Figure 1.
Phase Advances Induced by SFA and STD in the Network Model
(A) Scheme of the spiking neuron network employed (N = 300).
(B) Postsynaptic current Ipost in response to sinusoidal input current Iin for the networks: no SFA and no STD (−A−D, cyan), SFA only (+A−D, green), STD only (−A+D, blue), and SFA plus STD (+A+D, red). Last panel: mathematical derivative dIin/dt. All panels show Iin (gray) and dIin/dt (black). Signals are plotted rescaled by their s.d. (vertical scale bar = 1 s.d.).
(C) Cross-correlation functions between s.d.-rescaled Iin and Ipost showing the phase shift induced by the various mechanisms. The off-center location and the height of the central peak are measures of the phase advancement and the signal-to-noise ratio in Ipost, respectively.
Figure 2.
Fourier Analysis of the Network Model Output Ipost upon Stimulation with White Noise Input Current Iin
In a range of low frequencies, networks with STD act as differentiators, and networks with SFA have a low-pass cutoff at neighboring frequencies.
(A) Schematic representation of the model and the computation of its transfer function H(ω) from the Fourier transforms of Iin and of Ipost
. H(ω) is an imaginary quantity that is characterized by its magnitude and its phase.
(B) Phase and magnitude of the H(ω) for mathematical derivative (top) or time-shift (bottom) operators.
(C) H(ω) phase and (D) H(ω) magnitude for Γ = 0.5, τD = 400 ms, and τCa = 80 ms in the network models of Figure 1 (same color code). To reduce the variance of H(ω), a network with N = 1,000 presynaptic neurons was used and responses to 50 different white noise realizations were averaged together. Significant phase advancement with linear increasing amplitude occurs for frequencies below 10 Hz when the network includes STD. On a single trial basis, however, only-STD networks produce unsatisfactory, very noisy Ipost (Figures 1 and 4).
(D) Every curve is scaled independently to match the initial slopes of |H(ω)| and to allow for comparisons. Inset: low-frequency range on linear scales.
Figure 3.
Analytical Calculations Show That the Summation of Asynchronous STD Synapses Produces Derivative-Like Postsynaptic Currents in an Identified Range of Low Frequencies
This can be obtained from a white noise input rate (yielding the transfer function H(ω), solid lines) or from a sinusoidal input rate (dashed lines), through a different set of simplifying assumptions (see Protocol S1).
(A) Amplitude of postsynaptic current fluctuations (x-axis is Fourier frequency for solid line and frequency of stimulation sinusoid for dashed curve). Amplitude is plotted in arbitrary units, ignoring all dimensional prefactors. ωinf denotes the inflection point of the solid curve, around which a straight line is a very good approximation.
(B) Phase advancement with respect to input frequency ω. ωmax denotes the frequency at which the maximum phase advancement occurs; around this point the phase is relatively insensitive to frequency ω. Dashed regions correspond to the interval [1/τD,1/τe], in which amplitude grows linearly with ω and phase remains approximately constant. Parameters used were τD = 0.4 s, Γ = 0.4, r0 = 20 Hz.
Figure 4.
The Network Computed the Rate of Change of an Arbitrary Input, Ignoring High-Frequency Fluctuations
(A–E) Same network model as in Figure 1A. All presynaptic model cells receive the same Iin = Isignal + Inoise (gray), with Isignal the sum of low-frequency sinusoids (see Materials and Methods) and Inoise a Gaussian white noise. Panels show Ipost for different combinations of SFA and STD. Colors and symbols as in Figure 1. The derivative of Isignal is plotted (black) to compare with Ipost. Unlike the mathematical derivative shown in (D), SFA plus STD (E) acted as a pass-band filter differentiator.
(F) Smoothed Ipost for experimentally measured synapses (purple) is also very close to the derivative of Isignal (all signals equally smoothed for fair comparison). The response of an SFA-only network is superimposed for comparison (pale green). Vertical scale bars = 1 s.d.
Figure 5.
Experimentally, Real Cortical Synapses Produced Constant Phase Advance and Modulation Gain Proportional to Frequency
(A) Experiment scheme. Electric shocks were triggered by the sequentially chained spike trains of ∼400 presynaptic model cells. Postsynaptic responses in an intracellularly recorded neuron were analysed off-line, translated in synaptic conductances, and added together to simulate the putative total synaptic conductance evoked by the simultaneous activation of ∼400 presynaptic neurons. The voltage-clamped postsynaptic current Ipost is represented.
