Vincent Hakim

Exact solution of a 1d asymmetric exclusion model using a matrix formulation

Several recent works have shown that the one-dimensional fully asymmetric exclusion model, which describes a system of particles hopping in a preferred direction with hard core interactions, can be solved exactly in the case of open boundaries. Here the authors present a new approach based on representing the weights of each configuration in the steady state as a product of noncommuting matrices. With this approach the whole solution of the problem is reduced to finding two matrices and two vectors which satisfy very simple algebraic rules. They obtain several explicit forms for these non-commuting matrices which are, in the general case, infinite-dimensional. Their approach allows exact expressions to be derived for the current and density profiles. Finally they discuss briefly two possible generalizations of their results: the problem of partially asymmetric exclusion and the case of a mixture of two kinds of particles.

Fast global oscillations in networks of integrate-and-fire neurons with low firing rates

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.

Optimal Information Storage and the Distribution of Synaptic Weights: Perceptron versus Purkinje Cell

It is widely believed that synaptic modifications underlie learning and memory. However, few studies have examined what can be deduced about the learning process from the distribution of synaptic weights. We analyze the perceptron, a prototypical feedforward neural network, and obtain the optimal synaptic weight distribution for a perceptron with excitatory synapses. It contains more than 50% silent synapses, and this fraction increases with storage reliability: silent synapses are therefore a necessary byproduct of optimizing learning and reliability. Exploiting the classical analogy between the perceptron and the cerebellar Purkinje cell, we fitted the optimal weight distribution to that measured for granule cell-Purkinje cell synapses. The two distributions agreed well, suggesting that the Purkinje cell can learn up to 5 kilobytes of information, in the form of 40,000 input-output associations.

Laws of crack motion and phase-field models of fracture

Recently proposed phase-field models offer self-consistent descriptions of brittle fracture. Here, we analyze these theories in the quasistatic regime of crack propagation. We show how to derive the laws of crack motion either by using solvability conditions in a perturbative treatment for slight departure from the Griffith threshold or by generalizing the Eshelby tensor to phase-field models. The analysis provides a simple physical interpretation of the second component of the classic Eshelby integral in the limit of vanishing crack propagation velocity: it gives the elastic torque on the crack tip that is needed to balance the Herring torque arising from the anisotropic surface energy. This force-balance condition can be interpreted physically based on energetic considerations in the traditional framework of continuum fracture mechanics, in support of its general validity for real systems beyond the scope of phase-field models. The obtained law of crack motion reduces in the quasistatic limit to the principle of local symmetry in isotropic media and to the principle of maximum energy-release-rate for smooth curvilinear cracks in anisotropic media. Analytical predictions of crack paths in anisotropic media are validated by numerical simulations. Interestingly, for kinked cracks in anisotropic media, force-balance gives significantly different predictions from the principle of maximum energy-release-rate and the difference between the two criteria can be numerically tested. Simulations also show that predictions obtained from force-balance hold even if the phase-field dynamics is modified to make the failure process irreversible. Finally, the role of dissipative forces on the process zone scale as well as the extension of the results to motion of planar cracks under pure antiplane shear are discussed.

How Connectivity, Background Activity, and Synaptic Properties Shape the Cross-Correlation between Spike Trains

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.

Electrical Coupling Mediates Tunable Low-Frequency Oscillations and Resonance in the Cerebellar Golgi Cell Network

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.

High-Frequency Organization and Synchrony of Activity in the Purkinje Cell Layer of the Cerebellum

The cerebellum controls complex, coordinated, and rapid movements, a function requiring precise timing abilities. However, the network mechanisms that underlie the temporal organization of activity in the cerebellum are largely unexplored, because in vivo recordings have usually targeted single units. Here, we use tetrode and multisite recordings to demonstrate that Purkinje cell activity is synchronized by a high-frequency (approximately 200 Hz) population oscillation. We combine pharmacological experiments and modeling to show how the recurrent inhibitory connections between Purkinje cells are sufficient to generate these oscillations. A key feature of these oscillations is a fixed population frequency that is independent of the firing rates of the individual cells. Convergence in the deep cerebellar nuclei of Purkinje cell activity, synchronized by these oscillations, likely organizes temporally the cerebellar output.

Shape selection of Saffman-Taylor fingers

Among all problems of pattern selection the one posed by the Saffman-Taylor finger is yet unsolved, although numerics and experiments indicate the very simple property that the relative width of the finger tends to 1 2 in the low–surface-tension limit. We explain this by performing an expansion beyond all orders leading to the formulation of a nonlinear eigenvalue problem with a discrete set of solutions. In particular we predict that the relative width of the finger tends to 1 2 as the surface tension to the power 2 3 . PACS numbers: 47.10. + g, 03.40 Gc, 68.10−m

What can we learn from synaptic weight distributions

Much research effort into synaptic plasticity has been motivated by the idea that modifications of synaptic weights (or strengths or efficacies) underlie learning and memory. Here, we examine the possibility of exploiting the statistics of experimentally measured synaptic weights to deduce information about the learning process. Analysing distributions of synaptic weights requires a theoretical framework to interpret the experimental measurements, but the results can be unexpectedly powerful, yielding strong constraints on possible learning theories as well as information that is difficult to obtain by other means, such as the information storage capacity of a cell. We review the available experimental and theoretical techniques as well as important open issues.

Collective Cell Motion in an Epithelial Sheet Can Be Quantitatively Described by a Stochastic Interacting Particle Model

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.