Microscale models form a broad class of computational models that simulate fine-scale details, in contrast with macroscale models, which amalgamate details into select categories. [2] [3] Microscale and macroscale models can be used together to understand different aspects of the same problem.
Macroscale models can include ordinary, partial, and integro-differential equations, where categories and flows between the categories determine the dynamics, or may involve only algebraic equations. An abstract macroscale model may be combined with more detailed microscale models. Connections between the two scales are related to multiscale modeling. One mathematical technique for multiscale modeling of nanomaterials is based upon the use of multiscale Green's function.
In contrast, microscale models can simulate a variety of details, such as individual bacteria in biofilms, [4] individual pedestrians in simulated neighborhoods, [5] individual light beams in ray-tracing imagery, [6] individual houses in cities, [7] fine-scale pores and fluid flow in batteries, [8] fine-scale compartments in meteorology, [9] fine-scale structures in particulate systems, [10] and other models where interactions among individuals and background conditions determine the dynamics.
Discrete-event models, individual-based models, and agent-based models are special cases of microscale models. However, microscale models do not require discrete individuals or discrete events. Fine details on topography, buildings, and trees can add microscale detail to meteorological simulations and can connect to what is called mesoscale models in that discipline. [9] Square-meter-sized landscape resolution available from lidar images allows water flow across land surfaces to be modeled, for example, rivulets and water pockets, using gigabyte-sized arrays of detail. [11] Models of neural networks may include individual neurons but may run in continuous time and thereby lack precise discrete events. [12]
Ideas for computational microscale models arose in the earliest days of computing and were applied to complex systems that could not accurately be described by standard mathematical forms.
Two themes emerged in the work of two founders of modern computation around the middle of the 20th century. First, pioneer Alan Turing used simplified macroscale models to understand the chemical basis of morphogenesis, but then proposed and used computational microscale models to understand the nonlinearities and other conditions that would arise in actual biological systems. [13] Second, pioneer John von Neumann created a cellular automaton to understand the possibilities for self-replication of arbitrarily complex entities, [14] which had a microscale representation in the cellular automaton but no simplified macroscale form. This second theme is taken to be part of agent-based models, where the entities ultimately can be artificially intelligent agents operating autonomously.
By the last quarter of the 20th century, computational capacity had grown so far [15] [16] that up to tens of thousands of individuals or more could be included in microscale models, and that sparse arrays could be applied to also achieve high performance. [17] Continued increases in computing capacity allowed hundreds of millions of individuals to be simulated on ordinary computers with microscale models by the early 21st century.
The term "microscale model" arose later in the 20th century and now appears in the literature of many branches of physical and biological science. [5] [7] [8] [9] [18]
Figure 1 represents a fundamental macroscale model: population growth in an unlimited environment. Its equation is relevant elsewhere, such as compounding growth of capital in economics or exponential decay in physics. It has one amalgamated variable, , the number of individuals in the population at some time . It has an amalgamated parameter , the annual growth rate of the population, calculated as the difference between the annual birth rate and the annual death rate . Time can be measured in years, as shown here for illustration, or in any other suitable unit.
The macroscale model of Figure 1 amalgamates parameters and incorporates a number of simplifying approximations:
These approximations of the macroscale model can all be refined in analogous microscale models. On the first approximation listed above—that birth and death rates are constant—the macroscale model of Figure 1 is exactly the mean of a large number of stochastic trials with the growth rate fluctuating randomly in each instance of time. [19] Microscale stochastic details are subsumed into a partial differential diffusion equation and that equation is used to establish the equivalence.
To relax other assumptions, researchers have applied computational methods. Figure 2 is a sample computational microscale algorithm that corresponds to the macroscale model of Figure 1. When all individuals are identical and mutations in birth and death rates are disabled, the microscale dynamics closely parallel the macroscale dynamics (Figures 3A and 3B). The slight differences between the two models arise from stochastic variations in the microscale version not present in the deterministic macroscale model. These variations will be different each time the algorithm is carried out, arising from intentional variations in random number sequences.
