Roderick Edwards

Analysis of continuous-time switching networks

Abstract Models of a number of biological systems, including gene regulation and neural networks, can be formulated as switching networks, in which the interactions between the variables depend strongly on thresholds. An idealized class of such networks in which the switching takes the form of Heaviside step functions but variables still change continuously in time has been proposed as a useful simplification to gain analytic insight. These networks, called here Glass networks after their originator, are simple enough mathematically to allow significant analysis without restricting the range of dynamics found in analogous smooth systems. A number of results have been obtained before, particularly regarding existence and stability of periodic orbits in such networks, but important cases were not considered. Here we present a coherent method of analysis that summarizes previous work and fills in some of the gaps as well as including some new results. Furthermore, we apply this analysis to a number of examples, including surprising long and complex limit cycles involving sequences of hundreds of threshold transitions. Finally, we show how the above methods can be extended to investigate aperiodic behaviour in specific networks, though a complete analysis will have to await new results in matrix theory and symbolic dynamics.

Symbolic dynamics and computation in model gene networks.

We analyze a class of ordinary differential equations representing a simplified model of a genetic network. In this network, the model genes control the production rates of other genes by a logical function. The dynamics in these equations are represented by a directed graph on an n-dimensional hypercube (n-cube) in which each edge is directed in a unique orientation. The vertices of the n-cube correspond to orthants of state space, and the edges correspond to boundaries between adjacent orthants. The dynamics in these equations can be represented symbolically. Starting from a point on the boundary between neighboring orthants, the equation is integrated until the boundary is crossed for a second time. Each different cycle, corresponding to a different sequence of orthants that are traversed during the integration of the equation always starting on a boundary and ending the first time that same boundary is reached, generates a different letter of the alphabet. A word consists of a sequence of letters corresponding to a possible sequence of orthants that arise from integration of the equation starting and ending on the same boundary. The union of the words defines the language. Letters and words correspond to analytically computable Poincare maps of the equation. This formalism allows us to define bifurcations of chaotic dynamics of the differential equation that correspond to changes in the associated language. Qualitative knowledge about the dynamics found by integrating the equation can be used to help solve the inverse problem of determining the underlying network generating the dynamics. This work places the study of dynamics in genetic networks in a context comprising both nonlinear dynamics and the theory of computation. (c) 2001 American Institute of Physics.

The measurement of tremor using a velocity transducer: comparison to simultaneous recordings using transducers of displacement, acceleration and muscle activity

Precise kinematic measurements of tremor have historically been obtained using accelerometers. However, current technology permits precise measurements in velocity and displacement. The primary advantage of velocity recording is that only one step of integration or differentiation is required for either displacement or acceleration. A method is presented of measuring finger tremor using a laser system that transduces velocity precisely. Measurements of postural finger tremor thus obtained were compared to those simultaneously obtained from a laser system that transduces displacement, from an accelerometer and from surface electromyography (EMG) of the extensor digitorum communis. A range of amplitude and frequency content was obtained by testing control subjects and subjects with Parkinsons disease. The velocity transducer showed excellent correspondence of amplitude and frequency measurement with the displacement transducer. Measures of absolute and relative amplitude correlated well (r > or = 0.96 in amplitude measures in displacement, velocity and acceleration), and high coherence was found throughout the frequency range of interest. Measurements by the accelerometer generally showed poorer correspondence with those of the other instruments. EMG measurements showed good correspondence in some trials but poorer correspondence in others, attributed to the low level of muscle activity required in the task. Precise kinematic measurements appear to be highly sensitive to neuromotor impairment.

Delayed Coupling Between Two Neural Network Loops

Coupled loops with time delays are common in physiological systems such as neural networks. We study a Hopfield-type network that consists of a pair of one-way loops each with three neurons and two-way coupling (of either excitatory or inhibitory type) between a single neuron of each loop. Time delays are introduced in the connections between loops, and the effects of coupling strengths and delays on the network dynamics are investigated. These effects depend strongly on whether the coupling is symmetric (of the same type in both directions) or asymmetric (inhibitory in one direction and excitatory in the other). The network of six delay differential equations is studied by linear stability analysis and bifurcation theory. Loops having inherently stable zero solutions cannotbe destabilized by weak coupling, regardless of the delay. Asymmetric coupling is weakly stabilizing but easily upset by delays. Symmetric coupling (if not too weak) can destabilize an inherently stable zero solution, leading to nontri...

