Dynamical Entropy Production in Spiking Neuron Networks in the Balanced State

We demonstrate deterministic extensive chaos in the dynamics of large sparse networks of theta neurons in the balanced state. The analysis is based on numerically exact calculations of the full spectrum of Lyapunov exponents, the entropy production rate, and the attractor dimension. Extensive chaos is found in inhibitory networks and becomes more intense when an excitatory population is included. We find a strikingly high rate of entropy production that would limit information representation in cortical spike patterns to the immediate stimulus response.

Figure 1

Characteristics of the balanced state in inhibitory and excitatory-inhibitory networks of theta neurons: (a),(c) Spike patterns of 40 random neurons, (b),(d) voltage traces of one random neuron, (e) firing rate distributions, (f) coefficient of variation distributions, and (g) stationary voltage distributions (parameters: $N=10000$, $K=100$, $\overline{\nu }=1\mathrm{Hz}$, $J_0=1$, ${\tau }_m=10\mathrm{ms}$, $\epsilon =0.3$, $N_E=4N_I$).

Figure 2

Extensive deterministic chaos in balanced inhibitory networks: (a) Full Lyapunov spectra for different average firing rates $\overline{\nu }$, collapsed for different network sizes (see 13), (b) maximal Lyapunov exponent, (c) mean Lyapunov exponent, (d) entropy production rate $H$ and attractor dimension $D$ for various network sizes $N$, (e) average entropy production per spike per neuron, (f) attractor dimension density (parameters: $N=2000$, $K=100$, $\overline{\nu }=1\mathrm{Hz}$, $J_0=1$, ${\tau }_m=10\mathrm{ms}$).

Figure 3

Chaotic dynamics in balanced excitatory-inhibitory networks at different excitatory feedback loop activations $\epsilon$: (a) Partial Lyapunov spectra, (b) maximal Lyapunov exponent, (c) entropy production per spike per neuron, (d) attractor dimension density (dashed lines: isolated inhibitory networks with $N=N_I$, parameters: $N_I=2000$, $N_E=8000$, $K=100$, $\overline{\nu }=1\mathrm{Hz}$, $J_0=1$, ${\tau }_m=10\mathrm{ms}$).

Figure 4

Temporal network chaos in balanced inhibitory networks: (a) First Lyapunov vector $\delta \overrightarrow{\varphi }(t)$ (marked black if $|\delta {\varphi }_i(t)|>1/\sqrt{N}$), (b) participation ratio $P(t)$ of first Lyapunov vector, (c) voltage distribution of neurons with largest Lyapunov vector elements (dotted line: stationary distribution), (d),(e) neurons’ chaos indices $c_i$ versus firing rates ${\nu }_i$ and coefficients of variation ${\mathrm{cv}}_i$ (straight line: mean, dashed lines: $\pm 2$ std), (f) average participation ratio $\overline{P}$ vs network size $N$ (dotted line: guide to the eye for $\overline{P}\sim N$) (parameters: $N=2000$, $K=100$, $\overline{\nu }=1\mathrm{Hz}$, $J_0=1$, ${\tau }_m=10\mathrm{ms}$).