solvent model represented by stochastic forces has been tested against molecular dynamics results. The behaviour of the mean-square end-to-end distance 〈R ²〉 and 〈l ₁ ³ S ²〉 with increasing temperature are shown. where l ₁ is the longest eigenvalue of the moment of inertia tensor and S is the radius of gyration. All the chain models studied show different behaviour of these quantities at low temperature compared to high temperature where the shape of the tetramer is temperature insensitive. The main solvent effect is to pospone the transition to higher temperature. The stochastic solvent model qualitatively agrees with molecular dynamics results.     

Proteolysis and flash photolysis of bacteriorhodopsin in purple membrane fragments

Pronase treatment of aqueous suspensions of purple membrane fragments from H. halobium leads to the cleavage of bacteriorhodopsin. The protein fragments remaining in the membrane after treatment with relatively small concentrations of enzyme (2% w/w) in normal daylight range in molecular weight from 20,000-21,000 daltons, indicating that cleavage occurs mainly near the extremities of the protein chain. At higher enzyme concentrations the relative amounts of protein fragments having smaller molecular weight increase. Generally, the relative loss of retinal chromophore is larger than that of protein and thus the retinal binding site seems to be located near one of the chain ends that is cleaved off by enzyme.Irradiation with white light during the time of proteolysis (at both low and high enzyme concentrations) results in extensive cleavage, so that under certain conditions no high molecular weight components can be detected in SDS-polyacrylamide gels. It, therefore, appears that parts of the bacteriorhodopsin chain become more exposed to enzyme digestion when the purple membrane is illuminated.Enzyme treated aqueous purple membrane fragment suspensions still show photocycle activity. The main consequence of proteolysis is a pronounced appearance of biphasicity in the decay of M₄₁₂ and the regeneration of bR₅₇₀. Simultaneously the yield of O₆₆₀ is reduced. As with untreated purple membrane, the correlation between the rates of decay of M₄₁₂ and regeneration of bR₅₇₀ is greatest when the yield of O₆₆₀ is lowest.     

A new method for the partition of allelic diversity within and between subpopulations

A method is proposed for the analysis of allelic diversity in the context of subdivided populations. The definition of an allelic distance between subpopulations allows for the partition of total allelic diversity into within- and between-subpopulation components, in a way analogous to the classical partition of gene diversity. A new definition of allelic differentiation, A _ST , between subpopulations results from this partition, and is contrasted with the concept of allelic richness differentiation. The partition of allelic diversity makes it possible to establish the relative contribution of each subpopulation to within and between-subpopulation components of diversity with implications in priorisation for conservation. A comparison between this partition and that corresponding to allelic richness is illustrated with an example. Computer simulations are used to investigate the behaviour of the new statistic A _ST in comparison with F _ST for a finite island model under a range of mutation and migration rates. A _ST has less dependence on migration rate than F _ST for large values of migration rate, but the opposite occurs for low migration rates. In addition, the variance in the estimates of A _ST is higher than that of F _ST for low mutation rates, but the opposite for high mutation rates.     

Pacemaking mechanism of the afterdischarge of the ovulation hormone-producing caudo-dorsal cells in the gastropod mollusc Lymnaea stagnalis

The ovulation hormone-producing caudo-dorsal cells (CDC) of Lymnaea stagnalis have three states of excitability (active, inhibited, and resting), which are related to the egg-laying cycle. Active state CDC produce a firing pattern of prolonged spiking activity (1 spike/2 s), which in the animal occurs shortly before egg laying. In preparations it is evoked as an afterdischarge upon repetitive stimulation of CDC. The afterdischarge is not synaptically driven, but rests on a pacemaking mechanism. CDC are silent in the inhibited and resting states, which follow egg laying. In these states the membrane potential is mainly dependent on [K⁺]₀. In the active state the ratio of the K⁺, Na⁺, and Ca²⁺ permeabilities has changed considerably, probably resulting from an increased permeability to Na⁺ and Ca²⁺. The firing rate in the afterdischarge is dependent on the membrane potential, which is confirmed experimentally by varying [K⁺]₀.[Na⁺]₀ and [Ca²⁺]₀ directly influence the firing rate. Firing stops in Na⁺-free saline, but is enhanced by Ca²⁺-free or high-Mg²⁺ saline. TTX does not affect firing. Relatively high concentrations of Co²⁺ and La³⁺ (2 × 10^?3M) strongly inhibit CDC. Regular firing can be changed into bursting by various means, such as high K⁺ or addition of 1 mM Ba²⁺. Bursting normally occurs at the beginning of the afterdischarge. Postburst hyperpolarizations are reduced in Ca²⁺-free saline and by low Co²⁺ (10^?4-5 10^?4M). Active CDC are driven by a pacemaking mechanism constituted by a voltage-dependent Na⁺/Ca²⁺ channel and a Ca²⁺-dependent K⁺ channel, thus resembling that of bursting pacemakers. The pacemaking mechanism is inactive in the resting and inhibited state.     

