Recurrent interactions in spiking networks with arbitrary topology

Volker Pernice, Benjamin Staude, Stefano Cardanobile, and Stefan Rotter

Phys. Rev. E 85, 031916 – Published 29 March 2012

Abstract

The population activity of random networks of excitatory and inhibitory leaky integrate-and-fire neurons has been studied extensively. In particular, a state of asynchronous activity with low firing rates and low pairwise correlations emerges in sparsely connected networks. We apply linear response theory to evaluate the influence of detailed network structure on neuron dynamics. It turns out that pairwise correlations induced by direct and indirect network connections can be related to the matrix of direct linear interactions. Furthermore, we study the influence of the characteristics of the neuron model. Interpreting the reset as self-inhibition, we examine its influence, via the spectrum of single-neuron activity, on network autocorrelation functions and the overall correlation level. The neuron model also affects the form of interaction kernels and consequently the time-dependent correlation functions. We find that a linear instability of networks with Erdös-Rényi topology coincides with a global transition to a highly correlated network state. Our work shows that recurrent interactions have a profound impact on spike train statistics and provides tools to study the effects of specific network topologies.

Received 10 November 2011

DOI:https://doi.org/10.1103/PhysRevE.85.031916

Authors & Affiliations

Volker Pernice^*, Benjamin Staude, Stefano Cardanobile, and Stefan Rotter

Bernstein Center Freiburg and Faculty of Biology, University of Freiburg Hansastraße 9a, 79104 Freiburg, Germany

^*pernice@bcf.uni-freiburg.de

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Vol. 85, Iss. 3 — March 2012

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Images

Figure 1
(a) Spike trains in a random LIF network ( $N = 1250, γ = 5, J_{E} = 0.1 mV$ , and simulation time $5 \times 10^{4} s$ ). Covariances and fluctuations of population spike counts (bottom) are small. (b) Autocovariance of neurons with low and high rate. Normalization over bin size $Δ t$ was chosen such that the integral of the peak corresponds to the firing rate. (c) Histogram of the response to external excitatory and inhibitory input spikes, averaged across neurons, as well as fits to a function $\cos (ν τ) e^{- τ / α}$ (dashed lines).Reuse & Permissions
Figure 2
(a) Scatter plot of covariances between neuron pairs. Dark gray (red) denotes $C_{k j} (ω = 0 kHz)$ , black denotes $Re [C_{k j}] (ω = 0.31 kHz)$ , and light gray (cyan) denotes $Im [C_{k j}] (ω = 0.31 kHz)$ . The inset shows the standard deviation between prediction and simulation, normalized by the standard deviation of the covariance distribution, which increases with frequency. (b) Typical examples for simulated (gray) and predicted (black) pairwise covariance functions in the time domain. The parameters are the same as in Fig. 1.Reuse & Permissions
Figure 3
Larger reset decreases (a) effective and network autocovariances, (b) integrated impulse response, and (c) mean and standard deviation of the distribution of integrated covariances in a random network. The parameters are the same as in Fig. 1.Reuse & Permissions
Figure 4
(a) Semianalytical ${\hat{g}}_{E, I}$ in random networks of size $N$ for $J_{E} = 0.1 (dot - dashed line), J_{E} = 0.4 (dashed line), and J_{E} = 0.6$ (solid line) and $γ = 4$ . (b) Bulk spectrum radius $ρ$ of the corresponding coupling matrices for $γ = 4 and 4.5$ . (c) Semianalytical couplings (dot-dashed lines) and values from simulations with $J_{E} = 0.1$ . Circles denote $γ = 4.5$ and squares denote $γ = 6$ . Open symbols denote excitatory couplings ${\hat{g}}_{E}$ for $γ = 4.5 and 6$ and solid symbols denote the corresponding inhibitory couplings $| {\hat{g}}_{I} |$ . (d) Resulting values for $ρ$ for same parameters as in (c).Reuse & Permissions
Figure 5
(a) Average of covariance functions across pairs ( $N = 25 000$ and $J_{E} = 0.4$ ). (b) Integrated covariances normalized by average rates over $ρ (N)$ for $J_{E}$ as in Fig. 4. Lines for different $J_{E}$ partially overlap. The simulation time is $10^{2} s$ and other parameters are the same as in Fig. 1. (c) Eigenvalues of the integrated coupling matrix [light gray (green)] and the matrix rescaled by $\hat{g} (s_{0}) / \hat{g} (0)$ at resonance $s_{0}$ determined from a fit to the excitatory kernel in Fig. 1 [dark gray (blue)]. (d) Normalized Laplace transformed impulse response $| L [\hat{g} (s)] | / \hat{g} (0)$ for kernel fit $\cos (ν τ) e^{- τ / α}$ (left) and exponential kernel ( $ν = 0$ ).Reuse & Permissions
Figure 6
Average of covariances over pairs of equal distance for (a) $ω = 0 kHz$ and (b) $ω = 0.11 kHz$ (real part). Covariances in the direct simulation are well reproduced by the linear approximation. Covariances in the linear model fluctuate around the analytical profile due to the random realization of the network. (c) and (d) Contributions of different orders can be delineated in the analytic expression (16). Also shown are the distance and frequency dependence of the covariance (real part) in (e) direct simulations and (f) the linear approximation. The simulation time was $2 \times 10^{4} s$ , $p = 0.05, g = 6.5, σ_{E} = 312, σ_{I} = 156, and N = 2500$ .Reuse & Permissions

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