Sparse chaos in cortical circuits

TL;DR

The paper demonstrates that single-neuron spike initiation biophysics, captured by a tunable spike onset rapidness $r$ in the rapid theta model, qualitatively reshapes the chaotic dynamics of cortical networks. Using ergodic-theory tools, it reveals a transition from dense to sparse chaos as $r$ increases, marked by a peak in the largest Lyapunov exponent and localization of the leading covariant Lyapunov vector, and a concomitant dramatic reduction in attractor dimension and dynamical entropy rate. This transition is tied to the breakdown of the diffusion approximation and the emergence of shot-noise-dominated dynamics, with robust results across random, layered, and motif-structured networks. The findings connect single-cell biophysics to network controllability and information processing, predicting that cortical circuits may tune spike onset rapidness to optimize information flow and state control across layers. Overall, the work establishes a concrete link between microscopic spike initiation and macroscopic neural computation in large-scale cortical circuits, with implications for modeling, interpretation of chaos in the brain, and potential design principles for controllable neural systems.

Abstract

Nerve impulses, the currency of information flow in the brain, are generated by an instability of the neuronal membrane potential dynamics. Neuronal circuits exhibit collective chaos that appears essential for learning, memory, sensory processing, and motor control. However, the factors controlling the nature and intensity of collective chaos in neuronal circuits are not well understood. Here we use computational ergodic theory to demonstrate that basic features of nerve impulse generation profoundly affect collective chaos in neuronal circuits. Numerically exact calculations of Lyapunov spectra, Kolmogorov-Sinai-entropy, and upper and lower bounds on attractor dimension show that changes in nerve impulse generation in individual neurons moderately impact information encoding rates but qualitatively transform phase space structure. Specifically, we find a drastic reduction in the number of unstable manifolds, Kolmogorov-Sinai entropy, and attractor dimension. Beyond a critical point, marked by the simultaneous breakdown of the diffusion approximation, a peak in the largest Lyapunov exponent, and a localization transition of the leading covariant Lyapunov vector, networks exhibit sparse chaos: prolonged periods of near stable dynamics interrupted by short bursts of intense chaos. Analysis of large, more realistically structured networks supports the generality of these findings. In cortical circuits, biophysical properties appear tuned to this regime of sparse chaos. Our results reveal a close link between fundamental aspects of single-neuron biophysics and the collective dynamics of cortical circuits, suggesting that nerve impulse generation mechanisms are adapted to enhance circuit controllability and information flow.

Figures (45)