Effect doublet impulse sequences on the transient and steady responses in the computer-simulated nerve cell

The effects of doublet impulse sequences of the excitatory axon on the output response as firing probability (pr.) in the computer-simulated nerve cell were examined. A simple model was formulated to simulate the nerve cell, including the property that the resetting potential is influenced by the final membrane potential in the previous stage before firing. The relationship between input sequences with alternating long and short interval at the same mean rate and the transient and steady responses of the nerve cell was investigated. In this simulation, three summarized results were obtained: i) The responses were very sensitive to changing small size of excitatory post-synaptic potential (EPSP), especially in the firing stage of the transient state. ii) In the transient state, the size of characteristic area of responses was depending upon the size of absolute refractory period (ARP). The rise for shorter intervals was faster than that for longer intervals, agreeing well with part of the experimental results from the crayfish claw opener muscles. The transient responses were almost finished before the fifth firing. iii) In the steady state, the doublet impulse sequences usually produced the minimum response or valley-like response at which the doublet interval T _dwas 20 and/or 25 ms. These effects related to the characteristic areas in the transient responses.     

Using interspike intervals to quantify noise effects on spike trains in temperature encoding neurons

This paper examines how noise interacts with the non-linear dynamical mechanisms of neuronal stimulus. We study the spike trains generated by a minimal Hodgkin-Huxley type model of a cold receptor neuron. The distributions of interspike intervals(ISIs) of purely deterministic simulations exhibit considerable differences compared to the noisy ones. We quantify the effect of noise using ISI return plots and the ISI-distance recently proposed by Kreuz et al. (J Neurosci Meth, 165:151–161, 2007). It is shown that the spike trains of a cold receptor neuron are more strongly affected by noise for low temperatures than for high temperatures. This trend is also observed in both regimes of cold receptors: tonic firing(which occurs for low and high temperatures) and bursting (which occurs for intermediate temperatures).     

Mathematical models of threshold phenomena in the nerve membrane

The types of mathematical model which have been used to represent all-or-none behavior in the nerve membrane may be classified as follows: (1) thediscontinuous threshold phenomenon, in which differential equations with discontinuous functions provide both a discontinuity of response as a function of stimulus intensity at threshold and a finite maximum latency, (2) thesingular-point threshold phenomenon which exists in a phase space having analytic functions in its differential equations and having a singular point with one characteristic root positive and the rest with negative real parts, the latency being unbounded, and (3) thequasi threshold phenomenon, which has a finite maximum latency and continuous functions, but neither a true discontinuity in response nor an exact threshold. Several models of the nerve membrane in the literature are classified accordingly, and the applicability of the different types of threshold phenomena to the membrane is discussed, including an extension to a stochastic model.     

