Decorrelation of neural-network activity by inhibitory feedback

Abstract

Correlations in spike-train ensembles can seriously impair the encoding of information by their spatio-temporal structure. An inevitable source of correlation in finite neural networks is common presynaptic input to pairs of neurons. Recent studies demonstrate that spike correlations in recurrent neural networks are considerably smaller than expected based on the amount of shared presynaptic input. Here, we explain this observation by means of a linear network model and simulations of networks of leaky integrate-and-fire neurons. We show that inhibitory feedback efficiently suppresses pairwise correlations and, hence, population-rate fluctuations, thereby assigning inhibitory neurons the new role of active decorrelation. We quantify this decorrelation by comparing the responses of the intact recurrent network (feedback system) and systems where the statistics of the feedback channel is perturbed (feedforward system). Manipulations of the feedback statistics can lead to a significant increase in the power and coherence of the population response. In particular, neglecting correlations within the ensemble of feedback channels or between the external stimulus and the feedback amplifies population-rate fluctuations by orders of magnitude. The fluctuation suppression in homogeneous inhibitory networks is explained by a negative feedback loop in the one-dimensional dynamics of the compound activity. Similarly, a change of coordinates exposes an effective negative feedback loop in the compound dynamics of stable excitatory-inhibitory networks. The suppression of input correlations in finite networks is explained by the population averaged correlations in the linear network model: In purely inhibitory networks, shared-input correlations are canceled by negative spike-train correlations. In excitatory-inhibitory networks, spike-train correlations are typically positive. Here, the suppression of input correlations is not a result of the mere existence of correlations between excitatory (E) and inhibitory (I) neurons, but a consequence of a particular structure of correlations among the three possible pairings (EE, EI, II).

Figure 1. Spiking activity in excitatory-inhibitory LIF networks with intact (left column; feedback scenario) and opened feedback loop (right column; feedforward scenario).

A,B: Network sketches for the feedback (A) and feedforward scenario (B). C,D: Spiking activity (top panels) and population averaged firing rate (bottom panels) of the local presynaptic populations. E,F: Response spiking activity (top panels) and population averaged response rate (bottom panels). In the top panels of C–F, each pixel depicts the number of spikes (gray coded) of a subpopulation of neurons in a time interval. In both the feedback and the feedforward scenario, the neuron population is driven by the same realization of an uncorrelated white-noise ensemble; local input is fed to the population through the same connectivity matrix . The in-degrees, the synaptic weights and the shared-input statistics are thus exactly identical in the two scenarios. In the feedback case (A), local presynaptic spike-trains are provided by the network's response , i.e. the pre- (C) and postsynaptic spike-train ensembles (E) are identical. In the feedforward scenario (B), the local presynaptic spike-train population is replaced by an ensemble of independent realizations of a Poisson point process (D). Its rate is identical to the time- and population-averaged firing rate in the feedback case. See Table 1 and Table 2 for details on network models and parameters.

Figure 2. Suppression of low-frequency fluctuations in recurrent LIF networks with purely inhibitory (A, C) and mixed excitatory-inhibitory coupling (B, D) for instantaneous synapses with delay (A, B) and low-pass synapses with (C, D).

Power-spectra of population rates for the feedback (black) and the feedforward case (gray; cf. Fig. 1). See Table 1 and Table 2 for details on network models and parameters. In C and D, local synaptic inputs are modeled as currents with -function shaped kernel with time constant ( denotes Heaviside function). (Excitatory) Synaptic weights are set to (see Table 1 for details). Simulation time . Single-trial spectra smoothed by moving average (frame size ).

Figure 3. Partial canceling of fluctuations in a linear system by inhibitory feedback.

