Global Isochrons and Phase Sensitivity of Bursting Neurons

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 (two-dimensional) 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

Sensitivity to initial conditions is usually associated with chaotic dynamics and strange attractors. However, even systems with (quasi)periodic dynamics can exhibit it. In this context we report on the fractal properties of the isochrons of some continuous-time asymptotically periodic systems. We define a global measure of phase sensitivity that we call the phase sensitivity coefficient and show that it is an invariant of the system related to the capacity dimension of the isochrons. Similar results are also obtained with discrete-time systems. As an illustration of the framework, we compute the phase sensitivity coefficient for popular models of bursting neurons, suggesting that some elliptic bursting neurons are characterized by isochrons of high fractal dimensions and exhibit a very sensitive (unreliable) phase response.Comment: 32 page

Interactions of multiple rhythms in a biophysical network of neurons

Neural oscillations, including rhythms in the beta1 band (12–20 Hz), are important in various cognitive functions. Often neural networks receive rhythmic input at frequencies different from their natural frequency, but very little is known about how such input affects the network’s behavior. We use a simplified, yet biophysical, model of a beta1 rhythm that occurs in the parietal cortex, in order to study its response to oscillatory inputs. We demonstrate that a cell has the ability to respond at the same time to two periodic stimuli of unrelated frequencies, firing in phase with one, but with a mean firing rate equal to that of the other. We show that this is a very general phenomenon, independent of the model used. We next show numerically that the behavior of a different cell, which is modeled as a high-dimensional dynamical system, can be described in a surprisingly simple way, owing to a reset that occurs in the state space when the cell fires. The interaction of the two cells leads to novel combinations of properties for neural dynamics, such as mode-locking to an input without phase-locking to it.Published versio

Synchronization properties and functional implications of parietal beta1 rhythm

Neural oscillations, including rhythms in the beta1 band (12-20 Hz), are important in various cognitive functions. Often brain networks receive rhythmic input at frequencies different than their natural frequency, so understanding how neural networks process rhythmic input is important for understanding their function in the brain. In the current thesis we study a beta1 rhythm that appears in the parietal cortex, focusing on the way it interacts with other incoming rhythms, and the implications of this interaction for cognition. The main part of the thesis consists of two stand-alone chapters, both using as a basis a biophysical neural network model that has been previously proposed to model the parietal beta1 rhythm and validated with in vitro experiments. In the first chapter we use a reduced version of this model, in order to study its dynamics, applying both analytic and numerical methods from dynamical systems. We show that a cell can respond at the same time to two periodic stimuli of unrelated frequencies, firing in phase with one, but with a mean firing rate equal to the other, a consequence of general properties of the dynamics of the network. We next show numerically that the behavior of a different cell, which is modeled as a high-dimensional dynamical system, can be described in a surprisingly simple way, owing to a reset that occurs in the state space when the cell fires. The interaction of the two cells leads to novel combinations of properties for neural dynamics, such as mode-locking to an input without phase-locking to it. In the second chapter, we study the ability of the beta1 model to support memory functions, in particular working memory. Working memory is a highly distributed component of the brain's memory systems, partially based in the parietal cortex. We show numerically that the parietal beta1 rhythm can provide an anatomical substrate for an episodic buffer of working memory. Specifically, it can support flexible and updatable representations of sensory input which are sensitive to distractors, allow for a read-out mechanism, and can be modulated or terminated by executive input