Stochastic Differential Equation Model for Cerebellar Granule Cell Excitability

Event Abstract Back to Event Stochastic modeling of neuronal signaling Jukka Intosalmi1*, Tiina Manninen1, Keijo Ruohonen1 and Marja-Leena Linne1 1 Tampere University of Technology, Finland The time evolution of biochemical systems in neurons is traditionally modeled using deterministic ordinary differential equations (ODEs). Chemical reactions, however, are random in nature, and the deterministic approach is valid only for a restricted class of systems. Stochastic models take random fluctuations into account and are thus more realistic. Biochemical reactions can be modeled stochastically using numerous different methods. An ideal model would have the following three important properties. First, the model should be as realistic as possible, second, the mathematical method should be easily implementable as a computer algorithm, and third, the algorithm should be computationally effective. Naturally, all these three conditions cannot be fulfilled at the same time. Some realistic modeling approaches can be derived directly from chemical kinetics without making any approximations. Such approaches are called exact. A good example of an exact modeling approach is the stochastic simulation algorithm (SSA) developed by Gillespie. The SSA is applicable when the molecular populations in the system are small, but it becomes computationally inefficient when the numbers of molecules increase. In order to construct stochastic models that can be effectively simulated, new mathematical approaches have to be explored. As an approximate method also stochastic differential equations (SDEs) have been considered a promising way to model biochemical reactions stochastically. The SDE approach is attractive especially if we consider a system for which the SSA is computationally inefficient and the traditional deterministic ODE approach cannot be used as a good approximation. SDE models treat the chemical populations as real numbers and the construction of the model is based on the law of mass action, similarly as the construction of the traditional deterministic modeling approach. The SDE model, however, takes the random fluctuations into account by describing the time evolution of the system with the stochastic Itô process instead of the deterministic set of ODEs. The Itô process is basically a set of coupled stochastic differential equations. Although SDE modeling usually leads to equations that cannot be solved analytically, the solutions can be approximated numerically using different numerical integration methods. Thus, SDEs provide a mathematically rigorous way to construct models that can be simulated effectively. In order to investigate how well the SDE approach models biochemical systems, the results have to be compared with experimental data or with the simulation results from some exact simulation procedure. In this study, we use the methodology of spectral analysis to compare the simulation results of the SDE model and the SSA. As case studies, we consider e.g. calcium binding and protein kinase C (PKC) signal transduction pathway. We investigate the nature of noise in different modeling approaches and try to interpret the meaning of the noise in real biological systems. The main goal of our study is to find the most essential components and parameters in the SDE model and to adjust them so that the model is capable of giving similar results as the SSA. The simulations are carried out using the distributed computing resources (GRID) provided by Techila Technologies Ltd. Conference: Neuroinformatics 2008, Stockholm, Sweden, 7 Sep - 9 Sep, 2008. Presentation Type: Poster Presentation Topic: Computational Neuroscience Citation: Intosalmi J, Manninen T, Ruohonen K and Linne M (2008). Stochastic modeling of neuronal signaling. Front. Neuroinform. Conference Abstract: Neuroinformatics 2008. doi: 10.3389/conf.neuro.11.2008.01.023 Copyright: The abstracts in this collection have not been subject to any Frontiers peer review or checks, and are not endorsed by Frontiers. They are made available through the Frontiers publishing platform as a service to conference organizers and presenters. The copyright in the individual abstracts is owned by the author of each abstract or his/her employer unless otherwise stated. Each abstract, as well as the collection of abstracts, are published under a Creative Commons CC-BY 4.0 (attribution) licence (https://creativecommons.org/licenses/by/4.0/) and may thus be reproduced, translated, adapted and be the subject of derivative works provided the authors and Frontiers are attributed. For Frontiers’ terms and conditions please see https://www.frontiersin.org/legal/terms-and-conditions. Received: 28 Jul 2008; Published Online: 28 Jul 2008. * Correspondence: Jukka Intosalmi, Tampere University of Technology, Tampere, Finland, jukka.intosalmi@tut.fi Login Required This action requires you to be registered with Frontiers and logged in. To register or login click here. Abstract Info Abstract The Authors in Frontiers Jukka Intosalmi Tiina Manninen Keijo Ruohonen Marja-Leena Linne Google Jukka Intosalmi Tiina Manninen Keijo Ruohonen Marja-Leena Linne Google Scholar Jukka Intosalmi Tiina Manninen Keijo Ruohonen Marja-Leena Linne PubMed Jukka Intosalmi Tiina Manninen Keijo Ruohonen Marja-Leena Linne Related Article in Frontiers Google Scholar PubMed Abstract Close Back to top Javascript is disabled. Please enable Javascript in your browser settings in order to see all the content on this page.