Dose-rate dependent stochastic effects in radiation cell-survival models

Summary When cells are subjected to ionizing radiation the specific energy rate (microscopic analog of dose-rate) varies from cell to cell. Within one cell, this rate fluctuates during the course of time; a crossing of a sensitive cellular site by a high energy charged particle produces many ionizations almost simultaneously, but during the interval between events no ionizations occur. In any cell-survival model one can incorporate the effect of such fluctuations without changing the basic biological assumptions. Using stochastic differential equations and Monte Carlo methods to take into account stochastic effects we calculated the dose-survival relationships in a number of current cell survival models. Some of the models assume quadratic misrepair; others assume saturable repair enzyme systems. It was found that a significant effect of random fluctuations is to decrease the theoretically predicted amount of dose-rate sparing. In the limit of low dose-rates neglecting the stochastic nature of specific energy rates often leads to qualitatively misleading results by overestimating the surviving fraction drastically. In the opposite limit of acute irradiation, analyzing the fluctuations in rates merely amounts to analyzing fluctuations in total specific energyvia the usual microdosimetric specific energy distribution function, and neglecting fluctuations usually underestimates the surviving fraction. The Monte Carlo methods interpolate systematically between the low dose-rate and high dose-rate limits. As in other approaches, the slope of the survival curve at low dose-rates is virtually independent of dose and equals the initial slope of the survival curve for acute radiation.     

Regulation of bacteriorhodopsin synthesis by growth rate in continuous cultures of Halobacterium halobium

The independent effects of oxygen tension and growth rate on bacteriorhodopsin synthesis in Halobacterium halobium have been studied in chemostat cultures. Bacteriorhodopsin synthesis occurs only at low growth rates and is stimulated by low oxygen tension. Fast growth rates override the stimulatory effects of oxygen tension, with the result that bacteriorhodopsin can scarcely be detected. Illumination of cultures maintained at low growth rate and low oxygen tension significantly increases the steady state cell yield. This finding suggests that under these conditions the purple membrane proton pump is coupled to energy transduction.     

Functional characteristics of traps of aquatic carnivorous Utricularia species

Firing and resetting of traps in aquatic Utricularia species are associated with water flow and trap volume changes. In this study, trap thickness was used as a measure of water flow and was monitored automatically using an electronic position sensor. The basic characteristics of mechanically stimulated firing and resetting were measured in isolated traps from 13 aquatic Utricularia species and in two trap size categories in Utricularia reflexa. This allowed to study the relationship between these trap functions and trap thickness and length (criteria of trap size). Additionally, the characteristics of spontaneous firings (without any mechanical stimulation) were compared for U. reflexa traps fed or denied prey during a 1-day period. On the absolute scale, the 13 Utricularia species differed considerably in their firing and resetting rates. Significant interspecific differences were also found in the magnitude of firing (in total 3.7–4.2 times) and resetting rates (10–24 times) per unit trap thickness or length. Overall, traps of Utricularia australis, Utricularia stellaris and Utricularia inflata showed the greatest firing and resetting rates. The relative magnitude of firing per unit trap thickness or length showed a highly significant negative correlation with both trap thickness and length and the same also held for the relative resetting rates. Smaller and narrower traps are thus relatively more effective at trap firing and resetting than larger traps. Neither firing nor resetting characteristics were significantly different between unfed and prey-fed traps of U. reflexa and this was also true for the occurrence of spontaneous firings. A strict linear resetting rate, without any lag-period, was found during the first 3 min after trap firing in U. reflexa. This suggests that water is pumped out of the trap continuously and probably recirculates. Given the concepts of spontaneous firing and water recirculation, an ecological model based on the literature data has been devised to quantify the daily N and P gain from the ambient water by Utricularia traps devoid of animal prey. The model shows that the total N and P content estimated in the trap fluid is too high to be accumulated from only the ambient water. This implies that the prey-free traps do not take up N or P from the trap fluid but rather exude a quantity of these nutrients to the fluid to support the microbial community. Therefore, the trap microorganisms behave more as parasites than commensals and represent an additional ecological cost for trap maintenance.     

Slow Oscillations of the Purkinje Cell Firing Rate Induced by Electrical Stimulation of the Locus Coeruleus in Rats