When not all individuals are identical, the microscale dynamics can differ significantly from the macroscale dynamics, simulating more realistic situations than can be modeled at the macroscale (Figures 3C and 3D). The microscale model does not explicitly incorporate the differential equation, though for large populations it simulates it closely. When individuals differ from one another, the system has a well-defined behavior but the differential equations governing that behavior are difficult to codify. The algorithm of Figure 2 is a basic example of what is called an equation-free model. [20]
When mutations are enabled in the microscale model (), the population grows more rapidly than in the macroscale model (Figures 3C and 3D). Mutations in parameters allow some individuals to have higher birth rates and others to have lower death rates, and those individuals contribute proportionally more to the population. All else being equal, the average birth rate drifts to higher values and the average death rate drifts to lower values as the simulation progresses. This drift is tracked in the data structures named beta and delta of the microscale algorithm of Figure 2.
The algorithm of Figure 2 is a simplified microscale model using the Euler method. Other algorithms such as the Gillespie method [21] and the discrete event method [17] are also used in practice. Versions of the algorithm in practical use include efficiencies such as removing individuals from consideration once they die (to reduce memory requirements and increase speed) and scheduling stochastic events into the future (to provide a continuous time scale and to further improve speed). [17] Such approaches can be orders of magnitude faster.
The complexity of systems addressed by microscale models leads to complexity in the models themselves, and the specification of a microscale model can be tens or hundreds of times larger than its corresponding macroscale model. (The simplified example of Figure 2 has 25 times as many lines in its specification as does Figure 1.) Since bugs occur in computer software and cannot completely be removed by standard methods such as testing, [22] and since complex models often are neither published in detail nor peer-reviewed, their validity has been called into question. [23] Guidelines on best practices for microscale models exist [24] but no papers on the topic claim a full resolution of the problem of validating complex models.
Computing capacity is reaching levels where populations of entire countries or even the entire world are within the reach of microscale models, and improvements in the census and travel data allow further improvements in parameterizing such models. Remote sensors from Earth-observing satellites and ground-based observatories such as the National Ecological Observatory Network (NEON) provide large amounts of data for calibration. Potential applications range from predicting and reducing the spread of disease to helping understand the dynamics of the earth.
Figure 1.One of the simplest of macroscale models: an ordinary differential equation describing continuous exponential growth. is the size of the population at the time and is the rate of change through time in a single dimension . is the initial population, is the birth rate per time unit, and is a death rate per time unit. At the left is the differential form; at the right is the explicit solution in terms of standard mathematical functions, which follows in this case from the differential form. Almost all macroscale models are more complex than this example, in that they have multiple dimensions, lack explicit solutions in terms of standard mathematical functions, and must be understood from their differential forms.
Figure 2.A basic algorithm applying the Euler method to an individual-based model. See text for discussion. The algorithm, represented in pseudocode, begins with invocation of procedure , which uses the data structures to carry out the simulation according to the numbered steps described at the right. It repeatedly invokes function , which returns its parameter perturbed by a random number drawn from a uniform distribution with standard deviation defined by the variable . (The square root of 12 appears because the standard deviation of a uniform distribution includes that factor.) Function in the algorithm is assumed to return a uniformly distributed random number . The data are assumed to be reset to their initial values on each invocation of .
Figure 3.Graphical comparison of the dynamics of macroscale and microscale simulations of Figures 1 and 2, respectively.
In mathematics, time-scale calculus is a unification of the theory of difference equations with that of differential equations, unifying integral and differential calculus with the calculus of finite differences, offering a formalism for studying hybrid systems. It has applications in any field that requires simultaneous modelling of discrete and continuous data. It gives a new definition of a derivative such that if one differentiates a function defined on the real numbers then the definition is equivalent to standard differentiation, but if one uses a function defined on the integers then it is equivalent to the forward difference operator.