Response of a linear system with impulse response (1st-order low-pass, cutoff frequency ) to Gaussian white noise input with amplitude for three local-input scenarios. A (light gray): No feedback (local input ). B (black): Negative feedback () with strength . The fluctuations of the weighted local input (B) are anticorrelated to the external drive (B). C (dark gray): Feedback in B is replaced by uncorrelated feedforward input with the same auto-statistics as the response in B. The local input is constructed by assigning a random phase to each Fourier component of the response in B. Fluctuations in C and C are uncorrelated. A, B, C: Network sketches. A , B , C : External input . A , B , C : Weighted local input . A , B , C : Responses . D, E: Response auto-correlation functions (D) and power-spectra (E) for the three cases shown in A,B,C (same gray coding as in A,B,C; inset in D: normalized auto-correlations).

Figure 4. Suppression of low-frequency (LF) population-rate fluctuations in linearized homogeneous random networks with purely inhibitory (A) and mixed excitatory-inhibitory coupling (B).

Dependence of the zero-frequency power ratio on the effective coupling strength (solid curves: full solutions; dashed lines: strong-coupling approximations). The power ratio represents the ratio between the low-frequency population-rate power in the recurrent networks (A: Fig. 3 B; B: Fig. 5 A,B) and in networks where the feedback channels are replaced by uncorrelated feedforward input (A: Fig. 3 C; B, black: Fig. 5 C,D; B, gray: Fig. 5 D′). Dotted curves in B depict power ratio of the sum modes and (see text). B: Balance factor .

Figure 5. Sketch of the 2D (excitatory-inhibitory) model for the feedback (A,B) and the feedforward scenario (C,D) in normal (A,C) and Schur-basis representation (B,D).

A: Original 2D recurrent system. B: Schur-basis representation of the system shown in A. C: Feedforward scenario: Excitatory and inhibitory feedback connections of the original network (A) are replaced by feedforward input from populations with rates , , respectively. D: Schur-basis representation of the system shown in C. D′: Alternative feedforward scenario: Here, the feedforward channel (weight ) of the original system in Schur basis (B) remains intact. Only the inhibitory feedback (weight ) is replaced by feedforward input .

Figure 6. Dependence of population averaged correlations and population-rate fluctuations on the effective coupling in a linearized homogeneous network with excitatory-inhibitory coupling.

A: Spike-train variances (black) and (gray) of excitatory and inhibitory neurons. B: Spike-train covariances (black solid), (dark gray solid) and (light gray solid) for excitatory-excitatory, excitatory-inhibitory and inhibitory-inhibitory neuron pairs in the recurrent network, respectively, and shared-input contribution (black dotted curve; ‘feedforward case’). C: Decomposition of the total input covariance (light gray) into shared-input covariance (black) and weighted spike-train covariance (dark gray). Covariances in A, B and C are given in units of the noise variance . D: Input-correlation coefficient in the recurrent network (black solid curve). In the feedforward case, the input-correlation coefficient is identical to the network connectivity (horizontal dotted line). E: Spike-train correlation coefficients (black), (dark gray) and (solid light gray curve) for excitatory-excitatory, excitatory-inhibitory and inhibitory-inhibitory neuron pairs, respectively. Thick dashed curves represent approximate solutions assuming . F: Low-frequency (LF) power ratios (black), (dark gray), (solid light gray) for the population rate and the excitatory and inhibitory subpopulation rates and , respectively. The LF power ratio represents the ratio between the LF spectra in the recurrent network and for the case where the feedback channels are replaced by feedforward input with (cf. Fig. 5 C). Thick dashed curves in F show power ratios obtained by assuming that the auto-correlations are identical in the feedback and the feedforward scenario (see main text). Vertical dotted lines mark the stability limit of the linear model (see “ Methods : Linearized network model”). A–F: , , , , , .

Figure 7. Comparison between predictions of the linear theory (thick gray curves) and direct simulation of the LIF-network model (symbols and thin lines).

