Spatially distributed stochastic systems: Equation-free and equation-assisted preconditioned computations

Qiao, Liang; Erban, Radek; Kelley, C. T.; Kevrekidis, Ioannis G.

doi:10.1063/1.2372492

Cited by 18 publications

(20 citation statements)

References 32 publications

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“…It is worth noting that for the case of state-dependent noise, the local maxima of the probability density do not exactly correspond to zeros of µ(V ) and one needs instead to find fixed points of the effective potential given in (9) (zeros of the right hand side of (9) differentiated with respect to V ). For the scalar case, one can implement secant-type iterative methods to converge to the zeros of the appropriate function using function estimates only; for the multivariable case matrix-free iterative techniques like Newton-Krylov GMRES can be used (see [22,33]). In the cases we study µ(V ) is well approximated by a cubic function; we take advantage of this simplification by estimating µ(V ) for just four values of V , uniquely defining this cubic whose zeros we can then easily find.…”

Section: Bifurcationsmentioning

confidence: 99%

Coarse-grained dynamics of an activity bump in a neural field model

2007

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Abstract.We study a stochastic nonlocal PDE, arising in the context of modelling spatially distributed neural activity, which is capable of sustaining stationary and moving spatially-localized "activity bumps". This system is known to undergo a pitchfork bifurcation in bump speed as a parameter (the strength of adaptation) is changed; yet increasing the noise intensity effectively slowed the motion of the bump. Here we revisit the system from the point of view of describing the high-dimensional stochastic dynamics in terms of the effective dynamics of a single scalar "coarse" variable. We show that such a reduced description in the form of an effective Langevin equation characterized by a double-well potential is quantitatively successful. The effective potential can be extracted using short, appropriately-initialized bursts of direct simulation. We demonstrate this approach in terms of (a) an experience-based "intelligent" choice of the coarse observable and (b) an observable obtained through data-mining direct simulation results, using a diffusion map approach.

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Section: Bifurcationsmentioning

confidence: 99%

Coarse-grained dynamics of an activity bump in a neural field model

2007

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“…They have also been used to examine stochastic simulations of (bio)chemical reaction networks in simple [16], [17], [29]–[31] and somewhat more complex cases [18]–[20]. Here we extend this work, by applying equation-free techniques to Gillespie algorithm simulations of a realistic biochemical reaction network of moderate complexity, which represents a significant computational challenge to the method.…”

Section: Resultsmentioning

confidence: 94%

Equation-Free Analysis of Two-Component System Signalling Model Reveals the Emergence of Co-Existing Phenotypes in the Absence of Multistationarity

2012

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Phenotypic differences of genetically identical cells under the same environmental conditions have been attributed to the inherent stochasticity of biochemical processes. Various mechanisms have been suggested, including the existence of alternative steady states in regulatory networks that are reached by means of stochastic fluctuations, long transient excursions from a stable state to an unstable excited state, and the switching on and off of a reaction network according to the availability of a constituent chemical species. Here we analyse a detailed stochastic kinetic model of two-component system signalling in bacteria, and show that alternative phenotypes emerge in the absence of these features. We perform a bifurcation analysis of deterministic reaction rate equations derived from the model, and find that they cannot reproduce the whole range of qualitative responses to external signals demonstrated by direct stochastic simulations. In particular, the mixed mode, where stochastic switching and a graded response are seen simultaneously, is absent. However, probabilistic and equation-free analyses of the stochastic model that calculate stationary states for the mean of an ensemble of stochastic trajectories reveal that slow transcription of either response regulator or histidine kinase leads to the coexistence of an approximate basal solution and a graded response that combine to produce the mixed mode, thus establishing its essential stochastic nature. The same techniques also show that stochasticity results in the observation of an all-or-none bistable response over a much wider range of external signals than would be expected on deterministic grounds. Thus we demonstrate the application of numerical equation-free methods to a detailed biochemical reaction network model, and show that it can provide new insight into the role of stochasticity in the emergence of phenotypic diversity.

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“…Time-scale separation decomposes the model into subcomponents that operate on different time scales thus allowing the slower time scale subcomponents to be simulated less often. This separation might be performed due to prior knowledge about multiple scales in the system, but there are also attempts to determine this separation from the system dynamics [4,5]. Dynamic agent compression aggregates sets of homogeneous agents into a container object which then acts for the agents as a whole [6,7].…”

Section: Introductionmentioning

confidence: 99%

Integrative multicellular biological modeling: a case study of 3D epidermal development using GPU algorithms

et al. 2010

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Background: Simulation of sophisticated biological models requires considerable computational power. These models typically integrate together numerous biological phenomena such as spatially-explicit heterogeneous cells, cell-cell interactions, cell-environment interactions and intracellular gene networks. The recent advent of programming for graphical processing units (GPU) opens up the possibility of developing more integrative, detailed and predictive biological models while at the same time decreasing the computational cost to simulate those models. Results: We construct a 3D model of epidermal development and provide a set of GPU algorithms that executes significantly faster than sequential central processing unit (CPU) code. We provide a parallel implementation of the subcellular element method for individual cells residing in a lattice-free spatial environment. Each cell in our epidermal model includes an internal gene network, which integrates cellular interaction of Notch signaling together with environmental interaction of basement membrane adhesion, to specify cellular state and behaviors such as growth and division. We take a pedagogical approach to describing how modeling methods are efficiently implemented on the GPU including memory layout of data structures and functional decomposition. We discuss various programmatic issues and provide a set of design guidelines for GPU programming that are instructive to avoid common pitfalls as well as to extract performance from the GPU architecture. Conclusions: We demonstrate that GPU algorithms represent a significant technological advance for the simulation of complex biological models. We further demonstrate with our epidermal model that the integration of multiple complex modeling methods for heterogeneous multicellular biological processes is both feasible and computationally tractable using this new technology. We hope that the provided algorithms and source code will be a starting point for modelers to develop their own GPU implementations, and encourage others to implement their modeling methods on the GPU and to make that code available to the wider community.

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Spatially distributed stochastic systems: Equation-free and equation-assisted preconditioned computations

Cited by 18 publications

References 32 publications

Coarse-grained dynamics of an activity bump in a neural field model

Coarse-grained dynamics of an activity bump in a neural field model

Equation-Free Analysis of Two-Component System Signalling Model Reveals the Emergence of Co-Existing Phenotypes in the Absence of Multistationarity

Integrative multicellular biological modeling: a case study of 3D epidermal development using GPU algorithms

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