Canard-induced complex oscillations in an excitatory network

Ersöz, Elif Köksal; Desroches, Mathieu; Guillamón, Antoni; Rinzel, John; Tabak, Joël

doi:10.1007/s00285-020-01490-1

Cited by 10 publications

(8 citation statements)

References 50 publications

(109 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Slow-fast dynamics near a folded node provide a key mechanism to induce another type of complex oscillations, namely folded-node-induced mixed-mode oscillations [ 10 , 34 ]. Hence, the new bursting patterns proposed here are a combination of fold-initiated bursting oscillations (definition given in the next section) and mixed-mode oscillations (MMOs), for which isolated examples were constructed in our previous work [ 35 , 36 ], and which we shall generalize and classify in the present work. This extended framework is well suited to revisit a number of biological datasets where the mechanisms underpinning bursting activity may have not been fully unraveled; see Fig 3 for an example of such data extracted from [ 31 ].…”

Section: Introductionmentioning

confidence: 99%

Classification of bursting patterns: A tale of two ducks

2022

Self Cite

View full text Add to dashboard Cite

Bursting is one of the fundamental rhythms that excitable cells can generate either in response to incoming stimuli or intrinsically. It has been a topic of intense research in computational biology for several decades. The classification of bursting oscillations in excitable systems has been the subject of active research since the early 1980s and is still ongoing. As a by-product, it establishes analytical and numerical foundations for studying complex temporal behaviors in multiple timescale models of cellular activity. In this review, we first present the seminal works of Rinzel and Izhikevich in classifying bursting patterns of excitable systems. We recall a complementary mathematical classification approach by Bertram and colleagues, and then by Golubitsky and colleagues, which, together with the Rinzel-Izhikevich proposals, provide the state-of-the-art foundations to these classifications. Beyond classical approaches, we review a recent bursting example that falls outside the previous classification systems. Generalizing this example leads us to propose an extended classification, which requires the analysis of both fast and slow subsystems of an underlying slow-fast model and allows the dissection of a larger class of bursters. Namely, we provide a general framework for bursting systems with both subthreshold and superthreshold oscillations. A new class of bursters with at least 2 slow variables is then added, which we denote folded-node bursters, to convey the idea that the bursts are initiated or annihilated via a folded-node singularity. Key to this mechanism are so-called canard or duck orbits, organizing the underpinning excitability structure. We describe the 2 main families of folded-node bursters, depending upon the phase (active/spiking or silent/nonspiking) of the bursting cycle during which folded-node dynamics occurs. We classify both families and give examples of minimal systems displaying these novel bursting patterns. Finally, we provide a biophysical example by reinterpreting a generic conductance-based episodic burster as a folded-node burster, showing that the associated framework can explain its subthreshold oscillations over a larger parameter region than the fast subsystem approach.

show abstract

Section: Introductionmentioning

confidence: 99%

Classification of bursting patterns: A tale of two ducks

2022

Self Cite

View full text Add to dashboard Cite

show abstract

“…Later on, transitions between the neuronal excitability types was shown to be induced by the inhibitory and excitatory autapse in the Morris-Lecar model [70]. Folded singularities and corresponding canard solutions in higher dimensional systems also have been shown to be shaping systems' excitability properties [24,28,33,34,[63][64][65].…”

Section: Canard-mediated Transitions and Excitabilitymentioning

confidence: 99%

“…Geometric singular perturbation theory (GSPT) is a key tool for understanding the interaction between the geometry of the system and the emerging multiple time-scale dynamics. In particular, canard solutions, which can exist in multiple time-scale systems with a folded geometry, appear as building blocks of complex oscillations in both phenomenological and neurophysiologically plausible models ranging from single cell [23][24][25][26] to neural networks [27,28]. The canard phenomenon in such systems has been related to neural excitability [29], excitability thresholds [23,[30][31][32][33][34], and boundaries between different type of solutions, such as subthreshold oscillations and large amplitude spiking/bursting oscillations [24,28,[35][36][37][38][39][40][41][42][43].…”