The Lotka–Volterra equations, also known as the Lotka–Volterra predator–prey model, are a pair of first-order nonlinear differential equations, frequently used to describe the dynamics of biological systems in which two species interact, one as a predator and the other as prey. The populations change through time according to the pair of equations:
In numerical analysis, stochastic tunneling (STUN) is an approach to global optimization based on the Monte Carlo method-sampling of the function to be objective minimized in which the function is nonlinearly transformed to allow for easier tunneling among regions containing function minima. Easier tunneling allows for faster exploration of sample space and faster convergence to a good solution.
In plasma physics, the particle-in-cell (PIC) method refers to a technique used to solve a certain class of partial differential equations. In this method, individual particles in a Lagrangian frame are tracked in continuous phase space, whereas moments of the distribution such as densities and currents are computed simultaneously on Eulerian (stationary) mesh points.
Compartmental models are a very general modelling technique. They are often applied to the mathematical modelling of infectious diseases. The population is assigned to compartments with labels – for example, S, I, or R,. People may progress between compartments. The order of the labels usually shows the flow patterns between the compartments; for example SEIS means susceptible, exposed, infectious, then susceptible again.
The Level-set method (LSM) is a conceptual framework for using level sets as a tool for numerical analysis of surfaces and shapes. LSM can perform numerical computations involving curves and surfaces on a fixed Cartesian grid without having to parameterize these objects. LSM makes it easier to perform computations on shapes with sharp corners and shapes that change topology. These characteristics make LSM effective for modeling objects that vary in time, such as an airbag inflating or a drop of oil floating in water.
Multiscale modeling or multiscale mathematics is the field of solving problems that have important features at multiple scales of time and/or space. Important problems include multiscale modeling of fluids, solids, polymers, proteins, nucleic acids as well as various physical and chemical phenomena.
In probability theory, the Gillespie algorithm generates a statistically correct trajectory of a stochastic equation system for which the reaction rates are known. It was created by Joseph L. Doob and others, presented by Dan Gillespie in 1976, and popularized in 1977 in a paper where he uses it to simulate chemical or biochemical systems of reactions efficiently and accurately using limited computational power. As computers have become faster, the algorithm has been used to simulate increasingly complex systems. The algorithm is particularly useful for simulating reactions within cells, where the number of reagents is low and keeping track of every single reaction is computationally feasible. Mathematically, it is a variant of a dynamic Monte Carlo method and similar to the kinetic Monte Carlo methods. It is used heavily in computational systems biology.
The kinetic Monte Carlo (KMC) method is a Monte Carlo method computer simulation intended to simulate the time evolution of some processes occurring in nature. Typically these are processes that occur with known transition rates among states. It is important to understand that these rates are inputs to the KMC algorithm, the method itself cannot predict them.
A stochastic simulation is a simulation of a system that has variables that can change stochastically (randomly) with individual probabilities.
In computational biology, a Cellular Potts model is a computational model of cells and tissues. It is used to simulate individual and collective cell behavior, tissue morphogenesis and cancer development. CPM describes cells as deformable objects with a certain volume, that can adhere to each other and to the medium in which they live. The formalism can be extended to include cell behaviours such as cell migration, growth and division, and cell signalling. The first CPM was proposed for the simulation of cell sorting by François Graner and James A. Glazier as a modification of a large-Q Potts model. CPM was then popularized by Paulien Hogeweg for studying morphogenesis. Although the model was developed to describe biological cells, it can also be used to model individual parts of a biological cell, or even regions of fluid.
In computational chemistry, a constraint algorithm is a method for satisfying the Newtonian motion of a rigid body which consists of mass points. A restraint algorithm is used to ensure that the distance between mass points is maintained. The general steps involved are: (i) choose novel unconstrained coordinates, (ii) introduce explicit constraint forces, (iii) minimize constraint forces implicitly by the technique of Lagrange multipliers or projection methods.
Mean-field game theory is the study of strategic decision making by small interacting agents in very large populations. It lies at the intersection of game theory with stochastic analysis and control theory. The use of the term "mean field" is inspired by mean-field theory in physics, which considers the behavior of systems of large numbers of particles where individual particles have negligible impacts upon the system. In other words, each agent acts according to his minimization or maximization problem taking into account other agents’ decisions and because their population is large we can assume the number of agents goes to infinity and a representative agent exists.
