Boolean Dynamics with Random Couplings

This paper reviews a class of generic dissipative dynamical systems called N-K models. In these models, the dynamics of N elements, defined as Boolean variables, develop step by step, clocked by a discrete time variable. Each of the N Boolean elements at a given time is given a value which depends upon K elements in the previous time step. We review the work of many authors on the behavior of the models, looking particularly at the structure and lengths of their cycles, the sizes of their basins of attraction, and the flow of information through the systems. In the limit of infinite N, there is a phase transition between a chaotic and an ordered phase, with a critical phase in between. We argue that the behavior of this system depends significantly on the topology of the network connections. If the elements are placed upon a lattice with dimension d, the system shows correlations related to the standard percolation or directed percolation phase transition on such a lattice. On the other hand, a very different behavior is seen in the Kauffman net in which all spins are equally likely to be coupled to a given spin. In this situation, coupling loops are mostly suppressed, and the behavior of the system is much more like that of a mean field theory. We also describe possible applications of the models to, for example, genetic networks, cell differentiation, evolution, democracy in social systems and neural networks.Comment: 69 pages, 16 figures, Submitted to Springer Applied Mathematical Sciences Serie

Regularity of center-of-pressure trajectories depends on the amount of attention invested in postural control

The influence of attention on the dynamical structure of postural sway was examined in 30 healthy young adults by manipulating the focus of attention. In line with the proposed direct relation between the amount of attention invested in postural control and regularity of center-of-pressure (COP) time series, we hypothesized that: (1) increasing cognitive involvement in postural control (i.e., creating an internal focus by increasing task difficulty through visual deprivation) increases COP regularity, and (2) withdrawing attention from postural control (i.e., creating an external focus by performing a cognitive dual task) decreases COP regularity. We quantified COP dynamics in terms of sample entropy (regularity), standard deviation (variability), sway-path length of the normalized posturogram (curviness), largest Lyapunov exponent (local stability), correlation dimension (dimensionality) and scaling exponent (scaling behavior). Consistent with hypothesis 1, standing with eyes closed significantly increased COP regularity. Furthermore, variability increased and local stability decreased, implying ineffective postural control. Conversely, and in line with hypothesis 2, performing a cognitive dual task while standing with eyes closed led to greater irregularity and smaller variability, suggesting an increase in the “efficiency, or “automaticity” of postural control”. In conclusion, these findings not only indicate that regularity of COP trajectories is positively related to the amount of attention invested in postural control, but also substantiate that in certain situations an increased internal focus may in fact be detrimental to postural control

History-Dependent Excitability as a Single-Cell Substrate of Transient Memory for Information Discrimination

Neurons react differently to incoming stimuli depending upon their previous history of stimulation. This property can be considered as a single-cell substrate for transient memory, or context-dependent information processing: depending upon the current context that the neuron “sees” through the subset of the network impinging on it in the immediate past, the same synaptic event can evoke a postsynaptic spike or just a subthreshold depolarization. We propose a formal definition of History-Dependent Excitability (HDE) as a measure of the propensity to firing in any moment in time, linking the subthreshold history-dependent dynamics with spike generation. This definition allows the quantitative assessment of the intrinsic memory for different single-neuron dynamics and input statistics. We illustrate the concept of HDE by considering two general dynamical mechanisms: the passive behavior of an Integrate and Fire (IF) neuron, and the inductive behavior of a Generalized Integrate and Fire (GIF) neuron with subthreshold damped oscillations. This framework allows us to characterize the sensitivity of different model neurons to the detailed temporal structure of incoming stimuli. While a neuron with intrinsic oscillations discriminates equally well between input trains with the same or different frequency, a passive neuron discriminates better between inputs with different frequencies. This suggests that passive neurons are better suited to rate-based computation, while neurons with subthreshold oscillations are advantageous in a temporal coding scheme. We also address the influence of intrinsic properties in single-cell processing as a function of input statistics, and show that intrinsic oscillations enhance discrimination sensitivity at high input rates. Finally, we discuss how the recognition of these cell-specific discrimination properties might further our understanding of neuronal network computations and their relationships to the distribution and functional connectivity of different neuronal types