Mathematical models are routinely calibrated to experimental data, with goals ranging from building predictive models to quantifying parameters that cannot be measured. Whether or not reliable parameter estimates are obtainable from the available data can easily be overlooked. Such issues of parameter identifiability have important ramifications for both the predictive power of a model, and the mechanistic insight that can be obtained. Identifiability analysis is well-established for deterministic, ordinary differential equation (ODE) models, but there are no commonly adopted methods for analysing identifiability in stochastic models. We provide an accessible introduction to identifiability analysis and demonstrate how existing ideas for analysis of ODE models can be applied to stochastic differential equation (SDE) models through four practical case studies. To assess structural identifiability, we study ODEs that describe the statistical moments of the stochastic process using open-source software tools. Using practically motivated synthetic data and Markov chain Monte Carlo methods, we assess parameter identifiability in the context of available data. Our analysis shows that SDE models can often extract more information about parameters than deterministic descriptions. All code used to perform the analysis is available on Github.

Background: Growth trajectories are highly variable between children, making epidemiological analyses challenging both to the identification of malnutrition interventions at the population level and also risk assessment at individual level. We introduce stochastic differential equation (SDE) models into child growth research. SDEs describe flexible dynamic processes comprising: drift - gradual smooth changes – such as physiology or gut microbiome, and diffusion - sudden perturbations, such as illness or infection. Methods: We present a case study applying SDE models to child growth trajectory data from the Haydom, Tanzania and Venda, South Africa sites within the MAL-ED cohort. These data comprise n=460 children aged 0-24 months. A comparison with classical curve fitting (linear mixed models) is also presented. Results: The SDE models offered a wide range of new flexible shapes and parameterizations compared to classical additive models, with performance as good or better than standard approaches. The predictions from the SDE models suggest distinct longitudinal clusters that form distinct ‘streams’ hidden by the large between-child variability. Conclusions: Using SDE models to predict future growth trajectories revealed new insights in the observed data, where trajectories appear to cluster together in bands, which may have a future risk assessment application. SDEs offer an attractive approach for child growth modelling and potentially offer new insights.

It is shown how different but equivalent Itô stochastic differential equation models of random dynamical systems can be constructed. Advantages and disadvantages of the different models are described. Stochastic differential equation models are derived for problems in chemistry, textile engineering, and epidemiology. Computational comparisons are made between the different stochastic models.

Stochastic Differential Equation (SDE) models are often used to model the dynamics of complex biological systems. The stochastic nature of these models means that some behaviors are more likely than others. It is often the case that a model's primary purpose is to study rare but interesting or important behaviors, such as the formation of a tumor, or the failure of a cyber-physical system. Unfortunately, due to the limited availability of analytic methods for SDEs, stochastic simulations are the most common means for estimating (or bounding) the probability of rare behaviors. Naturally, the cost of stochastic simulations increases with the rarity of the behavior under consideration. To address this problem, we introduce a new algorithm, RESERCHE, that is specifically designed to quantify the likelihood of rare but interesting behaviors in SDE models. Our approach relies on the use of temporal logics for specifying rare behaviors of possible interest, and on the ability of bit-vector decision procedures to reason exhaustively about fixed precision arithmetic. We also compute the probability of an observed behavior under the assumption of Gaussian noise.

Stochastic differential equation (SDE) models are the foundation for pricing and hedging financial derivatives. The drift and volatility functions in SDE models are typically chosen to be algebraic functions with a small number ( < 5 ) of parameters which can be calibrated to market data. A more flexible approach is to use neural networks to model the drift and volatility functions, which provides more degrees-of-freedom to match observed market data. Training of models requires optimizing over an SDE, which is computationally challenging. For European options, we develop a fast stochastic gradient descent (SGD) algorithm for training the neural network-SDE model. Our SGD algorithm uses two independent SDE paths to obtain an unbiased estimate of the direction of steepest descent. For American options, we optimize over the corresponding Kolmogorov partial differential equation (PDE). The neural network appears as coefficient functions in the PDE. Models are trained on large datasets (many contracts), requiring either large simulations (many Monte Carlo samples for the stock price paths) or large numbers of PDEs (a PDE must be solved for each contract). Numerical results are presented for real market data including S&P 500 index options, S&P 100 index options, and single-stock American options. The neural-network-based SDE models are compared against the Black-Scholes model, the Dupire's local volatility model, and the Heston model. Models are evaluated in terms of how accurate they are at pricing out-of-sample financial derivatives, which is a core task in derivative pricing at financial institutions. Specifically, we calibrate a neural network-SDE model to market data for a financial derivative on an asset with price S t with a payoff function g ( s ) , and we then evaluate its generalization accuracy for a financial derivative on the same asset S t but with a different payoff function f ( s ) . In addition to comparing out-of-sample pricing accuracy, we evaluate the hedging performance of the neural network-SDE model.