• Figure 1: High spike onset rapidness $r$ increases population encoding bandwidth.a Single neuron dynamics have two fixed points: a stable fixed point (filled circle, resting potential) and an unstable fixed point (white circle, spike threshold). The slope at resting potential is $-1/\tau_{m}$, and the slope at the spike threshold $r/\tau_{m}$. b Voltage traces of the neuron model with constant input currents for varying rapidness $r$. c Same as in panel b, but with fluctuating input currents. Note that the spike waveform and initiation depend strongly on $r$, while the subthreshold dynamics are insensitive to $r$. The inset shows a magnified window of the voltage traces and the corresponding fluctuating input (gray) d Firing rates of an ensemble of rapid theta neurons with low and high rapidness for fluctuating input currents. Note that high rapidness enables the ensemble to accurately track the high-frequency components of the input. e Linear firing rate response for different values of rapidness, with direct numerical simulations (shaded lines) and Fokker Planck solutions (dashed lines) superimposed. ($\nu_{0}=1\textrm{\,\ Hz}$). f Mutual information rate in the Gaussian channel approximation based on spectral coherence, comparing Fokker Planck solutions (solid lines) and direct numerical simulations (squares) for different mean ensemble firing rates ($\nu_{0}=1$, $2$, $5$ Hz) (parameters: $\tau_{m}=10\textrm{ ms}$).
• Figure 2: High spike onset rapidness spike $r$ dramatically reduces chaos and dynamical entropy rate.a Spike trains of 50 random neurons for low (upper panel) and high (lower panel) rapidness. b Distribution of firing rates (upper panel) and coefficients of variation (lower panel) for different values of rapidness (ordered by time-averaged single neuron firing rate) c Lyapunov spectra reorganize with increasing rapidness (inset: full Lyapunov spectra) d Largest Lyapunov exponent and e entropy rate $h$ as a function of rapidness for different mean firing rates ($\bar{\nu}=1$, $2$, $5$ Hz), f Entropy rate $h$ for different network sizes (upper panel) and g different strengths of the scaling $\epsilon$ of the excitatory couplings (parameters: $N_{I}=2000$, $N_{E}=8000$, $K=100$, $\bar{\nu}=1\textrm{ Hz}$, $J_{0}=1$, $\tau_{m}=10\textrm{ ms}$).
• Figure 3: Localization of Lyapunov vectors for high spike onset rapidness $r$ reveals two types of network chaos. a First Lyapunov vector (marked black if $|\delta\phi_{i}(t)|>1/\sqrt{N}$) b First local Lyapunov exponent $\lambda_{1}^{\textrm{local}}(t)$ and c participation ratio $P(t)$ of the first Lyapunov vector as a function of time for $r\approx1.33$. d, e, f Same as a, b, c for $r\approx31.6$. Note that large local Lyapunov exponents are followed by low participation ratio (red stars). g Average participation ratio $\bar{P}$ and h largest Lyapunov exponent vs. spike rapidness $r$ for different network size $N$, i Power-law scaling exponent $\alpha$ from $\bar{P}\sim N^{\alpha}$ decreases approximately logarithmically as a function of $r$ and shows localization above peak rapidness $r_{\mathrm{peak}}$. j Poincaré section of the phases of neurons 2 and 3 whenever neuron 1 spikes for $r=1$. The first local Lyapunov exponent (LLE) at each point is color-coded, red colors indicate local instability, blue indicates local stability k same as j for $r=4$ (parameters: $N_{I}=1000$, $K=100$, $\bar{\nu}=3\textrm{ Hz}$, $J_{0}=1$, $\tau_{m}=10\textrm{ ms}$).
• Figure 4: Reduction of attractor dimensionality in the asynchronous state despite low pairwise spike count correlations.a Attractor dimension for different mean firing rates and varying spike rapidness $r$, dotted line: dimensionality estimate based on principal components of pairwise spike count correlations matrix (Supplementary Information), solid line: Kaplan-Yorke (KY) attractor dimension, dashed line: lower bound on attractor dimension (fraction of positive exponents) b KY Attractor dimension grows linearly with network size $N$c same as a for different scaling $\epsilon$ of the excitatory couplings d, h cross sections of basis on attraction in a plane perpendicular to the trajectory for $r=250,\:500$. Colors indicate basins of attraction of different trajectories ($N=200$, $K=100$$r_{c}\approx203$) e Mean pairwise spike count correlations for different values of rapidness $r$ between excitatory (E) and inhibitory (I) neurons, excitatory-excitatory pairs (EE) in green, inhibitory-inhibitory pairs (II) in red, mixed pairs in yellow f Mean pairwise spike count correlations decay $\propto1/N$, their standard deviations decays $\propto1/\sqrt{N}$, (color code as in b) g histograms of spike count correlations for different rapidness $r$ (EE-pairs)(color code as in b)(parameters as in Fig. \ref{['fig:fig2']}, spike count window $20$ ms)
• Figure 5: High spike onset rapidness $r$ reduces chaos and entropy rate in cortical circuit models:a Multilayered cortical column network model with layer- and cell type specific connection probabilities, 77,169 neurons, $\sim$285 Million synapses b spike raster illustrating layer-specific firing rates c largest Lyapunov exponent vs. spike onset rapidness $r$d entropy rate $h$ vs. spike onset rapidness $r$, the gray area indicate a physiologically plausible regime of spike onset rapidness $r$, e positive Lyapunov exponents of multilayered model. f Second order network motif overrepresentation estimated from experiments g dynamical entropy rate for random and realistic second order motif structure at different values of spike onset rapidness $r$.