Computational inference in systems biology

Parameter inference in mathematical models of biological pathways, expressed as coupled ordinary differential equations (ODEs), is a challenging problem. The computational costs associated with repeatedly solving the ODEs are often high. Aimed at reducing this cost, new concepts using gradient matching have been proposed. This paper combines current adaptive gradient matching approaches, using Gaussian processes, with a parallel tempering scheme, and conducts a comparative evaluation with current methods used for parameter inference in ODEs

Limits and dynamics of stochastic neuronal networks with random heterogeneous delays

Realistic networks display heterogeneous transmission delays. We analyze here the limits of large stochastic multi-populations networks with stochastic coupling and random interconnection delays. We show that depending on the nature of the delays distributions, a quenched or averaged propagation of chaos takes place in these networks, and that the network equations converge towards a delayed McKean-Vlasov equation with distributed delays. Our approach is mostly fitted to neuroscience applications. We instantiate in particular a classical neuronal model, the Wilson and Cowan system, and show that the obtained limit equations have Gaussian solutions whose mean and standard deviation satisfy a closed set of coupled delay differential equations in which the distribution of delays and the noise levels appear as parameters. This allows to uncover precisely the effects of noise, delays and coupling on the dynamics of such heterogeneous networks, in particular their role in the emergence of synchronized oscillations. We show in several examples that not only the averaged delay, but also the dispersion, govern the dynamics of such networks.Comment: Corrected misprint (useless stopping time) in proof of Lemma 1 and clarified a regularity hypothesis (remark 1

Triggering synchronized oscillations through arbitrarily weak diversity in close-to-threshold excitable media

It is shown that arbitrarily weak (frozen) heterogeneity can induce global synchronized oscillations in excitable media close to threshold. The work is carried out on networks of coupled van der Pol-FitzHugh-Nagumo oscillators. The result is shown to be robust against the presence of internal dynamical noise.Comment: 4 pages (RevTeX 3 style), 5 EPS figures, submitted to Phys. Rev. E (16 aug 2001

Dynamics of lattice spins as a model of arrhythmia

We consider evolution of initial disturbances in spatially extended systems with autonomous rhythmic activity, such as the heart. We consider the case when the activity is stable with respect to very smooth (changing little across the medium) disturbances and construct lattice models for description of not-so-smooth disturbances, in particular, topological defects; these models are modifications of the diffusive XY model. We find that when the activity on each lattice site is very rigid in maintaining its form, the topological defects - vortices or spirals - nucleate a transition to a disordered, turbulent state.Comment: 17 pages, revtex, 3 figure

Steady and Stable: Numerical Investigations of Nonlinear Partial Differential Equations

Excerpt: Mathematics is a language which can describe patterns in everyday life as well as abstract concepts existing only in our minds. Patterns exist in data, functions, and sets constructed around a common theme, but the most tangible patterns are visual. Visual demonstrations can help undergraduate students connect to abstract concepts in advanced mathematical courses. The study of partial differential equations, in particular, benefits from numerical analysis and simulation

General theory of instabilities for patterns with sharp interfaces in reaction-diffusion systems

An asymptotic method for finding instabilities of arbitrary $d$-dimensional large-amplitude patterns in a wide class of reaction-diffusion systems is presented. The complete stability analysis of 2- and 3-dimensional localized patterns is carried out. It is shown that in the considered class of systems the criteria for different types of instabilities are universal. The specific nonlinearities enter the criteria only via three numerical constants of order one. The performed analysis explains the self-organization scenarios observed in the recent experiments and numerical simulations of some concrete reaction-diffusion systems.Comment: 21 pages (RevTeX), 8 figures (Postscript). To appear in Phys. Rev. E (April 1st, 1996

Heterogeneous Delays in Neural Networks

We investigate heterogeneous coupling delays in complex networks of excitable elements described by the FitzHugh-Nagumo model. The effects of discrete as well as of uni- and bimodal continuous distributions are studied with a focus on different topologies, i.e., regular, small-world, and random networks. In the case of two discrete delay times resonance effects play a major role: Depending on the ratio of the delay times, various characteristic spiking scenarios, such as coherent or asynchronous spiking, arise. For continuous delay distributions different dynamical patterns emerge depending on the width of the distribution. For small distribution widths, we find highly synchronized spiking, while for intermediate widths only spiking with low degree of synchrony persists, which is associated with traveling disruptions, partial amplitude death, or subnetwork synchronization, depending sensitively on the network topology. If the inhomogeneity of the coupling delays becomes too large, global amplitude death is induced

Dynamical mean-field theory of spiking neuron ensembles: response to a single spike with independent noises

Dynamics of an ensemble of $N$-unit FitzHugh-Nagumo (FN) neurons subject to white noises has been studied by using a semi-analytical dynamical mean-field (DMF) theory in which the original $2 N$-dimensional {\it stochastic} differential equations are replaced by 8-dimensional {\it deterministic} differential equations expressed in terms of moments of local and global variables. Our DMF theory, which assumes weak noises and the Gaussian distribution of state variables, goes beyond weak couplings among constituent neurons. By using the expression for the firing probability due to an applied single spike, we have discussed effects of noises, synaptic couplings and the size of the ensemble on the spike timing precision, which is shown to be improved by increasing the size of the neuron ensemble, even when there are no couplings among neurons. When the coupling is introduced, neurons in ensembles respond to an input spike with a partial synchronization. DMF theory is extended to a large cluster which can be divided into multiple sub-clusters according to their functions. A model calculation has shown that when the noise intensity is moderate, the spike propagation with a fairly precise timing is possible among noisy sub-clusters with feed-forward couplings, as in the synfire chain. Results calculated by our DMF theory are nicely compared to those obtained by direct simulations. A comparison of DMF theory with the conventional moment method is also discussed.Comment: 29 pages, 2 figures; augmented the text and added Appendice

Stochastic Resonance in Nonpotential Systems

We propose a method to analytically show the possibility for the appearance of a maximum in the signal-to-noise ratio in nonpotential systems. We apply our results to the FitzHugh-Nagumo model under a periodic external forcing, showing that the model exhibits stochastic resonance. The procedure that we follow is based on the reduction to a one-dimensional dynamics in the adiabatic limit, and in the topology of the phase space of the systems under study. Its application to other nonpotential systems is also discussed.Comment: Submitted to Phys. Rev.

Emergent global oscillations in heterogeneous excitable media: The example of pancreatic beta cells

Using the standard van der Pol-FitzHugh-Nagumo excitable medium model I demonstrate a novel generic mechanism, diversity, that provokes the emergence of global oscillations from individually quiescent elements in heterogeneous excitable media. This mechanism may be operating in the mammalian pancreas, where excitable beta cells, quiescent when isolated, are found to oscillate when coupled despite the absence of a pacemaker region.Comment: See home page http://lec.ugr.es/~julya