Jack Heidel

Nonlinear dynamics indicates aging affects variability during gait

OBJECTIVE To investigate the nature of variability present in time series generated from gait parameters of two different age groups via a nonlinear analysis. DESIGN Measures of nonlinear dynamics were used to compare kinematic parameters between elderly and young females. BACKGROUND Aging may lead to changes in motor variability during walking, which may explain the large incidence of falls in the elderly. METHODS Twenty females, 10 younger (20-37 yr) and 10 older (71-79 yr) walked on a treadmill for 30 consecutive gait cycles. Time series from selected kinematic parameters of the right lower extremity were analyzed using nonlinear dynamics. The largest Lyapunov exponent and the correlation dimension of all time series, and the largest Lyapunov exponent of the original time series surrogated were calculated. Standard deviations and coefficient of variations were also calculated for selected discrete points from each gait cycle. Independent t-tests were used for statistical comparisons. RESULTS The Lyapunov exponents were found to be significantly different from their surrogate counterparts. This indicates that the fluctuations observed in the time series may reflect deterministic processes by the neuromuscular system. The elderly exhibited significantly larger Lyapunov exponents and correlation dimensions for all parameters evaluated indicating local instability. The linear measures indicated that the elderly demonstrated significantly higher variability. CONCLUSIONS The nonlinear analysis revealed that fluctuations in the time series of certain gait parameters are not random but display a deterministic behavior. This behavior may degrade with physiologic aging resulting in local instability. RELEVANCE Elderly show increased local instability or inability to compensate to the natural stride-to-stride variations present during locomotion. We hypothesized that this may be the one of the reasons for the increases in falling due to aging. Future efforts should attempt to evaluate this hypothesis by making comparisons to pathological subjects (i.e. elderly fallers), and examine the sensitivity and specificity of the nonlinear methods used in this study to aid clinical assessment.

Scalar equations for synchronous Boolean networks with biological applications

One way of coping with the complexity of biological systems is to use the simplest possible models which are able to reproduce at least some nontrivial features of reality. Although two value Boolean models have a long history in technology, it is perhaps a little bit surprising that they can also represent important features of living organizms. In this paper, the scalar equation approach to Boolean network models is further developed and then applied to two interesting biological models. In particular, a linear reduced scalar equation is derived from a more rudimentary nonlinear scalar equation. This simpler, but higher order, two term equation gives immediate information about both cycle and transient structure of the network.

Emergent decision-making in biological signal transduction networks

The complexity of biochemical intracellular signal transduction networks has led to speculation that the high degree of interconnectivity that exists in these networks transforms them into an information processing network. To test this hypothesis directly, a large scale model was created with the logical mechanism of each node described completely to allow simulation and dynamical analysis. Exposing the network to tens of thousands of random combinations of inputs and analyzing the combined dynamics of multiple outputs revealed a robust system capable of clustering widely varying input combinations into equivalence classes of biologically relevant cellular responses. This capability was nontrivial in that the network performed sharp, nonfuzzy classifications even in the face of added noise, a hallmark of real-world decision-making.

Non-chaotic behaviour in three-dimensional quadratic systems

It is shown that three-dimensional dissipative quadratic systems of ordinary differential equations with a total of four terms on the right-hand side of the equations do not exhibit chaos. This complements recent work of Sprott who has given many examples of chaotic quadratic systems with as few as five terms on the right-hand side of the equations. PACS Number: 0545

Nonchaotic behaviour in three-dimensional quadratic systems II. The conservative case

It is shown that almost all three-dimensional conservative quadratic systems of ordinary differential equations with a total of four terms on the right-hand side of the equations do not exhibit chaos. A previous paper showed the same thing for dissipative systems. PACS Number: 0545

An analysis of a fractal Michaelis-Menten curve

Simple chemical reactions can be described by the Michaelis Menten response curve relating the velocity V of the reaction and the concentration [S] of the substrate S. To handle more complicated reactions without introducing general polynomial response curves, the rate constants can be considered to be scale dependent. This leads to a new response curve with characteristic sigmoidal shape. But not all sigmoidal curves can be accurately fit with three parameters. In order to get an accurate fit, the lower part of the ∫ shaped curve cannot be too shallow and the upper part can’t be too steep. This paper determines an exact mathematical expression for the steepness and shallowness allowed.

The existence of periodic orbits of the tent map

Abstract For the tent map Tμ(x) = μx for 0 ⩽ x ⩽ 1 2 and = μ(1 − x) for 1 2 ⩽ x ⩽ 1 , the μ values are determined for which periodic orbits of each order exist. Except for period 2n orbits which exist for all μ ⩾ 1, the Sarkovskii ordering is vividly illustrated. After the μ values for odd periodic orbits are determined, the remaining even periodic orbits are handled by topological conjugacy.

NONCHAOTIC AND CHAOTIC BEHAVIOR IN THREE-DIMENSIONAL QUADRATIC SYSTEMS: FIVE-ONE CONSERVATIVE CASES

In this paper we study the nonchaotic and chaotic behavior of all 3D conservative quadratic ODE systems with five terms on the right-hand side and one nonlinear term (5-1 systems). We prove a theorem which provides sufficient conditions for solutions in 3D autonomous systems being nonchaotic. We show that all but five of these systems: (15a, 15b), (18b), (41)(A = ∓1), (43b), and (49a, 49b) are nonchaotic. Numerical simulations show that only one of the five systems, (43b), really appears to be chaotic. If proved to be true, it will be the simplest ODE system having chaos.

Combinatorial Fractal Geometry with a Biological Application

The solution to a deceptively simple combinatorial problem on bit strings results in the emergence of a fractal related to the Sierpinski Gasket. The result is generalized to higher dimensions and applied to the study of global dynamics in Boolean network models of complex biological systems.

A reverse Hölder type inequality for the logarithmic mean and generalizations

An inequality involving the logarithmic mean is established. Specifically, we show that L(c, x) ln(c/x) ln(c/a) L(x, a) ln(x/a) ln(c/a) < L(c, a) (1) where 0 < a < x < c and L(x, y) = y−x ln y−ln x , 0 < x < y. Then several generalizations are given.