Structured chaos shapes spike-response noise entropy in balanced neural networks

Abstract

Large networks of sparsely coupled, excitatory and inhibitory cells occur throughout the brain. For many models of these networks, a striking feature is that their dynamics are chaotic and thus, are sensitive to small perturbations. How does this chaos manifest in the neural code? Specifically, how variable are the spike patterns that such a network produces in response to an input signal? To answer this, we derive a bound for a general measure of variability-spike-train entropy. This leads to important insights on the variability of multi-cell spike pattern distributions in large recurrent networks of spiking neurons responding to fluctuating inputs. The analysis is based on results from random dynamical systems theory and is complemented by detailed numerical simulations. We find that the spike pattern entropy is an order of magnitude lower than what would be extrapolated from single cells. This holds despite the fact that network coupling becomes vanishingly sparse as network size grows-a phenomenon that depends on "extensive chaos," as previously discovered for balanced networks without stimulus drive. Moreover, we show how spike pattern entropy is controlled by temporal features of the inputs. Our findings provide insight into how neural networks may encode stimuli in the presence of inherently chaotic dynamics.

Keywords: chaotic networks; network dynamics; neural excitability; neural variability; spiking stimulus responses.

Figure 1

(Color online) (A) Sketch of equivalent dynamics between Quadratic-Integrate-and-Fire and the θ-neuron. (B) Cartoon representation of a network driven by a quenched collection of inputs I(t) = {I_i(t)}_{i = 1, …, N}. (C) Example of a single θ-neuron θ_i in response to a (quenched) input I_i(t) (η = −0.5, ε = 0.5). Red dots mark spike times.

Figure 2

(Color online) (A) Top: Raster plot of spike output for 100 randomly selected neurons on a single trial (dots are spikes). Bottom: Illustration of binary S_KL-word. (B) Raster plot of one randomly selected cell's spike output on 2000 trials where only network initial conditions change. (C) Single cell H^1L_noise estimates for different choices of “surrogate” noise (round markers); see text. From top to bottom: homogeneous poisson (blue), inhomogeneous poisson (red), network interactions (black). The bottom curve is a computation of $\frac{1}{2}{H}_{noise}^{2L}$ from a cell pair (diamond markers). Abscissa scale is 1/L to better visualize extrapolation of extensive regime to L → ∞ (left square marker). For all panels: η = −0.5, ε = 0.5, N = 500.

Figure 3

(Color online) (A) Typical histogram of noise correlation coefficient c_ij(t_l) between all neuron pairs for a fixed time. Inset shows c_ij(t_l) for the first 5000 pairs. (B) Histogram of noise correlation coefficient c_ij(t_l) between two connected cells across 10, 000 tu. Inset shows c_ij(t_l) for 100 tu. (C) Network-wide noise entropy estimates in bits/tu as a function of N. Slope 〈H¹〉 averaged over 20 random cells in a network with N = 500. Shaded area shows two standard errors of the mean. Markers show direct samples from single cells for various network sizes (ie NH¹). H_KS: square markers shows estimates from Lyapunov spectra for a range of N; black line is a linear fit. (D) Plot of first 10% of Lyap spectrum for N = 500, 1000, and 2000. For all panels: η = −0.5, ε = 0.5.

Figure 4

(A) Comparison of trajectories for single cells, for models (1) and (9); initial conditions and inputs are fixed. (B) First 60 Lyapunov exponents of models (1) and (9). (C) Empirical noise entropy bounds NH¹ and H_KS for models (1) and (9). For all panels, η = −0.5, ε = 0.5, Δt = 0.05. For (B,C), N = 500, κ = 20.

Figure 5

(Color online) (A) Heat map of excitatory population mean firing rate for a range of input amplitude ε and input mean η. Line is the contour curve for fixed firing rate of 0.820 spikes/tu ± 0.003, parameterized by numerical interpolation. Arrow shows direction of parametrization. Markers: square: η = −1, ε = 0.69, star: η = −0.5, ε = 0.5, circle: η = 0.07, ε = 0. (B) Lyapunov spectra along contour curve from (A). (C) H_KS bounds evaluated along contour curve from (A). (D) Network noise entropy bounds N〈H¹〉 and H_KS for square and circle marker parameters in (A). Slope 〈H¹〉 averaged over 20 random cells. Shaded area shows two standard errors of the mean. Both 〈H¹〉 and H_KS extrapolated from a network with N = 500, as are quantities from all other panels.