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Asynchronous Rate Chaos in Spiking Neuronal Circuits.

Harish O, Hansel D - PLoS Comput. Biol. (2015)

Bottom Line: Another feature of this regime is that the population-averaged firing rate is substantially smaller in the excitatory population than in the inhibitory population.This is not necessarily the case in the I-I mechanism.Finally, we discuss the neurophysiological and computational significance of our results.

View Article: PubMed Central - PubMed

Affiliation: Center for Neurophysics, Physiology and Pathologies, CNRS UMR8119 and Institute of Neuroscience and Cognition, Université Paris Descartes, Paris, France.

ABSTRACT
The brain exhibits temporally complex patterns of activity with features similar to those of chaotic systems. Theoretical studies over the last twenty years have described various computational advantages for such regimes in neuronal systems. Nevertheless, it still remains unclear whether chaos requires specific cellular properties or network architectures, or whether it is a generic property of neuronal circuits. We investigate the dynamics of networks of excitatory-inhibitory (EI) spiking neurons with random sparse connectivity operating in the regime of balance of excitation and inhibition. Combining Dynamical Mean-Field Theory with numerical simulations, we show that chaotic, asynchronous firing rate fluctuations emerge generically for sufficiently strong synapses. Two different mechanisms can lead to these chaotic fluctuations. One mechanism relies on slow I-I inhibition which gives rise to slow subthreshold voltage and rate fluctuations. The decorrelation time of these fluctuations is proportional to the time constant of the inhibition. The second mechanism relies on the recurrent E-I-E feedback loop. It requires slow excitation but the inhibition can be fast. In the corresponding dynamical regime all neurons exhibit rate fluctuations on the time scale of the excitation. Another feature of this regime is that the population-averaged firing rate is substantially smaller in the excitatory population than in the inhibitory population. This is not necessarily the case in the I-I mechanism. Finally, we discuss the neurophysiological and computational significance of our results.

No MeSH data available.


Related in: MedlinePlus

DMFT for the inhibitory rate model with threshold-linear transfer function.A: The PAC amplitude, σ0 − σ∞, is plotted against J0. At fixed point σ0 − σ∞ = 0 (blue). When  (black dot, BP) a bifurcation occurs and the chaotic state appears. For J0 > Jc, the fixed point is unstable (dashed blue) and the network settles in the chaotic state (σ0 − σ∞ > 0, black). Red: Perturbative solution in the limit J0 → Jc (see S4 Text). Inset:  plotted against δ = J0 − Jc showing that σ0 − σ∞ vanishes quadratically when δ → 0+. Black: Full numerical solution of the DMFT equations. Red: Perturbative solution at the leading order, O(δ). B: (σ − σ∞)/δ2 is plotted for different values of δ > 0 to show the convergence to the asymptotic function derived perturbatively in S4 Text. Inset: The function (σ(τ) − σ∞)/δ2 (black) can be well fitted to A/cosh(x/xdec) (red dots, A = 12.11, xdec = 2.84). C: Decorrelation time, τdec vs. PAC amplitude (blue). The function σ(τ) − σ∞ was obtained by integrating numerically Eq (29) and τdec was estimated by fitting this function to A/cosh(τ/τdec). Red: In the whole range of J0 considered (J0 ∈ [1.4, 1.9] the relation between τdec and σ0 − σ∞ can be well approximated by . Inset: The PAC computed by solving the DMFT equations for J0 = 1.81 (blue dots) and the fit to 0.93/cosh(τ/4.6). D: The PAC for J0 = 2 and K = 1200. Blue: Numerical integration of Eq (29). Red: Numerical simulations for N = 256,000.
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pcbi.1004266.g007: DMFT for the inhibitory rate model with threshold-linear transfer function.A: The PAC amplitude, σ0 − σ∞, is plotted against J0. At fixed point σ0 − σ∞ = 0 (blue). When (black dot, BP) a bifurcation occurs and the chaotic state appears. For J0 > Jc, the fixed point is unstable (dashed blue) and the network settles in the chaotic state (σ0 − σ∞ > 0, black). Red: Perturbative solution in the limit J0 → Jc (see S4 Text). Inset: plotted against δ = J0 − Jc showing that σ0 − σ∞ vanishes quadratically when δ → 0+. Black: Full numerical solution of the DMFT equations. Red: Perturbative solution at the leading order, O(δ). B: (σ − σ∞)/δ2 is plotted for different values of δ > 0 to show the convergence to the asymptotic function derived perturbatively in S4 Text. Inset: The function (σ(τ) − σ∞)/δ2 (black) can be well fitted to A/cosh(x/xdec) (red dots, A = 12.11, xdec = 2.84). C: Decorrelation time, τdec vs. PAC amplitude (blue). The function σ(τ) − σ∞ was obtained by integrating numerically Eq (29) and τdec was estimated by fitting this function to A/cosh(τ/τdec). Red: In the whole range of J0 considered (J0 ∈ [1.4, 1.9] the relation between τdec and σ0 − σ∞ can be well approximated by . Inset: The PAC computed by solving the DMFT equations for J0 = 1.81 (blue dots) and the fit to 0.93/cosh(τ/4.6). D: The PAC for J0 = 2 and K = 1200. Blue: Numerical integration of Eq (29). Red: Numerical simulations for N = 256,000.

