Evolution of aging: individual life history trade-offs and population heterogeneity account for mortality patterns across species.

A broad range of mortality patterns has been documented across species, some even including decreasing mortality over age. Whether there exist a common denominator to explain both similarities and differences in these mortality patterns remains an open question. The disposable soma theory, an evolutionary theory of aging, proposes that universal intracellular trade-offs between maintenance/lifespan and reproduction would drive aging across species. The disposable soma theory has provided numerous insights concerning aging processes in single individuals. Yet, which specific population mortality patterns it can lead to is still largely unexplored. In this article, we propose a model exploring the mortality patterns which emerge from an evolutionary process including only the disposable soma theory core principles. We adapt a well-known model of genomic evolution to show that mortality curves producing a kink or mid-life plateaus derive from a common minimal evolutionary framework. These mortality shapes qualitatively correspond to those of Drosophila melanogaster, Caenorhabditis elegans, medflies, yeasts and humans. Species evolved in silico especially differ in their population diversity of maintenance strategies, which itself emerges as an adaptation to the environment over generations. Based on this integrative framework, we also derive predictions and interpretations concerning the effects of diet changes and heat-shock treatments on mortality patterns.

Excitability of the Clay model for squid giant axon.

The squid giant axon is the canonical experimental membrane prototype for the study of action potential generation. This work is concerned with Clay's model for this preparation, which implements the nonlinear dependence of sodium and potassium currents on voltage, a multicompartmental description of sodium channel kinetics that takes into account the dependence between activation and inactivation, revised potassium activation function, and potassium accumulation in the axoplasm and its uptake by glial cells. This model accounts better than the standard Hodgkin-Huxley (HH) model for the response of squid giant axons to various stimuli. We systematically compare the responses of the Clay model and the standard HH model to pulse-like and constant current stimuli. We also analyze hybrid models that combine features from both models. These studies reveal that the differences between the sodium currents account for the main difference between the two models, namely the lower excitability of the Clay model.

An analysis of the reliability phenomenon in the FitzHugh-Nagumo model.

The reliability of single neurons on realistic stimuli has been experimentally confirmed in a wide variety of animal preparations. We present a theoretical study of the reliability phenomenon in the FitzHugh-Nagumo model on white Gaussian stimulation. The analysis of the model's dynamics is performed in three regimes-the excitable, bistable, and oscillatory ones. We use tools from the random dynamical systems theory, such as the pullbacks and the estimation of the Lyapunov exponents and rotation number. The results show that for most stimulus intensities, trajectories converge to a single stochastic equilibrium point, and the leading Lyapunov exponent is negative. Consequently, in these regimes the discharge times are reliable in the sense that repeated presentation of the same aperiodic input segment evokes similar firing times after some transient time. Surprisingly, for a certain range of stimulus intensities, unreliable firing is observed due to the onset of stochastic chaos, as indicated by the estimated positive leading Lyapunov exponents. For this range of stimulus intensities, stochastic chaos occurs in the bistable regime and also expands in adjacent parts of the excitable and oscillating regimes. The obtained results are valuable in the explanation of experimental observations concerning the reliability of neurons stimulated with broad-band Gaussian inputs. They reveal two distinct neuronal response types. In the regime where the first Lyapunov has negative values, such inputs eventually lead neurons to reliable firing, and this suggests that any observed variance of firing times in reliability experiments is mainly due to internal noise. In the regime with positive Lyapunov exponents, the source of unreliable firing is stochastic chaos, a novel phenomenon in the reliability literature, whose origin and function need further investigation.

Comment on "Dynamics of some neural network models with delay".

Based upon numerical evidence, Ruan et al. [J. Ruan, L. Li, and W. Lin, Phys. Rev. E 63, 051906 (2001)] suggest that the delay differential equation dx/dt(t)=-x(t)+A tanh[x(t)]+B tanh[x(t-tau)] may display chaotic dynamics. As mentioned by Pakdaman and Malta [IEEE Trans. Neural Netw. 9, 231 (1998)], this equation presents a monotonic delayed feedback, so that it satisfies a Poincaré-Bendixson-like theorem, ruling out the existence of complex aperiodic dynamics.

White-noise stimulation of the Hodgkin-Huxley model.

We studied the combined influence of noise and constant current stimulations on the Hodgkin-Huxley neuron model through time and frequency analysis of the membrane-potential dynamics. We observed that, in agreement with experimental data (Guttman et al. 1974), at low noise and low constant current stimulation the behavior of the model is well approximated by that of the linearized Hodgkin-Huxley system. Conversely, nonlinearities due to firing dominate at large noise or current stimulations. The transition between the two regimes is abrupt, and takes place in the same range of noise and current intensities as the noise-induced transition characterized by the qualitative change in the stationary distribution of the membrane potential (Tanabe and Pakdaman 2001a). The implications of these results are discussed.

