On a method for determining limit cycles in nonlinear circuits

Savov, V. N.; Todorov, G.

doi:10.1080/00207210050028760

Cited by 6 publications

(3 citation statements)

References 13 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In our mathematical analyses, we focus on characterizing the system's fixed points and inferring from them the presence of oscillatory behavior associated with limit cycles. Directly identifying such limit cycles is a mathematically arduous process (Savov and Todorov, 2000) unnecessary for the conclusions drawn from our analyses. However, considering the behavior of our spiking networks remains ''bounded'' (see Figure S3B), we can confidently infer that such limit cycles exist, as is typical when a supercritical Hopf bifurcation yields an unstable fixed point.…”

Section: Dynamical Differences In Network With Varying Levels Of Hete...mentioning

confidence: 99%

Loss of neuronal heterogeneity in epileptogenic human tissue impairs network resilience to sudden changes in synchrony

et al. 2022

View full text Add to dashboard Cite

Section: Dynamical Differences In Network With Varying Levels Of Hete...mentioning

confidence: 99%

Loss of neuronal heterogeneity in epileptogenic human tissue impairs network resilience to sudden changes in synchrony

et al. 2022

View full text Add to dashboard Cite

“…In our mathematical analyses, we focus on characterizing the system’s fixed points and inferring from them the presence of oscillatory behavior associated with limit cycles. Directly identifying such limit cycles is a mathematically arduous process (Savov & Todorov, 2000) unnecessary for the conclusions drawn from our analyses. However, considering the behavior of our spiking networks remains “bounded” (see Supplementary Figure S3 (b) ), we can confidently infer that such limit cycles exist, as is typical when a supercritical Hopf bifurcation yields an unstable fixed point.…”

Section: Resultsmentioning

confidence: 99%

“…We also emphasize that, in our mathematical analyses, we focus on characterizing the system's fixed points and inferring from them the presence of oscillatory behavior associated with limit cycles. Directly identifying such limit cycles is a mathematically arduous process (Savov & Todorov, 2000) unnecessary for these analyses, where our primary interest is differentiating the mathematical structure of these four exemplar networks. However, considering the behavior of our spiking networks remains "bounded" (i.e., consistent oscillatory activity is associated with unstable fixed points with imaginary eigenvalues; see Supplementary Figure S2(b)), we can confidently infer that such limit cycles exist, as is typical when a bifurcation yields an unstable fixed point.…”

Section: Dynamical Differences In Network With Varying Levels Of Heterogeneity Are Explained By Their Distinct Mathematical Structuresmentioning

confidence: 99%

Resilience through diversity: Loss of neuronal heterogeneity in epileptogenic human tissue impairs network resilience to sudden changes in synchrony

Rich

Chameh

Lefebvre

et al. 2021

Preprint

View full text Add to dashboard Cite

A myriad of pathological changes associated with epilepsy can be recast as decreases in cell and circuit heterogeneity. We propose that epileptogenesis can be recontextualized as a process where reduction in cellular heterogeneity renders neural circuits less resilient to transitions into information-poor, over-correlated seizure states. We provide in vitro, in silico, and mathematical support for this hypothesis. Patch clamp recordings from human layer 5 (L5) cortical neurons demonstrate significantly decreased biophysical heterogeneity of excitatory neurons in seizure generating areas (epilepetogenic zone). This decreased heterogeneity renders model neural circuits prone to sudden dynamical transitions into synchronous, hyperactive states (paralleling ictogenesis) while also explaining counter-intuitive differences in population activation functions (i.e., FI curves) between epileptogenic and non-epileptogenic tissue. Mathematical analyses based in mean-field theory reveal clear distinctions in the dynamical structure of networks with low and high heterogeneity, providing the theoretical undergird for how ictogenic dynamics accompany a reduction in biophysical heterogeneity.

show abstract

Analysis and synthesis of perturbed Duffing oscillators

Savov

Georgiev

Тодоров

2006

Circuit Theory & Apps

View full text Add to dashboard Cite

Analysis and synthesis of perturbed Duffing oscillators have been presented. The oscillations in such systems are regarded as limit cycles in perturbed Hamiltonian systems under polynomial perturbations of sixth degree and are analysed by using the Melnikov function. It has been proved that there exists a polynomial perturbation depending on the zeros of the Melnikov function so that the system considered can have either two simple limit cycles, or one limit cycle of multiplicity 2, or one simple limit cycle. A synthesis of such oscillators based on the Melnikov's theory has been proposed. Copyright © 2006 John Wiley & Sons, Ltd.

show abstract

On a method for determining limit cycles in nonlinear circuits

Cited by 6 publications

References 13 publications

Loss of neuronal heterogeneity in epileptogenic human tissue impairs network resilience to sudden changes in synchrony

Loss of neuronal heterogeneity in epileptogenic human tissue impairs network resilience to sudden changes in synchrony

Resilience through diversity: Loss of neuronal heterogeneity in epileptogenic human tissue impairs network resilience to sudden changes in synchrony

Analysis and synthesis of perturbed Duffing oscillators

Contact Info

Product

Resources

About