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Characterization of K-complexes and slow wave activity in a neural mass model.

Weigenand A, Schellenberger Costa M, Ngo HV, Claussen JC, Martinetz T - PLoS Comput. Biol. (2014)

Bottom Line: While the underlying dynamics on the neuronal level is well known and can be easily measured, the resulting behavior on the macroscopic population level remains unclear.Importantly, there is a canard phenomenon arising from a homoclinic bifurcation, whose orbit determines the shape of large amplitude SOs.The model generates both time series and spectra that strikingly resemble real electroencephalogram data and points out possible differences between the different stages of natural sleep.

View Article: PubMed Central - PubMed

Affiliation: Institute for Neuro- and Bioinformatics, University of Lübeck, Lübeck, Germany; Graduate School for Computing in Medicine and Life Science, University of Lübeck, Lübeck, Germany.

ABSTRACT
NREM sleep is characterized by two hallmarks, namely K-complexes (KCs) during sleep stage N2 and cortical slow oscillations (SOs) during sleep stage N3. While the underlying dynamics on the neuronal level is well known and can be easily measured, the resulting behavior on the macroscopic population level remains unclear. On the basis of an extended neural mass model of the cortex, we suggest a new interpretation of the mechanisms responsible for the generation of KCs and SOs. As the cortex transitions from wake to deep sleep, in our model it approaches an oscillatory regime via a Hopf bifurcation. Importantly, there is a canard phenomenon arising from a homoclinic bifurcation, whose orbit determines the shape of large amplitude SOs. A KC corresponds to a single excursion along the homoclinic orbit, while SOs are noise-driven oscillations around a stable focus. The model generates both time series and spectra that strikingly resemble real electroencephalogram data and points out possible differences between the different stages of natural sleep.

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Related in: MedlinePlus

One-dimensional bifurcation diagrams for different gain levels .Low gain corresponds to high values of . Thick black lines depict stable fixed points, dashed lines unstable fixed points and red lines stable periodic solutions. The gray dashed lines mark bifurcations and separate the different regimes. (a) Two saddle-node bifurcations lead to excitability in region VI and bistability in region V. (b) A Hopf bifurcation appears (between I and III) in addition to the two saddle-nodes. The initial small amplitude limit cycle transitions into a high-amplitude relaxation cycle via a canard explosion. The high-amplitude periodic solutions vanish at the left saddle-node via a homoclinic bifurcation. The period of the relaxation oscillations goes to infinity as one approaches the homoclinic bifurcation. (c) Only the Hopf bifurcation remains, after the saddle nodes disappeared via a cusp bifurcation. Within region II there is no canard anymore.
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pcbi-1003923-g003: One-dimensional bifurcation diagrams for different gain levels .Low gain corresponds to high values of . Thick black lines depict stable fixed points, dashed lines unstable fixed points and red lines stable periodic solutions. The gray dashed lines mark bifurcations and separate the different regimes. (a) Two saddle-node bifurcations lead to excitability in region VI and bistability in region V. (b) A Hopf bifurcation appears (between I and III) in addition to the two saddle-nodes. The initial small amplitude limit cycle transitions into a high-amplitude relaxation cycle via a canard explosion. The high-amplitude periodic solutions vanish at the left saddle-node via a homoclinic bifurcation. The period of the relaxation oscillations goes to infinity as one approaches the homoclinic bifurcation. (c) Only the Hopf bifurcation remains, after the saddle nodes disappeared via a cusp bifurcation. Within region II there is no canard anymore.

Mentions: As can be seen in Figure 2 the dynamics of the system is shaped by two bifurcations. The first one is a fold created by two saddle node bifurcations (black), that vanishes in a cusp. Between the two saddle nodes there are three equilibrium states, leading to bistability or excitability, see Figure 3a or Figure 3b. This is in good agreement with [31] and [70], as in the case of a fixed sodium concentration is constant, and an increase in acts as a decrease in resting potential.


Characterization of K-complexes and slow wave activity in a neural mass model.

