The time-delayed inverted pendulum: implications for human balance control

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.

Contribution of Individual Retinal Ganglion Cell Responses to Velocity and Acceleration Encoding

We investigate the capability of turtle retinal ganglion cell (RGC) ensembles to simultaneously encode multiple aspects of visual motion: speed, direction, and acceleration of moving patterns. Bayesian stimulus reconstruction reveals that the instantaneous firing rates of RGCs contain information about all of these stimulus properties. Stimulus velocity is mainly encoded by steady-state firing rates, whereas acceleration can be reconstructed from transient components in RGC activity induced by abrupt velocity changes. Therefore neurons in higher brain areas may in principle extract information about changing velocity from the instantaneous firing activity of RGCs, without the need to compare responses to present velocities to previous ones. However, reconstruction requires the estimation of a combined acceleration and velocity signal, indicating that RGC ensembles signal both properties simultaneously. In accordance with this conclusion, combined velocity/acceleration sensitivity enhances the similarity of artificial spike trains to experimental data by 50% compared with the case of pure velocity tuning. Decoding of motion direction in addition to speed and acceleration requires direction-sensitive cells, which generate higher firing rates for one of the motion directions and therefore show asymmetric velocity tuning. By dividing the entire ensemble of simultaneously recorded cells into one group of direction-sensitive cells and one group with symmetric tuning, we demonstrate that the population of direction-sensitive cells encodes a combination of motion speed, acceleration, and direction. However, estimation of velocity and acceleration is improved by including the larger group of RGC responses that are sensitive to speed but not to motion direction.

Distributed delays stabilize ecological feedback systems

We consider the effect of distributed delays in predator-prey models and ecological food webs. Whereas the occurrence of delays in population dynamics is usually regarded a destabilizing factor leading to the extinction of species, we here demonstrate complementarily that delay distributions yield larger stability regimes than single delays. Food webs with distributed delays closely resemble nondelayed systems in terms of ecological stability measures. Thus, we state that dependence of dynamics on multiple instances in the past is an important, but so far underestimated, factor for stability in dynamical systems.

Coding with noisy neurons: stability of tuning curve estimation strongly depends on the analysis method

An important issue in the neurosciences is a quantitative description of the relation between sensory stimuli presented to an animal and their representations in the nervous system. A standard technique is the construction of a neural tuning curve, that is, a neuron's average firing rate as a function of some parameter characterizing a family of stimuli. It is unavoidable that some of the response data are erroneously attributed to a cell, e.g., during spike sorting. However, the widely used method of statistical analysis based on the sample mean and least-squares approximation for the spike count can perform extremely badly if the noise distribution is not exactly normal, which is almost never the case in applications. Here, we present a method for constructing neural tuning curves that is especially suited for cases of high noise and the presence of outliers. Since it is usually not decidable if an outlier is faulty or not we limit the influence of far outlying points rather than try to identify and discard them. In contrast to traditional methods employing a point-by-point estimation of a tuning curve, we use all measured data from all different stimulus conditions at once in the construction. Given the measured data at only a finite number of stimulus conditions, a robust tuning curve is obtained that approximates the cell's ideal tuning curve optimally in all stimulus conditions with respect to a given distance measure. A measure that assesses the quality of this fitting method with respect to the traditional least-squares fitting method and to a median-based fitting method is introduced. The reliability of inference with respect to the encoding accuracy that can be achieved by a population of neurons is demonstrated in both artificially generated and experimentally recorded data from rat primary visual cortex. While the data shown in this paper are responses to orientation stimuli, the method of tuning curve construction is also viable and maintains its optimality properties for the case in which the stimulus is defined on a finite interval.

Extending the shine-through effect to classical masking paradigms

A vernier, presented for a short time, shines through a following grating if the grating contains nine and more elements but remains largely invisible for smaller gratings. Therefore, extended grating masks yield, surprisingly, less masking than smaller ones. Here, we show that this mask size effect is not unique to grating masks. Masking diminishes if the size of classical pattern-, noise-, light-, and metacontrast masks increases and if these masks are regular, i.e. highly ordered.

Local interactions in neural networks explain global effects in Gestalt processing and masking

One of the fundamental and puzzling questions in vision research is how objects are segmented from their backgrounds and how object formation evolves in time. The recently discovered shine-through effect allows one to study object segmentation and object formation of a masked target depending on the spatiotemporal Gestalt of the masking stimulus (Herzog & Koch, 2001). In the shine-through effect, a vernier (two abutting lines) precedes a grating for a very short time. For small gratings, the vernier remains invisible while it regains visibility as a shine-through element for extended and homogeneous gratings. However, even subtle deviations from the homogeneity of the grating diminish or even abolish shine-through. At first glance, these results suggest that explanations of these effects have to rely on high-level Gestalt terminology such as homogeneity rather than on low-level properties such as luminance (Herzog, Fahle, & Koch, 2001). Here, we show that a simple neural network model of the Wilson-Cowan type qualitatively and quantitatively explains the basic effects in the shine-through paradigm, although the model does not contain any explicit, global Gestalt processing. Visibility of the target vernier corresponds to transient activation of neural populations resulting from the dynamics of local lateral interactions of excitatory and inhibitory layers of neural populations.