Equation-free modeling is a method for multiscale computation and computer-aided analysis. It is designed for a class of complicated systems in which one observes evolution at a macroscopic, coarse scale of interest, while accurate models are only given at a finely detailed, microscopic, level of description. The framework empowers one to perform macroscopic computational tasks using only appropriately initialized microscopic simulation on short time and small length scales. The methodology eliminates the derivation of explicit macroscopic evolution equations when these equations conceptually exist but are not available in closed form; hence the term equation-free.
Mean-field particle methods are a broad class of interacting type Monte Carlo algorithms for simulating from a sequence of probability distributions satisfying a nonlinear evolution equation. These flows of probability measures can always be interpreted as the distributions of the random states of a Markov process whose transition probabilities depends on the distributions of the current random states. A natural way to simulate these sophisticated nonlinear Markov processes is to sample a large number of copies of the process, replacing in the evolution equation the unknown distributions of the random states by the sampled empirical measures. In contrast with traditional Monte Carlo and Markov chain Monte Carlo methods these mean-field particle techniques rely on sequential interacting samples. The terminology mean-field reflects the fact that each of the samples interacts with the empirical measures of the process. When the size of the system tends to infinity, these random empirical measures converge to the deterministic distribution of the random states of the nonlinear Markov chain, so that the statistical interaction between particles vanishes. In other words, starting with a chaotic configuration based on independent copies of initial state of the nonlinear Markov chain model, the chaos propagates at any time horizon as the size the system tends to infinity; that is, finite blocks of particles reduces to independent copies of the nonlinear Markov process. This result is called the propagation of chaos property. The terminology "propagation of chaos" originated with the work of Mark Kac in 1976 on a colliding mean-field kinetic gas model.
Multi-state modeling of biomolecules refers to a series of techniques used to represent and compute the behaviour of biological molecules or complexes that can adopt a large number of possible functional states.
Multiscale Green's function (MSGF) is a generalized and extended version of the classical Green's function (GF) technique for solving mathematical equations. The main application of the MSGF technique is in modeling of nanomaterials. These materials are very small – of the size of few nanometers. Mathematical modeling of nanomaterials requires special techniques and is now recognized to be an independent branch of science. A mathematical model is needed to calculate the displacements of atoms in a crystal in response to an applied static or time dependent force in order to study the mechanical and physical properties of nanomaterials. One specific requirement of a model for nanomaterials is that the model needs to be multiscale and provide seamless linking of different length scales.
In mathematics, the Leimkuhler-Matthews method is an algorithm for finding discretized solutions to the Brownian dynamics
Hybrid stochastic simulations are a sub-class of stochastic simulations. These simulations combine existing stochastic simulations with other stochastic simulations or algorithms. Generally they are used for physics and physics-related research. The goal of a hybrid stochastic simulation varies based on context, however they typically aim to either improve accuracy or reduce computational complexity. The first hybrid stochastic simulation was developed in 1985.
The spike response model (SRM) is a spiking neuron model in which spikes are generated by either a deterministic or a stochastic threshold process. In the SRM, the membrane voltage V is described as a linear sum of the postsynaptic potentials (PSPs) caused by spike arrivals to which the effects of refractoriness and adaptation are added. The threshold is either fixed or dynamic. In the latter case it increases after each spike. The SRM is flexible enough to account for a variety of neuronal firing pattern in response to step current input. The SRM has also been used in the theory of computation to quantify the capacity of spiking neural networks; and in the neurosciences to predict the subthreshold voltage and the firing times of cortical neurons during stimulation with a time-dependent current stimulation. The name Spike Response Model points to the property that the two important filters and of the model can be interpreted as the response of the membrane potential to an incoming spike (response kernel , the PSP) and to an outgoing spike (response kernel , also called refractory kernel). The SRM has been formulated in continuous time and in discrete time. The SRM can be viewed as a generalized linear model (GLM) or as an (integrated version of) a generalized integrate-and-fire model with adaptation.
