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Register a new userA structure preserving numerical scheme for Fokker-Planck equations of structured neural networks with learning rules
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Added by
Qing He
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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In this work, we are concerned with a Fokker-Planck equation related to the nonlinear noisy leaky integrate-and-fire model for biological neural networks which are structured by the synaptic weights and equipped with the Hebbian learning rule. The equation contains a small parameter $varepsilon$ separating the time scales of learning and reacting behavior of the neural system, and an asymptotic limit model can be derived by letting $varepsilonto 0$, where the microscopic quasi-static states and the macroscopic evolution equation are coupled through the total firing rate. To handle the endowed flux-shift structure and the multi-scale dynamics in a unified framework, we propose a numerical scheme for this equation that is mass conservative, unconditionally positivity preserving, and asymptotic preserving. We provide extensive numerical tests to verify the schemes properties and carry out a set of numerical experiments to investigate the models learning ability, and explore the solutions behavior when the neural network is excitatory.
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In this work we consider an extension of a recently proposed structure preserving numerical scheme for nonlinear Fokker-Planck-type equations to the case of nonconstant full diffusion matrices. While in existing works the schemes are formulated in a one-dimensional setting, here we consider exclusively the two-dimensional case. We prove that the proposed schemes preserve fundamental structural properties like nonnegativity of the solution without restriction on the size of the mesh and entropy dissipation. Moreover, all the methods presented here are at least second order accurate in the transient regimes and arbitrarily high order for large times in the hypothesis in which the flux vanishes at the stationary state. Suitable numerical tests will confirm the theoretical results.
We study a family of structure-preserving deterministic numerical schemes for Lindblad equations, and carry out detailed error analysis and absolute stability analysis. Both error and absolute stability analysis are validated by numerical examples.
Fractional Fokker-Planck equation plays an important role in describing anomalous dynamics. To the best of our knowledge, the existing discussions mainly focus on this kind of equation involving one diffusion operator. In this paper, we first derive the fractional Fokker-Planck equation with two-scale diffusion from the Levy process framework, and then the fully discrete scheme is built by using the $L_{1}$ scheme for time discretization and finite element method for space. With the help of the sharp regularity estimate of the solution, we optimally get the spatial and temporal error estimates. Finally, we validate the effectiveness of the provided algorithm by extensive numerical experiments.
The Poisson-Nernst-Planck equations with generalized Frumkin-Butler-Volmer boundary conditions (PNP-FBV) describe ion transport with Faradaic reactions and have applications in a wide variety of fields. Using an adaptive time-stepper based on a second-order variable step-size, semi-implicit, backward differentiation formula (VSSBDF2), we observe that when the underlying dynamics is one that would have the solutions converge to a steady state solution, the adaptive time-stepper produces solutions that nearly converge to the steady state and that, simultaneously, the time-step sizes stabilize at a limiting size $dt_infty$. Linearizing the SBDF2 scheme about the steady state solution, we demonstrate that the linearized scheme is conditionally stable and that this is the cause of the adaptive time-steppers behaviour. Mesh-refinement, as well as a study of the eigenvectors corresponding to the critical eigenvalues, demonstrate that the conditional stability is not due to a time-step constraint caused by high-frequency contributions. We study the stability domain of the linearized scheme and find that it can have corners as well as jump discontinuities.
In neuroscience, the distribution of a decision time is modelled by means of a one-dimensional Fokker--Planck equation with time-dependent boundaries and space-time-dependent drift. Efficient approximation of the solution to this equation is required, e.g., for model evaluation and parameter fitting. However, the prescribed boundary conditions lead to a strong singularity and thus to slow convergence of numerical approximations. In this article we demonstrate that the solution can be related to the solution of a parabolic PDE on a rectangular space-time domain with homogeneous initial and boundary conditions by transformation and subtraction of a known function. We verify that the solution of the new PDE is indeed more regular than the solution of the original PDE and proceed to discretize the new PDE using a space-time minimal residual method. We also demonstrate that the solution depends analytically on the parameters determining the boundaries as well as the drift. This justifies the use of a sparse tensor product interpolation method to approximate the PDE solution for various parameter ranges. The predicted convergence rates of the minimal residual method and that of the interpolation method are supported by numerical simulations.
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