Maciej Krupa

Mixed-Mode Oscillations in a Multiple Time Scale Phantom Bursting System

In this work we study mixed mode oscillations in a model of secretion of GnRH (Gonadotropin Releasing Hormone). The model is a phantom burster consisting of two feedforward coupled FitzHugh-Nagumo systems, with three time scales. The forcing system (Regulator) evolves on the slowest scale and acts by moving the slow nullcline of the forced system (Secretor). There are three modes of dynamics: pulsatility (transient relaxation oscillation), surge (quasi steady state) and small oscillations related to the passage of the slow nullcline through a fold point of the fast nullcline. We derive a variety of reductions, taking advantage of the mentioned features of the system. We obtain two results; one on the local dynamics near the fold in the parameter regime corresponding to the presence of small oscillations and the other on the global dynamics, more specifically on the existence of an attracting limit cycle. Our local result is a rigorous characterization of small canards and sectors of rotation in the case of folded node with an additional time scale, a feature allowing for a clear geometric argument. The global result gives the existence of an attracting unique limit cycle, which, in some parameter regimes, remains attracting and unique even during passages through a canard explosion.

Inflection, canards and excitability threshold in neuronal models

A technique is presented, based on the differential geometry of planar curves, to evaluate the excitability threshold of neuronal models. The aim is to determine regions of the phase plane where solutions to the model equations have zero local curvature, thereby defining a zero-curvature (inflection) set that discerns between sub-threshold and spiking electrical activity. This transition can arise through a Hopf bifurcation, via the so-called canard explosion that happens in an exponentially small parameter variation, and this is typical for a large class of planar neuronal models (FitzHugh–Nagumo, reduced Hodgkin–Huxley), namely, type II neurons (resonators). This transition can also correspond to the crossing of the stable manifold of a saddle equilibrium, in the case of type I neurons (integrators). We compute inflection sets and study how well they approximate the excitability threshold of these neuron models, that is, both in the canard and in the non-canard regime, using tools from invariant manifold theory and singularity theory. With the latter, we investigate the topological changes that inflection sets undergo upon parameter variation. Finally, we show that the concept of inflection set gives a good approximation of the threshold in both the so-called resonator and integrator neuronal cases.

A network model of the periodic synchronization process in the dynamics of calcium concentration in GnRH neurons.

Mathematical neuroendocrinology is a branch of mathematical neurosciences that is specifically interested in endocrine neurons, which have the uncommon ability of secreting neurohormones into the blood. One of the most striking features of neuroendocrine networks is their ability to exhibit very slow rhythms of neurosecretion, on the order of one or several hours. A prototypical instance is that of the pulsatile secretion pattern of GnRH (gonadotropin releasing hormone), the master hormone controlling the reproductive function, whose origin remains a puzzle issue since its discovery in the seventies. In this paper, we investigate the question of GnRH neuron synchronization on a mesoscopic scale, and study how synchronized events in calcium dynamics can arise from the average electric activity of individual neurons. We use as reference seminal experiments performed on embryonic GnRH neurons from rhesus monkeys, where calcium imaging series were recorded simultaneously in tens of neurons, and which have clearly shown the occurrence of synchronized calcium peaks associated with GnRH pulses, superposed on asynchronous, yet oscillatory individual background dynamics. We design a network model by coupling 3D individual dynamics of FitzHugh–Nagumo type. Using phase-plane analysis, we constrain the model behavior so that it meets qualitative and quantitative specifications derived from the experiments, including the precise control of the frequency of the synchronization episodes. In particular, we show how the time scales of the model can be tuned to fit the individual and synchronized time scales of the experiments. Finally, we illustrate the ability of the model to reproduce additional experimental observations, such as partial recruitment of cells within the synchronization process or the occurrence of doublets of synchronization.

Complex Oscillations in the Delayed FitzHugh–Nagumo Equation

Motivated by the dynamics of neuronal responses, we analyze the dynamics of the FitzHugh–Nagumo slow–fast system with delayed self-coupling. This system provides a canonical example of a canard explosion for sufficiently small delays. Beyond this regime, delays significantly enrich the dynamics, leading to mixed-mode oscillations, bursting and chaos. These behaviors emerge from a delay-induced subcritical Bogdanov–Takens instability arising at the fold points of the S-shaped critical manifold. Underlying the transition from canard-induced to delay-induced dynamics is an abrupt switch in the nature of the Hopf bifurcation.

A Multiple Time Scale Coupling of Piecewise Linear Oscillators. Application to a Neuroendocrine System

We analyze a four-dimensional slow-fast piecewise linear system consisting of two coupled McKean caricatures of the FitzHugh--Nagumo system. Each oscillator is a continuous slow-fast piecewise linear system with three zones of linearity. The coupling is one-way, that is, one subsystem evolves independently and is forcing the other subsystem. In contrast to the original FitzHugh--Nagumo system, we consider a negative slope of the linear nullcline in both the forcing and the forced system. In the forcing system, this lets us, by just changing one parameter, pass from a system having one equilibrium and a relaxation cycle to a system with three equilibria keeping the relaxation cycle. Thus, we can easily control the changes in the oscillation frequency of the forced system. The case with three equilibria and a linear slow nullcline is a new configuration of the McKean caricature, where the existence of the relaxation cycle was not studied previously. We also consider a negative slope of the

Weakly coupled two-slow–two-fast systems, folded singularities and mixed mode oscillations

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Heteroclinic cycles in Hopfield networks

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Canard explosion in delayed equations with multiple timescales

We study Mixed Mode Oscillations (MMOs) in systems of two weakly coupled slow/fast oscillators. We focus on the existence and properties of a folded singularity called FSN II that allows the emergence of MMOs in the presence of a suitable global return mechanism. As FSN II corresponds to a transcritical bifurcation for a desingularized reduced system, we prove that, under certain non-degeneracy conditions, such a transcritical bifurcation exists. We then apply this result to the case of two coupled systems of FitzHugh- Nagumo type. This leads to a non trivial condition on the coupling that enables the existence of MMOs.

Consecutive and non-consecutive heteroclinic cycles in Hopfield networks

It is widely believed that information is stored in the brain by means of the varying strength of synaptic connections between neurons. Stored patterns can be replayed upon the arrival of an appropriate stimulus. Hence, it is interesting to understand how an information pattern can be represented by the dynamics of the system. In this work, we consider a class of network neuron models, known as Hopfield networks, with a learning rule which consists of transforming an information string to a coupling pattern. Within this class of models, we study dynamic patterns, known as robust heteroclinic cycles, and establish a tight connection between their existence and the structure of the coupling.

Firing-rate, symbolic dynamics and frequency dependence in periodically driven spiking models: a piecewise-smooth approach

We analyze canard explosions in delay differential equations with a one-dimensional slow manifold. This study is applied to explore the dynamics of the van der Pol slow–fast system with delayed self-coupling. In the absence of delays, this system provides a canonical example of a canard explosion. We show that as the delay is increased a family of ‘classical’ canard explosions ends as a Bogdanov–Takens bifurcation occurs at the folds points of the S-shaped critical manifold.