Chaos and reliability in balanced spiking networks with temporal drive

Abstract

Biological information processing is often carried out by complex networks of interconnected dynamical units. A basic question about such networks is that of reliability: If the same signal is presented many times with the network in different initial states, will the system entrain to the signal in a repeatable way? Reliability is of particular interest in neuroscience, where large, complex networks of excitatory and inhibitory cells are ubiquitous. These networks are known to autonomously produce strongly chaotic dynamics-an obvious threat to reliability. Here, we show that such chaos persists in the presence of weak and strong stimuli, but that even in the presence of chaos, intermittent periods of highly reliable spiking often coexist with unreliable activity. We elucidate the local dynamical mechanisms involved in this intermittent reliability, and investigate the relationship between this phenomenon and certain time-dependent attractors arising from the dynamics. A conclusion is that chaotic dynamics do not have to be an obstacle to precise spike responses, a fact with implications for signal coding in large networks.

FIG. 1

(Color online) (A) Typical firing rate distributions for excitatory and inhibitory populations. (B) Typical interspike-interval (ISI) distribution of a single cell. The coefficient of variation (CV) is close to 1. (C) Invariant measure for an excitable cell (η < 0); inset: typical trajectory trace of an excitable cell where solid and dotted lines mark the stable and unstable fixed points. (D) Network raster plots for 250 randomly chosen cells. For all panels, η = −0.5, ε = 0.5.

FIG. 2

(Color online) (A) First 100 Lyapunov exponents of network with fixed parameters as in Fig. 1, as a function of ε. (B) Plot of λ₁ (right scale), M_λ/N: the fraction of λ_i > 0 (left scale) vs ε. (C) Raster plots show example spike times of an arbitrarily chosen cell in the network on 30 distinct trials, initialized with random ICs. Circle and star markers indicate ε values of 0.18 and 0.5, respectively, shown in panel (B). For all panels, η = −0.5.

FIG. 3

(Color online) (A) Snapshots of 1000 trajectories projected in two randomly chosen neural directions (θ₁,θ₂) at three distinct times. Upper and lower rows with the same parameters as in Fig 2 (C) and show a random sink and random strange attractor respectively. (B) Projections of the sample measure ${\mu }_{\zeta }^t$ onto the θ₁ neural direction at distinct moments. (C) Scatter plot of average support score 〈s_i(t)〉 vs. entropy of projected measure $h(p_{i,\zeta }^t)$ sampled over 2000 time points and 30 distinct cells. (D) Example histogram of 〈s_i〉 sampled across all cells in the network at a randomly chosen moment in time. Inset: snapshot of 〈s_i〉 vs. cell number i. (E) Example histogram of 〈s_i(t)〉 sampled across 2000 time points from a randomly chosen cell. Inset: sample time trace of 〈s_i(t)〉 vs. time. (F) and (G) Time evolution of distance between two distinct trajectories θ¹(t), θ²(t) (F) Green dashed (bottom): ${\Vert {\theta }_i^1(t)-{\theta }_i^2(t)\Vert }_{S^1}$ in a randomly chosen θ_i direction. Black solid (top): ${max}_j\{{\Vert {\theta }_j^1(t)-{\theta }_j^2(t)\Vert }_{S^1}\}$. (G) . For all panels except A (top), network parameters: η = −0.5, ε = 0.5 with λ₁ ≈ 2.5.

FIG. 4

(Colors online) (A) Top and middle: cartoon representations of the flux Φ_i(t). Bottom: sample Φ_i time trace for a randomly chosen cell approximated from 1000 trajectories. (B) Top: Illustration of spike event definition. Bottom: Distribution of spike event participation fraction f. (For (A) and (B): η = −0.5, ε = 0.5, λ₁ ≃ 2.5) (C) Curves of 1 − 〈f〉 (network), 1 − 〈f^shuffle〉 (single cell with shuffled input spike trains from networks simulations) and 1 − 〈f^poisson〉 (single cell with random poisson spike inputs) vs. ε. Also shown is the fraction of λ_i > 0, M_λ/N vs. ε. (D) Mean 1 − R_spike vs. ε curves for three threshold values. Error bars show one standard deviation of mean R_spike across all cells in the network. (For (C) and (D): η = −0.5)

FIG. 5

(Color online) For (A) through (E), t = 0 marks the spike time and rel/unrel indicates the identity of the spike used in the average. (A) and (B), Spike triggered averaged external signal εZ(θ_i)ζ_i(t) (black), excitatory (purple) and inhibitory (orange) network inputs Z(θ_i) Σ_j a_ijg(θ_j). (A) Triggered on reliable spikes. (B) Triggered on unreliable spikes. (C) Spike triggered support score S. (D) Spike triggered local expansion measure E. (E) Spike triggered average phase θ_i. For all panels: η = −0.5, ε = 0.5 with λ₁ ≈ 2.5. Shaded areas surrounding the computed averages show two standard errors of the mean.^a No shade indicates that the error is too small to visualize. ^a Computed standard deviations where verified by spot checks using the method of batched means with about 100 batches of size 1000.

FIG. 6

(Colors online) (A) Distinct terms of the single cell flow and Jacobian. Inset : synaptic coupling function g(o). For panels (B–F), t = 0 marks the spike time and rel/unrel indicate the identity of the spike used in the average. (B) Spike triggered average phase θ_i (same as in Fig 5 E). (C–F) Spike triggered average terms H₀(θ_i), H₁(θ_i), H₂(θ_i) and H₃(θ_i). Network parameters: η = −0.5, ε = 0.5, yielding λ₁ ≃ 2.5. For all panels except (A): shaded areas surrounding the computed averages show two standard errors of the mean. ^a No shade indicates that the error is too small to visualize. ^a Standard errors of the mean were verified via spot checks using the method of batched means, with about 100 batches of size 1000.