Several recent works have shown that the onedimensional fully asymmetric exclusion model, which describes a ystem of panicles hopping in a preferred direction with hard core interactions, can be solved exactb in the case of open boundaries. Here we present a new approach based on representing the weights of each mnfiguration in the steady state as a product of noncommuting matrices. Wth this approach the whole solution of the problem is reduced to finding two matrices and two vectors which satisQ ve'y simple algebraic NI= We obtain several explicit toms for these noncommuting matrices which are, in the general case. infinite-dimensional. Our approach allows exam expresions to be derived for the current and density profiles. Finally we discuss h'efly two possible generalizations of our results: the pmblem of panially asymmetric exclusion and the case of a mixture of two kinds of panicles.

We study analytically the dynamics of a network of sparsely connected inhibitory integrate-and-fire neurons in a regime where individual neurons emit spikes irregularly and at a low rate. In the limit when the number of neurons N → ∞, the network exhibits a sharp transition between a stationary and an oscillatory global activity regime where neurons are weakly synchronized. The activity becomes oscillatory when the inhibitory feedback is strong enough. The period of the global oscillation is found to be mainly controlled by synaptic times, but depends also on the characteristics of the external input. In large but finite networks, the analysis shows that global oscillations of finite coherence time generically exist both above and below the critical inhibition threshold. Their characteristics are determined as functions of systems parameters, in these two different regimes. The results are found to be in good agreement with numerical simulations.

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. Many types of neurons exhibit subthreshold resonance. However, little is known about whether this frequency preference influences spike emission. Here, the link between subthreshold resonance and firing rate is examined in the framework of conductance-based models. A classification of the subthreshold properties of a general class of neurons is first provided. In particular, a class of neurons is identified in which the input impedance exhibits a suppression at a nonzero low frequency as well as a peak at higher frequency. The analysis is then extended to the effect of subthreshold resonance on the dynamics of the firing rate. The considered input current comprises a background noise term, mimicking the massive synaptic bombardment in vivo. Of interest is the modulatory effect an additional weak oscillating current has on the instantaneous firing rate. When the noise is weak and firing regular, the frequency most preferentially modulated is the firing rate itself. Conversely, when the noise is strong and firing irregular, the modulation is strongest at the subthreshold resonance frequency. These results are demonstrated for two specific conductance-based models and for a generalization of the integrate-and-fire model that captures subthreshold resonance. They suggest that resonant neurons are able to communicate their frequency preference to postsynaptic targets when the level of noise is comparable to that prevailing in vivo.

Tonic motor control involves oscillatory synchronization of activity at low frequency (5-30 Hz) throughout the sensorimotor system, including cerebellar areas. We investigated the mechanisms underpinning cerebellar oscillations. We found that Golgi interneurons, which gate information transfer in the cerebellar cortex input layer, are extensively coupled through electrical synapses. When depolarized in vitro, these neurons displayed low-frequency oscillatory synchronization, imposing rhythmic inhibition onto granule cells. Combining experiments and modeling, we show that electrical transmission of the spike afterhyperpolarization is the essential component for oscillatory population synchronization. Rhythmic firing arises in spite of strong heterogeneities, is frequency tuned by the mean excitatory input to Golgi cells, and displays pronounced resonance when the modeled network is driven by oscillating inputs. In vivo, unitary Golgi cell activity was found to synchronize with low-frequency LFP oscillations occurring during quiet waking. These results suggest a major role for Golgi cells in coordinating cerebellar sensorimotor integration during oscillatory interactions.