Mentions: We studied the dynamics in detail for γ = 1. The DMFT predicts that for all I0 and K (K large). As already mentioned, the simulations agree well with this result (Fig 5B). We studied analytically the dynamics for J0 close to this transition (Fig 7A-7C). To this end, we solved the self-consistent DMFT equations in the limit δ = J0 − Jc → 0+. The perturbative calculation, explained in S4 Text, is less straightforward than in the case of a sigmoid transfer function. This stems from the fact that at the threshold, the threshold-linear transfer function is only differentiable once. It yields that σ − σ∞ ∼ δασs(τ/δβ) with α = 2, β = −1/2 and the function σs(x)) has to be determined numerically. The function σs is plotted in Fig 7B. It can be well fitted to the function A[cosh(x/xdec)]−1 with A = 12.11 and xdec = 2.84 (see Fig 7B, inset). In particular, for small δ, the amplitude and the decorrelation time of the PAC are related by τdec ∝ 1/(σ0 − σ∞)1/4. Note that the amplitude of the PAC vanishes more rapidly (α = 2) than for sigmoidal transfer functions (α = 1) whereas the decorrelation time diverges with the same critical exponent (β = −1/2) in the two cases.


Asynchronous Rate Chaos in Spiking Neuronal Circuits.

Harish O, Hansel D - PLoS Comput. Biol. (2015)

DMFT for the inhibitory rate model with threshold-linear transfer function.A: The PAC amplitude, σ0 − σ∞, is plotted against J0. At fixed point σ0 − σ∞ = 0 (blue). When  (black dot, BP) a bifurcation occurs and the chaotic state appears. For J0 > Jc, the fixed point is unstable (dashed blue) and the network settles in the chaotic state (σ0 − σ∞ > 0, black). Red: Perturbative solution in the limit J0 → Jc (see S4 Text). Inset:  plotted against δ = J0 − Jc showing that σ0 − σ∞ vanishes quadratically when δ → 0+. Black: Full numerical solution of the DMFT equations. Red: Perturbative solution at the leading order, O(δ). B: (σ − σ∞)/δ2 is plotted for different values of δ > 0 to show the convergence to the asymptotic function derived perturbatively in S4 Text. Inset: The function (σ(τ) − σ∞)/δ2 (black) can be well fitted to A/cosh(x/xdec) (red dots, A = 12.11, xdec = 2.84). C: Decorrelation time, τdec vs. PAC amplitude (blue). The function σ(τ) − σ∞ was obtained by integrating numerically Eq (29) and τdec was estimated by fitting this function to A/cosh(τ/τdec). Red: In the whole range of J0 considered (J0 ∈ [1.4, 1.9] the relation between τdec and σ0 − σ∞ can be well approximated by . Inset: The PAC computed by solving the DMFT equations for J0 = 1.81 (blue dots) and the fit to 0.93/cosh(τ/4.6). D: The PAC for J0 = 2 and K = 1200. Blue: Numerical integration of Eq (29). Red: Numerical simulations for N = 256,000.
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Related In: Results  -  Collection

