TY - JOUR
T1 - Canard-induced complex oscillations in an excitatory network
AU - Köksal Ersöz, Elif
AU - Desroches, Mathieu
AU - Guillamon, Antoni
AU - Rinzel, John
AU - Tabak, Joël
N1 - Funding Information:
EKE was supported by the ERC Advanced Grant NerVi No. 227747. AG was funded by the Spanish grant MINECO-FEDER-UE MTM-2015-71509-C2-2-R and the Catalan Grant 2017SGR1049. AG also wants to thank the MathNeuro Team at Inria for their hospitality during part of the development of this project.
Publisher Copyright:
© 2020, Springer-Verlag GmbH Germany, part of Springer Nature.
PY - 2020/6/1
Y1 - 2020/6/1
N2 - In Neuroscience, mathematical modelling involving multiple spatial and temporal scales can unveil complex oscillatory activity such as excitable responses to an input current, subthreshold oscillations, spiking or bursting. While the number of slow and fast variables and the geometry of the system determine the type of the complex oscillations, canard structures define boundaries between them. In this study, we use geometric singular perturbation theory to identify and characterise boundaries between different dynamical regimes in multiple-timescale firing rate models of the developing spinal cord. These rate models are either three or four dimensional with state variables chosen within an overall group of two slow and two fast variables. The fast subsystem corresponds to a recurrent excitatory network with fast activity-dependent synaptic depression, and the slow variables represent the cell firing threshold and slow activity-dependent synaptic depression, respectively. We start by demonstrating canard-induced bursting and mixed-mode oscillations in two different three-dimensional rate models. Then, in the full four-dimensional model we show that a canard-mediated slow passage creates dynamics that combine these complex oscillations and give rise to mixed-mode bursting oscillations (MMBOs). We unveil complicated isolas along which MMBOs exist in parameter space. The profile of solutions along each isola undergoes canard-mediated transitions between the sub-threshold regime and the bursting regime; these explosive transitions change the number of oscillations in each regime. Finally, we relate the MMBO dynamics to experimental recordings and discuss their effects on the silent phases of bursting patterns as well as their potential role in creating subthreshold fluctuations that are often interpreted as noise. The mathematical framework used in this paper is relevant for modelling multiple timescale dynamics in excitable systems.
AB - In Neuroscience, mathematical modelling involving multiple spatial and temporal scales can unveil complex oscillatory activity such as excitable responses to an input current, subthreshold oscillations, spiking or bursting. While the number of slow and fast variables and the geometry of the system determine the type of the complex oscillations, canard structures define boundaries between them. In this study, we use geometric singular perturbation theory to identify and characterise boundaries between different dynamical regimes in multiple-timescale firing rate models of the developing spinal cord. These rate models are either three or four dimensional with state variables chosen within an overall group of two slow and two fast variables. The fast subsystem corresponds to a recurrent excitatory network with fast activity-dependent synaptic depression, and the slow variables represent the cell firing threshold and slow activity-dependent synaptic depression, respectively. We start by demonstrating canard-induced bursting and mixed-mode oscillations in two different three-dimensional rate models. Then, in the full four-dimensional model we show that a canard-mediated slow passage creates dynamics that combine these complex oscillations and give rise to mixed-mode bursting oscillations (MMBOs). We unveil complicated isolas along which MMBOs exist in parameter space. The profile of solutions along each isola undergoes canard-mediated transitions between the sub-threshold regime and the bursting regime; these explosive transitions change the number of oscillations in each regime. Finally, we relate the MMBO dynamics to experimental recordings and discuss their effects on the silent phases of bursting patterns as well as their potential role in creating subthreshold fluctuations that are often interpreted as noise. The mathematical framework used in this paper is relevant for modelling multiple timescale dynamics in excitable systems.
KW - Canard solutions
KW - Excitability
KW - Mixed-mode bursting oscillations
KW - Multiple timescale systems
KW - Rate models
UR - http://www.scopus.com/inward/record.url?scp=85083372020&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85083372020&partnerID=8YFLogxK
U2 - 10.1007/s00285-020-01490-1
DO - 10.1007/s00285-020-01490-1
M3 - Article
C2 - 32266428
AN - SCOPUS:85083372020
SN - 0303-6812
VL - 80
SP - 2075
EP - 2107
JO - Journal Of Mathematical Biology
JF - Journal Of Mathematical Biology
IS - 7
ER -