Weigenand A, Schellenberger Costa M, Ngo HV, Claussen JC, Martinetz T - PLoS Comput. Biol. (2014)

One-dimensional bifurcation diagrams for different gain levels .Low gain corresponds to high values of . Thick black lines depict stable fixed points, dashed lines unstable fixed points and red lines stable periodic solutions. The gray dashed lines mark bifurcations and separate the different regimes. (a) Two saddle-node bifurcations lead to excitability in region VI and bistability in region V. (b) A Hopf bifurcation appears (between I and III) in addition to the two saddle-nodes. The initial small amplitude limit cycle transitions into a high-amplitude relaxation cycle via a canard explosion. The high-amplitude periodic solutions vanish at the left saddle-node via a homoclinic bifurcation. The period of the relaxation oscillations goes to infinity as one approaches the homoclinic bifurcation. (c) Only the Hopf bifurcation remains, after the saddle nodes disappeared via a cusp bifurcation. Within region II there is no canard anymore.
© Copyright Policy
Related In: Results  -  Collection

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Show All Figures
getmorefigures.php?uid=PMC4230734&req=5

pcbi-1003923-g003: One-dimensional bifurcation diagrams for different gain levels .Low gain corresponds to high values of . Thick black lines depict stable fixed points, dashed lines unstable fixed points and red lines stable periodic solutions. The gray dashed lines mark bifurcations and separate the different regimes. (a) Two saddle-node bifurcations lead to excitability in region VI and bistability in region V. (b) A Hopf bifurcation appears (between I and III) in addition to the two saddle-nodes. The initial small amplitude limit cycle transitions into a high-amplitude relaxation cycle via a canard explosion. The high-amplitude periodic solutions vanish at the left saddle-node via a homoclinic bifurcation. The period of the relaxation oscillations goes to infinity as one approaches the homoclinic bifurcation. (c) Only the Hopf bifurcation remains, after the saddle nodes disappeared via a cusp bifurcation. Within region II there is no canard anymore.
Mentions: As can be seen in Figure 2 the dynamics of the system is shaped by two bifurcations. The first one is a fold created by two saddle node bifurcations (black), that vanishes in a cusp. Between the two saddle nodes there are three equilibrium states, leading to bistability or excitability, see Figure 3a or Figure 3b. This is in good agreement with [31] and [70], as in the case of a fixed sodium concentration is constant, and an increase in acts as a decrease in resting potential.

Bottom Line: While the underlying dynamics on the neuronal level is well known and can be easily measured, the resulting behavior on the macroscopic population level remains unclear.Importantly, there is a canard phenomenon arising from a homoclinic bifurcation, whose orbit determines the shape of large amplitude SOs.The model generates both time series and spectra that strikingly resemble real electroencephalogram data and points out possible differences between the different stages of natural sleep.

View Article: PubMed Central - PubMed

Affiliation: Institute for Neuro- and Bioinformatics, University of Lübeck, Lübeck, Germany; Graduate School for Computing in Medicine and Life Science, University of Lübeck, Lübeck, Germany.

ABSTRACT
NREM sleep is characterized by two hallmarks, namely K-complexes (KCs) during sleep stage N2 and cortical slow oscillations (SOs) during sleep stage N3. While the underlying dynamics on the neuronal level is well known and can be easily measured, the resulting behavior on the macroscopic population level remains unclear. On the basis of an extended neural mass model of the cortex, we suggest a new interpretation of the mechanisms responsible for the generation of KCs and SOs. As the cortex transitions from wake to deep sleep, in our model it approaches an oscillatory regime via a Hopf bifurcation. Importantly, there is a canard phenomenon arising from a homoclinic bifurcation, whose orbit determines the shape of large amplitude SOs. A KC corresponds to a single excursion along the homoclinic orbit, while SOs are noise-driven oscillations around a stable focus. The model generates both time series and spectra that strikingly resemble real electroencephalogram data and points out possible differences between the different stages of natural sleep.

Show MeSH
Related in: MedlinePlus