The reliability of the stochastic active rotator.

The reliability of firing of excitable-oscillating systems is studied through the response of the active rotator, a neuronal model evolving on the unit circle, to white gaussian noise. A stochastic return map is introduced that captures the behavior of the model. This map has two fixed points: one stable and the other unstable. Iterates of all initial conditions except the unstable point tend to the stable fixed point for almost all input realizations. This means that to a given input realization, there corresponds a unique asymptotic response. In this way, repetitive stimulation with the same segment of noise realization evokes, after possibly a transient time, the same response in the active rotator. In other words, this model responds reliably to such inputs. It is argued that this results from the nonuniform motion of the active rotator around the unit circle and that similar results hold for other neuronal models whose dynamics can be approximated by phase dynamics similar to the active rotator.

Response of a pacemaker neuron model to stochastic pulse trains.

We investigated the response of a pacemaker neuron model to trains of inhibitory stochastic impulsive perturbations. The model captures the essential aspect of the dynamics of pacemaker neurons. Especially, the model reproduces linearization by stochastic pulse trains, that is, the disappearance of the paradoxical segments in which the output firing rate of pacemaker neurons increases with inhibition rate, as the coefficient of variation of the input pulse train increases. To study the response of the model to stochastic pulse trains, we use a Markov operator governing the phase transition. We show how linearization occurs based on the spectral analysis of the Markov operator. Moreover, using Lyapunov exponents, we show that variable inputs evoke reliable firing, even in situations where periodic stimulation with the same mean rate does not.

Random dynamics of the Hodgkin-Huxley neuron model.

Noise can alter the response of neurons, enhancing their ability to detect weak inputs. We analyze how the Hodgkin-Huxley equations, a canonical neuron model, respond to white noise stimulation. We show that this model possesses a stochastic attractor, reduced to a unique stochastic equilibrium point that attracts all trajectories.

Dynamical behavior of a pacemaker neuron model with fixed delay stimulation.

In physiological and pathological conditions, many biological oscillators, such as pacemaker cells, operate under the influence of feedbacks. Fixed delay stimulation is a standard preparation to evaluate the effects of such influences. Through the study of the Hodgkin-Huxley model, we show that such recurrent excitation can lead to regular and irregular discharge trains with interdischarge intervals that are up to several multiples of the period of the oscillator. In other words, we show that recurrent excitation can considerably slow down the firings of the pacemaker. This result contrasts with previous studies of similar preparations that have reported that fixed delay stimulation leads to a bursting pattern in which regimes of high-frequency firing alternate with periods of quiescence. We elucidate the mechanisms underlying the behavior of the oscillator under fixed delay perturbation through the analysis of the dynamics of a well-known two-dimensional oscillator, namely, the Poincaré oscillator.

Noise-enhanced neuronal reliability.

This work shows that noise can enhance the discharge time reliability in Hodgkin-Huxley neuron models stimulated by weak periodic and aperiodic inputs. By expanding the Fokker-Planck equation of an elementary model for excitable systems, the dependence of the optimal noise intensity on input characteristics is discussed.

Noise-induced transition in excitable neuron models.

We studied the influence of noisy stimulation on the Hodgkin-Huxley neuron model. Rather than examining the noise-related variability of the discharge times of the model--as has been done previously--our study focused on the effect of noise on the stationary distributions of the membrane potential and gating variables of the model. We observed that a gradual increase in the noise intensity did not result in a gradual change of the distributions. Instead, we could identify a critical intermediate noise range in which the shapes of the distributions underwent a drastic qualitative change. Namely, they moved from narrow unimodal Gaussian-like shapes associated with low noise intensities to ones that spread widely at large noise intensities. In particular, for the membrane potential and the sodium activation variable, the distributions changed from unimodal to bimodal. Thus, our investigation revealed a noise-induced transition in the Hodgkin-Huxley model. In order to further characterize this phenomenon, we considered a reduced one-dimensional model of an excitable system, namely the active rotator. For this model, our analysis indicated that the noise-induced transition is associated with a deterministic bifurcation of approximate equations governing the dynamics of the mean and variance of the state variable. Finally, we shed light on the possible functional importance of this noise-induced transition in neuronal coding by determining its effect on the spike timing precision in models of neuronal ensembles.

External noise synchronizes forced oscillators.

Periodic pulsatile perturbation of nonlinear oscillators generates phase-locking, quasiperiodic, and chaotic responses. This work shows that the application of external noise to ensembles of such forced systems can synchronize oscillations, even in regimes where neither the noise nor the periodic forcing, when applied alone, would lead to such a phenomenon.

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 Vm, the potassium fast inactivation Vn for the sodium fast inactivation Vh, and the sodium slow inactivation Vl for the potassium slow inactivation Vr. 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.

Variability of the electric organ discharge interval duration in resting Gymnotus carapo.

