Dynamic Flux Tubes Form Reservoirs of Stability in Neuronal Circuits

Abstract

Neurons in cerebral cortical circuits interact by sending and receiving electrical impulses called spikes. The ongoing spiking activity of cortical circuits is fundamental to many cognitive functions including sensory processing, working memory, and decision making. London et al. [Sensitivity to Perturbations In Vivo Implies High Noise and Suggests Rate Coding in Cortex, Nature (London) 466, 123 (2010).] recently argued that even a single additional spike can cause a cascade of extra spikes that rapidly decorrelate the microstate of the network. Here, we show theoretically in a minimal model of cortical neuronal circuits that single-spike perturbations trigger only a very weak rate response. Nevertheless, single-spike perturbations are found to rapidly decorrelate the microstate of the network, although the dynamics is stable with respect to small perturbations. The coexistence of stable and unstable dynamics results from a system of exponentially separating dynamic flux tubes around stable trajectories in the network’s phase space. The radius of these flux tubes appears to decrease algebraically with neuron number N and connectivity K , which implies that the entropy of the circuit’s repertoire of state sequences scales as Nln(KN) .

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Plain Language Summary

Neurons in the brain communicate by exchanging tiny electrical impulses called spikes. Every second more than 100 billion spikes are generated in the human brain. It is often thought that each individual spike among this astronomical number plays only a negligible role in the greater picture of information processing. In this paper, however, we show, in an idealized model of cerebral circuits, that stopping a single spike from reaching its targets typically sets the entire neural network on a different dynamical path. We also present computational techniques to characterize this type of dynamical sensitivity and outline how it may enhance the computational capabilities of biological and artificial neural circuits.

In general, a conceptually clarifying way of describing the states and the temporal evolution of a system of nonlinear dynamics is through the phase-space perspective. Each state of the system is represented by a point in the phase space, and the temporal evolution of the system from any state onward draws out a trajectory in this space. How different trajectories are distributed in the phase space reveals telltale signs about the dynamics of the system, in particular, the dynamical sensitivity to small changes in the starting state. In the neuronal-circuit models we have investigated, we have uncovered that the high sensitivity to single spikes results from the occurrence of a previously unknown structure of organization of the phase space of the models, which we call exponentially separating dynamical flux tubes.

The nature of the high sensitivity that we have seen is novel and highly nontrivial. Classically, Lyapunov exponents, the rates of exponential state separation, measure the sensitive dependence on initial conditions. Positive Lyapunov exponents essentially define the notion of deterministic chaos, or high sensitivity to small perturbations. In the neuronal-circuit models studied in this paper, all of the Lyapunov exponents are negative, and the system is, in fact, formally stable and nonchaotic. The exponential state separation we have revealed originates from a fundamentally different mechanism: The phase space is organized into a complex landscape of stability basins of dynamical flux tubes, separated by exponentially separating borders. Single spike perturbations that can take the system across a border lead to the observed dynamical sensitivity. From this new understanding, we have also been able to obtain information on the neural-circuit models that were previously unavailable.

We are confident that not only will future studies building on this work lead to the discovery of ways to shape the landscape of flux tubes to enhance the power of network computations, but the new finding of dynamical flux tubes will also have considerable impact on nonlinear-dynamics research in general.