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In this study, the group of finite cyclic lamplighter states is reinterpreted as the novel lamplighter puzzle. The rules of the puzzle are outlined and related back to properties of the lamplighter group with specific interest placed upon the discussion of which puzzle instances are solvable. The paper shows that, through the use of algebra, many puzzle instances can be identified as solvable without the use of an exhaustive search algorithm. Solvability depends upon the creation of irregular generating sets for subgroups of the finite cyclic lamplighter group and the cosets formed by these subgroups. Further possible generalizations of the lamplighter puzzle are also discussed in closing.

A column model of the Arctic atmosphere is developed including the nonlinear positive feedback responses of surface albedo and water vapour to temperature. The atmosphere is treated as a grey gas and the flux of longwave radiation is governed by the two-stream Schwarzschild equations. Water vapour concentration is determined by the Clausius–Clapeyron equation. Representative concentration pathways (RCPs) are used to model carbon dioxide concentrations into the future. The resulting 9D two-point boundary value problem is solved under various RCPs and the solutions analysed. The model predicts that under the highest carbon pathway, the Arctic climate will undergo an irreversible bifurcation to a warm steady state, which would correspond to annually ice-free conditions. Under the lowest carbon pathway, corresponding to very aggressive carbon emission reductions, the model exhibits only a mild increase in Arctic temperatures. Under the two intermediate carbon pathways, temperatures increase more substantially, and the system enters a region of bistability where external perturbations could possibly cause an irreversible switch to a warm, ice-free state.

Urine output measurement is critical for the management of hospitalized cats and their underlying conditions. Ultrasound-guided estimation of urinary bladder volume (UBV) is a non-invasive surrogate measurement that can provide important clinical information. The purpose of this study was to validate the accuracy of a novel 3D computation method in estimating UBV using 2D point-of-care ultrasonographic images. Bladder volume estimation was performed using coordinates from bladder circumference tracings on paired longitudinal and transverse ultrasonographic images (n = 359) aligned in 3D space for mathematical algorithmic computation. Ultrasonographic images were obtained by 2 different observers at 18 different time points on 10 healthy, purpose-bred male cats under general anesthesia in sternal recumbency. Actual urine volumes were measured via urinary catheterization and compared to UBV estimations using Lin’s concordance correlation coefficient and Bland-Altman analysis. Estimation of UBV using the 3D computational bladder circumference tracing method showed moderate strength-of-agreement with actual bladder volume (ρ(c) = 0.94 to 0.95) with clinically insignificant bias (3D computation-derived minus actual volume) of −1.96 mL (IQR = −3.89 to −0.57 mL, P < 0.001) and −2.42 mL (IQR = −4.64 to −0.66 mL, P < 0.001) for the 2 observers, respectively. Our study demonstrated acceptable accuracy of 3D computation method for UBV estimation in healthy cats. This method may provide a bridging alternative until 3D ultrasound becomes more readily accessible.

A new energy balance model (EBM) is presented and is used to study paleoclimate transitions. While most previous EBMs only dealt with the globally averaged climate, this new EBM has three variants: Arctic, Antarctic and tropical climates. The EBM incorporates the greenhouse warming effects of both carbon dioxide and water vapour, and also includes ice–albedo feedback and evapotranspiration. The main conclusion to be inferred from this EBM is that the climate system may possess multiple equilibrium states, both warm and frozen, which coexist mathematically. While the actual climate can exist in only one of these states at any given time, the EBM suggests that climate can undergo transitions between the states via mathematical saddle-node bifurcations. This paper proposes that such bifurcations have actually occurred in Paleoclimate transitions. The EBM is applied to the study of the Pliocene paradox, the glaciation of Antarctica and the so-called warm, equable climate problem of both the mid-Cretaceous Period and the Eocene Epoch. In all cases, the EBM is in qualitative agreement with the geological record.

Three methods for obtaining uniform sampling on the standard simplex are summarized and a new derivation of one method is provided, which only uses basic notions from calculus, difference equations, and differential equations.