Combinatorial explosion in model gene networks

The explosive growth in knowledge of the genome of humans and other organisms leaves open the question of how the functioning of genes in interacting networks is coordinated for orderly activity. One approach to this problem is to study mathematical properties of abstract network models that capture the logical structures of gene networks. The principal issue is to understand how particular patterns of activity can result from particular network structures, and what types of behavior are possible. We study idealized models in which the logical structure of the network is explicitly represented by Boolean functions that can be represented by directed graphs on n-cubes, but which are continuous in time and described by differential equations, rather than being updated synchronously via a discrete clock. The equations are piecewise linear, which allows significant analysis and facilitates rapid integration along trajectories. We first give a combinatorial solution to the question of how many distinct logical structures exist for n-dimensional networks, showing that the number increases very rapidly with n. We then outline analytic methods that can be used to establish the existence, stability and periods of periodic orbits corresponding to particular cycles on the n-cube. We use these methods to confirm the existence of limit cycles discovered in a sample of a million randomly generated structures of networks of 4 genes. Even with only 4 genes, at least several hundred different patterns of stable periodic behavior are possible, many of them surprisingly complex. We discuss ways of further classifying these periodic behaviors, showing that small mutations (reversal of one or a few edges on the n-cube) need not destroy the stability of a limit cycle. Although these networks are very simple as models of gene networks, their mathematical transparency reveals relationships between structure and behavior, they suggest that the possibilities for orderly dynamics in such networks are extremely rich and they offer novel ways to think about how mutations can alter dynamics. (c) 2000 American Institute of Physics.

Using frequency domain characteristics to discriminate physiologic and parkinsonian tremors.

The manner in which characteristics of time series in the frequency domain can enhance discrimination between physiologic and parkinsonian tremor when tremor amplitude is low was examined. Rest tremor and postural tremor with and without visual feedback were recorded twice in the two hands of a group of patients with Parkinsons disease (PD) (n = 21) and a group of healthy control subjects (n = 30) using displacement laser systems. Recordings were analyzed quantitatively using amplitude and seven frequency domain characteristics. Postural tremor with no visual feedback allowed the most efficient discrimination between the two groups of subjects especially in velocity and acceleration (derived from displacement) and allowed identification of more patients with PD as separate from the range observed in the control group. Moreover, the frequency domain characteristics that were investigated identified the majority of the patients even when amplitude did not. After eliminating redundant (correlated) characteristics, it was found that the frequency composition of tremor in PD can be described adequately with four characteristics, which are the most reliable, independent, and discriminative elements for detecting early or subtle modifications in tremor. Also, a series of finger flexions was found to enhance physiologic tremor but not tremor in PD. Discrimination of low-amplitude tremor in PD from normal physiologic tremor is enhanced by examining the median frequency of oscillations, the concentration of power in the power spectrum, and the distribution of power in particular ranges. Tremor measurement should not be limited to acceleration data as some information is more visible in velocity time series.

Stiffness and postural stability in adults with Down syndrome.

The purpose of this study was to characterize postural sway in quiet standing under eyes-open and eyes-closed conditions, and to obtain a measure of postural stiffness during quiet standing in adults with Down syndrome (DS) versus control subjects. We obtained descriptive measures from centre-of-pressure (COP) data and analysed and compared COP trajectories and postural stiffness estimates from two stochastic models, the “pinned polymer” (PP) and “inverted pendulum” (IP) models. These estimates were correlated with clinical measures of muscle tone. Our results showed that overall, estimated values for postural stiffness from both models were larger for the DS group than for normal controls. In addition, average stiffness measures were greater under the eyes-closed condition than under the eyes-open condition for the DS group. The IP model detected significant trends over trials whereas the PP model did not. Clinical assessment of muscle tone for the DS group ranged from low to high-normal and there was no significant correlation with the postural stiffness measures obtained from either model. These results suggest that individuals with DS have the ability to modulate their underlying “stiffness” under conditions of quiet standing. Furthermore, there appears to be no strong relationship between clinical measures of muscle tone and postural stiffness measures under dynamic conditions.

