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. 2003 Dec 17;23(37):11628-40.
doi: 10.1523/JNEUROSCI.23-37-11628.2003.

How spike generation mechanisms determine the neuronal response to fluctuating inputs

Nicolas Fourcaud-Trocmé  1 , David Hansel, Carl van Vreeswijk, Nicolas Brunel
Affiliations

How spike generation mechanisms determine the neuronal response to fluctuating inputs

Nicolas Fourcaud-Trocmé et al. J Neurosci. .

Abstract

This study examines the ability of neurons to track temporally varying inputs, namely by investigating how the instantaneous firing rate of a neuron is modulated by a noisy input with a small sinusoidal component with frequency (f). Using numerical simulations of conductance-based neurons and analytical calculations of one-variable nonlinear integrate-and-fire neurons, we characterized the dependence of this modulation on f. For sufficiently high noise, the neuron acts as a low-pass filter. The modulation amplitude is approximately constant for frequencies up to a cutoff frequency, fc, after which it decays. The cutoff frequency increases almost linearly with the firing rate. For higher frequencies, the modulation amplitude decays as C/falpha, where the power alpha depends on the spike initiation mechanism. For conductance-based models, alpha = 1, and the prefactor C depends solely on the average firing rate and a spike "slope factor," which determines the sharpness of the spike initiation. These results are attributable to the fact that near threshold, the sodium activation variable can be approximated by an exponential function. Using this feature, we propose a simplified one-variable model, the "exponential integrate-and-fire neuron," as an approximation of a conductance-based model. We show that this model reproduces the dynamics of a simple conductance-based model extremely well. Our study shows how an intrinsic neuronal property (the characteristics of fast sodium channels) determines the speed with which neurons can track changes in input.

