Fig 1.
Cerebellar Purkinje cells show inverse stochastic resonance (ISR).
A. Whole-cell patch-clamp recording from a Purkinje cell in a cerebellar slice, showing current injection of 1 s noise waveform periods with increasing amplitude, and recorded membrane potential Vm. Holding current is I = −290 pA. The firing rate of the Purkinje cell (PC) is reduced for intermediary noise amplitude. B. Firing frequency during 1 s noise injection vs. noise amplitude σ corresponding to the trace in A. Error bars indicate standard deviation. The firing rate is minimal for σ = 100 pA. C. ISR is observed in all Purkinje cells tested. Summary of optimal noise amplitude σ = 152.60 ± 64.42 pA (n = 19). D. ISR curve of a different PC, generated with a current injection protocol of continuously changing noise amplitude and for a series of holding currents, exploring the full range of the f-I curve (E). The firing rate is most reduced when the cell is hyperpolarized to the edge of the f-I curve step. The optimal noise amplitude for inhibition of firing is σ = 200 pA. E. Frequency vs. current generated with 1 s step current injections. The color code corresponds to the region explored for the ISR curve in D. F. Membrane potential distributions computed from a somatic whole-cell patch-clamp recording from a Purkinje cell during injection of a stimulus current evoking transitions between spiking and silent states (A).
Fig 2.
Experimental characterization of Purkinje cell bistability.
A. Whole-cell patch-clamp recording from a Purkinje cell, showing a representative hysteresis measurement with slow current ramp injection (0.9 nA/s) ascending (red) and descending (blue), and the resulting PC membrane potential response. B. Instantaneous firing frequency and current for each spike. Linear fits of the ascending ramp (red) and the descending ramp (blue) are averages of 10 trials. C. Characterization of the hysteresis using the difference in frequency between first and last spike Δf and difference in current ΔI. The color code illustrates the region explored for the ISR curve. Red corresponds to both the hysteresis and the optimal ISR region (Fig 1D and 1E). D. Distribution of hysteresis parameters across the population of recorded Purkinje cells. E. Correlation between the width of the hysteresis range ΔI and the optimal noise level for ISR σopt. Error bars indicate standard deviation, R2 = 0.79 (n = 19).
Fig 3.
aEIF model fitting procedure to Purkinje cell experimental data.
A. Double somatic whole-cell patch-clamp recording from a representative Purkinje cell: one electrode for current injection and one for voltage recording (scale bar, 100 μ m). B. Traces of injected noise current Iin(t), recorded membrane potential Vm(t), in a spontaneously active PC, calculated membrane current Im(t), and calculated spike-dependent adaptation current wsp(t). C. Im vs. Vm and dynamic IV curve as the average over Vm. Error bars indicate SEM. Inset, the distribution of data points at Vm = -52 mV is Gaussian. D. Fitting the dynamic IV curve F(V) = −Idyn/C with the EIF model function. Parameters are: resting potential Em, membrane time constant τm, threshold potential VT, and spike slope factor ΔT. Error bars indicate SD. Inset, capacitance determination by minimizing the variance of Im. E. Spike triggered adaptation wsp(t) plotted versus time after the last spike. Error bars indicate SEM. Inset, the distribution of data points at t = 12 ms is Gaussian. F. The spike-triggered adaptation is fitted to a single exponential, with time constant τw and wsp (t = 0) = b. Error bars indicate SD.
Fig 4.
Hysteresis and ISR of the aEIF model.
A Voltage response of the aEIF model to Ornstein-Uhlenbeck current noise injection with increasing amplitude. B. Mean firing rate of the aEIF model in response to current noise stimulation with amplitude σ and mean I (color code). C. Hysteresis of the aEIF model. Top, voltage response to ascending (red) and descending (blue) ramp of current. Bottom, instantaneous firing rate vs. instantaneous injected current. Color code is the same as in B. D. Parameter space of the rescaled aEIF model, white region: type I excitability, gray region: type II excitability. The 7 fitted cells are in the type II region. E, F. Dependence of the hysteresis size ΔI on the parameters T τw/τm (E) and A = a/gL (F). G. Membrane potential distribution in the aEIF model during spiking and silent states.
Fig 5.
Bifurcation diagram and phase plane of the aEIF model.
A. Phase-plane of the model. Gray lines are the null-clines of the model. Drop-like set (black) corresponds to the basin of attraction of a stable fixed point. Red and blue trajectories correspond to rest and spiking respectively (inset). B. Bifurcation diagram of the aEIF model. Solid and thin lines represent stable and unstable fixed points for different values of I. Inset shows the intersection of the null-clines before and after the Andronov-Hopf bifurcation (HB) point. Point corresponds to HB. C. Probability of spiking during the stimulation by noise with various means (compare Fig 4B). D, E, F. Phase plane of the model with the corresponding trajectories and voltage traces (inset) when stimulated by Ornstein-Uhlenbeck noise for 1000 ms with mean I = −150 pA and noise amplitudes σ = 10 pA, 30 pA and 60 pA.
Fig 6.
ISR transforms brief inputs into long-term firing states depending on background noise.
A, B, C. Characteristic voltage traces of the aEIF model in response to a single synaptic excitatory input in the presence of different levels of background noise with amplitude σ = 10 pA, 30 pA and 60 pA. Bottom, corresponding probability of spiking for a range of input amplitudes (color code). D. Maximal probability of state transition vs. synaptic input amplitude for 3 background noise amplitudes, E. Decay time constant for the duration of the spiking state induced by a synaptic input of 100 pA. Remark: two data points corresponding to σ = 0 pA, 100 pA are not shown because for σ = 0 pA the duration of stimulus-induced spiking state is infinite, while for σ = 100 pA the duration of this state could not be distinguished from the firing baseline.
Fig 7.
ISR leads to local optimum of mutual information between the input and output spike train.
A, B, C. Voltage traces (top) of the aEIF model in response to series of excitatory synaptic inputs (middle, amplitude Am = 100 pA), in addition to background stimulation by noise with amplitude σ = 10 pA, 30 pA and 60 pA (bottom). D. Mutual Information (MI) of the input and output spike train of the aEIF model. E. MI as a function of the decay time constant (duration of a spiking state). Remark: two data points corresponding to σ = 0 pA, 100 pA are not shown (as in Fig 6E). For σ = 0 pA the duration of the stimulus-induced spiking state is infinite, while for σ = 100 pA the duration of this state could not be distinguished from the firing baseline.