On the stability of a model of testosterone dynamics

We prove the global asymptotic stability of a well-known delayed negative-feedback model of testosterone dynamics, which has been proposed as a model of oscillatory behavior. We establish stability (and hence the impossibility of oscillations) even in the presence of delays of arbitrary length.Supported in part by AFOSR Grant F49620-01-1-0063, NIH Grant P20 GM64375, and Dimacs.E.D. Sontag: Supported in part by AFOSR Grant F49620-01-1-0063 and NIH Grant R01 GM46383.Acknowledgement We would like to thank Augusto Ponce for useful suggestions.     

A predator prey model with age structure

A general predator-prey model is considered in which the predator population is assumed to have an age structure which significantly affects its fecundity. The model equations are derived from the general McKendrick equations for age structured populations. The existence, stability and destabilization of equilibria are studied as they depend on the prey's natural carrying capacity and the maturation periodm of the predator. The main result of the paper is that for a broad class of maturation functions positive equilibria are either unstable for smallm or are destabilized asm decreases to zero. This is in contrast to the usual rule of thumb that increasing (not decreasing) delays in growth rate responses cause instabilities.Research supported by National Science Foundation Grant No. MCS-7901307-01Research supported by National Scholarship for Study Abroad No. EDN/S-59/80 from the government of India     

Optimal filtering of nerve singnals

Recording from multiple electrodes at different sites along a peripheral nerve permits the application of powerful filtering methods to extract the activity of populations of fibres within the nerve which differ in temporal or spectral characteristics. The design of optimal linear filters is initially treated as a general problem in the calculus of variations in which the signals from one population of nerve fibres are extracted so as to minimize those from a second population of nerve fibres or from other sources (noise). A particularly important application arises when the signals at two electrodes are related by weighting functions. In the simplest example the weighting function represents the time delay for nerve impulses to conduct from one electrode to the other, but explicit results are also derivable when there are a range of conduction delays with probabilities distributed according to well-known functions such as the sinc² function.This work was partially supported by the National Research Council of Canada and Medical Research Council of Canada by Grant NRC A-4345 to MNO and Grant MRC MA-3307 to RBS through the University of Alberta     

Integral equation models for endemic infectious diseases

Summary Endemic infectious diseases for which infection confers permanent immunity are described by a system of nonlinear Volterra integral equations of convolution type. These constant-parameter models include vital dynamics (births and deaths), immunization and distributed infectious period. The models are shown to be well posed, the threshold criteria are determined and the asymptotic behavior is analysed. It is concluded that distributed delays do not change the thresholds and the asymptotic behaviors of the models.This work was partially supported by NIH Grant AI 13233.     

Numerical solution of a nonlinear advance-delay-differential equation from nerve conduction theory

A functional differential equation which is nonlinear and involves forward and backward deviating arguments is solved numerically. The equation models conduction in a myelinated nerve axon in which the myelin completely insulates the membrane, so that the potential change jumps from node to node. The equation is of first order with boundary values given at t=±. The problem is approximated via a difference scheme which solves the problem on a finite interval by utilizing an asymptotic representation at the endpoints, cubic interpolation and iterative techniques to approximate the delays, and a continuation method to start the procedure. The procedure is tested on a class of problems which are solvable analytically to access the scheme's accuracy and stability, then applied to the problem that models propagation in a myelinated axon. The solution's dependence on various model parameters of physical interest is studied. This is the first numerical study of myelinated nerve conduction in which the advance and delay terms are treated explicitly.Supported in part by NSF Grant MCS8301724 and by a Biomedical Research Support Grant 2SO7RR0706618 from NIH     

Transient persistence of neural activity after periodic stimulation in the cat LGN

Oscillatory neural activity recorded from gross concentric electrodes implanted in the lateral geniculate nucleus (LGN) of cat was investigated after trains of periodic electrical shock stimulation were applied to the contralateral optic nerve. Spectral analysis using repeated trials and spectral averaging revealed statistically significant spectral peaks (p<0.05) at the stimulus frequency when there was a 0.5 to 1.5 s delay between the last pulse of the stimulus train and the beginning of the data analysis window. This suggests that a loosely coupled neuronal net may allow for the continuation of activity far longer than the time accounted for by neuronal time constants and delays.Supported by NIH Training Grant No. GM 01455 and NSF Grant No. ENG 7515736     

