Random Sequential Dynamics Research Articles

Sequential and efficient neural-population coding of complex task information

Recent work has highlighted that many types of variables are represented in each neocortical area. How can these many neural representations be organized together without interference and coherently maintained/updated through time? We recorded from excitatory neural populations in posterior cortices as mice performed a complex, dynamic task involving multiple interrelated variables. The neural encoding implied that highly correlated task variables were represented by less-correlated neural population modes, while pairs of neurons exhibited a spectrum of signal correlations. This finding relates to principles of efficient coding, but notably utilizes neural population modes as the encoding unit and suggests partial whitening of task-specific information where different variables are represented with different signal-to-noise levels. Remarkably, this encoding function was multiplexed with sequential neural dynamics yet reliably followed changes in task-variable correlations throughout the trial. We suggest that neural circuits can implement time-dependent encodings in a simple way using random sequential dynamics as a temporal scaffold.

Exact solution to the random sequential dynamics of a message passing algorithm.

We analyze the random sequential dynamics of a message passing algorithm for Ising models with random interactions in the large system limit. We derive exact results for the two-time correlation functions and the speed of convergence. The de Almedia-Thouless stability criterion of the static problem is found to be necessary and sufficient for the global convergence of the random sequential dynamics.

Three is much more than two in coarsening dynamics of cyclic competitions.

The classical game of rock-paper-scissors has inspired experiments and spatial model systems that address the robustness of biological diversity. In particular, the game nicely illustrates that cyclic interactions allow multiple strategies to coexist for long-time intervals. When formulated in terms of a one-dimensional cellular automata, the spatial distribution of strategies exhibits coarsening with algebraically growing domain size over time, while the two-dimensional version allows domains to break and thereby opens the possibility for long-time coexistence. We consider a quasi-one-dimensional implementation of the cyclic competition, and study the long-term dynamics as a function of rare invasions between parallel linear ecosystems. We find that increasing the complexity from two to three parallel subsystems allows a transition from complete coarsening to an active steady state where the domain size stays finite. We further find that this transition happens irrespective of whether the update is done in parallel for all sites simultaneously or done randomly in sequential order. In both cases, the active state is characterized by localized bursts of dislocations, followed by longer periods of coarsening. In the case of the parallel dynamics, we find that there is another phase transition between the active steady state and the coarsening state within the three-line system when the invasion rate between the subsystems is varied. We identify the critical parameter for this transition and show that the density of active boundaries has critical exponents that are consistent with the directed percolation universality class. On the other hand, numerical simulations with the random sequential dynamics suggest that the system may exhibit an active steady state as long as the invasion rate is finite.

Analytical Approach to the One-Dimensional Disordered Exclusion Process with Open Boundaries and Random Sequential Dynamics

A one-dimensional disordered particle hopping rate asymmetric exclusion process (ASEP) with open boundaries and a random sequential dynamics is studied analytically. Combining the exact results of the steady states in the pure case with a perturbative mean field-like approach the broken particle-hole symmetry is highlighted and the phase diagram is studied in the parameter space (α,β), where α and β represent respectively the injection rate and the extraction rate of particles. The model displays, as in the pure case, high-density, low-density and maximum-current phases. All critical lines are determined analytically showing that the high-density low-density first order phase transition occurs at α≠β. We show that the maximum-current phase extends its stability region as the disorder is increased and the usual \(1/\sqrt{\ell}\) -decay of the density profile in this phase is universal. Assuming that some exact results for the disordered model on a ring hold for a system with open boundaries, we derive some analytical results for platoon phase transition within the low-density phase and we give an analytical expression of its corresponding critical injection rate α *. As it was observed numerically (Bengrine et al. J. Phys. A: Math. Gen. 32:2527, [1999]), we show that the quenched disorder induces a cusp in the current-density relation at maximum flow in a certain region of parameter space and determine the analytical expression of its slope. The results of numerical simulations we develop agree with the analytical ones.

