Khashayar Pakdaman

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 ρ = 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.

Effect of delay on the boundary of the basin of attraction in a system of two neurons

The behavior of neural networks may be influenced by transmission delays and many studies have derived constraints on parameters such as connection weights and output functions which ensure that the asymptotic dynamics of a network with delay remains similar to that of the corresponding system without delay. However, even when the delay does not affect the asymptotic behavior of the system, it may influence other important features in the systems dynamics such as the boundary of the basin of attraction of the stable equilibria. In order to better understand such effects, we study the dynamics of a system constituted by two neurons interconnected through delayed excitatory connections. We show that the system with delay has exactly the same stable equilibrium points as the associated system without delay, and that, in both the network with delay and the corresponding one without delay, most trajectories converge to these stable equilibria. Thus, the asymptotic behavior of the network with delay and that of the corresponding system without delay are similar. We obtain a theoretical characterization of the boundary separating the basins of attraction of two stable equilibria, which enables us to estimate the boundary. Our numerical investigations show that, even in this simple system, the boundary separting the basins of attraction of two stable equilibrium points depends on the value of the delays. The extension of these results to networks with an arbritrary number of units is discussed.

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.

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

Abstract. 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.

Coherence resonance and discharge time reliability in neurons and neuronal models.

Neurons are subject to internal and external noise that have been known to modify the way they process incoming signals. Recent studies have suggested that such alterations have functional roles and can also be used in biomedical applications. The present work goes over experimental and theoretical descriptions of the response of neurons to white noise stimulation. It examines various forms of noise related behavior in a standard neuronal model, namely the leaky integrate and fire. This clarifies the conditions under which specific noise induced changes occur in neurons, and consequently can help in determining whether nervous systems operate under similar circumstances.

Global attractor and asymptotic dynamics in the Kuramoto model for coupled noisy phase oscillators

We study the dynamics of the large N limit of the Kuramoto model of coupled phase oscillators, subject to white noise. We introduce the notion of shadow inertial manifold and we prove their existence for this model, supporting the fact that the long-term dynamics of this model is finite dimensional. Following this, we prove that the global attractor of this model takes one of two forms. When coupling strength is below a critical value, the global attractor is a single equilibrium point corresponding to an incoherent state. Otherwise, when coupling strength is beyond this critical value, the global attractor is a two-dimensional disc composed of radial trajectories connecting a saddle-point equilibrium (the incoherent state) to an invariant closed curve of locally stable equilibria (partially synchronized state). Our analysis hinges, on the one hand, upon sharp existence and uniqueness results and their consequence for the existence of a global attractor, and, on the other hand, on the study of the dynamics in the vicinity of the incoherent and coherent (or synchronized) equilibria. We prove in particular nonlinear stability of each synchronized equilibrium, and normal hyperbolicity of the set of such equilibria. We explore mathematically and numerically several properties of the global attractor, in particular we discuss the limit of this attractor as noise intensity decreases to zero.

Relaxation and self-sustained oscillations in the time elapsed neuron network model

The time elapsed model describes the firing activity of a homogenous assembly of neurons thanks to the distribution of times elapsed since the last discharge. It gives a mathematical description of the probability density of neurons structured by this time. In an earlier work, based on generalized relative entropy methods, it is proved that for highly or weakly connected networks the model exhibits relaxation to the steady state and for moderately connected networks numerical evidence is obtained of the appearance of self-sustained periodic solutions. Here, we go further and, using the particular form of the model, we quantify the regime where relaxation to a stationary state occurs in terms of the network connectivity. To introduce our methodology, we first consider the case where the neurons are not connected and we give a new statement showing that total asynchronous firing of neurons appears asymptotically. In a second step, we consider the case with connections and give a low connectivity condition t...

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.