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Stabilizing synchrony by inhomogeneity.

Bolhasani E, Valizadeh A - Sci Rep (2015)

Bottom Line: We show that for two weakly coupled identical neuronal oscillators with strictly positive phase resetting curve, isochronous synchrony can only be seen in the absence of noise and an arbitrarily weak noise can destroy entrainment and generate intermittent phase slips.Small inhomogeneity-mismatch in the intrinsic firing rate of the neurons-can stabilize the phase locking and lead to more precise relative spike timing of the two neurons.The results can explain how for a class of neuronal models, including leaky integrate-fire model, inhomogeneity can increase correlation of spike trains when the neurons are synaptically connected.

View Article: PubMed Central - PubMed

Affiliation: Institute for Advanced Studies in Basic Sciences, Department of physics, Zanjan, 45137-66731, Iran.

ABSTRACT
We show that for two weakly coupled identical neuronal oscillators with strictly positive phase resetting curve, isochronous synchrony can only be seen in the absence of noise and an arbitrarily weak noise can destroy entrainment and generate intermittent phase slips. Small inhomogeneity-mismatch in the intrinsic firing rate of the neurons-can stabilize the phase locking and lead to more precise relative spike timing of the two neurons. The results can explain how for a class of neuronal models, including leaky integrate-fire model, inhomogeneity can increase correlation of spike trains when the neurons are synaptically connected.

No MeSH data available.


Related in: MedlinePlus

(A,B) Representative examples of the evolution of the phase difference of two neurons for three different values of mismatch in intrinsic frequencies. Larger values of mismatch have led to fewer phase slips. In (A) neurons are phase oscillators with canonical type-I phase sensitivity and in (B) the results are presented for LIF neurons. (C) The mean escape time is plotted against frequency mismatch. Increasing effective coupling constant Δg = g1 − g2 the maximum escape time is seen in larger values of frequency mismatch. In (D) the ratio of the firing rates of the coupled neurons is plotted. For large values of mismatch the fixed point of 1:1 locking vanishes.
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f1: (A,B) Representative examples of the evolution of the phase difference of two neurons for three different values of mismatch in intrinsic frequencies. Larger values of mismatch have led to fewer phase slips. In (A) neurons are phase oscillators with canonical type-I phase sensitivity and in (B) the results are presented for LIF neurons. (C) The mean escape time is plotted against frequency mismatch. Increasing effective coupling constant Δg = g1 − g2 the maximum escape time is seen in larger values of frequency mismatch. In (D) the ratio of the firing rates of the coupled neurons is plotted. For large values of mismatch the fixed point of 1:1 locking vanishes.

Mentions: The fixed point of Eq. (3) with D = 0 is the solution of Δω = g12H(ϕ) − g21H(−ϕ). For QIF oscillators Z(ϕ) = 1 − cos(ϕ) with asymmetric connections Δg ≠ 0, when the oscillators are identical Δω = 0, the zero-lag synchrony ϕ ≠ 0 is a fixed point, but the system is in the point of a saddle-node bifurcation. Such fixed points are unstable, but in , when no other fixed points are present in the phase space, they attract all the points in the phase space in infinite time52. Consequently, in the absence of noise the oscillators can synchronize isochronously but a weak noise can destroy synchrony and lead to phase slips. Mismatch in the intrinsic firing rates of the neurons, stabilizes the fixed point through a saddle-node bifurcation while moves the fixed point away from zero. For small mismatch, this provides an asymmetric basin of attraction which is vulnerable to sufficiently large perturbations in one direction around the fixed point. In the presence of noise the system shows epochs of intermittent locking between which the relative phase of the oscillators slips by one cycle, while the mean escape time from locked state increases with frequency mismatch (see Fig. 1A,B). This increase in the escape time, is related to expansion of the effective basin of attraction of the stable fixed point of the system. The maximum mean escape time is seen in a certain value of mismatch (Fig. 1C) and further increase of the mismatch shrinks the basin of the attraction of the fixed point in the opposite side which decreases the escape time. For larger mismatch , the fixed point corresponding to 1:1 locked state will disappear through another saddle-node bifurcation (see supplementary material Fig. S1). Two points are worthy of note about generality of the above arguments: First, for the biological neuronal models, PRC is zero near the spike time of the neuron and for type-I neurons with a well-behaved non-negative PRC, this means that the slope of the PRC has opposite signs in the two sides of the spike time which is usually is taken as phase zero. Therefore, for a coupled pair of such neurons (when the neurons are identical), the synchronized state is a fixed point which is attracting but is not stable and the above arguments hold for all the neuronal models which have this property, e.g., Wang-Bazsaki (WB) neurons53. Second, for the type-II oscillators the PRC has negative and positive parts and this means that a well-behaved PRC has at least one zero-crossing point with negative slope and there exists a stable phase-locked fixed point for coupled identical type-II oscillators. For such systems small inhomogeneity does not change the stability of the fixed point but it can expand or shrink its basin of attraction.


