arXiv subjects

Yi Ming Lai

Publications and source records attributed to Yi Ming Lai.

5 recordsLinked to original sources

arXiv2023

Oscillatory networks: Insights from piecewise-linear modeling

There is enormous interest -- both mathematically and in diverse applications -- in understanding the dynamics of coupled oscillator networks. The real-world motivation of such networks arises from studies of the brain, the heart, ecology, and more. It is common to describe the rich emergent behavior in these systems in terms of complex patterns of network activity that reflect both the connectivity and the nonlinear dynamics of the network components. Such behavior is often organized around phase-locked periodic states and their instabilities. However, the explicit calculation of periodic orbits in nonlinear systems (even in low dimensions) is notoriously hard, so network-level insights often require the numerical construction of some underlying periodic component. In this paper, we review powerful techniques for studying coupled oscillator networks. We discuss phase reductions, phase-amplitude reductions, and the master stability function for smooth dynamical systems. We then focus in particular on the augmentation of these methods to analyze piecewise-linear systems, for which one can readily construct periodic orbits. This yields useful insights into network behavior, but the cost is that one needs to study nonsmooth dynamical systems. The study of nonsmooth systems is well-developed when focusing on the interacting units (i.e., at the node level) of a system, and we give a detailed presentation of how to use \textit{saltation operators}, which can treat the propagation of perturbations through switching manifolds, to understand dynamics and bifurcations at the network level. We illustrate this merger of tools and techniques from network science and nonsmooth dynamical systems with applications to neural systems, cardiac systems, networks of electro-mechanical oscillators, and cooperation in cattle herds.

math.DS↗

arXiv2020

Models Currently Implemented in MIIND

This is a living document that will be updated when appropriate. MIIND [1, 2] is a population-level neural simulator. It is based on population density techniques, just like DIPDE [3]. Contrary to DIPDE, MIIND is agnostic to the underlying neuron model used in its populations so any 1, 2 or 3 dimensional model can be set up with minimal effort. The resulting populations can then be grouped into large networks, e.g. the Potjans-Diesmann model [4]. The MIIND website http://miind.sf.net contains training materials, and helps to set up MIIND, either by using virtual machines, a DOCKER image, or directly from source code.

q-bio.NC↗

arXiv2017

Population Density Equations for Stochastic Processes with Memory Kernels

We present a novel method for solving population density equations (PDEs), where the populations can be subject to non-Markov noise for arbitrary distributions of jump sizes. The method combines recent developments in two different disciplines that traditionally have had limited interaction: computational neuroscience and the theory of random networks. The method uses a geometric binning scheme, based on the method of characteristics, to capture the deterministic neurodynamics of the population, separating the deterministic and stochastic process cleanly. We can independently vary the choice of the deterministic model and the model for the stochastic process, leading to a highly modular numerical solution strategy. We demonstrate this by replacing the Master equation implicit in many formulations of the PDE formalism, by a generalization called the generalized Montroll-Weiss equation - a recent result from random network theory - describing a random walker subject to transitions realized by a non-Markovian process. We demonstrate the method for leaky- (LIF) and quadratic-integrate and fire (QIF) neurons subject to spike trains with Poisson and gamma distributed spike intervals. We are able to model jump responses for both models accurately to both excitatory and inhibitory input under the assumption that all inputs are generated by one renewal process.

physics.bio-ph↗

arXiv2013

Noise-Induced Synchronization, Desynchronization, and Clustering in Globally Coupled Nonidentical Oscillators

We study ensembles of globally coupled, nonidentical phase oscillators subject to correlated noise, and we identify several important factors that cause noise and coupling to synchronize or desychronize a system. By introducing noise in various ways, we find a novel estimate for the onset of synchrony of a system in terms of the coupling strength, noise strength, and width of the frequency distribution of its natural oscillations. We also demonstrate that noise alone is sufficient to synchronize nonidentical oscillators. However, this synchrony depends on the first Fourier mode of a phase-sensitivity function, through which we introduce common noise into the system. We show that higher Fourier modes can cause desychronization due to clustering effects, and that this can reinforce clustering caused by different forms of coupling. Finally, we discuss the effects of noise on an ensemble in which antiferromagnetic coupling causes oscillators to form two clusters in the absence of noise.

nlin.AO↗

arXiv2010

Nonlinear Waves in Disordered Diatomic Granular Chains

We investigate the propagation and scattering of highly nonlinear waves in disordered granular chains composed of diatomic (two-mass) units of spheres that interact via Hertzian contact. Using ideas from statistical mechanics, we consider each diatomic unit to be a "spin", so that a granular chain can be viewed as a spin chain composed of units that are each oriented in one of two possible ways. Experiments and numerical simulations both reveal the existence of two different mechanisms of wave propagation: In low-disorder chains, we observe the propagation of a solitary pulse with exponentially decaying amplitude. Beyond a critical level of disorder, the wave amplitude instead decays as a power law, and the wave transmission becomes insensitive to the level of disorder. We characterize the spatio-temporal structure of the wave in both propagation regimes and propose a simple theoretical interpretation for such a transition. Our investigation suggests that an elastic spin chain can be used as a model system to investigate the role of heterogeneities in the propagation of highly nonlinear waves.

nlin.PS↗