Universality of the power-law approach to the jamming limit in random sequential adsorption dynamics

Random sequential adsorption (RSA), on a two-dimensional continuum substrate, of different types of zero area objects that disallow domain formation and hence lead to jamming, is examined by simulation. In all the cases, in the asymptotic time regime, the approach of the number density rho(t) at instant t to jamming density rho(infinity) is found to exhibit power law rho(infinity)-rho(t) approximately t{-p} as that for RSA of finite area objects. These results suggest the possibility of the power law being universal for all jamming systems in RSA on a continuum substrate. A generalized analytical treatment is also proposed.

Factorized steady states in mass transport models on an arbitrary graph

We study a general mass transport model on an arbitrary graph consisting of $L$ nodes each carrying a continuous mass. The graph also has a set of directed links between pairs of nodes through which a stochastic portion of mass, chosen from a site-dependent distribution, is transported between the nodes at each time step. The dynamics conserves the total mass and the system eventually reaches a steady state. This general model includes as special cases various previously studied models such as the Zero-range process and the Asymmetric random average process. We derive a general condition on the stochastic mass transport rules, valid for arbitrary graph and for both parallel and random sequential dynamics, that is sufficient to guarantee that the steady state is factorisable. We demonstrate how this condition can be achieved in several examples. We show that our generalized result contains as a special case the recent results derived by Greenblatt and Lebowitz for $d$-dimensional hypercubic lattices with random sequential dynamics.

Dyck paths, Motzkin paths and traffic jams

It has recently been observed that the normalization of a one-dimensional out-of-equilibriummodel, the asymmetric exclusion process (ASEP) with random sequential dynamics, isexactly equivalent to the partition function of a two-dimensional lattice path model ofone-transit walks, or equivalently Dyck paths. This explains the applicabilityof the Lee–Yang theory of partition function zeros to the ASEP normalization.In this paper we consider the exact solution of the parallel-update ASEP, a special case ofthe Nagel–Schreckenberg model for traffic flow, in which the ASEP phase transitions can beinterpreted as jamming transitions, and find that Lee–Yang theory still applies. We showthat the parallel-update ASEP normalization can be expressed as one of severalequivalent two-dimensional lattice path problems involving weighted Dyck or Motzkinpaths. We introduce the notion of thermodynamic equivalence for such paths andshow that the robustness of the general form of the ASEP phase diagram undervarious update dynamics is a consequence of this thermodynamic equivalence.

Phase transition in a spatial Lotka-Volterra model.

Spatial evolution is investigated in a simulated system of nine competing and mutating bacterium strains, which mimics the biochemical war among bacteria capable of producing two different bacteriocins (toxins) at most. Random sequential dynamics on a square lattice is governed by very symmetrical transition rules for neighborhood invasions of sensitive strains by killers, killers by resistants, and resistants by sensitives. The community of the nine possible toxicity/resistance types undergoes a critical phase transition as the uniform transmutation rates between the types decreases below a critical value P(c) above that all the nine types of strains coexist with equal frequencies. Passing the critical mutation rate from above, the system collapses into one of three topologically identical (degenerated) states, each consisting of three strain types. Of the three possible final states each accrues with equal probability and all three maintain themselves in a self-organizing polydomain structure via cyclic invasions. Our Monte Carlo simulations support that this symmetry-breaking transition belongs to the universality class of the three-state Potts model.

A simulation study of an asymmetric exclusion model with open boundaries and random rates

Using numerical simulations, we study the asymmetric exclusion model with open boundaries, particlewise disorder and parallel dynamics. At each time step, particles are injected at the left boundary with probability , removed on the right with probability , and jump in the bulk with probability , where is a random rate associated with each injected particle . The parameter interpolates between fully parallel and random sequential dynamics. The phase diagram in the -plane displays high-density, low-density and maximum-current phases, with the first-order transition line between high- and low-density phases shifted away from the line . Within the low-density phase a platoon phase transition occurs, many features of which can be explained using exact results for asymmetric exclusion with particlewise disorder on the ring. In a certain region of parameter space the disorder induces a cusp in the current-density relation at maximum flow. Our simulations indicate that this does not affect the topology of the phase diagram, nor the familiar -decay of the density profile in the maximum-current phase.

