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Dynamic Control of Synchronous Activity in Networks of Spiking Neurons

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ABSTRACT

Oscillatory brain activity is believed to play a central role in neural coding. Accumulating evidence shows that features of these oscillations are highly dynamic: power, frequency and phase fluctuate alongside changes in behavior and task demands. The role and mechanism supporting this variability is however poorly understood. We here analyze a network of recurrently connected spiking neurons with time delay displaying stable synchronous dynamics. Using mean-field and stability analyses, we investigate the influence of dynamic inputs on the frequency of firing rate oscillations. We show that afferent noise, mimicking inputs to the neurons, causes smoothing of the system’s response function, displacing equilibria and altering the stability of oscillatory states. Our analysis further shows that these noise-induced changes cause a shift of the peak frequency of synchronous oscillations that scales with input intensity, leading the network towards critical states. We lastly discuss the extension of these principles to periodic stimulation, in which externally applied driving signals can trigger analogous phenomena. Our results reveal one possible mechanism involved in shaping oscillatory activity in the brain and associated control principles.

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Network stability and equilibrium are shaped by noise.A. Fixed point of the system as per Eq (11) as a function of increasing noise intensity. Noise generically decreases the equilibrium, due to an increased recruitment of recurrent connections. B. Network susceptibility as a function of noise intensity. A gradual shift towards the critical susceptibility Rc occurs under the action of noise, causing the system to transit from slow non-linear oscillations to fast linear oscillations. C. System’s eigenvalues for moderate (D = 0.01) and strong (D = 0.1) noise levels. The eigenvalues gradually shift towards the left hand side of the imaginary plane. Critical eigenvalues (pairs of roots inside the black boxes) translate towards the imaginary axis (Re(λ) = 0) i.e. closer to the critical state. Other parameters are α = 100Hz, β = 2500/mV, g = −2mV/Hz, s = 4mV/Hz, τ = 25ms.
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pone.0161488.g003: Network stability and equilibrium are shaped by noise.A. Fixed point of the system as per Eq (11) as a function of increasing noise intensity. Noise generically decreases the equilibrium, due to an increased recruitment of recurrent connections. B. Network susceptibility as a function of noise intensity. A gradual shift towards the critical susceptibility Rc occurs under the action of noise, causing the system to transit from slow non-linear oscillations to fast linear oscillations. C. System’s eigenvalues for moderate (D = 0.01) and strong (D = 0.1) noise levels. The eigenvalues gradually shift towards the left hand side of the imaginary plane. Critical eigenvalues (pairs of roots inside the black boxes) translate towards the imaginary axis (Re(λ) = 0) i.e. closer to the critical state. Other parameters are α = 100Hz, β = 2500/mV, g = −2mV/Hz, s = 4mV/Hz, τ = 25ms.

Mentions: To investigate the frequency tuning observed in Fig 2, we take a closer look at the solutions of Eq (10). Synchronous activity in our network emerges as a consequence of an expected supercritical Hopf bifurcation commonly seen in recurrent delayed nets [31]. The single fixed point of result from Eq (10) by setting u¯o=w¯2(1+erf[u¯o2D])≈−πDg2g−2πD,(11)where the last equation assumes small noise intensity D. Fig 3A shows that the mean-field fixed point decreases with increasing noise level.


Dynamic Control of Synchronous Activity in Networks of Spiking Neurons
Network stability and equilibrium are shaped by noise.A. Fixed point of the system as per Eq (11) as a function of increasing noise intensity. Noise generically decreases the equilibrium, due to an increased recruitment of recurrent connections. B. Network susceptibility as a function of noise intensity. A gradual shift towards the critical susceptibility Rc occurs under the action of noise, causing the system to transit from slow non-linear oscillations to fast linear oscillations. C. System’s eigenvalues for moderate (D = 0.01) and strong (D = 0.1) noise levels. The eigenvalues gradually shift towards the left hand side of the imaginary plane. Critical eigenvalues (pairs of roots inside the black boxes) translate towards the imaginary axis (Re(λ) = 0) i.e. closer to the critical state. Other parameters are α = 100Hz, β = 2500/mV, g = −2mV/Hz, s = 4mV/Hz, τ = 25ms.
© Copyright Policy
Related In: Results  -  Collection

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Show All Figures
getmorefigures.php?uid=PMC5036852&req=5

pone.0161488.g003: Network stability and equilibrium are shaped by noise.A. Fixed point of the system as per Eq (11) as a function of increasing noise intensity. Noise generically decreases the equilibrium, due to an increased recruitment of recurrent connections. B. Network susceptibility as a function of noise intensity. A gradual shift towards the critical susceptibility Rc occurs under the action of noise, causing the system to transit from slow non-linear oscillations to fast linear oscillations. C. System’s eigenvalues for moderate (D = 0.01) and strong (D = 0.1) noise levels. The eigenvalues gradually shift towards the left hand side of the imaginary plane. Critical eigenvalues (pairs of roots inside the black boxes) translate towards the imaginary axis (Re(λ) = 0) i.e. closer to the critical state. Other parameters are α = 100Hz, β = 2500/mV, g = −2mV/Hz, s = 4mV/Hz, τ = 25ms.
Mentions: To investigate the frequency tuning observed in Fig 2, we take a closer look at the solutions of Eq (10). Synchronous activity in our network emerges as a consequence of an expected supercritical Hopf bifurcation commonly seen in recurrent delayed nets [31]. The single fixed point of result from Eq (10) by setting u¯o=w¯2(1+erf[u¯o2D])≈−πDg2g−2πD,(11)where the last equation assumes small noise intensity D. Fig 3A shows that the mean-field fixed point decreases with increasing noise level.

View Article: PubMed Central - PubMed

ABSTRACT

Oscillatory brain activity is believed to play a central role in neural coding. Accumulating evidence shows that features of these oscillations are highly dynamic: power, frequency and phase fluctuate alongside changes in behavior and task demands. The role and mechanism supporting this variability is however poorly understood. We here analyze a network of recurrently connected spiking neurons with time delay displaying stable synchronous dynamics. Using mean-field and stability analyses, we investigate the influence of dynamic inputs on the frequency of firing rate oscillations. We show that afferent noise, mimicking inputs to the neurons, causes smoothing of the system’s response function, displacing equilibria and altering the stability of oscillatory states. Our analysis further shows that these noise-induced changes cause a shift of the peak frequency of synchronous oscillations that scales with input intensity, leading the network towards critical states. We lastly discuss the extension of these principles to periodic stimulation, in which externally applied driving signals can trigger analogous phenomena. Our results reveal one possible mechanism involved in shaping oscillatory activity in the brain and associated control principles.

No MeSH data available.


Related in: MedlinePlus