Periodic Reporting for period 2 - NEUROMORPH (Emerging Network Structures and Neuromorphic Applications)
Reporting period: 2023-04-01 to 2024-09-30
More than 50 years ago, G. Moore predicted that the number of transistors on a silicon computer chip would double every two years. The exponential growth of computing power is ending today because of physical limits, such that new technologies are needed. Neuromorphic computing seems to be a promising avenue. This concept, developed by C. Mead in the late 1980s, describes the use of electronic analog devices and circuits to mimic neurobiological architectures present in the nervous system. A key aspect is to understand the morphology of neurons and circuits with the aim to develop novel low-power and high-density computer networks. This project aims to make a major contribution by analyzing neuromorphic network structures from a mathematical viewpoint.
Neuromorphic structures are inspired from biological systems like neuronal networks in the brain and vascular networks (blood vessels). The aim of the project is to understand how such network structures emerge and how they develop. The novelty is the consideration of the multiple components of the structures, like different ion species and signaling molecules. Mathematically, this leads to multicomponent systems, whose mathematical analysis and numerical simulation is extremely challenging. These issues will be overcome by novel mathematical concepts inspired from thermodynamics and the theory of partial differential equations.
This project is a mathematical journey starting from modeling issues, over theoretical aspects of diffusion systems, to numerical schemes that reflect the structure of the equations, completed by feedback loops. The project culminates in the numerical simulation of bio-inspired neuromorphic networks. In particular, we wish to explore artificial-to-biological synaptors, which describe a simple implantable neuronal interface, and learning rules in neuromorphic networks, realized by new semiconductor devices.
Neuromorphic structures are inspired from biological systems like neuronal networks in the brain and vascular networks (blood vessels). The aim of the project is to understand how such network structures emerge and how they develop. The novelty is the consideration of the multiple components of the structures, like different ion species and signaling molecules. Mathematically, this leads to multicomponent systems, whose mathematical analysis and numerical simulation is extremely challenging. These issues will be overcome by novel mathematical concepts inspired from thermodynamics and the theory of partial differential equations.
This project is a mathematical journey starting from modeling issues, over theoretical aspects of diffusion systems, to numerical schemes that reflect the structure of the equations, completed by feedback loops. The project culminates in the numerical simulation of bio-inspired neuromorphic networks. In particular, we wish to explore artificial-to-biological synaptors, which describe a simple implantable neuronal interface, and learning rules in neuromorphic networks, realized by new semiconductor devices.
During the first period of the project from October 2021 until March 2024, we have working on Work Packages A1-A4, B2, and C1.
In Work Package A1-A4, we have investigated several diffusion systems modeling segregating populations, multicomponent mixtures, and vasculogenesis. Most of our results concern the diffusive transport of (charged) particles, analyzed from different mathematical viewpoints like the global existence of weak solutions, weak-strong uniqueness property, large-time behavior of solutions, and finite-volume discretizations. We also started to include methodology from stochastic analysis. As a result, we obtained the existence of global martingale solutions to stochastic cross-diffusion systems in cell biology and population dynamics and the large-time decay of time-continuous Markov chains using coupling methods. The intention is to combine techniques from the two different research fields "PDE theory" and "Stochastic analysis", and currently we continue this research direction in the context of mean-field limits of interacting particle systems and fluctuations.
In Work Package B2, we have worked on a class of local and nonlocal cross-diffusion systems for populations and networks, which has been derived from stochastic interacting particle systems in the mean-field limit. We have proved the existence of global weak solutions to the generalized Busenberg-Travis model when the model coefficients form a positive definite matrix and devised structure-preserving numerical schemes for the nonlocal model. In the case that this matrix has zero eigenvalues, meaning that the network is not completely connected, we found a normal symmetric form of the hyperbolic-parabolic system and proved the existence of dissipative measure-valued solutions.