Diffusion approximations of the two-locus Wright-Fisher model

Diffusion approximations are established for the multiallelic, two-locus Wright-Fisher model for mutation, selection, and random genetic drift in a finite, panmictic, monoecious, diploid population. All four combinations of weak or strong selection and tight or loose linkage are treated, though the proof in the case of strong selection and loose linkage is incomplete. Under certain conditions, explicit formulas are obtained for the stationary distributions of the two diffusions with loose linkage.Supported in part by NSF Grant DMS-8704369Supported in part by NSF Grant BSR-8512844     

'Flapping valves' in brachiopods

M. J. S. Rudwick and others postulate 'rhythmic-flow' feeding for the Permian richtho-feniacean bi-achiopods, whereas R. E. Grant claims that they fed by normal ciliary action. Suspension-feeding has two components, current generation and food capture; normal brachiopod lophophores do both, but this is neither universal nor compulsory among animals. Opening and closing the richthofeniid shell generated a 'tidal-flow' current precisely analogous to respiratory currents in mammals; this is neither inefficient nor 'self-defeating', as Grant claims. Grant's analysis fails because he chose the wrong mechanical analogy (a pump). Morphological features of richthofeniids are better explained on a tidal-flow hypothesis than on a ciliary-flow model, and all the data adduced by Grant in rejecting the former is compatible with it or favorable to it. It explains morphological features that are bizarre mysteries on the ciliaiy-current model, and is therefore superior even though it implies that these Permian brachiopods were radically innovative.     

Absolute stability in predator-prey models

An equilibrium of a time-lagged population model is said to be absolutely stable if it remains locally stable regardless of the length of the time delay, and it is argued that the criteria for absolute stability provide a valuable guide to the behavior of population models. For example, it is sometimes assumed that time delays have a limited impact until they exceed the natural time scale of a system; here it is stressed that under some conditions very short time delays can have a marked (and often maximal) destabilizing effect. Consequently it is important that our understanding of population dynamics is robust to the inclusion of the short time delays present in all biological systems. The absolute stability criteria are ideally suited for this role. Another important reason for using the criteria for absolute stability rather than using criteria which depend upon the details of a time delay is that biological time delays are unlikely to be constant. For example, a time delay due to maturation inevitably varies between individuals and the mean may itself vary over time. Here it is shown that the criteria for absolute stability are generally robust in the presence of distributed delays and of varying delays. The analysis presented is based upon a general predator-prey model and it is shown that absolute stability can be expected under a broad range of parameter values whenever the time delay is due to the maturation time of either the predator or the prey or of both. This stability occurs because of the interaction between delayed and undelayed dynamic features of the model. A time-delayed process, when viewed across all possible delays, always reduces stability and this effect occurs regardless of whether the process would act to stabilize or destabilize an undelayed system. Opposing the destabilization due to a time delay and making absolute stability a possibility are a number of processes which act without delay. Some of these processes can be identified as stabilizing from the analysis of undelayed models (for example, the type 3 functional response) but other cannot (for example, the nonreproductive numerical response of predators).     

Task Partitioning in Insect Societies. II. Use of Queueing Delay Information in Recruitment

The collection and handling of colony resources such as food, water, and nest-construction material is often divided into subtasks in which the material is passed from one worker to another. This is known as task partitioning. If tasks are partitioned with direct transfer of material between foragers and receivers, queueing delays can occur as individuals search or wait for a transfer partner. Changes in environmental conditions and relative number of foragers and receivers affect these delays as well as colony ergonomic efficiency. These delays are used in recruitment in both honeybees and Polybia wasps. This study investigates the distribution of queueing delays and the information content and quality of those delays using a stochastic-simulation model. Information quality increases with colony size. When the relative proportions of foragers and receivers are suboptimal, the group in excess has better information. Individuals can increase information quality of delays by two mechanisms: averaging over consecutive trips and averaging over multiple transfers within a trip where direct transfer occurs. We suggest that multiple transfer occurs in the honeybee in order to improve information quality.     