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pcbi.1004266.g007: DMFT for the inhibitory rate model with threshold-linear transfer function.A: The PAC amplitude, σ0 − σ∞, is plotted against J0. At fixed point σ0 − σ∞ = 0 (blue). When (black dot, BP) a bifurcation occurs and the chaotic state appears. For J0 > Jc, the fixed point is unstable (dashed blue) and the network settles in the chaotic state (σ0 − σ∞ > 0, black). Red: Perturbative solution in the limit J0 → Jc (see S4 Text). Inset: plotted against δ = J0 − Jc showing that σ0 − σ∞ vanishes quadratically when δ → 0+. Black: Full numerical solution of the DMFT equations. Red: Perturbative solution at the leading order, O(δ). B: (σ − σ∞)/δ2 is plotted for different values of δ > 0 to show the convergence to the asymptotic function derived perturbatively in S4 Text. Inset: The function (σ(τ) − σ∞)/δ2 (black) can be well fitted to A/cosh(x/xdec) (red dots, A = 12.11, xdec = 2.84). C: Decorrelation time, τdec vs. PAC amplitude (blue). The function σ(τ) − σ∞ was obtained by integrating numerically Eq (29) and τdec was estimated by fitting this function to A/cosh(τ/τdec). Red: In the whole range of J0 considered (J0 ∈ [1.4, 1.9] the relation between τdec and σ0 − σ∞ can be well approximated by . Inset: The PAC computed by solving the DMFT equations for J0 = 1.81 (blue dots) and the fit to 0.93/cosh(τ/4.6). D: The PAC for J0 = 2 and K = 1200. Blue: Numerical integration of Eq (29). Red: Numerical simulations for N = 256,000.
Mentions: We studied the dynamics in detail for γ = 1. The DMFT predicts that for all I0 and K (K large). As already mentioned, the simulations agree well with this result (Fig 5B). We studied analytically the dynamics for J0 close to this transition (Fig 7A-7C). To this end, we solved the self-consistent DMFT equations in the limit δ = J0 − Jc → 0+. The perturbative calculation, explained in S4 Text, is less straightforward than in the case of a sigmoid transfer function. This stems from the fact that at the threshold, the threshold-linear transfer function is only differentiable once. It yields that σ − σ∞ ∼ δασs(τ/δβ) with α = 2, β = −1/2 and the function σs(x)) has to be determined numerically. The function σs is plotted in Fig 7B. It can be well fitted to the function A[cosh(x/xdec)]−1 with A = 12.11 and xdec = 2.84 (see Fig 7B, inset). In particular, for small δ, the amplitude and the decorrelation time of the PAC are related by τdec ∝ 1/(σ0 − σ∞)1/4. Note that the amplitude of the PAC vanishes more rapidly (α = 2) than for sigmoidal transfer functions (α = 1) whereas the decorrelation time diverges with the same critical exponent (β = −1/2) in the two cases.

Bottom Line: Another feature of this regime is that the population-averaged firing rate is substantially smaller in the excitatory population than in the inhibitory population.This is not necessarily the case in the I-I mechanism.Finally, we discuss the neurophysiological and computational significance of our results.

View Article: PubMed Central - PubMed

Affiliation: Center for Neurophysics, Physiology and Pathologies, CNRS UMR8119 and Institute of Neuroscience and Cognition, Université Paris Descartes, Paris, France.

ABSTRACT
The brain exhibits temporally complex patterns of activity with features similar to those of chaotic systems. Theoretical studies over the last twenty years have described various computational advantages for such regimes in neuronal systems. Nevertheless, it still remains unclear whether chaos requires specific cellular properties or network architectures, or whether it is a generic property of neuronal circuits. We investigate the dynamics of networks of excitatory-inhibitory (EI) spiking neurons with random sparse connectivity operating in the regime of balance of excitation and inhibition. Combining Dynamical Mean-Field Theory with numerical simulations, we show that chaotic, asynchronous firing rate fluctuations emerge generically for sufficiently strong synapses. Two different mechanisms can lead to these chaotic fluctuations. One mechanism relies on slow I-I inhibition which gives rise to slow subthreshold voltage and rate fluctuations. The decorrelation time of these fluctuations is proportional to the time constant of the inhibition. The second mechanism relies on the recurrent E-I-E feedback loop. It requires slow excitation but the inhibition can be fast. In the corresponding dynamical regime all neurons exhibit rate fluctuations on the time scale of the excitation. Another feature of this regime is that the population-averaged firing rate is substantially smaller in the excitatory population than in the inhibitory population. This is not necessarily the case in the I-I mechanism. Finally, we discuss the neurophysiological and computational significance of our results.

No MeSH data available.


Related in: MedlinePlus