Dependence of the spike-train and population-rate statistics on the synaptic weight (PSP amplitude) in a recurrent excitatory-inhibitory network (‘feedback system’, ‘FB’) and in a population of unconnected neurons receiving randomized feedforward input (‘feedforward system’, ‘FF’) from neurons in the recurrent network. Average presynaptic firing rates and shared-input structure are identical in the two systems. In the FF case, the average correlations between presynaptic spike-trains are homogenized (i.e. ) as a result of the random reassignment of presynaptic neuron types. The mapping of the LIF dynamics to the linear reduced dynamics (“ Methods : Response kernel of the LIF model”) relates the PSP amplitude to the effective coupling strength by (45), as shown in Fig. 8 B. A: Average firing rates in the FB (black up-triangles: excitatory neurons; gray down-triangles: inhibitory neurons) and in the FF system (open circles). Analytical prediction (??) (gray curve). B: Spike-train correlation coefficients (black up-triangles), (gray squares) and (gray down-triangles) for excitatory-excitatory, excitatory-inhibitory, and inhibitory-inhibitory neuron pairs, respectively, in the FB system. Analytical prediction (19) (gray curves). Spike-train correlation coefficient (open circles) in the FF system with homogenized presynaptic spike-train correlations. Analytical prediction (86) (underlying gray curve). C: Shared-input (; black up-triangles) and spike-correlation contribution (FB: gray down-triangles; FF: open circles) to the input correlation (normalized by ). Analytical predictions (20). D: Low-frequency (LF) power ratio of the compound activity. Vertical dotted lines in A–D mark the stability limit of the linear model (see “ Methods : Linearized network model”). , , , . Size of postsynaptic population in the FF case: . Simulation time: .

Figure 8. Linear response and relation between synaptic weight and effective coupling strength .

A: Firing-rate deflection of a LIF neuron caused by an incoming spike event of postsynaptic amplitude . B: Integral of the firing rate deflection shown in A as a function of the postsynaptic amplitude (simulation: black dots; analytical approximation (45) : gray curve). The neuron receives constant synaptic background input with , , and rates , resulting in a first and second moment (42) and . Simulation results are obtained by averaging over trials of duration each with input impulses on average. For further parameters of the neuron model, see Table 1 and Table 2.

Figure 9. Amplification of population-rate fluctuations by different types of feedback manipulations in a random network of excitatory and inhibitory LIF neurons (simulation results).

Top row (A–C): Unperturbed feedback (FB; black), shuffling of spike-train senders across entire network (Shuff1D; dark gray) and within each subpopulation (E,I) separately (Shuff2D; light gray). Bottom row (D–F): Unperturbed feedback (FB; black), replacement of spike trains by realizations of inhomogeneous (PoissI; dark gray) and homogeneous Poisson processes (PoissH; light gray). In the PoissI (PoissH) case, the (time averaged) subpopulation rates are approximately preserved. A, D: Compound power-spectra of input spike-train ensembles. B, E: Power-spectra of population-response rates. C, F: Low-frequency (LF; –) power ratio (increase in LF power relative to the unperturbed case [FB]; logarithmic scaling). Note that in A, the compound-input spectra (FB, Shuff1D, Shuff2D) are identical. In D, the input spectra for the intact recurrent network (FB) and the inhomogeneous-Poisson case (PoissI) are barely distinguishable. See Table 1 and Table 2 for details on the network model and parameters. Simulation time . Single-trial spectra smoothed by moving average (frame size ).

Figure 10. Recurrent network dynamics stabilizes dynamics of embedded synfire chains.

Spiking activity in a synfire chain ( layers, layer width ) receiving background input from an excitatory-inhibitory network (A, cf. Fig. 1 C) or from a finite pool of excitatory and inhibitory Poisson processes (B, cf. Fig. 1 D). Average input firing rates, in-degrees and amount of shared input are identical in both cases. Neurons of the first synfire layer (neuron ids ) are stimulated by current pulses at times and . Each neuron in layer receives inputs from all neurons in the preceding layer (synaptic weights , spike transmission delays ), and and excitatory and inhibitory background inputs, respectively, randomly drawn from the presynaptic populations. Neurons in the first layer receive and excitatory and inhibitory background inputs, respectively. Note that there is no feedback from the synfire chain to the embedding network. See Table 2 for network parameters.