Fig 1.
Critical slowing down before neuronal spiking in a pyramidal neuron.
a, Current stimulation protocol, gray areas mark segments from which variance and autocorrelation were calculated, black areas segments used to determine recovery rates. b, Time course of the membrane potential subject to brief perturbations by current injections on top of a slowly depolarizing step current. The inset shows a magnification of the voltage response to a short current injection and an exponential fit to its recovery (red line). c, Recovery rates λ after perturbations, variance and lag-50ms autocorrelation in the subthreshold voltage, in this case for a pyramidal neuron.
Fig 2.
Illustration of stochastic scaling laws near the saddle-node (fold) bifurcation in a model system.
a, Phase space with a single stochastic sample path (black) of a saddle-node bifurcation (eq. 10) for the initial condition (V(0), y(0)) = (−4, 1.6) with σ1 = 0.001, ϵ = 0.001 and small perturbations of size σ2(ti) = 0.1 with ti = 60. The bifurcation occurs at (Vc, yc) = (0, 0) (red dot). The gray curves are the system equilibria (for ϵ = 0). b, Sample path Vd plotted as a time series where the equilibrium values have been subtracted (i.e. detrending along the equilibrium branch). c, Scaling of recovery rate λ, variance v and autocorrelation as dynamics approaches the bifurcation point (red vertical line). Recovery rate and variance follow a power-law scaling with exponents ±0.5 illustrated by black dashed lines.
Fig 3.
Illustration of stochastic scaling laws near the subcritical Hopf bifurcation in a model system.
a, Phase space with a single stochastic sample path (black) of a Hopf bifurcation (eq. 12) for the initial condition (V1(0), V2(0), y0) = (0, 0, −2) with σ1, 2 = 0.001, ϵ = 0.001 and small perturbations of size σ3(ti) = 0.005 with ti = 60. The bifurcation occurs at (V1c, V2c, yc) = (0, 0, 0) (red dot). b, Sample path V1 plotted as a time series used for further analysis. c, Scaling of recovery rate λ, variance v and autocorrelation as dynamics approaches the bifurcation point (red vertical line). Recovery rate and variance follow a power-law scaling with exponents ±1 illustrated by black dashed lines.
Fig 4.
Scaling analysis of indicators related to critical slowing down in pyramidal neurons.
a, photomicrograph of a neuron with pyramidal morphology and typical responses to depolarizing and hyperpolarizing currents. b, Recovery rate as a function of ΔI, the distance to the bifurcation point, for all trials combined and fitted exponents averaged over individual trials and for different minimal values ΔImin for normal conditions (right, black markers, standard deviation) and after bath application of tetrodotoxin (right, gray markers, standard deviation). c, Variance. d, Autocorrelation. Grey dashed lines on the left side show power-laws with exponent 0.5 for recovery rate, -0.5 for variance and -0.27 for autocorrelation.
Fig 5.
Scaling analysis of indicators related to critical slowing down in fast-spiking (FS) neurons.
a, Photomicrograph of a typical FS neuron with round morphology and responses to depolarizing and hyperpolarizing currents. Right: the f-I relationship shows a discontinuity in frequency at the onset of spiking. Different markers correspond to different neurons; for comparability the injected current has been normalized to the onset of spiking. b, FS neurons (red markers) could be distinguished from pyramidal neurons (blue markers) by shorter spike width and greater afterhyperpolarization (AHP) values. c, Exponents (mean ± standard deviation) for recovery rate (θ, round markers) and variance (τ, diamonds) for different minimal values ΔImin of the fit.
Fig 6.
Prediction of the spiking threshold using scaling relations of critical slowing down.
a, The critical voltage Vc in pyramidal neurons was determined as a fit parameter by fitting recovery rates λ (red markers) excluding the last five measurements (blue markers) to voltage by λ = a(Vc−V)θ. ΔVp is the difference between the fitted critical voltage (red line) and the last value included in the fit (green line); ΔVm, respectively, refers to the difference between the measured voltage at the onset of spiking (blue line) and the last value used in the fit (green line). b, Predicted ΔVp and measured ΔVm exhibit a significant correlation when fitted with exponent θ = 0.5 but not when fitted with exponent θ = 1.0 (c). P and R values refer to the linear regression analysis (solid black lines).