Chaos Theory seminars

Over the last 20 years, neuroimaging and electrophysiology techniques have become central to understanding the mechanisms that accompany loss and recovery of consciousness. Much of this research is performed in the context of healthy individuals with neurotypical brain dynamics. Yet, a true understanding of how consciousness emerges from the joint action of neurons has to account for how severely pathological brains, often showing phenotypes typical of unconsciousness, can nonetheless generate a subjective viewpoint. In this presentation, I will start from the context of Disorders of Consciousness and will discuss recent work aimed at finding generalizable signatures of consciousness that are reliable across a spectrum of brain electrophysiological phenotypes focusing in particular on the notion of edge-of-chaos criticality.

Neural circuits exhibit complex activity patterns, both spontaneously and in response to external stimuli. Information encoding and learning in neural circuits depend on the ability of time-varying stimuli to control spontaneous network activity. In particular, variability arising from the sensitivity to initial conditions of recurrent cortical circuits can limit the information conveyed about the sensory input. Spiking and firing rate network models can exhibit such sensitivity to initial conditions that are reflected in their dynamic entropy rate and attractor dimensionality computed from their full Lyapunov spectrum. I will show how chaos in both spiking and rate networks depends on biophysical properties of neurons and the statistics of time-varying stimuli. In spiking networks, increasing the input rate or coupling strength aids in controlling the driven target circuit, which is reflected in both a reduced trial-to-trial variability and a decreased dynamic entropy rate. With sufficiently strong input, a transition towards complete network state control occurs. Surprisingly, this transition does not coincide with the transition from chaos to stability but occurs at even larger values of external input strength. Controllability of spiking activity is facilitated when neurons in the target circuit have a sharp spike onset, thus a high speed by which neurons launch into the action potential. I will also discuss chaos and controllability in firing-rate networks in the balanced state. For these, external control of recurrent dynamics strongly depends on correlations in the input. This phenomenon was studied with a non-stationary dynamic mean-field theory that determines how the activity statistics and the largest Lyapunov exponent depend on frequency and amplitude of the input, recurrent coupling strength, and network size. This shows that uncorrelated inputs facilitate learning in balanced networks. The results highlight the potential of Lyapunov spectrum analysis as a diagnostic for machine learning applications of recurrent networks. They are also relevant in light of recent advances in optogenetics that allow for time-dependent stimulation of a select population of neurons.

The remarkable properties of information-processing of biological and of artificial neuronal networks alike arise from the interaction of large numbers of neurons. A central quest is thus to characterize their collective states. The directed coupling between pairs of neurons and their continuous dissipation of energy, moreover, cause dynamics of neuronal networks outside thermodynamic equilibrium. Tools from non-equilibrium statistical mechanics and field theory are thus instrumental to obtain a quantitative understanding. We here present progress with this recent approach [1]. On the experimental side, we show how correlations between pairs of neurons are informative on the dynamics of cortical networks: they are poised near a transition to chaos [2]. Close to this transition, we find prolongued sequential memory for past signals [3]. In the chaotic regime, networks offer representations of information whose dimensionality expands with time. We show how this mechanism aids classification performance [4]. Together these works illustrate the fruitful interplay between theoretical physics, neuronal networks, and neural information processing.