In anesthetized Wistar rats, we studied the effect of electrical stimulation of the locus coeruleus (LC) on the firing rates of Purkinje cells using spectral analysis. The frequency of extracellularly recorded activity of Purkinje cells was measured before and during the 1st, 5th, 6th, and 11th min after cessation of 10-sec-long LC stimulations. Spectral analysis of the Purkinje cell firing rates (imp./bin, the bin duration was 2^-8 sec) for 60- to 120-sec-long intervals was performed using fast Fourier transformation after digital conversion of unitary spikes. Mean power spectra of the Purkinje cell firing rates (derived from 8-sec-long consecutive epochs at a sampling rate of 256 sec^-1) showed an increase in the slow frequency range (0.1-1.0 Hz) after LC stimulation, particularly due to the slowest components (below 0.5 Hz). This effect lasted more than 1 min and usually less than 6 min after cessation of LC stimulation and could be interpreted as the development of slow oscillations in the Purkinje cell firing. Our results suggest that slow oscillations of the firing rate of cerebellar output neurons, induced by LC stimulation, reflect a specific coordination of the cerebellar neuronal activities (important for a central norepinephrine influence) in regulation of different pathological states.     

Stochastic simulations of a multi-group compartmental model for Johne's disease on US dairy herds with test-based culling intervention

Infection elimination may be an important goal of control programs. Only in stochastic infection models can true infection elimination be observed as a fadeout. The phenomena of fadeout and variable prevalence are important in understanding the transmission dynamics of infectious diseases and these phenomena are essential to evaluate the effectiveness of control measures. To investigate the stochastic dynamics of Mycobacterium avium subsp. paratuberculosis (MAP) infection on US dairy herds with test-based culling intervention, we developed a multi-group stochastic compartmental model (a continuous time Markov chain model) with both horizontal and vertical transmission. The stochastic model predicted fadeout and within-herd prevalence to have a large variance. Although test-based culling intervention generally decreased prevalence over time, it took longer than desired by producers to eliminate the endemic MAP infection from a herd. Uncertainty analysis showed that, using annual culture test and culling of only high shedders or culling of both low and high shedders with a 12-month delay in culling of low shedders, MAP infection persisted in many herds beyond 20 years. While using semi-annual culture test and culling of low and high shedders with a 6-month delay in culling of low shedders, MAP infection in many herds would be extinct within 20 years. Sensitivity analysis of the cumulative density function of fadeout suggested that combining test-based culling intervention and reduction of transmission rates through improved management between susceptible calves and shedding animals may be more effective than either alone in eliminating endemic MAP infection. We also discussed the effects of other factors such as herd size, heifer replacement, and adult cow infection on the probability of fadeout.     

Novel moment closure approximations in stochastic epidemics

Moment closure approximations are used to provide analytic approximations to non-linear stochastic population models. They often provide insights into model behaviour and help validate simulation results. However, existing closure schemes typically fail in situations where the population distribution is highly skewed or extinctions occur. In this study we address these problems by introducing novel second-and third-order moment closure approximations which we apply to the stochastic SI and SIS epidemic models. In the case of the SI model, which has a highly skewed distribution of infection, we develop a second-order approximation based on the beta-binomial distribution. In addition, a closure approximation based on mixture distribution is developed in order to capture the behaviour of the stochastic SIS model around the threshold between persistence and extinction. This mixture approximation comprises a probability distribution designed to capture the quasi-equilibrium probabilities of the system and a probability mass at 0 which represents the probability of extinction. Two third-order versions of this mixture approximation are considered in which the log-normal and the beta-binomial are used to model the quasi-equilibrium distribution. Comparison with simulation results shows: (1) the beta-binomial approximation is flexible in shape and matches the skewness predicted by simulation as shown by the stochastic SI model and (2) mixture approximations are able to predict transient and extinction behaviour as shown by the stochastic SIS model, in marked contrast with existing approaches. We also apply our mixture approximation to approximate a likehood function and carry out point and interval parameter estimation.     