This article presents the results of a bifurcation analysis of a simple energy balance model (EBM) for the future climate of the Earth. The main focus is on the following question: can the nonlinear processes intrinsic to atmospheric physics, including natural positive feedback mechanisms, cause a mathematical bifurcation of the climate state, as a consequence of continued anthropogenic forcing by rising greenhouse gas emissions? Our analysis shows that such a bifurcation could cause an abrupt change to a drastically different climate state in the EBM, which is warmer and more equable than any climate existing on Earth since the Pliocene epoch. In previous papers, with this EBM adapted to paleoclimate conditions, it was shown to exhibit saddle-node and cusp bifurcations, as well as hysteresis. The EBM was validated by the agreement of its predicted bifurcations with the abrupt climate changes that are known to have occurred in the paleoclimate record, in the Antarctic at the Eocene–Oligocene transition (EOT) and in the Arctic at the Pliocene–Paleocene transition (PPT). In this paper, the EBM is adapted to fit Anthropocene climate conditions, with emphasis on the Arctic and Antarctic climates. The four Representative Concentration Pathways (RCP) considered by the IPCC (Intergovernmental Panel on Climate Change) are used to model future CO2 concentrations, corresponding to different scenarios of anthropogenic activity. In addition, the EBM investigates four naturally occurring nonlinear feedback processes which magnify the warming that would be caused by anthropogenic CO2 emissions alone. These four feedback mechanisms are ice–albedo feedback, water vapour feedback, ocean heat transport feedback, and atmospheric heat transport feedback. The EBM predicts that a bifurcation resulting in a catastrophic climate change, to a pre-Pliocene-like climate state, will occur in coming centuries for an RCP with unabated anthropogenic forcing, amplified by these positive feedbacks. However, the EBM also predicts that appropriate reductions in carbon emissions may limit climate change to a more tolerable continuation of what is observed today. The globally averaged version of this EBM has an equilibrium climate sensitivity (ECS) of 4.34 K, near the high end of the likely range reported by the IPCC.

Human illness due to contamination of food by pathogenic strains of Escherichia coli is a serious public health concern and can cause significant economic losses in the food industry. Recent outbreaks of such illness sourced from ground beef production motivates the work in this paper. Most ground beef is produced in large facilities where many carcasses are butchered and various pieces of them are ground together in sequential batches. Assuming that the source of contamination is a single carcass and that downstream from the production facility ground beef from a particular batch has been identified as contaminated by E. coli, the probability that previous and subsequent batches are also contaminated is modelled. This model may help the beef industry to identify the likelihood of contamination in other batches and potentially save money by not needing to cook or recall unaffected batches of ground beef.

Climate models predict that the climate of the Earth is warming and will continue to warm in coming centuries, if there is no mitigation. A recent energy balance model [Kypke et al., Nonlin. Process. Geophys. 27 (2020) 391–409] forecasts that, if the current increase of carbon dioxide in the atmosphere continues unabated, then in the next century the climate of the Earth will not only get warmer, but will transition abruptly via a bifurcation, to a warm equable climate unlike any climate seen on Earth since the Pliocene. This transition to a new climate state is a topological change. That model includes the effects of water vapour feedback and ice albedo feedback, as well as ocean and atmospheric heat transport. This paper adds to that model further amplification by permafrost feedback. That is, as the Arctic warms, permafrost will thaw, releasing large amounts of the greenhouse gases carbon dioxide and methane, which cause further warming. Since knowledge of permafrost stores and release rates is limited, a range of permafrost carbon release sensitivities (Q 10) is considered. The model predicts that permafrost feedback accelerates the timing and increases the likelihood of a topological climate change in the Arctic, and reinforces the view that permafrost feedback should not be ignored in Anthropocene climate models.

This paper presents an algorithm for parameter range reduction from partial data in systems of differential algebraic equations. Interval arithmetic and linear multistep discretizations are used to prune regions of parameter space that are inconsistent with available time series data. Methods of splitting parameter space to ensure further progress are discussed. The algorithm outputs a reduced range and an approximation for each parameter value. These outputs are shown to have beneficial effect as input to traditional parameter estimation schemes. The range reduction algorithm is tested on three systems with partial data sets.