Reassessing mechanisms of low-frequency sound localisation

Interaural time differences (ITDs) are the dominant cues for human localisation of low-frequency sounds. Although a mechanism for ITD processing proposed in 1948 seems applicable to birds, and is consistent with many aspects of the responses found in mammals, recent data suggest that key tenets of the model might need to be reconsidered. The model requires, at every frequency, a distribution of cells with firing rate peaks across all ITD values within the animal's physiological range. The ITD tuning relies on internal delays in the form of a neural delay line. The evidence for such a delay line structure in mammals is not as convincing as it is in birds and, in some small animals the full range of physiological ITDs are not fully represented by peak firing of neurones at every frequency channel. Alternative means of achieving internal delays such as inhibitory inputs or the delays associated with cochlear filtering are being considered.     

Common source epidemics I: A stochastic model

A discrete time stochastic model is formulated for the spread of a disease which is transmitted to an uninfected but susceptible individual through an environmental source and not through contact (either direct or indirect) with infected individuals. The model incorporates both exposure and infection components. The exposure component includes consideration of the introduction of an infectious agent into the environment and the subsequent diffusion of the agent. It also includes time and location patterns for visits by individuals in the target population to the affected environment. The infection component incorporates physiological responses of exposed individuals to the infectious agent. The goal of the model is to provide a method for developing a predicted epidemic curve. Comments are given on an application of the model to the study of an outbreak of toxoplasmosis in Atlanta, Georgia, in 1977. This work was partially supported by BRSG Grant S07 RR0731 awarded by the Biomedical Research Support Grant Program, Division of Research Resources, National Institutes of Health.     

Encoding of spatio-temporal input characteristics by a CA1 pyramidal neuron model

The in vivo activity of CA1 pyramidal neurons alternates between regular spiking and bursting, but how these changes affect information processing remains unclear. Using a detailed CA1 pyramidal neuron model, we investigate how timing and spatial arrangement variations in synaptic inputs to the distal and proximal dendritic layers influence the information content of model responses. We find that the temporal delay between activation of the two layers acts as a switch between excitability modes: short delays induce bursting while long delays decrease firing. For long delays, the average firing frequency of the model response discriminates spatially clustered from diffused inputs to the distal dendritic tree. For short delays, the onset latency and inter-spike-interval succession of model responses can accurately classify input signals as temporally close or distant and spatially clustered or diffused across different stimulation protocols. These findings suggest that a CA1 pyramidal neuron may be capable of encoding and transmitting presynaptic spatiotemporal information about the activity of the entorhinal cortex-hippocampal network to higher brain regions via the selective use of either a temporal or a rate code.     

Evoked potentials in artificial neural nets

A neural net model of discrete populations of formal neurons have been constructed to study evoked potentials based on our previous simulation studies (Anninos, 1972). Some interesting results came up from the examination of our findings regarding the latencies and the period of the cyclic activity of the evoked potentials. In fact, the different successive latencies for the five identical stimuli and the different periods for each of the cyclic activities, all are consequences of inhibitory and excitatory influences from a large neuronal population. Furthermore, such behavior of the system is not only related to the unknown neuronal population but was also substantially altered by what occurred in other systems at the time of stimulus, or prior to it.Computation assistance was provided by the Health Sciences Computing Facility, UCLA, sponsored by NIH Special Research Resources Grant RR-3. Research was sponsored by NSF Grant GB 30498 and NIH Grant NS-8498.     

Mean residence time in multicompartmental models with time delays

Calculation of the mean residence time (MRT) of a drug in a stationary compartmental model is classically carried out from several expressions. Nevertheless, one or more time delays between compartments modify the mean residence times. It is the aim of this paper to propose a general method for MRT calculations, in any n-compartmental models which may include time delays. As examples, catenary and mammillary models are considered.     