The charge transport in memristors has been analyzed in Work Package C1. We proved the existence of weak solutions to drift-diffusion equations for memristor devices. Because of (physical) incompatibilities of the boundary conditions, the regularity of the solutions is very low. Allowing for nonlinear diffusivities (modeling the high-density regime), we have been able to improve the regularity up to bounded weak solutions. Moreover, we have investigated spin diffusion models since spintronic devices may have similar properties as memristors.
In Work Package A1-A4, we have investigated several diffusion systems modeling segregating populations, multicomponent mixtures, and vasculogenesis. Most of our results concern the diffusive transport of (charged) particles, analyzed from different mathematical viewpoints like the global existence of weak solutions, weak-strong uniqueness property, large-time behavior of solutions, and finite-volume discretizations. We also started to include methodology from stochastic analysis. As a result, we obtained the existence of global martingale solutions to stochastic cross-diffusion systems in cell biology and population dynamics and the large-time decay of time-continuous Markov chains using coupling methods. The intention is to combine techniques from the two different research fields "PDE theory" and "Stochastic analysis", and currently we continue this research direction in the context of mean-field limits of interacting particle systems and fluctuations.
In Work Package B2, we have worked on a class of local and nonlocal cross-diffusion systems for populations and networks, which has been derived from stochastic interacting particle systems in the mean-field limit. We have proved the existence of global weak solutions to the generalized Busenberg-Travis model when the model coefficients form a positive definite matrix and devised structure-preserving numerical schemes for the nonlocal model. In the case that this matrix has zero eigenvalues, meaning that the network is not completely connected, we found a normal symmetric form of the hyperbolic-parabolic system and proved the existence of dissipative measure-valued solutions.
The charge transport in memristors has been analyzed in Work Package C1. We proved the existence of weak solutions to drift-diffusion equations for memristor devices. Because of (physical) incompatibilities of the boundary conditions, the regularity of the solutions is very low. Allowing for nonlinear diffusivities (modeling the high-density regime), we have been able to improve the regularity up to bounded weak solutions. Moreover, we have investigated spin diffusion models since spintronic devices may have similar properties as memristors.
Major progress beyond the state of the art has been achieved in four different directions. First, we have extended the boundedness-by-entropy method for cross-diffusion systems to finite-volume approximations. The main novelty is the derivation of a vector-valued discrete chain rule. Second, we developed a novel approximation strategy by means of an abstract regularization operator for the solution of cross-diffusion systems with multiplicative noise, avoiding the usually used implicit Euler approximation. Third, we advanced the theory of hyperbolic-parabolic systems for a diffusion system for segregating systems not satisfying the nullspace invariance condition of Kawashima and Shizuta. Fourth, we analyzed for the first time a drift-diffusion system for memristor devices, showing numerically that the model is able to produce memristive effects.
In the second half of the project, we intend to deepen and strengthen the results obtained so far. First, we wish to finish our program on the derivation of fluctuations around the mean-field limit for diffusion equations and their systems. The aim is to justify noise terms in diffusion systeme which up to now have been added only heuristically. Second, we wish to develop self-wiring models, which extend models for axon growth and vasculogenesis to full network systems. Our focus will be on thermodynamically consistent models. Third, we continue our analysis for drift-diffusion systems for memristors, now coupled to electric circuits. The aim is to design and analyze, for instance, learning rules in neuromorphic networks.
In the second half of the project, we intend to deepen and strengthen the results obtained so far. First, we wish to finish our program on the derivation of fluctuations around the mean-field limit for diffusion equations and their systems. The aim is to justify noise terms in diffusion systeme which up to now have been added only heuristically. Second, we wish to develop self-wiring models, which extend models for axon growth and vasculogenesis to full network systems. Our focus will be on thermodynamically consistent models. Third, we continue our analysis for drift-diffusion systems for memristors, now coupled to electric circuits. The aim is to design and analyze, for instance, learning rules in neuromorphic networks.