Adaptation and fatigue model for neuron networks and large time asymptotics in a nonlinear fragmentation equation.

Motivated by a model for neural networks with adaptation and fatigue, we study a conservative fragmentation equation that describes the density probability of neurons with an elapsed time s after its last discharge. In the linear setting, we extend an argument by Laurençot and Perthame to prove exponential decay to the steady state. This extension allows us to handle coefficients that have a large variation rather than constant coefficients. In another extension of the argument, we treat a weakly nonlinear case and prove total desynchronization in the network. For greater nonlinearities, we present a numerical study of the impact of the fragmentation term on the appearance of synchronization of neurons in the network using two "extreme" cases. Mathematics Subject Classification (2000)2010: 35B40, 35F20, 35R09, 92B20.

Reproduction cost reduces demographic stochasticity and enhances inter-individual compatibility.

A population׳s survival depends on its ability to adapt to constraints impinging upon it. As such, adaptation is at the heart of an increasing number of theoretical models. In this paper, we propose a bottom-up evolutionary model to explore the relationship between individual evolutionary dynamics and population-level survival. To do so, we extend a well-established model of gene network evolution by introducing a cost for reproduction. As a result population sizes fluctuate and populations can even go extinct. We find that if a population survives a small and critical number of generations, it will reach a quasi-stationary state which ensures long-term survival. In a constant environment, individual adaptation occurs in response to changes in a populations genetic composition. We show that genetic compatibility increases over generations as a by-product of selection for robustness, thus preventing extinction. We also demonstrate that the number of reproductive opportunities per individual, initial population size, and mutation rates all influence population survival. Finally, mixing different populations reveals that individual properties of gene networks co-evolve with the genetic composition of the population in order to maximize an individuals reproductive success.

Commuter mobility and the spread of infectious diseases: application to influenza in France.

Commuting data is increasingly used to describe population mobility in epidemic models. However, there is little evidence that the spatial spread of observed epidemics agrees with commuting. Here, using data from 25 epidemics for influenza-like illness in France (ILI) as seen by the Sentinelles network, we show that commuting volume is highly correlated with the spread of ILI. Next, we provide a systematic analysis of the spread of epidemics using commuting data in a mathematical model. We extract typical paths in the initial spread, related to the organization of the commuting network. These findings suggest that an alternative geographic distribution of GP accross France to the current one could be proposed. Finally, we show that change in commuting according to age (school or work commuting) impacts epidemic spread, and should be taken into account in realistic models.

Synchronization in random balanced networks.

Characterizing the influence of network properties on the global emerging behavior of interacting elements constitutes a central question in many areas, from physical to social sciences. In this article we study a primary model of disordered neuronal networks with excitatory-inhibitory structure and balance constraints. We show how the interplay between structure and disorder in the connectivity leads to a universal transition from trivial to synchronized stationary or periodic states. This transition cannot be explained only through the analysis of the spectral density of the connectivity matrix. We provide a low-dimensional approximation that shows the role of both the structure and disorder in the dynamics.

How evolving heterogeneity distributions of resource allocation strategies shape mortality patterns.

It is well established that individuals age differently. Yet the nature of these inter-individual differences is still largely unknown. For humans, two main hypotheses have been recently formulated: individuals may experience differences in aging rate or aging timing. This issue is central because it directly influences predictions for human lifespan and provides strong insights into the biological determinants of aging. In this article, we propose a model which lets population heterogeneity emerge from an evolutionary algorithm. We find that whether individuals differ in (i) aging rate or (ii) timing leads to different emerging population heterogeneity. Yet, in both cases, the same mortality patterns are observed at the population level. These patterns qualitatively reproduce those of yeasts, flies, worms and humans. Such findings, supported by an extensive parameter exploration, suggest that mortality patterns across species and their potential shapes belong to a limited and robust set of possible curves. In addition, we use our model to shed light on the notion of subpopulations, link population heterogeneity with the experimental results of stress induction experiments and provide predictions about the expected mortality patterns. As biology is moving towards the study of the distribution of individual-based measures, the model and framework we propose here paves the way for evolutionary interpretations of empirical and experimental data linking the individual level to the population level.

Phenotype-genotype relation in Wagner's canalization model.

Theoretical works have shed light on the impact of natural selection in shaping phenotypes and genotypes. Wagner's canalization model (Wagner, 1996) is one of the well-established models which describe emergent properties of evolving gene networks. In this paper, we propose a deeper theoretical understanding of this well-studied model and we extend its conclusions by characterizing new emergent properties of evolving networks. We start with the review of the Wagner model and its applications to robustness of gene networks, gene duplication and evolution of sexual reproduction. Then, we perform a mathematical analysis to gain a better understanding of the model evolutionary dynamics. Doing so paves the way to study systematically the impact of mutation rates on compatibility of genotypes, variability of phenotypes and viability of offspring in evolving populations. Finally, we derive new observations concerning two emergent properties concerning evolved genomes robustness. First, we show that selecting for development towards a specific phenotype also contributes to enhance the stability of other alternative phenotypes which can be revealed under stress. Second, we find that this generalized canalization also renders gene networks more robust towards gene deletion, loss of interactions, perturbations of regulation activity and mutations. Therefore, not only evolution selects for individuals robust to types of perturbation they have faced in previous generations, but also robust to types of perturbations they have never experienced.