Section: Introductionmentioning

confidence: 99%

“…In particular, canard solutions, which can exist in multiple time-scale systems with a folded geometry, appear as building blocks of complex oscillations in both phenomenological and neurophysiologically plausible models ranging from single cell [23][24][25][26] to neural networks [27,28]. The canard phenomenon in such systems has been related to neural excitability [29], excitability thresholds [23,[30][31][32][33][34], and boundaries between different type of solutions, such as subthreshold oscillations and large amplitude spiking/bursting oscillations [24,28,[35][36][37][38][39][40][41][42][43]. While such canard-organized fine structures have been shown in a wide range of two-time-scale models, recent studies started to explore canard-mediated processes in systems with three or more time-scales [44][45][46].…”

Section: Introductionmentioning

confidence: 99%

“…Canards of folded node and FSN II singularities support mixed mode oscillations [27,36,44]. FSN II singularities have been identified in neuronal models where the exchange from an excitable to a relaxation oscillatory state is accompanied by subthreshold oscillations [24,28,42,62]. Folded-saddle canards have been shown to be sculpting firing threshold manifolds, as well [33,34,[63][64][65].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Canard solutions in neural mass models: consequences on critical regimes

Ersöz

Wendling

2021

J. Math. Neurosc.

Self Cite

View full text Add to dashboard Cite

Mathematical models at multiple temporal and spatial scales can unveil the fundamental mechanisms of critical transitions in brain activities. Neural mass models (NMMs) consider the average temporal dynamics of interconnected neuronal subpopulations without explicitly representing the underlying cellular activity. The mesoscopic level offered by the neural mass formulation has been used to model electroencephalographic (EEG) recordings and to investigate various cerebral mechanisms, such as the generation of physiological and pathological brain activities. In this work, we consider a NMM widely accepted in the context of epilepsy, which includes four interacting neuronal subpopulations with different synaptic kinetics. Due to the resulting three-time-scale structure, the model yields complex oscillations of relaxation and bursting types. By applying the principles of geometric singular perturbation theory, we unveil the existence of the canard solutions and detail how they organize the complex oscillations and excitability properties of the model. In particular, we show that boundaries between pathological epileptic discharges and physiological background activity are determined by the canard solutions. Finally we report the existence of canard-mediated small-amplitude frequency-specific oscillations in simulated local field potentials for decreased inhibition conditions. Interestingly, such oscillations are actually observed in intracerebral EEG signals recorded in epileptic patients during pre-ictal periods, close to seizure onsets.

show abstract

Bursting in a next generation neural mass model with synaptic dynamics: a slow–fast approach

2022

Self Cite

View full text Add to dashboard Cite

We report a detailed analysis on the emergence of bursting in a recently developed neural mass model that includes short-term synaptic plasticity. Neural mass models can mimic the collective dynamics of large-scale neuronal populations in terms of a few macroscopic variables like mean membrane potential and firing rate. The present one is particularly important, as it represents an exact meanfield limit of synaptically coupled quadratic integrate and fire (QIF) neurons. Without synaptic dynamics, a periodic external current with slow frequency $$\varepsilon $$ ε can lead to burst-like dynamics. The firing patterns can be understood using singular perturbation theory, specifically slow–fast dissection. With synaptic dynamics, timescale separation leads to a variety of slow–fast phenomena and their role for bursting becomes inordinately more intricate. Canards are crucial to understand the route to bursting. They describe trajectories evolving nearby repelling locally invariant sets of the system and exist at the transition between subthreshold dynamics and bursting. Near the singular limit $$\varepsilon = 0$$ ε = 0 , we report peculiar jump-on canards, which block a continuous transition to bursting. In the biologically more plausible $$\varepsilon $$ ε -regime, this transition becomes continuous and bursts emerge via consecutive spike-adding transitions. The onset of bursting is complex and involves mixed-type-like torus canards, which form the very first spikes of the burst and follow fast-subsystem repelling limit cycles. We numerically evidence the same mechanisms to be responsible for bursting emergence in the QIF network with plastic synapses. The main conclusions apply for the network, owing to the exactness of the meanfield limit.

show abstract

Canard-induced complex oscillations in an excitatory network

Cited by 10 publications

References 50 publications

Classification of bursting patterns: A tale of two ducks

Classification of bursting patterns: A tale of two ducks

Canard solutions in neural mass models: consequences on critical regimes

Bursting in a next generation neural mass model with synaptic dynamics: a slow–fast approach

Contact Info

Product

Resources

About