Finite-size effects of avalanche dynamics

We study the avalanche dynamics of a system of globally coupled threshold elements receiving random input. The model belongs to the same universality class as the random-neighbor version of the Olami-Feder-Christensen stick-slip model. A closed expression for avalanche size distributions is derived for arbitrary system sizes N using geometrical arguments in the system's configuration space. For finite systems, approximate power-law behavior is obtained in the nonconservative regime, whereas for N--> infinity, critical behavior with an exponent of -3/2 is found in the conservative case only. We compare these results to the avalanche properties found in networks of integrate-and-fire neurons, and relate the different dynamical regimes to the emergence of synchronization with and without oscillatory components.

Recurrent inhibitory dynamics: the role of state-dependent distributions of conduction delay times

We have formulated and analysed a dynamic model for recurrent inhibition that takes into account the state dependence of the delayed feedback signal (due to the variation in threshold of fibres with their size) and the distribution of these delays (due to the distribution of fibre diameters in the feedback pathway). Using a combination of analytic and numerical tools, we have analysed the behaviour of this model. Depending on the parameter values chosen, as well as the initial preparation of the system, there may be a spectrum of post-synaptic firing dynamics ranging from stable constant values through periodic bursting (limit cycle) behaviour and chaotic firing as well as bistable behaviours. Using detailed parameter estimation for a physiologically motivated example (the CA3-basket cell-mossy fibre system in the hippocampus), we present some of these numerical behaviours. The numerical results corroborate the results of the analytic characterization of the solutions. Namely, for some parameter values the model has a single stable steady state while for the others there is a bistability in which the eventual behaviour depends on the magnitude of stimulation (the initial function).

Representational accuracy of stochastic neural populations

Fisher information is used to analyze the accuracy with which a neural population encodes D stimulus features. It turns out that the form of response variability has a major impact on the encoding capacity and therefore plays an important role in the selection of an appropriate neural model. In particular, in the presence of baseline firing, the reconstruction error rapidly increases with D in the case of Poissonian noise but not for additive noise. The existence of limited-range correlations of the type found in cortical tissue yields a saturation of the Fisher information content as a function of the population size only for an additive noise model. We also show that random variability in the correlation coefficient within a neural population, as found empirically, considerably improves the average encoding quality. Finally, the representational accuracy of populations with inhomogeneous tuning properties, either with variability in the tuning widths or fragmented into specialized subpopulations, is superior to the case of identical and radially symmetric tuning curves usually considered in the literature.

Population coding of motion patterns in the early visual system

Using extracellular recordings and computational modeling, we study the responses of a population of turtle (Pseudemys scripta elegans) retinal ganglion cells to different motion patterns. The onset of motion of a bright bar is signaled by a rise of the population activity that occurs within less than 100 ms. Correspondingly, more complex stimulus movement patterns are reflected by rapid variations of the firing rate of the retinal ganglion cell population. This behavior is reproduced by a computational model that generates ganglion cell activity from the spatio-temporal stimulus pattern using a Wiener model complemented by a non-linear contrast gain control feedback loop responsible for the sharp transients in response to motion onset. This study demonstrates that contrast gain control strongly influences the temporal course of retinal population activity, and thereby plays a major role in the formation of a population code for stimulus movement patterns.

Multidimensional encoding strategy of spiking neurons

Neural responses in sensory systems are typically triggered by a multitude of stimulus features. Using information theory, we study the encoding accuracy of a population of stochastically spiking neurons characterized by different tuning widths for the different features. The optimal encoding strategy for representing one feature most accurately consists of narrow tuning in the dimension to be encoded, to increase the single-neuron Fisher information, and broad tuning in all other dimensions, to increase the number of active neurons. Extremely narrow tuning without sufficient receptive field overlap will severely worsen the coding. This implies the existence of an optimal tuning width for the feature to be encoded. Empirically, only a subset of all stimulus features will normally be accessible. In this case, relative encoding errors can be calculated that yield a criterion for the function of a neural population based on the measured tuning curves.

Coarse coding: calculation of the resolution achieved by a population of large receptive field neurons

Electrophysiological studies in various sensory systems of different species show that many neurons involved in object localization have large receptive fields. This seems to contradict the high sensory resolution and the behavioral precision observed in localization experiments. Assuming a coarse coding mechanism, the resolution obtained by an ensemble of neurons is analytically calculated as a function of receptive field size. It is shown that particularly large receptive fields yield a high resolution.