Bradykinesia in patients with Parkinson's disease having levodopa-induced dyskinesias

We investigated the likelihood that bradykinesia coexisted with levodopa-induced dyskinesias (LID) in 10 dyskinetic Parkinsons disease patients (DPD). Their motor performance was compared to that of 10 age/gender-matched non-dyskinetic patients (NDPD) and 10 healthy controls. Whole-body movement (WBM) and rapid alternating movements (RAM) at the wrist were recorded simultaneously using 6-degree of freedom magnetic motion tracker and forearm rotational sensors, respectively. WBM was recorded prior to, and while subjects performed pronation-supination movements of their dominant hand with maximal rotational excursion, and as fast as possible for 10s. RANGE, VELOCITY and IRREGULARITY of pronation-supination cycles were quantified to assess motor performance. Results show that DPD patients had greater WBM than NDPD and controls during rest and RAM performance, as expected. There were no differences in motor performance between DPD and NDPD groups for RANGE and VELOCITY, despite significantly longer disease duration for the DPD group (15.5+/-6.2 years versus 6.6+/-2.6 years). However, both the NDPD and DPD groups showed significantly lower RANGE and reduced VELOCITY compared to controls, suggesting the presence of bradykinesia. For IRREGULARITY, DPD patients showed increased fluctuations in pronation-supination cycle amplitude compared to NDPD and controls. However, the lack of correlation between WBM magnitude and IRREGULARITY within the DPD group (Spearmans rank order, rho = 0.31, p > 0.05), suggests that LID were not the primary cause of increased IRREGULARITY. In conclusion, our results demonstrated that bradykinesia can coexist with dyskinesias, suggestive of distinct neural circuits. Our results also demonstrated that the occurrence of LID is not inevitably accompanied with worsening of motor performance.

Movement patterns of peak-dose levodopa-induced dyskinesias in patients with Parkinson's disease.

The present study characterized involuntary movements associated with levodopa-induced dyskinesias (LID) in patients with Parkinsons disease. We used amplitude, proportional energy, frequency dispersion and sample entropy to determine whether LID movement patterns are truly random, as clinical description seems to suggest, or possess some underlying pattern that is not visible to the naked eye. LID was captured using a magnetic tracker system, which provided 3D rendering of whole-body LID. Patients were instructed to maintain a standing position, with arms extended in front of them. We compared the measurements of the dyskinetic PD group (DPD) with 10 patients without dyskinesias (NDPD) and 10 control subjects. In comparison to the other two groups, movement patterns from the DPD group had significantly higher amplitude, confirming the presence of dyskinesias. In addition, higher frequency components in the power spectrum of velocity were detected, suggestive of higher velocity in LID movement. Furthermore, there was a concentration in narrow frequency bands, which suggested stable oscillatory activity. Finally, sample entropy revealed more regularity in the DPD group. Although not statistically significant, we found that the amplitude from the NDPD group had a tendency to be smaller than those of controls. As well, the spectra were often more dispersed for the NDPD group. In conclusion, the present results suggest that LID cannot be considered as purely random movement since they possess some deterministic pattern of motion. This may provide a way for patients to adapt to these involuntary movements while performing voluntary motor acts.

Tremor in Cree subjects exposed to methylmercury: a preliminary study.

Rest, postural, and kinetic tremors were recorded in Cree subjects (n = 36) exposed to low levels of methylmercury (MeHg) and control subjects (n = 30) using lasers designed to measure displacement. Displacement and derived velocity and acceleration time series were analyzed using quantitative characteristics in time and frequency domains. We found: 1) relatively low agreement between our results and those of a clinical examination done at the time of testing (r = 0.31 for postural tremor); 2) best discrimination with velocity of static (postural with visual feedback) and kinetic tremors for which significant differences between the two groups are found in many of the characteristics examined; 3) myoclonic-like finger microdisplacements in tremor of Cree subjects; 4) significant changes in static tremor of Cree subjects following a kinetic task; 5) significant differences among three age-matched subgroups of six subjects each (Cree with higher and lower MeHg level, and a control group) in characteristics that reflect difficulty with tracking, myoclonic-like microdisplacements and change in frequency composition of tremor after the tracking task. The subtle differences detected in this preliminary study suggest that further work is warranted to determine whether they can be unambiguously associated with exposure to MeHg.