Human illness due to contamination of food by pathogenic strains of Escherichia coli is a serious public health concern and can cause significant economic losses in the food industry. Recent outbreaks of such illness sourced from ground beef production motivates the work in this paper. Most ground beef is produced in large facilities where many carcasses are butchered and various pieces of them are ground together in sequential batches. Assuming that the source of contamination is a single carcass and that downstream from the production facility ground beef from a particular batch has been identified as contaminated by E. coli, the probability that previous and subsequent batches are also contaminated is modelled. This model may help the beef industry to identify the likelihood of contamination in other batches and potentially save money by not needing to cook or recall unaffected batches of ground beef.

Influenza A virus (IAV) in swine is a pathogen that causes a threat to the health as well as to the production of swine. Moreover, swine can spread this virus to other species including humans. The virus persists in different types of swine farms as evident in a number of studies. The core objectives of this study are (i) to analyze the dynamics of influenza infection of a farrow-to-finish swine farm, (ii) to explore the reinfection at the farm level, and finally (iii) to examine the effectiveness of two control strategies: vaccination and reduction of indirect contact. The analyses are conducted using a deterministic Susceptible-Exposed-Infectious-Recovered (SEIR) model. Simulation results show that the disease is maintained in gilts and piglets because of new susceptible pigs entering the population on a weekly basis. A sensitivity analysis shows that the results are not sensitive to variation in the parameters. The results of the reinfection simulation indicate that the virus persists in the entire farm. The control strategies studied in this work are not successful in eliminating the virus within the farm.

Cheyne-Stokes respiration is a distinct breathing pattern consisting of periods of hyperpnea followed by apnoeas, with unknown aetiology. One in two patients with congestive heart failure suffers from this condition. Researchers hypothesize that key factors in CSR are the fluid shift from the standing to supine position and the differences between genders. A mathematical model of the cardiorespiratory system was constructed using parameter values from real data. Hopf bifurcation analysis was used to determine regions of steady versus oscillatory breathing patterns. In the model, Cheyne-Stokes respiration is more likely to occur while in the supine position and males are more likely to develop Cheyne-Stokes than females. These findings, which are in agreement with clinical experience, suggest that both gender and fluid shift contribute to the pathogenesis of Cheyne-Stokes respiration and that physical quantities such as blood volumes and neural feedback may be sufficient to explain the observations of CSR.

An algorithm for enclosing a given set of time series data inside a continuous piecewise linear band of varying height subject to certain constraints is presented. The band is defined by two piecewise linear curves that lie above and below the data respectively. Segments of these curves are constrained to start and end at one of the data points, and those whose slope does not lie between its neighbors’ slopes are required to be at least as wide as a user-specified value. The algorithm yields a band which accurately preserves the general trends of the data, while enclosing the inherent measurement noise. This band is typically obtained in $$O(n\log K)$$ time, where $$n$$ is the number of data points and $$K$$ is the number of linear segments. The algorithm is described and its capabilities are tested on four data sets. Comparisons are made with alternative algorithms.

A previous physiologically-structured model for a fish population based on individual-level characteristics is studied. The foraging rate is generalized to include a size-dependent functional response and the energy distribution of adults is generalized to permit both reproduction and growth. Equilibria are determined and their stability studied along with a discussion of harvesting strategies. The model with these generalizations is shown to give different predictions than the original model regarding age distribution of fish and harvesting strategies.

This paper analyzes the effectiveness of various monotonic discretizations of an ODE in a parameter range reduction algorithm. Several properties of discretizations are given, and five classes of discretizations are defined for various step numbers $s$. The range reduction algorithm that employs these discretizations is described. Using both analytical results based on the prototypical model $x'=\lambda x$, and empirical results based on two more complicated models, it is shown that one particular class of discretizations (the A1OUT class) results in the tightest bounds on the parameters. This result is shown to be attributed to a certain characteristic value, $A_0$, of the discretization. Accumulation of these discretizations is also defined, and its usefulness in the range reduction algorithm is described.

