Page 390 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 390
TOPOLOGICAL METHODS IN DYNAMICAL SYSTEMS (MS-65)
Zero topological entropy and invariant measures in dimension one
Piotr Oprocha, oprocha@agh.edu.pl
AGH University of Science and Technology, Poland
In this talk I will survey some recent results on special properties of maps with zero entropy on
one-dimensional continua (mostly interval and topological graphs). I will be interested in rela-
tions between Li-Yorke pairs, sequence entropy and how they influence properties of invariant
measures. The talk will be based, in part, on recent joint works with Jian Li, Guohua Zhang and
Xianjuan Liang.
Dissipative flows and bifurcations of global attractors
José M. R. Sanjurjo, jose_sanjurjo@mat.ucm.es
Universidad Complutense de Madrid, Spain
Coauthor: Héctor Barge
We study bifurcations of dissipative flows in which a global attractor becomes a non-global
attractor after a small perturbation of the flow. Using Conley’s index theory, we examine the
mechanism which produces this phenomenon and we identify a dynamical property which is
responsible for this bifurcation and analyze the topological features of some isolated invariant
sets generated throughout this process. On the other hand we show that global attractors con-
tinue to global attractors if and only if the family of flows is uniformly dissipative. We illustrate
this property with the flow induced by the Lorenz equations and their global attractors.
Dynamical mechanisms of Type III responses in a nonlinear hybrid
neuron model
Justyna Signerska-Rynkowska, justyna.signerska@pg.edu.pl
Gdan´sk University of Technology, Poland
Coauthors: Jonathan Rubin, Jonathan Touboul
The dynamic mechanisms shaping neurons’ responses to transient inputs can bear significant
physiological relevance and is connected, among others, with such phenomena as post-inhibitory
facilitation (PIF), where an otherwise subthreshold excitatory input can induce a spike if it is ap-
plied with proper timing after an inhibitory pulse, and slope detection, in which a neuron spikes
to a transient input only when the input’s rate of change is in a specific, bounded range. These
phenomena have been previously associated with so-called Type III neurons (in Hodgkin’s clas-
sification) which are those neurons that never exhibit continuous firing in response to sustained
excitatory currents.
In our study we analyse responses to transient inputs in nonlinear adaptive hybrid models
and provide a geometric characterization of dynamical structures associated with PIF and an
analytical study of slope detection for tent inputs. In particular, our proofs show that PIF and
slope-detection do not always require pure Type III regime.
388
Zero topological entropy and invariant measures in dimension one
Piotr Oprocha, oprocha@agh.edu.pl
AGH University of Science and Technology, Poland
In this talk I will survey some recent results on special properties of maps with zero entropy on
one-dimensional continua (mostly interval and topological graphs). I will be interested in rela-
tions between Li-Yorke pairs, sequence entropy and how they influence properties of invariant
measures. The talk will be based, in part, on recent joint works with Jian Li, Guohua Zhang and
Xianjuan Liang.
Dissipative flows and bifurcations of global attractors
José M. R. Sanjurjo, jose_sanjurjo@mat.ucm.es
Universidad Complutense de Madrid, Spain
Coauthor: Héctor Barge
We study bifurcations of dissipative flows in which a global attractor becomes a non-global
attractor after a small perturbation of the flow. Using Conley’s index theory, we examine the
mechanism which produces this phenomenon and we identify a dynamical property which is
responsible for this bifurcation and analyze the topological features of some isolated invariant
sets generated throughout this process. On the other hand we show that global attractors con-
tinue to global attractors if and only if the family of flows is uniformly dissipative. We illustrate
this property with the flow induced by the Lorenz equations and their global attractors.
Dynamical mechanisms of Type III responses in a nonlinear hybrid
neuron model
Justyna Signerska-Rynkowska, justyna.signerska@pg.edu.pl
Gdan´sk University of Technology, Poland
Coauthors: Jonathan Rubin, Jonathan Touboul
The dynamic mechanisms shaping neurons’ responses to transient inputs can bear significant
physiological relevance and is connected, among others, with such phenomena as post-inhibitory
facilitation (PIF), where an otherwise subthreshold excitatory input can induce a spike if it is ap-
plied with proper timing after an inhibitory pulse, and slope detection, in which a neuron spikes
to a transient input only when the input’s rate of change is in a specific, bounded range. These
phenomena have been previously associated with so-called Type III neurons (in Hodgkin’s clas-
sification) which are those neurons that never exhibit continuous firing in response to sustained
excitatory currents.
In our study we analyse responses to transient inputs in nonlinear adaptive hybrid models
and provide a geometric characterization of dynamical structures associated with PIF and an
analytical study of slope detection for tent inputs. In particular, our proofs show that PIF and
slope-detection do not always require pure Type III regime.
388
