Phase-Amplitude Response Functions for Transient-State Stimuli

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.

Section: Discussionmentioning

confidence: 75%

Global Isochrons and Phase Sensitivity of Bursting Neurons

Mauroy¹,

Rhoads²,

Moehlis³

et al. 2014

“…On one hand, the value of the PRC not only on the limit cycle (s = 0) but also in a neighborhood of it (s > 0) immediately follows from K. So, the effects of the perturbation on the normal variable s are also known. This allows us to study perturbations that displace the trajectory away from the limit cycle as well as perturbations that occur when the system is not on the limit cycle [35,7,46].…”

Section: The Morris-lecar Modelmentioning

confidence: 99%

Computation of Limit Cycles and Their Isochrons: Fast Algorithms and Their Convergence

Huguet¹,

Llave²

2013

Self Cite

We present efficient algorithms to compute limit cycles and their isochrons (i.e., the sets of points with the same asymptotic phase) for planar vector fields. We formulate a functional equation for the parameterization of the invariant cycle and its isochrons, and we show that it can be solved by means of a Newton method. Using the right transformations, we can solve the equation of the Newton step efficiently. The algorithms are efficient in the sense that if we discretize the functions using N points, a Newton step requires O(N ) storage and O(N log N ) operations in Fourier discretization or O(N ) operations in other discretizations. We prove convergence of the algorithms and present a validation theorem in an a posteriori format. That is, we show that if there is an approximate solution of the invariance equation that satisfies some some mild nondegeneracy conditions, then there is a true solution nearby. Thus, our main theorem can be used to validate numerically computed solutions. The theorem also shows that the isochrons are analytic and depend analytically on the base point. Moreover, it establishes smooth dependence of the solutions on parameters and provides efficient algorithms to compute perturbative expansions with respect to external parameters. We include a discussion on the numerical implementation of the algorithms as well as numerical results for representative examples.

“…Note that there are other phase reductions that can be utilized (see [60]); we focus on the case where the oscillators return to the limit cycle quickly with weak perturbations as described in Teramae, Nakao, and Ermentrout [54]. Recent work has addressed the case of strong perturbations [56,10,29] requiring more variables than just the phase, which can lead to different dynamics [43].…”

Section: Introductionmentioning

confidence: 98%

Dynamics of Coupled Noisy Neural Oscillators with Heterogeneous Phase Resetting Curves

Ly¹

2014

Pulse-coupled phase oscillators have been utilized in a variety of contexts. Motivated by neuroscience, we study a network of pulse-coupled phase oscillators receiving independent and correlated noise. An additional physiological attribute, heterogeneity, is incorporated in the phase-resetting curve (PRC), which is a vital entity for modeling the biophysical dynamics of oscillators. An accurate probability density or mean field description is large-dimensional, requiring reduction methods for tractability. We present a reduction method to capture the pairwise synchrony via the probability density of the phase differences, and we explore the robustness of the method. We find the reduced methods can capture some of the synchronous dynamics in these networks. The variance of the noisy period (or spike times) in this network is also considered. In particular, we find that phase oscillators with predominately positive PRCs (type I) have larger variance with inhibitory pulse-coupling than PRCs with a larger negative region (type II), but with excitatory pulse-coupling the opposite happens-type I oscillators have lower variability than type II. Analysis of this phenomenon is provided via an asymptotic approximation with weak noise and weak coupling, where we demonstrate how the individual PRC alters variability with pulse-coupling. We compare the phase oscillators to full oscillator networks and discuss utility and shortcomings.