arXiv subjects

Morten Gram Pedersen

Publications and source records attributed to Morten Gram Pedersen.

5 recordsLinked to original sources

arXiv2026

Dissecting emerging slow rhythms in delay-coupled neural oscillators

Synaptic transmission delays are ubiquitous in neural circuits and can alter the dynamical repertoire of coupled oscillators quantitatively and qualitatively. Here, we demonstrate that delayed coupling in inhibitory networks introduces an effective slow-fast structure in the phase-difference dynamics, generating low-frequency components that are not due to intrinsic cellular properties, and we show that this behavior is not specific to a particular model structure. The origin of this generic phenomenon is analyzed by numerical continuation and bifurcation analysis, which provides a systematic approach to find such delay-induced slow modulating rhythms. We employ phase reduction based on phase response curves to derive a phase-difference model with delay for mutually inhibitory coupled oscillators, where the individual units are given by the FitzHugh-Nagumo model, the Morris-Lecar model, or a next-generation neural mass model derived from quadratic integrate-and-fire neurons. We use phase planes to study multistability and limit cycles, which correspond to slow modulation of fast oscillations in the full model. Treating the synaptic delay as a bifurcation parameter, we apply numerical continuation to construct delay-dependent bifurcation diagrams. The analysis reveals Hopf, heteroclinic, and saddle-node-of-periodics bifurcations that cause and organize slow rhythmic behavior. Our analysis provides a systematic approach to the search for limit cycles in phase-reduction models corresponding to delay-induced slow rhythms in the original model.

nlin.CD↗

arXiv2026

Cusped singularities organize mixed-mode oscillations in mutually inhibitory slow-fast systems

Mutual inhibition is a common motif in neural systems. Here, we establish that cusped singularities - folded singularities located at cusp points of critical manifolds - provide a universal organizing mechanism for mixed-mode oscillations (MMOs) in coupled slow-fast systems with mutual inhibition. We show that the geometric setup of these systems generically satisfies the conditions required by established geometric singular perturbation theory and blow-up methods, guaranteeing that such cusped singularities yield small-amplitude oscillations (SAOs). MMOs appear from the SAOs combined with an appropriate return mechanism. Further, we show that the geometric presence of a cusped singularity is strictly related to occurrence of a nearby singular Hopf bifurcation. We demonstrate the efficacy of this framework in two distinct neuronal models: the Curtu rate model of mutually inhibitory neural populations and coupled Morris-Lecar neurons with synaptic inhibition. In both cases, pushing the full system equilibrium near the cusped singularity triggers SAOs as the system passes near the cusp and approaches a full-system saddle-focus related to the singular Hopf bifurcation. Large-amplitude oscillations appear as the system spirals away from the saddle-focus, leading to MMOs, which may exhibit distinctive alternating patterns, in contrast to standard saddle-node induced MMOs. Our results establish cusped singularities as a generic, biologically relevant mechanism for complex oscillatory dynamics in inhibitory neural networks as well as for other inhibitory slow-fast systems.

math.DS↗

arXiv2025

Near-Hamiltonian dynamics and energy-like quantities of next-generation neural mass models

Neural mass models describe the mean-field dynamics of populations of neurons. In this work we illustrate how fundamental ideas of physics, such as energy and conserved quantities, can be explored for such models. We show that time-rescaling renders recent next-generation neural mass models Hamiltonian in the limit of a homogeneous population or strong coupling. The corresponding energy-like quantity provides considerable insight into the model dynamics even in the case of heterogeneity, and explain for example why orbits are near-ellipsoidal and predict spike amplitude during bursting dynamics. We illustrate how these energy considerations provide a possible link between neuronal population behavior and energy landscape theory, which has been used to analyze data from brain recordings. Our introduction of near-Hamiltonian descriptions of neuronal activity could permit the application of highly developed physics theory to get insight into brain behavior.

q-bio.NC↗

arXiv2023

Geometric slow-fast analysis of a hybrid pituitary cell model with stochastic ion channel dynamics

To obtain explicit understanding of the behavior of dynamical systems, geometrical methods and slow-fast analysis have proved to be highly useful. Such methods are standard for smooth dynamical systems, and increasingly used for continuous, non-smooth dynamical systems. However, they are much less used for random dynamical systems, in particular for hybrid models with discrete, random dynamics. Indeed, the analysis of such systems has typically been done by studying the corresponding deterministic system and considering how noise perturbs the deterministic geometrical structures. Here we propose a geometrical method that works directly with the hybrid system. We illustrate our approach through an application to a hybrid pituitary cell model in which the stochastic dynamics of very few active large-conductance potassium (BK) channels is coupled to a deterministic model of the other ion channels and calcium dynamics. To employ our geometric approach, we exploit the slow-fast structure of the model. The random fast subsystem is analyzed by considering discrete phase planes, corresponding to the discrete number of open BK channels, and stochastic events correspond to jumps between these planes. The evolution within each plane can be understood from nullclines and limit cycles, and the overall dynamics, e.g., whether the model produces a spike or a burst, is determined by the location at which the system jumps from one plane to another. Our approach is generally applicable to other scenarios to study discrete random dynamical systems defined by hybrid stochastic-deterministic models.

q-bio.QM↗

arXiv2022

Mixed-mode oscillations in coupled FitzHugh-Nagumo oscillators: blowup analysis of cusped singularities

In this paper, we use geometric singular perturbation theory and blowup, as our main technical tool, to study the mixed-mode oscillations (MMOs) that occur in two coupled FitzHugh-Nagumo units with symmetric and repulsive coupling. In particular, we demonstrate that the MMOs in this model are not due to generic folded singularities, but rather due to singularities at a cusp -- not a fold -- of the critical manifold. Using blowup, we determine the number of SAOs analytically, showing -- as for the folded nodes -- that they are determined by the Weber equation and the ratio of eigenvalues. We also show that the model undergoes a (symmetric) saddle-node bifurcation in the desingularized reduced problem, which -- although resembling a folded saddle-node (type II) at this level -- also occurs on a cusp, and not a fold. We demonstrate that this bifurcation is associated with the emergence of an invariant cylinder, the onset of SAOs, as well as SAOs of increasing amplitude. We relate our findings with numerical computations and find excellent agreement.

math.DS↗