(B) Control recording for periodic stimulation at 10 Hz (purple) and model trace given by Γ = 0.65 (gray) (Γ is the ratio of peaks of contiguous postsynaptic currents at very high stimulation rates).
(C) Smoothed postsynaptic current phase advance in the model with SFA only (green) was enhanced when replacing nondepressing model synapses by real cortical synapses (purple). Results for two different Iin frequencies are shown (top: 3.5 Hz, bottom: 5 Hz). Signals are plotted normalized to s.d. (computed from triangles onward to avoid transients).
(D) Cross-correlation functions between Iin and non-s.d.–normalized experimental Ipost for the two frequencies (dashed: 3.5 Hz, solid: 5 Hz) overlapped when plotted versus phase and when normalized by stimulation frequency (inset). All data shown here come from the same neuron.
Figure 6.
In a Network Scheme with Both SFA and STD, Basic Integration and Spiking Mechanisms in the Postsynaptic Membrane Allow the Postsynaptic Neuron to Encode the Temporal Derivative
(A) Current injected into presynaptic neurons (Iin, solid black line).
(B) Synaptic conductance opening in the postsynaptic neuron Gsyn(t).
(C) Membrane voltage modulations resulting from synaptic currents and synaptic conductance changes when postsynaptic spiking is inactivated.
(D) Sample spike trains (above) and trial-averaged firing rate (below) of the postsynaptic neuron subject to the presynaptic network activity. In all panels, the dashed black curves trace the mathematical derivative of the input, rescaled by the s.d. and recentered by the mean of the plotted signal to allow a direct comparison. The network is exactly as in Figure 4E, with enabled synaptic integration and spiking mechanisms in the postsynaptic neuron. The firing rate curve in (D) was obtained from 400 different simulations and by averaging together normalized bell-curve of s.d. 10 ms centered at the time of spike occurrence. Average postsynaptic firing rate was 64 Hz.
Figure 7.
Anticipatory Responses through a Taylor-Approximation–Inspired Neurophysiological Circuit
(A) Model scheme for a biological implementation of the first-order Taylor approximation.
(B) Anticipatory Ipost response to Iin = Isignal of Figure 4 (both currents equally smoothed for readability). Dashed square zoomed in inset. Vertical scale bar represents 0.5 μA/cm2.
(C) Cross-correlation function between Ipost and Iin (black) superimposed on autocorrelation function of Iin displaced by 16 ms (gray). Inset shows the displacement of the central peak (dashed square).
Figure 8.
Slightly Degraded Anticipatory Response from a Disordered, Heterogeneous Population of Adapting Neurons and Depressing Synapses
(A) Model scheme for a less constrained anticipatory architecture: presynaptic neurons have different values of the IKCa conductance (gKCa) responsible for SFA, and their synapses have varying degrees of STD (parameter Γ).
(B) Exponential distributions of gKCa and Γ, by which these parameters are randomly and independently distributed in presynaptic neurons and synapses, respectively.
(C) Cross-correlation function between Ipost and Iin superimposed on autocorrelation function of Iin displaced by 17 ms. The output is a slightly degraded version of an anticipation of the input. Inset: zoom on central peak (dashed square).
Figure 9.
Physiological Implementations of an Anticipatory Network Reproduce Advanced Sensory Responses to Moving Versus Flashed Visual Stimuli
(A) Stimulus configurations as in [21]. Incoming current Iin depended on the position of the moving or flashed bar in the receptive field (top row). For a bar moving with speed v, Iin was a Gaussian of s.d. L/v, (L = receptive field radius, ∼3° in primary visual cortex).
(B) Superposition of the Iin's for the two stimuli in (A) (gray lines) and corresponding outputs from the model presented in Figure 7 (Ipost's, black traces) show advanced response to moving with respect to flashed stimuli (Δt = 29 ms). Inset: Apparent spatial shift between moving and flashed stimulus from neural responses (vΔt) for increasing speed v.
(C) Same as in (B) but for the model presented in Figure 8. Inset: zoom on the inputs (gray) and responses (black) around the time of the flash onset, showing the advanced response to moving with respect to flashed stimuli.
Vertical scale bar in (B) for (B) and (C) represents 0.5 μA/cm2.