Reduction of stochastic conductance-based neuron models with time-scales separation.

We introduce a method for systematically reducing the dimension of biophysically realistic neuron models with stochastic ion channels exploiting time-scales separation. Based on a combination of singular perturbation methods for kinetic Markov schemes with some recent mathematical developments of the averaging method, the techniques are general and applicable to a large class of models. As an example, we derive and analyze reductions of different stochastic versions of the Hodgkin Huxley (HH) model, leading to distinct reduced models. The bifurcation analysis of one of the reduced models with the number of channels as a parameter provides new insights into some features of noisy discharge patterns, such as the bimodality of interspike intervals distribution. Our analysis of the stochastic HH model shows that, besides being a method to reduce the number of variables of neuronal models, our reduction scheme is a powerful method for gaining understanding on the impact of fluctuations due to finite size effects on the dynamics of slow fast systems. Our analysis of the reduced model reveals that decreasing the number of sodium channels in the HH model leads to a transition in the dynamics reminiscent of the Hopf bifurcation and that this transition accounts for changes in characteristics of the spike train generated by the model. Finally, we also examine the impact of these results on neuronal coding, notably, reliability of discharge times and spike latency, showing that reducing the number of channels can enhance discharge time reliability in response to weak inputs and that this phenomenon can be accounted for through the analysis of the reduced model.

Intrinsic variability of latency to first-spike.

The assessment of the variability of neuronal spike timing is fundamental to gain understanding of latency coding. Based on recent mathematical results, we investigate theoretically the impact of channel noise on latency variability. For large numbers of ion channels, we derive the asymptotic distribution of latency, together with an explicit expression for its variance. Consequences in terms of information processing are studied with Fisher information in the Morris-Lecar model. A competition between sensitivity to input and precision is responsible for favoring two distinct regimes of latencies.

Random dynamics of the Morris-Lecar neural model.

Determining the response characteristics of neurons to fluctuating noise-like inputs similar to realistic stimuli is essential for understanding neuronal coding. This study addresses this issue by providing a random dynamical system analysis of the Morris-Lecar neural model driven by a white Gaussian noise current. Depending on parameter selections, the deterministic Morris-Lecar model can be considered as a canonical prototype for widely encountered classes of neuronal membranes, referred to as class I and class II membranes. In both the transitions from excitable to oscillating regimes are associated with different bifurcation scenarios. This work examines how random perturbations affect these two bifurcation scenarios. It is first numerically shown that the Morris-Lecar model driven by white Gaussian noise current tends to have a unique stationary distribution in the phase space. Numerical evaluations also reveal quantitative and qualitative changes in this distribution in the vicinity of the bifurcations of the deterministic system. However, these changes notwithstanding, our numerical simulations show that the Lyapunov exponents of the system remain negative in these parameter regions, indicating that no dynamical stochastic bifurcations take place. Moreover, our numerical simulations confirm that, regardless of the asymptotic dynamics of the deterministic system, the random Morris-Lecar model stabilizes at a unique stationary stochastic process. In terms of random dynamical system theory, our analysis shows that additive noise destroys the above-mentioned bifurcation sequences that characterize class I and class II regimes in the Morris-Lecar model. The interpretation of this result in terms of neuronal coding is that, despite the differences in the deterministic dynamics of class I and class II membranes, their responses to noise-like stimuli present a reliable feature.

The electric fish Brachyhypopomus pinnicaudatus produces jamming avoidance responses to signals that are harmonically related to its own discharges.

Jamming avoidance responses (JARs) are exhibited by pairs of pulse type electric fish that discharge with similar frequencies whenever their individual pulses are about to coincide: responses consist of the transient shortenings in inter-discharge intervals in the fish with the higher frequency. This study describes and models novel forms of JARs observed in sexually mature male or female Brachyhypopomus pinnicaudatus. One novel JAR was observed in male-female pairs in their natural habitat. It happened when the baseline frequencies were not similar but, rather, when one was almost twice that of the other; moreover, the transient interval shortenings occurred not in the fish with the higher frequency but in the slower one. Transient interval shortenings similar to those in all natural JARs were observed in individual fish in tanks and submitted to periodic electrical pulse trains. They happened not only when pulse frequencies were slightly lower than the unperturbed frequency emitted by the fish but also when slightly lower than the frequency's sub- or higher harmonics (e.g. one half or twice). The proposed model satisfactorily reproduces all experimental observations. In it, forthcoming inter-pulse intervals reflect the differences between the cophases of pulses that arrive within the 'sensitive windows' belonging to either consecutive (i.e. one and the next) or alternating (e.g. every other, every three) intervals. Paired pulse fish embody interacting oscillators, and, in particular, JARs embody either quasiperiodic phase walk-throughs and intermittencies or periodic and locked forms. Hence, their study would profit by the powerful theories and approaches advanced by nonlinear dynamics.

Influenza epidemics in the United States, France, and Australia, 1972-1997.