Garden of Eden states in traffic models

We investigate the allowed configurations in the stationary state of the cellular automaton model for single-lane traffic. It is found that certain states in the configuration space cannot be reached if one uses parallel dynamics. These so-called Garden of Eden (GoE) states do not exist for random-sequential dynamics and are responsible for the strong short-ranged correlations found in parallel dynamics. By eliminating the GoE states we obtain a simple and effective approximative description of the model. For the exact solution is recovered. For this elimination leads to much higher values of the flux compared to the mean-field result which are in good agreement with Monte Carlo simulations.

Exact steady states of disordered hopping particle models with parallel and ordered sequential dynamics

A one-dimensional driven lattice gas with disorder in the particle hopping probabilities is considered. It has previously been shown that in the version of the model with random sequential updating, a phase transition occurs from a low density inhomogeneous phase to a high density congested phase. Here the steady states for both parallel (fully synchronous) updating and ordered sequential updating are solved exactly and the phase transition shown to persist in both cases. For parallel dynamics and forward ordered sequential dynamics the phase transition occurs at the same density but for backward ordered sequential dynamics it occurs at a higher density. In both cases the critical density is higher than that for random sequential dynamics. In all the models studied the steady state velocity is related to the fugacity of a Bose system suggesting a principle of minimisation of velocity. A generalisation of the dynamics where the hopping probabilities depend on the number of empty sites in front of the particles, is also solved exactly in the case of parallel updating. The models have natural interpretations as simplistic descriptions of traffic flow. The relation to more sophisticated traffic flow models is discussed.

Oscillations and chaos in neural networks: an exactly solvable model.

We consider a randomly diluted higher-order network with noise, consisting of McCulloch-Pitts neurons that interact by Hebbian-type connections. For this model, exact dynamical equations are derived and solved for both parallel and random sequential updating algorithms. For parallel dynamics, we find a rich spectrum of different behaviors including static retrieving and oscillatory and chaotic phenomena in different parts of the parameter space. The bifurcation parameters include first- and second-order neuronal interaction coefficients and a rescaled noise level, which represents the combined effects of the random synaptic dilution, interference between stored patterns, and additional background noise. We show that a marked difference in terms of the occurrence of oscillations or chaos exists between neural networks with parallel and random sequential dynamics.

Synchronous or asynchronous parallel dynamics. Which is more efficient?

The convergence time of parallel dynamics is compared analytically with that of random sequential dynamics. In parallel dynamics all elements are updated simultaneously, whereas in random sequential dynamics the elements are updated only on average at the same speed. One-dimensional and infinite-range interactions are studied. The relevance of the results to neural networks and Monte Carlo simulations is discussed.

On the relation between random coupling and sequential intramolecular dynamics

The relation between various recently studied random coupling models for intramolecular dynamics is examined. It is shown that the frequently made assumption of factorable coupling is in fact highly restrictive, that it implies strong correlations among the values of specific coupling matrix elements, and that it can sometimes lead to time-dependent behavior inconsistent with that expected for real systems. A specific random-coupling limit inconsistent with factorable coupling is rigorously shown equivalent to regarding each intermediate state to be coupled independently to its own quasicontinuum, and this limit therefore implies the expected sequential behavior. A general theory of the relation between uncorrelated matrix elements and sequential molecular dynamics (without the factorable coupling assumption) is then presented and shown to yield the expected sequential behavior under conditions of random coupling. The theory is extended by developing, within the same framework, a scheme for treating the correlated couplings as a perturbation on the uncorrelated couplings, and additional uncorrelated-coupling conditions sufficient to assure sequential behavior are obtained.