Natural extension of fast-slow decomposition for dynamical systems

Abstract

Modeling and parameter estimation to capture the dynamics of physical systems are often challenging because many parameters can range over orders of magnitude and are difficult to measure experimentally. Moreover, selecting a suitable model complexity requires a sufficient understanding of the model's potential use, such as highlighting essential mechanisms underlying qualitative behavior or precisely quantifying realistic dynamics. We present an approach that can guide model development and tuning to achieve desired qualitative and quantitative solution properties. It relies on the presence of disparate time scales and employs techniques of separating the dynamics of fast and slow variables, which are well known in the analysis of qualitative solution features. We build on these methods to show how it is also possible to obtain quantitative solution features by imposing designed dynamics for the slow variables in the form of specified two-dimensional paths in a bifurcation-parameter landscape.

Classification of bursting patterns: A tale of two ducks

“…This implies that, in principle, the Rinzel–Izhikevich and the Bertram–Golubitsky approaches both lead to the same number of bursting oscillation cases. Finally, one can consider more complicated slow paths in the fast subsystem’s parameter space, which may induce more than 2 crossings of bifurcation curves; see, e.g., [ 55 ]. However, this will likely not increase the number of possible bursting patterns captured.…”

Correlation Analysis of Synchronization Type and Degree in Respiratory Neural Network

“…The somatic subsystem is affected by calcium-activated nonspecific cationic (CAN) current, while the calcium subsystem is independent of the former. In this paper, we consider a parameterized path in the plane ([Ca], l ), which is of the form of an ellipse with the principal axis along the [Ca] axis and l axis (see also [ 49 ] for similar ideas). The variation along the ellipse can also be regarded as the solution of the equation, as follows: where [Ca] is the intracellular calcium concentration, l is the fraction of IP3(inositol triphosphate) channels in the endoplasmic reticulum membrane that have not been inactivated, [Ca] c and l c define the center of the ellipse, d is its aspect ratio, and ε is the speed with which the ellipse is traced.…”

“…( 15 – 17 ), and the values used in this paper are listed in Table 1 . f is the hill function related to calcium concentration: The description of the somatic subsystem can be referred to references [ 29 , 30 ], and the derivation and explanation of the imposed path about the calcium subsystem are shown in reference [ 49 ]. The numerical software in this paper is mainly XPPAUT and MATLAB, and the fourth-order Runge–Kutta algorithm is used with a step size of 0.1.…”

Dynamics Analysis of Firing Patterns in Pre-Bötzinger Complex Neurons Model

“… Rubin (2006) studied the network model of synaptic connections through excitable intermediate neurons and explained firing phenomenon of PBC. Bertram and Rubin (2017) and Rubin et al (2018) studied the parameter range that affects bursting in PBC neural network by using fast-slow analysis and bifurcation theory, classified different bursting patterns in parameter space, and explained various bursting in detail by combining physical and biological systems. Kiehn (2016) turned to study the effect of ion change on neuron firing by observing the activity of single PBC neuron under the change of potassium and calcium ions, which was proved that the bursting of mammalian medulla cells was the key to the generation of respiratory rhythms combined with clinical experiments.…”