ABSTRACTTwo Itô stochastic differential equation (SDE) systems are constructed for a Susceptible-Infected-Susceptible epidemic model with temporary vaccination. A constant number of new members enter the population and total size of the population is variable. Some conditions for disease extinction in the stochastic models are established and compared with conditions in deterministic one. It is shown that the two stochastic models are equivalent in the sense that their solutions come from same distribution. In addition, the SDE models are simulated and the equivalence of the two stochastic models is confirmed by numerical examples. The probability distribution for extinction is also obtained numerically, provided there exists a probability for disease persistence whereas the expected duration of epidemic is acquired when extinction occurs with probability 1.

Modeling and simulation of reactor kinetics under the point model approximation can be roughly divided into two categories: deterministic models, which consider the flux and other state variables (delayed neutrons, temperature, etc.) as time-dependent functions, and stochastic models, which study the flux as a stochastic process. Since the neutron reactions and fission chain are stochastic by nature, the stochastic models are a better reflection of the physical system. On the other hand, due to the complexity of the stochastic models and the large number of reactions, most common analysis is done in the deterministic setting via the widely spread point reactor kinetic (PRK) equation. In the most basic setting, when we assume that all reaction rates are constant in time, the PRK can be generalized to the stochastic setting via the master equation (Chapman–Kolmogorov equation). The resulting model is often implemented in reactor noise experiments and the Feynman-α method, and the interpretation of the deterministic model as a mean field approximation of the stochastic model is very well understood. Once we include nonlinear feedback into the equation (typically due to temperature influence on the cross-sections), the stochastic modeling becomes much more delicate, due to (at least) two issues: First, both energy and temperature are continuous, and thus the applicability of the master equation questionable. Second, in a nonlinear setting, the mean field approximation of the stochastic model is not equivalent to the PRK equation, and the two are unified only if we assume implicitly that the covariance of the power and the temperature is nullified, which is bound to create a systematic error. Thus, modeling and simulation of stochastic fission chains and temperature represent a central topic in reactor control and measurements. One of the most classic studies on stochastic fission chains in a reactor with thermal feedback is that of Harris (1958), where a discretization of the energy bins is performed to allow use of the master equation technique. The outline of the present study is to construct and study the properties of an Ito-type stochastic differential equation (SDE) counterpart to the classic Harris model. The contribution of the study will be divided into three consecutive steps: First, we will explicitly derive from first principles the parameters of the SDE. Then we will compute the covariance between the system variables and prove that the Harris model and the SDE model have a very strong common feature: They share the exact same second-moment equations. Finally, we will demonstrate the applicability of the SDE for a non-analog Monte Carlo scheme for sampling distribution of the different stochastic variables.

The problem of computing the maximum likelihood estimate of the parameters of a specific class of stochastic differential equation (SDE) models with linear drift whose sample paths are observed at discrete time points is considered. This estimate is obtained as in Cleur and Manfredi (1999) by discretizing the explicit expressions for the estimates which maximize the likelihood function in continuous time, by discretizing the likelihood function through a quadrature approximation before maximizing it, and by maximizing the likelihood function of the Euler scheme approximation to the underlying continuous process. Simulation results indicate that, for the constellation of parameter values considered, all three approaches lead to very similar results.

Parameter and uncertainty estimation in stochastic differential equation models with multi-rate data and nonstationary disturbances

Two new stochastic epidemic models, a continuous-time Markov chain model and a stochastic differential equation model, are formulated. These are based on a deterministic model that includes vaccination and is applicable to pertussis. For some parameter values, the deterministic model exhibits a backward bifurcation if the vaccine is imperfect. Thus a region of bistability exists in a subset of parameter space. The dynamics of the stochastic epidemic models are investigated in this region of bistability, and compared with those of the deterministic model. In this region the probability distribution associated with the infective population exhibits bimodality with one mode at the disease-free equilibrium and the other at the larger endemic equilibrium. For population sizes N=/>1000, the deterministic and stochastic models agree, but for small population sizes the stochastic models indicate that the backward bifurcation may have little effect on the disease dynamics.