Diffusion-driven instability in a predator-prey system with time-varying diffusivities

We consider an ecological model by Levin and Segel (1976) for predator-prey planktonic species, which consists of two reaction-diffusion equations, and extend it to plankton populations with time-varying diffusivities. The local stability of uniform equilibria is examined both analytically and numerically. It is found that diffusive instability is less likely to occur in systems with time-varying diffusivity than those with constant diffusivity. Contribution No. 803 of the Marine Sciences Research Center, State University of New York, Stony Brook Supported by the Danish Science Research Council (Grant nos. 11-7128, 11-8321), the Danish Research Academy (Grant nos. E-880011, V-890085) and a Travel Grant for Mathematicians (Rejselegat for Matematikere) Supported by Hudson River Foundation, Grant no. 01488AO37     

Inferring gene regulatory networks with time delays using a genetic algorithm

Recently a state-space model with time delays for inferring gene regulatory networks was proposed. It was assumed that each regulation between two internal state variables had multiple time delays. This assumption caused underestimation of the model with many current gene expression datasets. In biological reality, one regulatory relationship may have just a single time delay, and not multiple time delays. This study employs Boolean variables to capture the existence of the time-delayed regulatory relationships in gene regulatory networks in terms of the state-space model. As the solution space of time delayed relationships is too large for an exhaustive search, a genetic algorithm (GA) is proposed to determine the optimal Boolean variables (the optimal time-delayed regulatory relationships). Coupled with the proposed GA, Bayesian information criterion (BIC) and probabilistic principle component analysis (PPCA) are employed to infer gene regulatory networks with time delays. Computational experiments are performed on two real gene expression datasets. The results show that the GA is effective at finding time-delayed regulatory relationships. Moreover, the inferred gene regulatory networks with time delays from the datasets improve the prediction accuracy and possess more of the expected properties of a real network, compared to a gene regulatory network without time delays.     

Analogous mechanisms compensate for neural delays in the sensory and the motor pathways: evidence from motor flash-lag

Motor behaviors require animals to coordinate neural activity across different areas within their motor system. In particular, the significant processing delays within the motor system must somehow be compensated for. Internal models of the motor system, in particular the forward model, have emerged as important potential mechanisms for compensation. For motor responses directed at moving visual objects, there is, additionally, a problem of delays within the sensory pathways carrying crucial position information. The visual phenomenon known as the flash-lag effect has led to a motion-extrapolation model for compensation of sensory delays. In the flash-lag effect, observers see a flashed item colocalized with a moving item as lagging behind the moving item. Here, we explore the possibility that the internal forward model and the motion-extrapolation model are analogous mechanisms compensating for neural delays in the motor and the visual system, respectively. In total darkness, observers moved their right hand gripping a rod while a visual flash was presented at various positions in relation to the rod. When the flash was aligned with the rod, observers perceived it in a position lagging behind the instantaneous felt position of the invisible rod. These results suggest that compensation of neural delays for time-varying motor behavior parallels compensation of delays for time-varying visual stimulation.     

Population density models of integrate-and-fire neurons with jumps: well-posedness

In this paper we study the well-posedness of different models of population of leaky integrate-and-fire neurons with a population density approach. The synaptic interaction between neurons is modeled by a potential jump at the reception of a spike. We study populations that are self excitatory or self inhibitory. We distinguish the cases where this interaction is instantaneous from the one where there is a repartition of conduction delays. In the case of a bounded density of delays both excitatory and inhibitory population models are shown to be well-posed. But without conduction delay the solution of the model of self excitatory neurons may blow up. We analyze the different behaviours of the model with jumps compared to its diffusion approximation.     

The Freter model: A simple model of biofilm formation

A simple, conceptual model of biofilm formation, due to R. Freter et al. (1983), is studied analytically and numerically in both CSTR and PFR. Two steady state regimes are identified, namely, the complete washout of the microbes from the reactor and the successful colonization of both the wall and bulk fluid. One of these is stable for any particular set of parameter values and sharp and explicit conditions are given for the stability of each. The effects of adding an anti-microbial agent to the CSTR are examined.Supported by NSF Grant DMS 0107439 and UTA Grant REP 14748717Supported by NSF Grant DMS 0107160