Limitations of perturbative techniques in the analysis of rhythms and oscillations

Lin, Kevin K.; Wedgwood, Kyle C. A.; Coombes, Stephen; Young, Lai Sang

doi:10.1007/s00285-012-0506-0

Cited by 14 publications

(22 citation statements)

References 42 publications

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“…This is in contrast to high sensitivity to initial conditions usually reported in neuron models with an external input (e.g. shear-induced chaos in the periodically kicked Morris-Lecar neuron [15]). [4,5], which corresponds to the quiescent segment on the limit cycle.…”

Section: Application To Bursting Neuronscontrasting

confidence: 74%

Extreme phase sensitivity in systems with fractal isochrons

Mauroy

Mezić

2015

Physica D: Nonlinear Phenomena

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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.

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Section: Application To Bursting Neuronscontrasting

confidence: 74%

Extreme phase sensitivity in systems with fractal isochrons

Mauroy

Mezić

2015

Physica D: Nonlinear Phenomena

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“…There it was shown that a combined phase and amplitude analysis is needed to completely describe the phenomenon. In addition, [51] explores scenarios where geometry and phase-space structures play a critical role, so that perturbative approaches based on phase response curves can not predict the dynamical behavior. Since the approach of [19] does not apply to full spiking induced by noise, a corresponding theory is still needed for such a case.…”

Section: Discussionmentioning

confidence: 99%

A Computational Study of Spike Time Reliability in Two Types of Threshold Dynamics

Kuske

2013

J. Math. Neurosc.

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Spike time reliability (STR) refers to the phenomenon in which repetitive applications of a frozen copy of one stochastic signal to a neuron trigger spikes with reliable timing while a constant signal fails to do so. Observed and explored in numerous experimental and theoretical studies, STR is a complex dynamic phenomenon depending on the nature of external inputs as well as intrinsic properties of a neuron. The neuron under consideration could be either quiescent or spontaneously spiking in the absence of the external stimulus. Focusing on the situation in which the unstimulated neuron is quiescent but close to a switching point to oscillations, we numerically analyze STR treating each spike occurrence as a time localized event in a model neuron. We study both the averaged properties as well as individual features of spike-evoking epochs (SEEs). The effects of interactions between spikes is minimized by selecting signals that generate spikes with relatively long interspike intervals (ISIs). Under these conditions, the frequency content of the input signal has little impact on STR. We study two distinct cases, Type I in which the f–I relation (f for frequency, I for applied current) is continuous and Type II where the f–I relation exhibits a jump. STR in the two types shows a number of similar features and differ in some others. SEEs that are capable of triggering spikes show great variety in amplitude and time profile. On average, reliable spike timing is associated with an accelerated increase in the “action” of the signal as a threshold for spike generation is approached. Here, “action” is defined as the average amount of current delivered during a fixed time interval. When individual SEEs are studied, however, their time profiles are found important for triggering more precisely timed spikes. The SEEs that have a more favorable time profile are capable of triggering spikes with higher precision even at lower action levels.

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“…This space is assumed to be a finite dimensional compact manifold. The advantages of considering that dim(x i ) ≥ 1 (not necessarily 1) are, among others, the possibility of showing dynamical bifurcations between different rythms and oscillations that appear in some biological neurons [10], that would not appear if the mathematical model of all the neurons were necessarily one-dimensional.…”

Section: Model Of An Isolated Neuronmentioning

confidence: 99%

Dale’s Principle Is Necessary for an Optimal Neuronal Network’s Dynamics

Catsigeras¹

2013

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We study a mathematical model of biological neuronal networks composed by any finite number 2 N  of non-necessarily identical cells. The model is a deterministic dynamical system governed by finite-dimensional impulsive differential equations. The statical structure of the network is described by a directed and weighted graph whose nodes are certain subsets of neurons, and whose edges are the groups of synaptical connections among those subsets. First, we prove that among all the possible networks such as their respective graphs are mutually isomorphic, there exists a dynamical optimum. This optimal network exhibits the richest dynamics: namely, it is capable to show the most diverse set of responses (i.e. orbits in the future) under external stimulus or signals. Second, we prove that all the neurons of a dynamically optimal neuronal network necessarily satisfy Dale's Principle, i.e. each neuron must be either excitatory or inhibitory, but not mixed. So, Dale's Principle is a mathematical necessary consequence of a theoretic optimization process of the dynamics of the network. Finally, we prove that Dale's Principle is not sufficient for the dynamical optimization of the network.

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Limitations of perturbative techniques in the analysis of rhythms and oscillations

Cited by 14 publications

References 42 publications

Extreme phase sensitivity in systems with fractal isochrons

Extreme phase sensitivity in systems with fractal isochrons

A Computational Study of Spike Time Reliability in Two Types of Threshold Dynamics

Dale’s Principle Is Necessary for an Optimal Neuronal Network’s Dynamics

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