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Figures

Figure 1.
Figure 1.
A, B, f-I curves of the LIF, QIF, EIF, and WB neurons for a constant input current (A) and a noisy input current (B) (Gaussian white noise, σ = 5 mV). The parameters of the EIF model were chosen to match the f-I curve of the WB model. The parameters of the QIF model were chosen to match the behavior of the f-I curve of the WB model near firing onset. The range of firing rates in which the f-I curves of the QIF and WB models match is more restricted than for the EIF model. The f-I curve of the LIF neuron cannot be made to agree with the f-I curves of the other models at low firing rates because of the different qualitative dependence of the firing rate on the input current (logarithmic vs square-root). In contrast, the parameters of the LIF model can be determined to match the f-I curve of the WB model at high frequencies (see Materials and Methods for details on the determination of the model parameters).
Figure 2.
Figure 2.
Firing rate modulation of a conductance-based neuron. A, Deterministic part of the input current (dashed line) with input noise (solid line). B, Raster plot, 2000 repetitions of the input current with independent noise sources. C, Instantaneous firing rate of the neuron averaged over these repetitions. D, E, Gain (D) and phase shift (E) of the firing rate modulation of a WB neuron versus the frequency, f, of the oscillating input. Spiking threshold, Vs, is -20 mV (□) and 20 mV (○). The dashed line in D is obtained by fitting the simulated data, assuming a decay proportional to 1/f at high frequencies. The phase shift at high frequencies depends on the definition of the spike time. It increases linearly with f (dotted and dotted-dashed lines are linear fits at high frequencies). In both D and E, error bars are smaller than symbol size. Parameters: σ = 6.3 mV (white noise), ν0 = 40 Hz.
Figure 9.
Figure 9.
Comparison of the filters of the WB model (□) and the EIF model (○). We plot the gain (A, 1-2) and phase shift (B, 1-2) with high (A-1, B-1) and low noise (A-2, B-2). Note the good agreement in all regimes for the gain. The phase shifts of both models are very similar up to an input frequency at ∼100 Hz in which the WB model has an additional phase lag. This is a consequence of the fixed delay between EIF and WB spike time shown in Figure 3B.
Figure 3.
Figure 3.
A, Voltage traces for WB, EIF, QIF, and LIF neurons for the same realization of the noisy input current. B shows a higher resolution for a short time interval in which a spike has been generated in all models. The subthreshold traces are similar for all models; however, the dynamics of the spike are different on an msec time scale. When the fluctuation leads to a spike in all models, the LIF neuron spikes first. The EIF neuron spikes almost exactly at the spike onset of the WB. The QIF neuron fires much later. For details of the QIF and EIF parameters, see Materials and Methods. For the WB parameters, see Appendix A. Here, the LIF model has the leak current of the WB model, a reset potential of Vr = - 68 mV, and Vth = - 57 mV, to get the same average firing rate as the WB model. C, I-V curve of the EIF (solid line) and WB (dotted line) neurons. The threshold VT is defined as the minimum of the curve. The spike slope factor ΔT is proportional to the radius of the curvature of the I-V curve at its minimum.
Figure 4.
Figure 4.
The qualitative behaviors of the firing rate modulation of the EIF model at intermediate frequencies depend on the characteristics of the input current. Insets show a representative example of the corresponding behavior (gain vs input frequency in log-log scale). In all insets, we plot a simulation of the EIF model (squares, gain) and the EIF and QIF high input frequency regimes (solid and dotted-dashed lines). The dashed line gives the noise level below which there are resonant peaks at the average firing rate ν0 and possibly at integer multiples of ν0 in response. Above the dashed line, the EIF model behaves like a low-pass filter, with approximately constant gain at low frequencies and a 1/f attenuation for sufficiently high frequencies. The gain reaches the asymptotic behavior from above at low firing rates, whereas it reaches the asymptotic behavior from below for high firing rates and high noise (see differences in the two insets in the high noise region).
Figure 5.
Figure 5.
Influence of the spike sharpness on the EIF filter. A, B, Gain (A) and phase (B) of the firing rate modulation are plotted for different values of the spike slope factor ΔT, indicated in A. Other parameters: ν0 = 20 Hz, σ = 6.3 mV. Note that as ΔT decreases, the high frequency asymptotic regime is shifted to higher input frequencies. For large ΔT, the gain decays faster than 1/f, and the phase shift is larger than 90° in an intermediate frequency range.
Figure 6.
Figure 6.
Influence of noise correlation time on the gain of the EIF neuron. The stationary firing rate is ν0 = 30 Hz, and the noise level is adjusted so that the low-frequency response is constant as the synaptic decay time, τs, is varied from 0 msec (white noise) to 20 msec.
Figure 7.
Figure 7.
A, B, Cutoff frequency of the EIF filter in the high-noise regime as a function of the firing rate (A) and spike slope factor (B). In A, the parameters of the EIF model are chosen to match the WB model (ΔT = 3.48 mV). In B, δT is varied. In both panels, σ = 8 mV. A, The cutoff frequency is approximately proportional to the average firing rate ν0 (simulations with white noise). B, The cutoff frequency depends weakly on the slope factor ΔT for white noise but strongly increases when δT decreases for colored noise (ν0 = 24 Hz; values of the synaptic time constants are indicated in the legend).
Figure 8.
Figure 8.
Amplitude of the first three Fourier components of the response, normalized by the average firing rate, ν0, as a function of the strength of input modulation. Circles, First Fourier component; squares, second Fourier component; diamonds, third Fourier component. Parameters: σ = 6.3 mV; ν0 = 20 Hz; f = 20 Hz. The first Fourier component, ν1, is linear in I1 up to modulations of the firing rate of order ν0. The higher Fourier components are negligible up to approximately the same amount of modulation. Note that all simulations presented in this study (except in this figure) are done with ν1 /ν0 ≈ 0.25, well into the domain of validity of the linear approximation.
Figure 10.
Figure 10.
Comparison of the response of the WB (solid line) and EIF (dashed line) models to a current step (average, >1,600,000 repetitions; firing rate computed in 0.6 msec bins; σ = 6.3 mV). The time course of the responses of WB and EIF models is indistinguishable. The agreement between both models is excellent, confirming the close match of the linear filters shown in Figure 9.
Figure 11.
Figure 11.
Gain of the firing rate modulation of EIF and WB models with noninstantaneous sodium activation kinetics. The dashed line shows the asymptotic regime for an instantaneous activation. The activation kinetics attenuates the response slightly more only for frequencies beyond several hundred Hertz.
Figure 12.
Figure 12.
Amplitude of the firing rate modulation for a conductance-based model with additional currents. A, B, Response to inputs with strong (A) and weak (B) noise and various average firing rates (indicated in corresponding panels). The high frequency behavior is proportional to 1/f with a cutoff close to the average firing rate, ν0. The dashed line shows the EIF high-input frequency asymptotic regime with parameters matching the sodium activation curve of the conductance-based neuron. Note that in this figure, the frequency is normalized to the average firing rate, ν0, so that all response curves match approximately at high frequencies. The 1/f regime extends up to 1000 Hz as in the WB model.
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References

    1. Abbott LF, van Vreeswijk C ( 1993) Asynchronous states in a network of pulse-coupled oscillators. Phys Rev E Stat Phys Plasmas Fluids Relat Interdiscip Topics 48: 1483-1490. - PubMed
    1. Bair W, Koch C ( 1996) Temporal precision of spike trains in extrastriate cortex of the behaving macaque monkey. Neural Comput 8: 1185-1202. - PubMed
    1. Brunel N ( 2000) Dynamics of sparsely connected networks of excitatory and inhibitory spiking neurons. J Comput Neurosci 8: 183-208. - PubMed
    1. Brunel N, Hakim V ( 1999) Fast global oscillations in networks of integrate-and-fire neurons with low firing rates. Neural Comput 11: 1621-1671. - PubMed
    1. Brunel N, Chance F, Fourcaud N, Abbott L ( 2001) Effects of synaptic noise and filtering on the frequency response of spiking neurons. Phys Rev Lett 86: 2186-2189. - PubMed

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