Analysis of models for crustacean stretch receptors

 The mechanisms underlying the diverse responses to step current stimuli of models [Edman et al. (1987) J Physiol (Lond) 384: 649–669] of lobster slowly adapting stretch receptor organs (SAO) and fast-adapting stretch receptor organs (FAO) are analyzed. In response to a step current, the models display three distinct types of firing reflecting the level of adaptation to the stimulation. Low-amplitude currents evoke transient firing containing one to several action potentials before the system stabilizes to a resting state. Conversely, high-amplitude stimulations induce a high frequency transient burst that can last several seconds before the model returns to its quiescent state. In the SAO model, the transition between the two regimes is characterized by a sustained pacemaker firing at an intermediate stimulation amplitude. The FAO model does not exhibit such a maintained firing; rather, the duration of the transient firing increases at first with the stimulus intensity, goes through a maximum and then decreases at larger intensities. Both models comprise seven variables representing the membrane potential, the sodium fast activation, fast inactivation, slow inactivation, the potassium fast activation, slow inactivation gating variables, and the intra cellular sodium concentration. To elucidate the mechanisms of the firing adaptations, the seven-variable model for the lobster stretch receptor neuron is first reduced to a three-dimensional system by regrouping variables with similar time scales. More precisely, we substituted the membrane potential V for the sodium fast activation equivalent potential V _m , the potassium fast inactivation V _n for the sodium fast inactivation V _h , and the sodium slow inactivation V _l for the potassium slow inactivation V _r . Comparison of the responses of the reduced models to those of the original models revealed that the main behaviors of the system were preserved in the reduction process. We classified the different types of responses of the reduced SAO and FAO models to constant current stimulation. We analyzed the transient and stationary responses of the reduced models by constructing bifurcation diagrams representing the qualitatively distinct dynamics of the models and the transitions between them. These revealed that (1) the transient firings prior to reaching the stationary state can be accounted for by the sodium slow inactivation evolving more slowly than the other two variables, so that the changes during the transient firings reflect the bifurcations that the two-dimensional system undergoes when the sodium slow inactivation, considered as a parameter, is varied; and (2) the stationary behaviors of the models are captured by the standard bifurcations of a two-dimensional system formed by the membrane potential and the potassium fast inactivation. We found that each type of firing and the transitions between them is due to the interplay between essentially three variables: two fast ones accounting for the action potential generation and the post-discharge refractoriness, and a third slow one representing the adaptation. Received: 28 February 2000 / Accepted in revised form: 4 October 2000     

Origin of Self-Replicating Biopolymers: Autocatalytic Feedback Can Jump-Start the RNA World

Life is based on biopolymers that have the ability to replicate themselves. Here we consider how a self-replicating RNA system may have originated. We consider a reaction system in which polymerization is possible by the addition of an activated monomer to the end of a chain. We suppose that a small fraction of polymers longer than some minimum length L have the ability to act as polymerase ribozymes. Polymerization can occur spontaneously at a slow rate and can also be catalyzed by polymerase ribozymes, if these ribozymes exist. The system contains autocatalytic feedback: increasing the polymerization rate causes the ribozyme concentration to increase, which causes the polymerization rate to further increase. For an infinite volume, the dynamics are deterministic. There are two stable states: a ‘dead’ state with a very low concentration of ribozymes and a polymerization rate almost equal to the spontaneous rate, and a ‘living’ state with a high concentration of ribozymes and a high rate of polymerization occurring via ribozyme catalysis. In a finite volume, such as the interior of a lipid vesicle or other small compartment, the reaction dynamics is stochastic and concentration fluctuations can occur. Using a stochastic simulation, we show that if a small number of ribozymes is initially formed spontaneously, this can be enough to drive the system from the dead to the living state where ribozyme-catalyzed synthesis of large numbers of additional ribozymes occurs. This transition occurs most easily in volumes of intermediate size.     

Six‐year demographic study reveals threat of stochastic extinction for remnant populations of a threatened amphibian

Sustained demographic studies are essential for early detection of species decline in time for effective management response. A paucity of such background data hindered the potential for successful conservation during the global amphibian decline and remains problematic today. The current study analysed 6 years of mark‐recapture data to determine the vital demographic rates in three habitat precincts of the threatened frog, Litoria aurea (Hylidae) and to understand the underlying causes of variability in population size. Variability in population size of L. aurea was similar to many pond‐breeding species; however this level of fluctuation is rare among threatened amphibians. Highly variable populations are at greater risk of local extinction and the low level of connectivity between L. aurea populations means they are at a greater risk of further decline due to stochastic extinction events and incapacity to recolonize distant habitat. We recommend that management of this species should encourage recolonization through creation of habitat corridors and reintroduction of L. aurea to areas where stochastic extinction events are suspected.