Influenza epidemics occur once a year during the winter in temperate areas. Little is known about the similarities between epidemics at different locations. We have analyzed pneumonia and influenza deaths from 1972 to 1997 in the United States, France, and Australia to examine the correlation over space and time between the three countries. We found a high correlation in both areas between France and the United States (correlation in impact, Spearman's r= 0.76, p < 0.001, and test for synchrony in timing of epidemics, p < 0.001). We did not find a similar correlation between the United States and Australia or between France and Australia, when considering a systematic half-year lead or delay of influenza epidemics in Australia as compared with those in the United States or France. These results support a high correlation at the hemisphere level and suggest that the global interhemispheric circulation of epidemics follows an irregular pathway with recurrent changes in the leading hemisphere.

Association of influenza epidemics with global climate variability.

The reasons for the seasonality and annual changes in the impact of influenza epidemics remain poorly understood. We investigated the covariations between a major component of climate, namely the El Niño Southern Oscillation (ENSO), and indicators of the impact of influenza, as measured by morbidity, excess mortality and viral subtypes collected in France during the period 1971-2002. We show that both the circulating subtype and the magnitude of ENSO are associated with the impact of influenza epidemics. Recognition of this association could lead to better understanding of the mechanisms of emergence of influenza epidemics.

Interspike interval correlations, memory, adaptation, and refractoriness in a leaky integrate-and-fire model with threshold fatigue.

Neuronal adaptation as well as interdischarge interval correlations have been shown to be functionally important properties of physiological neurons. We explore the dynamics of a modified leaky integrate-and-fire (LIF) neuron, referred to as the LIF with threshold fatigue, and show that it reproduces these properties. In this model, the postdischarge threshold reset depends on the preceding sequence of discharge times. We show that in response to various classes of stimuli, namely, constant currents, step currents, white gaussian noise, and sinusoidal currents, the model exhibits new behavior compared with the standard LIF neuron. More precisely, (1) step currents lead to adaptation, that is, a progressive decrease of the discharge rate following the stimulus onset, while in the standard LIF, no such patterns are possible; (2) a saturation in the firing rate occurs in certain regimes, a behavior not seen in the LIF neuron; (3) interspike intervals of the noise-driven modified LIF under constant current are correlated in a way reminiscent of experimental observations, while those of the standard LIF are independent of one another; (4) the magnitude of the correlation coefficients decreases as a function of noise intensity; and (5) the dynamics of the sinusoidally forced modified LIF are described by iterates of an annulus map, an extension to the circle map dynamics displayed by the LIF model. Under certain conditions, this map can give rise to sensitivity to initial conditions and thus chaotic behavior.

Activity in sparsely connected excitatory neural networks: effect of connectivity.

The Nucleus Tractus Solitarius (NTS) of the brainstem contains a neural circuit with only excitatory connections displaying a spontaneous activity involved in the control of respiration. A model of a network with random connections is presented and is used to investigate a possible mechanism of spontaneous activity generation consisting of the amplification of a low-background activity by the excitatory connections. First, the steady states of the network model and its ability to amplify the activity are studied. Then, a low-background activity is introduced, and dynamics of simulated networks are examined. Low-tonic, slow-phasic and fast-tonic activities are successively observed when the mean number K of connections per neuron increases. The transition between the two first types of activity is progressive whereas the transition from slow-phasic to fast-tonic activity is sharp. Simulation results show that activities of low frequency can be obtained with the proposed mechanism of spontaneous activity generation only if the network connectivity is low.

Single Neuron with Recurrent Excitation: Effect of the Transmission Delay.

Recurrent excitatory circuits and the positive feedback they imply are assigned important roles in a variety of tasks in living organisms. Such networks obviously do not exhibit saturated behaviour in the sense of extremely fast rates and/or insensitivity to input variations, as artificial systems with positive feedback generally do. It is therefore important to identify how neural saturation is avoided. A single neuron that excites itself directly provides the simplest anatomical circuitry where this problem can be studied. Such a system was simulated experimentally by Diez-Martínez and Segundo using the pacemaker neuron in the crayfish stretch receptor organ. They showed that the feedback transmission time, called "delay", was strongly influential, and small changes led to markedly different outcomes. As the delay was increased the discharge patterns went from pacemaker spike trains to multiplets separated by silent intervals to still longer bursts and longer silent intervals. We hypothesized that neuronal sensitivity decreased along the rapid successive firings (adaptation) and prevented saturation and therefore played an important role in the observed behaviour. This hypothesis was tested using models of increasing complexity. The simplest model was an integrate and fire neuron without adaptation to repeated stimuli, in this case the dynamics were qualitatively different from the experimental data. The other models exhibited adaptation to repeated stimuli. Two were relatively simple: an integrate and fire and a leaky integrator. The last model was more complex, it included membrane conductances. Neither of these models exhibited saturation when recurrent excitation was introduced. Their dynamics were in fact similar to those in the crayfish preparation, both exhibiting pacemaker firing for short delays, and multiplets or burst for intermediate delays. Simulations were therefore compatible with the hypothesis that neuronal adaptation is important in preventing saturation. Copyright 1996 Elsevier Science Ltd