Motion analysis in three dimensions demonstrate that the fluctuations in the vertical displacement angle of a stick balanced at the fingertip obey a scaling law characteristic of on-off intermittency and that >98% of the corrective movements occur fast compared to the measured time delay. These experimental observations are reproduced by a model for an inverted pendulum with time-delayed feedback in which parametric noise forces a control parameter across a particular stability boundary. Our observations suggest that parametric noise is an essential, but up until now underemphasized, component of the neural control of balance.

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Multistable dynamical systems have important applications as pattern recognition and memory storage devices. Conditions under which time-delayed recurrent loops of spiking neurons exhibit multistability are presented. Our results are illustrated on both a simple integrate-and-fire neuron and a Hodgkin-Huxley-type neuron, whose recurrent inputs are delayed versions of their output spike trains. Two kinds of multistability with respect to initial spiking functions are found, depending on whether the neuron is excitable or repetitively firing in the absence of feedback.

State-dependent, or parametric, noise is an essential component of the neural control mechanism for stick balancing at the fingertip. High-speed motion analysis in three dimensions demonstrates that the controlling movements made by the fingertip during stick balancing can be described by a Lèvy flight. The Lèvy index, alpha, is approximately 0.9; a value close to optimal for a random search. With increased skill, the index alpha does not change. However, the tails of the Lèvy distribution become broader. These observations suggest a Lèvy flight that is truncated by the properties of the nervous and musculoskeletal system; the truncation decreasing as skill level increases. Measurements of the cross-correlation between the position of the tip of the stick and the fingertip demonstrate that the role of closed-loop feedback changes with increased skill. Moreover, estimation of the neural latencies for stick balancing show that for a given stick length, the latency increases with skill level. It is suggested that the neural control for stick balancing involves a mechanism in which brief intervals of consciously generated, corrective movements alternate with longer intervals of prediction-free control. With learning the truncation of the Lèvy flight becomes better optimized for balance control and hence the time between successive conscious corrections increases. These observations provide the first evidence that changes in a Lèvy flight may have functional significance for the nervous system. This work has implications for the control of balancing problems ranging from falling in the elderly to the design of two-legged robots and earthquake proof buildings.

A dynamical analogy supported by five scale-free statistics (the Gutenberg-Richter distribution of event sizes, the distribution of interevent intervals, the Omori and inverse Omori laws, and the conditional waiting time until the next event) is shown to exist between two classes of seizures ("focal" in humans and generalized in animals) and earthquakes. Increments in excitatory interneuronal coupling in animals expose the system's dependence on this parameter and its dynamical transmutability: moderate increases lead to power-law behavior of seizure energy and interevent times, while marked ones to scale-free (power-law) coextensive with characteristic scales and events. The coextensivity of power law and characteristic size regimes is predicted by models of coupled heterogeneous threshold oscillators of relaxation and underscores the role of coupling strength in shaping the dynamics of these systems.

Correlation functions with multiple scaling regions occur in the description of the fluctuations in the center of pressure during quiet standing. Postural sway is modeled as an inverted pendulum with a delayed feedback constructed such that for deviations beyond a spatial threshold a constant restoring force is engaged. In the absence of noise, two stable limit cycles coexist. The correlation function depends on the added noise intensity: at intermediate noise levels three scaling regions appear whereas only two occur for high noise levels. Our observations suggest that correlation functions with multiple scaling regions reflect noise-induced transitions in bistable dynamical systems.

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The inverted pendulum is frequently used as a starting point for discussions of how human balance is maintained during standing and locomotion. Here we examine three experimental paradigms of time-delayed balance control: (1) mechanical inverted time-delayed pendulum, (2) stick balancing at the fingertip, and (3) human postural sway during quiet standing. Measurements of the transfer function (mechanical stick balancing) and the two-point correlation function (Hurst exponent) for the movements of the fingertip (real stick balancing) and the fluctuations in the center of pressure (postural sway) demonstrate that the upright fixed point is unstable in all three paradigms. These observations imply that the balanced state represents a more complex and bounded time-dependent state than a fixed-point attractor. Although mathematical models indicate that a sufficient condition for instability is for the time delay to make a corrective movement, tau(n), be greater than a critical delay tau(c) that is proportional to the length of the pendulum, this condition is satisfied only in the case of human stick balancing at the fingertip. Thus it is suggested that a common cause of instability in all three paradigms stems from the difficulty of controlling both the angle of the inverted pendulum and the position of the controller simultaneously using time-delayed feedback. Considerations of the problematic nature of control in the presence of delay and random perturbations ("noise") suggest that neural control for the upright position likely resembles an adaptive-type controller in which the displacement angle is allowed to drift for small displacements with active corrections made only when theta exceeds a threshold. This mechanism draws attention to an overlooked type of passive control that arises from the interplay between retarded variables and noise.