- Sponsor
- Department of Mathematics
- Speaker
- Rainer Engelken (University of Illinois)
- Contact
- Daniel Brendan Cooney
- dbcoone2@illinois.edu
- Phone
- 914-563-4916
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Speaker: Rainer Engelken (University of Illinois)
Title: Breakdown of the Diffusion Approximation: Sparse Chaos and Attractor Geometry in Spiking Neural Networks
Abstract:
Nerve impulses, the currency of information flow in the brain, are generated by an instability of the neuronal membrane potential dynamics. While large neuronal circuits are often modeled using mean-field theories that approximate discrete synaptic inputs as a diffusion process (Gaussian white noise), we demonstrate here that this standard approximation breaks down when considering the biophysical rapidness of action potential onset. Using computational ergodic theory and numerically exact event-based simulations, we show that the interplay between the timescale of single-unit instability and the integration time of input shot noise profoundly affects collective chaos in neuronal circuits.We calculate the full spectrum of Lyapunov exponents, revealing that changes in the rapidness of nerve impulse generation qualitatively transform the 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 transition into a novel regime of sparse chaos. This regime is characterized by prolonged periods of near-stable dynamics interrupted by short bursts of intense, localized instability. We map the basins of attraction in the high-dimensional phase space, showing how the system transitions from a "flux tube" geometry to a complex strange attractor. These results reveal a close link between fundamental aspects of single-neuron biophysics and the collective dynamics of cortical circuits, suggesting that the discrete nature of pulse-coupling creates dynamical regimes invisible to standard continuous mean-field approximations.