Global Isochrons and Phase Sensitivity of Bursting Neurons

Abstract

Abstract. Phase sensitivity analysis is a powerful method for studying (asymptotically periodic) bursting neuron models. One popular way of capturing phase sensitivity is through the computation of isochrons-subsets of the state space that each converge to the same trajectory on the limit cycle. However, the computation of isochrons is notoriously difficult, especially for bursting neuron models. In [W. E. Sherwood and J. Guckenheimer, SIAM J. Appl. Dyn. Syst., 9 (2010), pp. 659-703], the phase sensitivity of the bursting Hindmarsh-Rose model is studied through the use of singular perturbation theory: cross sections of the isochrons of the full system are approximated by those of fast subsystems. In this paper, we complement the previous study, providing a detailed phase sensitivity analysis of the full (three-dimensional) system, including computations of the full (twodimensional) isochrons. To our knowledge, this is the first such computation for a bursting neuron model. This was made possible thanks to the numerical method recently proposed in [A. Mauroy and I. Mezić, Chaos, 22 (2012), 033112]-relying on the spectral properties of the so-called Koopman operator-which is complemented with the use of adaptive quadtree and octree grids. The main result of the paper is to highlight the existence of a region of high phase sensitivity called the almost phaseless set and to completely characterize its geometry. In particular, our study reveals the existence of a subset of the almost phaseless set that is not predicted by singular perturbation theory (i.e., by the isochrons of fast subsystems). We also discuss how the almost phaseless set is related to empirically observed phenomena such as addition/deletion of spikes and to extrema of the phase response of the system. Finally, through the same numerical method, we show that an elliptic bursting model is characterized by a very high phase sensitivity and other remarkable properties.

Extreme phase sensitivity in systems with fractal isochrons

“…Elliptic (E) bursting models considered in our analysis (in blue) are characterized by a high phase sensitivity coefficient β = 1 − α, and therefore by strong fractal properties. This is in agreement with the preliminary observations of [18]. In contrast, parabolic (P) bursting models (in red) have a phase sensitivity coefficient equal to 0, thereby exhibiting no fractal properties.…”

“…showing the sensitivity of neurons to external inputs. Motivated by preliminary observations presented in [18], we illustrate the framework based on the phase sensitivity coefficient on popular (periodic) bursting neuron models (see Appendix B). Our comparison of different models shows that some elliptic bursting models exhibit strong fractal properties associated with very high phase sensitivity.…”

“…The main goal of this paper is to propose a theoretical framework-and associated numerical methods-that complements the brief and empirical observations given in [18] on the fractal properties of the isochrons. To that end, we define the phase sensitivity coefficient that quantifies the overall uncertainty of the asymptotic phase under small perturbations.…”

Algorithmic design of a noise-resistant and efficient closed-loop deep brain stimulation system: A computational approach

“…In our analysis, we opt for a phase-reduced bursting neuron model [ 40 , 46 , 47 ]. Our motivation for this particular selection is twofold.…”

“…However, in this study, we focus on a computational model able to qualitatively capture pathological neuronal dynamics, i.e., bursting behavior. Accordingly, a major part of the phase-response dynamics of the reduced model has been determined based on the Hindmarsh-Rose model for bursting [ 46 , 47 ]. Importantly, the employed phase-reduced model, which simulates the effect of stimulation on pathological neuronal activity, is data-driven , i.e., microelectrode recordings (MERs) acquired during subthalamic nucleus (STN) DBS surgical interventions for PD and OCD are used to estimate the unknown model parameters off-line.…”