In this study, we propose a new method that is useful for estimating unknown parameter values of stochastic differential equation (SDE) models, based on probability density function (PDF) data measured from random dynamical systems. As our method does not require explicit description of PDF, it can be applied to the SDE models even when their PDFs are hardly derived in explicit forms due to multiplicative-noise terms, nonlinear terms, and so on. Therefore, our method is expected to provide a versatile tool to dynamically parameterize measured PDF data. In our proposed method, it is assumed that a measured PDF is obtained from a random dynamical system whose structure is described by a known SDE model with unknown parameter values. With the help of It\^o calculus, the Fokker-Planck equation (FPE) is derived from the SDE model. The measured PDF and a candidate of parameter values are substituted into the FPE to calculate a FPE residual. Our method is applied to two random vibration systems. Their FPE residuals tend to zero as the parameter values tend to exact values, showing that our proposed FPE residual can be utilized for unknown parameter estimation of SDE models.

Analysis of deterministic and stochastic mathematical models with resistant bacteria and bacteria debris for bacteriophage dynamics

&amp;lt;p&amp;gt;A complex system is a system composed of highly interconnected components in which the collective property of an underlying system cannot be described by dynamical behavior of the individual parts. Typically, complex systems are governed by nonlinear interactions and intricate fluctuations, thus to retrieve dynamics of a system, it is required to characterize and asses interactions between deterministic tendencies and random fluctuations.&amp;amp;#160;&amp;lt;/p&amp;gt;&amp;lt;p&amp;gt;For systems with large numbers of degrees of freedom, interacting across various time scales, deriving time-evolution equations from data is computationally expensive. A possible way to circumvent this problem is to isolate a small number of relatively slow degrees of freedom that may suffice to characterize the underlying dynamics and solve the governing motion equation for the reduced-dimension system in the framework of stochastic differential equations(SDEs). &amp;amp;#160;For some specific example settings, we have studied the performance of three stochastic dimension-reduction methods (Langevin equation(LE), generalized Langevin Equation(GLE) and Empirical Model Reduction(EMR)) to model various synthetic and real-world time series. In this study corresponding numerical simulations of all models have been examined by probability distribution function(PDF) and Autocorrelation function(ACF) of the average simulated time series as statistical benchmarks for assessing the differnt models' performance.&amp;amp;#160;&amp;lt;/p&amp;gt;&amp;lt;p&amp;gt;First we reconstruct the Ni&amp;amp;#241;o-3 monthly sea surface temperature (SST) indices averages across (5&amp;amp;#176;N&amp;amp;#8211;5&amp;amp;#176;S, 150&amp;amp;#176;&amp;amp;#8211;90&amp;amp;#176;W) from 1891 to 2015 using the three aforementioned stochastic models. We demonstrate that all these considered models can reproduce the same skewed and heavy-tailed distributions of Ni&amp;amp;#241;o-3 SST, comparing ACFs, GLE exhibits a tendency towards achieving a higher accuracy than LE and EMR. A particular challenge for deriving the underlying dynamics of complex systems from data is given by situations of abrupt transitions between alternative states. We show how the Kramers-Moyal approach to derive drift and diffusion terms for LEs can help in such situations. A prominent example of such 'Tipping Events' is given by the Dansgaard-Oeschger events during previous glacial intervals. We attempt to obtain the statistical properties of high-resolution, 20yr average, &amp;amp;#948;&amp;lt;sup&amp;gt;18&amp;lt;/sup&amp;gt;O and Ca&amp;lt;sup&amp;gt;+&amp;lt;/sup&amp;gt;&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt;&amp;amp;#160;collected from the same ice core from the NGRIP on the GICC05 time scale. Through extensive analyses of various systems, our results signify that stochastic differential equation models considering memory effects are comparatively better approaches for understanding &amp;amp;#160;complex systems.&amp;lt;/p&amp;gt;&amp;lt;p&amp;gt;&amp;amp;#160;&amp;lt;/p&amp;gt;

Following its introduction over 30 years ago, the Markov cohort state-transition model has been used extensively to model population trajectories over time in health decision modeling and cost-effectiveness analysis studies. We recently showed that a cohort model represents the average of a continuous-time stochastic process on a multidimensional integer lattice governed by a master equation, which represents the time-evolution of the probability function of an integer-valued random vector. By leveraging this theoretical connection, this study introduces an alternative modeling method using a stochastic differential equation (SDE) approach, which captures not only the mean behavior but also the variance of the population process. We show the derivation of an SDE model from first principles, describe an algorithm to construct an SDE and solve the SDE via simulation for use in practice, and demonstrate the two applications of an SDE in detail. The first example demonstrates that the population trajectories, and their mean and variance, from the SDE and other commonly used methods in decision modeling match. The second example shows that users can readily apply the SDE method in their existing works without the need for additional inputs beyond those required for constructing a conventional cohort model. In addition, the second example demonstrates that the SDE model is superior to a microsimulation model in terms of computational speed. In summary, an SDE model provides an alternative modeling framework which includes information on variance, can accommodate for time-varying parameters, and is computationally less expensive than a microsimulation for a typical cohort modeling problem.