Stabilizing synchrony by inhomogeneity.

Bolhasani E, Valizadeh A - Sci Rep (2015)

(A,B) Representative examples of the evolution of the phase difference of two neurons for three different values of mismatch in intrinsic frequencies. Larger values of mismatch have led to fewer phase slips. In (A) neurons are phase oscillators with canonical type-I phase sensitivity and in (B) the results are presented for LIF neurons. (C) The mean escape time is plotted against frequency mismatch. Increasing effective coupling constant Δg = g1 − g2 the maximum escape time is seen in larger values of frequency mismatch. In (D) the ratio of the firing rates of the coupled neurons is plotted. For large values of mismatch the fixed point of 1:1 locking vanishes.
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4559804&req=5

f1: (A,B) Representative examples of the evolution of the phase difference of two neurons for three different values of mismatch in intrinsic frequencies. Larger values of mismatch have led to fewer phase slips. In (A) neurons are phase oscillators with canonical type-I phase sensitivity and in (B) the results are presented for LIF neurons. (C) The mean escape time is plotted against frequency mismatch. Increasing effective coupling constant Δg = g1 − g2 the maximum escape time is seen in larger values of frequency mismatch. In (D) the ratio of the firing rates of the coupled neurons is plotted. For large values of mismatch the fixed point of 1:1 locking vanishes.
Mentions: The fixed point of Eq. (3) with D = 0 is the solution of Δω = g12H(ϕ) − g21H(−ϕ). For QIF oscillators Z(ϕ) = 1 − cos(ϕ) with asymmetric connections Δg ≠ 0, when the oscillators are identical Δω = 0, the zero-lag synchrony ϕ ≠ 0 is a fixed point, but the system is in the point of a saddle-node bifurcation. Such fixed points are unstable, but in , when no other fixed points are present in the phase space, they attract all the points in the phase space in infinite time52. Consequently, in the absence of noise the oscillators can synchronize isochronously but a weak noise can destroy synchrony and lead to phase slips. Mismatch in the intrinsic firing rates of the neurons, stabilizes the fixed point through a saddle-node bifurcation while moves the fixed point away from zero. For small mismatch, this provides an asymmetric basin of attraction which is vulnerable to sufficiently large perturbations in one direction around the fixed point. In the presence of noise the system shows epochs of intermittent locking between which the relative phase of the oscillators slips by one cycle, while the mean escape time from locked state increases with frequency mismatch (see Fig. 1A,B). This increase in the escape time, is related to expansion of the effective basin of attraction of the stable fixed point of the system. The maximum mean escape time is seen in a certain value of mismatch (Fig. 1C) and further increase of the mismatch shrinks the basin of the attraction of the fixed point in the opposite side which decreases the escape time. For larger mismatch , the fixed point corresponding to 1:1 locked state will disappear through another saddle-node bifurcation (see supplementary material Fig. S1). Two points are worthy of note about generality of the above arguments: First, for the biological neuronal models, PRC is zero near the spike time of the neuron and for type-I neurons with a well-behaved non-negative PRC, this means that the slope of the PRC has opposite signs in the two sides of the spike time which is usually is taken as phase zero. Therefore, for a coupled pair of such neurons (when the neurons are identical), the synchronized state is a fixed point which is attracting but is not stable and the above arguments hold for all the neuronal models which have this property, e.g., Wang-Bazsaki (WB) neurons53. Second, for the type-II oscillators the PRC has negative and positive parts and this means that a well-behaved PRC has at least one zero-crossing point with negative slope and there exists a stable phase-locked fixed point for coupled identical type-II oscillators. For such systems small inhomogeneity does not change the stability of the fixed point but it can expand or shrink its basin of attraction.

Bottom Line: We show that for two weakly coupled identical neuronal oscillators with strictly positive phase resetting curve, isochronous synchrony can only be seen in the absence of noise and an arbitrarily weak noise can destroy entrainment and generate intermittent phase slips.Small inhomogeneity-mismatch in the intrinsic firing rate of the neurons-can stabilize the phase locking and lead to more precise relative spike timing of the two neurons.The results can explain how for a class of neuronal models, including leaky integrate-fire model, inhomogeneity can increase correlation of spike trains when the neurons are synaptically connected.

View Article: PubMed Central - PubMed

Affiliation: Institute for Advanced Studies in Basic Sciences, Department of physics, Zanjan, 45137-66731, Iran.

ABSTRACT
We show that for two weakly coupled identical neuronal oscillators with strictly positive phase resetting curve, isochronous synchrony can only be seen in the absence of noise and an arbitrarily weak noise can destroy entrainment and generate intermittent phase slips. Small inhomogeneity-mismatch in the intrinsic firing rate of the neurons-can stabilize the phase locking and lead to more precise relative spike timing of the two neurons. The results can explain how for a class of neuronal models, including leaky integrate-fire model, inhomogeneity can increase correlation of spike trains when the neurons are synaptically connected.

No MeSH data available.


Related in: MedlinePlus