In the zero temperature Glauber dynamics of the ferromagnetic Ising or q-state Potts model, the size of domains is known to grow like t 1/2 . Recent simulations have shown that the fraction r(q, t) of spins which have never flipped up to time t decays like a power law r(q, t) ∼ t −θ(q) with a nontrivial dependence of the exponent θ(q) on q and on space dimension. By mapping the problem on an exactly soluble one-species coagulation model (A + A → A), we obtain the exact expression of θ(q) in dimension one. VERSION :October 9, 2018Phase ordering and domain growth in systems quenched from a disordered phase to an ordered phase has been a subject of much interest during the last fifteen years in fields ranging from metallurgy to cosmology [1]. It is well established that the pattern of growing domains is self-similar in time and that the characteristic domain size increases with a simple power law t ρ . For example, ρ = 1/2 holds for all systems with short range-interactions described by a scalar non-conserved order parameter. However, as noted for the auto-correlation function [2], correlations at different times are characterized by more complicated exponents. Recently, the fraction of spins r(q, t) which have never flipped up to time t has been measured in simulations of coarsening at zero-temperature for the Ising and for the q-state Potts models. This fraction decreases with time like a power law [3,4],and numerical data indicate that the exponent θ(q) varies both with q and the dimension of space. Since dr(q, t)/dt measures the probability that a given point is crossed for the first time at time t by a domain wall, θ(q) can be viewed as a first passage exponent [5]. The aim of the present letter is to give the exact expression of θ(q) in one dimension.This result fully agrees with previous numerical predictions based on MonteCarlo simulations [3,4] or on finite size scaling calculations [6]. It implies that for the Ising model θ(2) = 3/8 is exact. Note however that for other choices of q, the exponent θ(q) is in general irrational (for example θ(3) = .53795082..).To obtain (2), we are going to follow four main steps: first by using finite size scaling we will relate the exponent θ(q) to the large L behavior of the fraction ρ L (q) of spins which never flip between time 0 and time ∞ for a finite one dimensional system of L sites with periodic boundary conditions (for the zero temperature Glauber dynamics of the q-state Potts model); secondly, we will show that the calculation of ρ L (q) can be reduced to solving the steady state of a reaction-diffusion model (A + A → A) on a one dimensional lattice of L sites with a source of particles at the origin (i.e. at site 0 ≡ L); our third step will be the solution of the steady state of that reaction-diffusion model leading to the exact expression of ρ L (q) for arbitrary L and q; lastly, we will extract the exponent θ(q) from the large L behavior of this expression.In an infinite system, it is known [7,8] that starting with a random initial condition, the size of domai...

Modelling the displacement of thousands of cells that move in a collective way is required for the simulation and the theoretical analysis of various biological processes. Here, we tackle this question in the controlled setting where the motion of Madin-Darby Canine Kidney (MDCK) cells in a confluent epithelium is triggered by the unmasking of free surface. We develop a simple model in which cells are described as point particles with a dynamic based on the two premises that, first, cells move in a stochastic manner and, second, tend to adapt their motion to that of their neighbors. Detailed comparison to experimental data show that the model provides a quantitatively accurate description of cell motion in the epithelium bulk at early times. In addition, inclusion of model “leader” cells with modified characteristics, accounts for the digitated shape of the interface which develops over the subsequent hours, providing that leader cells invade free surface more easily than other cells and coordinate their motion with their followers. The previously-described progression of the epithelium border is reproduced by the model and quantitatively explained.

Functional interactions between neurons in vivo are often quantified by cross-correlation functions (CCFs) between their spike trains. It is therefore essential to understand quantitatively how CCFs are shaped by different factors, such as connectivity, synaptic parameters, and background activity. Here, we study the CCF between two neurons using analytical calculations and numerical simulations. We quantify the role of synaptic parameters, such as peak conductance, decay time, and reversal potential, and analyze how various patterns of connectivity influence CCF shapes. In particular, we find that the symmetry of the CCF distinguishes in general, but not always, the case of shared inputs between two neurons from the case in which they are directly synaptically connected. We systematically examine the influence of background synaptic inputs from the surrounding network that set the baseline firing statistics of the neurons and modulate their response properties. We find that variations in the background noise modify the amplitude of the cross-correlation function as strongly as variations of synaptic strength. In particular, we show that the postsynaptic neuron spiking regularity has a pronounced influence on CCF amplitude. This suggests an efficient and flexible mechanism for modulating functional interactions.