We recorded the electric organ discharges of resting Gymnotus carapo specimens. We analyzed the time series formed by the sequence of interdischarge intervals. Nonlinear prediction, false nearest neighbor analyses, and comparison between the performance of nonlinear and linear autoregressive models fitted to the data indicated that nonlinear correlations between intervals were absent, or were present to a minor extent only. Following these analyses, we showed that linear autoregressive models with combined Gaussian and shot noise reproduced the variability and correlations of the resting discharge pattern. We discuss the implications of our findings for the mechanisms underlying the timing of electric organ discharge generation. We also argue that autoregressive models can be used to evaluate the changes arising during a wide variety of behaviors, such as the modification in the discharge intervals during interaction between fish pairs.

Dynamics of moments of FitzHugh-Nagumo neuronal models and stochastic bifurcations.

For the study of the behavior of noisy neuronal models, Rodriguez and Tuckwell have introduced an elegant and systematic method which consists of replacing the system of stochastic differential equations with a system of deterministic equations representing the dynamics of the means, variances, and covariance of the state variables [R. Rodriguez and H.C. Tuckwell, Phys. Rev. E 54, 5585 (1996)]. In this work, we first report a modification of their method in the case of the FitzHugh-Nagumo model which enhances the accuracy of the approximation without including higher order moments. This method is then combined with a self-consistency argument in order to better characterize the behavior of the underlying stochastic processes through the computation of approximate auto- and cross-correlation functions of the state variables. Finally, we argue that the moments' equations can also reveal the existence of stochastic bifurcations, i.e., qualitative changes in the dynamics of stochastic systems.

Periodically forced leaky integrate-and-fire model.

The discharge pattern of periodically forced leaky integrate-and-fire models is studied. While previous analyses have been mainly concerned with the response of this model to sinusoidal stimulation, our results hold for arbitrary periodic inputs. It is shown that, for any periodic input, the map representing the relation between input phases at consecutive discharge times can be restricted to a piecewise continuous, orientation preserving circle map. This implies that (i) the rotation number is well defined and independent of the initial condition, and (ii) in the same way as for sinusoidal forcing, other forms of periodic stimuli can evoke only one of four types of response, namely, phase locking, quasiperiodic discharges, nonchaotic aperiodic firing, and termination of the discharge after a finite number of firings.

Complex ligand-protein systems: a globally convergent iterative method for the n x m case.

When n types of univalent ligands are competing for the binding to m types of protein sites, the determination of the system composition at equilibrium reduces to the solving of a non-linear system of n equations in C = [0; 1](n). We present an iterative method to solve such a system. We show that the sequence presented here is always convergent, regardless of the initial value in C. We also prove that the limit of this sequence is the unique solution in C of the non-linear system of equations.

A first-passage-time analysis of the periodically forced noisy leaky integrate-and-fire model.

We present a general method for the analysis of the discharge trains of periodically forced noisy leaky integrate-and-fire neuron models. This approach relies on the iterations of a stochastic phase transition operator that generalizes the phase transition function used for the study of periodically forced deterministic oscillators to noisy systems. The kernel of this operator is defined in terms of the the first passage time probability density function of the Ornstein Uhlenbeck process through a suitable threshold. Numerically, it is computed as the solution of a singular integral equation. It is shown that, for the noisy system, quantities such as the phase distribution (cycle histogram), the interspike interval distribution, the autocorrelation function of the intervals, the autocorrelogram and the power spectrum density of the spike train, as well as the input-output cross-correlation and cross-spectral density can all be computed using the stochastic phase transition operator. A detailed description of the numerical implementation of the method, together with examples, is provided.

Reconstructing bifurcation diagrams of dynamical systems using measured time series.

We present an algorithm for reconstructing the bifurcation structure of a dynamical system from time series. The method consists in finding a parameterized predictor function whose bifurcation structure is similar to that of the given system. Nonlinear autoregressive (NAR) models with polynomial terms are employed as predictor functions. The appropriate terms in the NAR models are obtained using a fast orthogonal search scheme. This scheme eliminates the problem of multiparameter optimization and makes the approach robust to noise. The algorithm is applied to the reconstruction of the bifurcation diagram (BD) of a neuron model from the simulated membrane potential waveforms. The reconstructed BD captures the different behaviors of the given system. Moreover, the algorithm also works well even for a limited number of time series.

Reconstructing bifurcation diagrams from noisy time series using nonlinear autoregressive models.

We introduce a formalism for the reconstruction of bifurcation diagrams from noisy time series. The method consists in finding a parametrized predictor function whose bifurcation structure is similar to that of the given system. The reconstruction algorithm is composed of two stages: model selection and bifurcation parameter identification. In the first stage, an appropriate model that best represents all the given time series is selected. A nonlinear autoregressive model with polynomial terms is employed in this study. The identification of the bifurcation parameters from among the many model parameters is done in the second stage. The algorithm works well even for a limited number of time series.