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# Multiscale modeling of brain dynamics: from single neurons and networks to mathematical tools

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The extreme complexity of the brain naturally requires mathematical modeling approaches on a large variety of scales; the spectrum ranges from single neuron dynamics over the behavior of groups of neurons to neuronal network activity. Thus, the connection between the microscopic scale (single neuron activity) to macroscopic behavior (emergent behavior of the collective dynamics) and vice versa is a key to understand the brain in its complexity. In this work, we attempt a review of a wide range of approaches, ranging from the modeling of single neuron dynamics to machine learning. The models include biophysical as well as data‐driven phenomenological models. The discussed models include Hodgkin–Huxley, FitzHugh–Nagumo, coupled oscillators (Kuramoto oscillators, Rössler oscillators, and the Hindmarsh–Rose neuron), Integrate and Fire, networks of neurons, and neural field equations. In addition to the mathematical models, important mathematical methods in multiscale modeling and reconstruction of the causal connectivity are sketched. The methods include linear and nonlinear tools from statistics, data analysis, and time series analysis up to differential equations, dynamical systems, and bifurcation theory, including Granger causal connectivity analysis, phase synchronization connectivity analysis, principal component analysis (PCA), independent component analysis (ICA), and manifold learning algorithms such as ISOMAP, and diffusion maps and equation‐free techniques. WIREs Syst Biol Med 2016, 8:438–458. doi: 10.1002/wsbm.1348

• Analytical and Computational Methods > Computational Methods
• Developmental Biology > Developmental Processes in Health and Disease
• Models of Systems Properties and Processes > Mechanistic Models
Overview of the presented modeling methods and mathematical tools for brain dynamics. The brain images were visualized with the BrainNet Viewer.
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Schematic of the Equation‐free multiscale modeling framework. Once the appropriate macroscopic observables have been identified, the following steps provide a bridge between the microscopic and macroscopic modeling scale: (1) Prescribe a macroscopic initial condition u (t 0). Define also a restriction operator M mapping the microscopic‐level description to the macroscopic one U; that is, u = M[U]. (2) Transform the macroscopic initial condition u (t 0) through a lifting operator, μ , into consistent microscopic realizations: U(t 0) = μ[u(t 0)]. (3) Evolve these realizations in time using the microscopic simulator for a short macroscopic time T generating the microscopic distribution U (t 0 + T). The choice of T is associated with the (estimated) spectral gap between micro and macro level. (4) Obtain the coarse‐grained values using the restriction operator $M:u t 0 + T =M U t 0 + T$. The above procedure defines a so‐called coarse timestepper from time t 0 to $t 0 +T$ between macroscopic state dynamics: $u t 0 + T = F T u t 0 , p$ where p R q is the parameter vector of the system. At this point, one can utilize bifurcation analysis tools as shells 'wrapped around' the coarse timestepper to trace branches of coarse‐grained equilibria or periodic solutions (even unstable ones) and perform stability and rare‐events analysis (e.g., compute transition rates between the apparent coarse‐grained states). Coupling Equation‐Free with manifold learning algorithms such as Diffusion Maps allows the identification of the dominant macroscopic states.
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