Double Hopf bifurcations have been studied prior to this work in the generic nonresonant case and in certain strongly resonant cases, including 1:1 resonance. In this paper, the case of symmetrically coupled identical oscillators, motivated by the classic problem of synchronization of Huygens' clocks, is studied using the codimension-three Elphick-Huygens equivariant normal form presented here. The focus is on the effect that the Huygens symmetry assumption has on the dynamic behavior of the system. Periodic solutions include the classical in-phase and anti-phase normal modes that are forced by the symmetry, as well as pairs of mixed mode phase-locked periodic solutions. The escapement paradox is explained. A theorem based on topological degree theory establishes the existence of quasiperiodic solutions in an invariant 3-torus that resembles a 2-torus slightly thickened to a solid toroidal shell, with the two principal radii of the 2-torus slowly modulated in time; that is, a \emph{toroidal breather}. Secondary bifurcations from the in-phase and anti-phase normal modes are explored, of codimension-one and -two, and it is shown that an Arnold tongue plays a fundamental role in the determination of whether secondary bifurcation gives birth to phase-locked periodic solutions or to quasiperiodic solutions. Detailed numerical analysis, using Matlab, extends the local bifurcation analysis to a more global picture that includes coexistence of multiple stable solutions and a "swallowtail" bifurcation of periodic solutions. Natural Sciences and Engineering Research Council of Canada

Human illness due to contamination of food by pathogenic strains of Escherichia coli is a serious public health concern and can cause significant economic losses in the food industry. Undercooked ground beef is the primary means of transmission of pathogenic E. coli to humans. In the Western world, most ground beef is produced in large facilities where many carcasses are butchered and various pieces of them are ground together in large batches. Assuming that the source of contamination is a single carcass, the primary determinant of how many batches of ground beef from a particular production cycle are affected is the manner in which pieces of that carcass are spread about in the raw sources that contribute to the ground beef batches. Assuming that ground beef from a particular batch has been identified at the consumer end as contaminated by E. coli, we model the probability that previous and subsequent batches generated in the same production cycle are also contaminated. This model may help the beef industry to identify the likelihood of contamination in other batches and potentially save money by not needing to cook or recall unaffected batches of ground beef.

Cytokine storms are a potentially fatal exaggerated immune response consisting of an uncontrolled positive feedback loop between immune cells and cytokines. The dynamics of cytokines are highly complex and little is known about specific interactions. Researchers at the Ontario Veterinary College have encountered cytokine storms during virotherapy. Multiple mouse trials were conducted where a virus was injected into mice whose leukocytes lacked expression of the type I interferon receptor. In each case a rapid, fatal cytokine storm occurred. A nonlinear differential equation model of the recorded cytokine amounts was produced to obtain some information on their mutual interactions. Results provide insight into the complex mechanism that drives the storm and possible ways to prevent such immune responses.

The application of manure is an important component of nutrient management in the production of field crops. The regulations governing the safe application of manure are based on laboratory data, which may or may not accurately reflect the environmental fluctuations seen in field conditions.

This paper presents an algorithm for finding a piecewise linear band which bounds given two-dimensional data. The band is of constant height, and the distance between adjacent slope discontinuities in opposite directions is restricted by two user-specified parameters.

Multi-state models of ion channel gating have been used extensively, but choosing optimally small yet sufficiently complex models to describe particular experimental data remains a difficult task. In order to provide some insight into appropriate model selection, this paper presents some basic results about the behavior of solutions of multi-state models, particularly those arranged in a chain formation. Some properties of the eigenvalues and eigenvectors of constant-rate multi-state models are presented. A geometric description of a three-state chain is given and, in particular, differences between a chain equivalent to an Hodgkin--Huxley model and a chain with identical rates are analyzed. One distinguishing feature between these two types of systems is that decay from the open state in the Hodgkin--Huxley model is dominated by the most negative eigenvalue while the identical rate chain displays a mix of modes over all eigenvalues. Natural Sciences and Engineering Research Council of Canada. D.N.: University of Guelph, Undergraduate Research Assistantship.

The eigenvalue problem for a certain tridiagonal matrix with complex coefficients is considered. The eigenvalues and eigenvectors are shown to be expressible in terms of solutions of a certain scalar trigonometric equation. Explicit solutions of this equation are obtained for several special cases, and further analysis of this equation in several other cases provides information about the distribution of eigenvalues.

We consider fitting an ODE model to time series data of the system variables. We assume that the parameters of the model have some initial range of possible values and the goal is to reduce these ranges to produce a smaller parameter region from which to start a global nonlinear optimization algorithm. We introduce the class of cumulative backward differentiation formulas (CBDFs) and show that they inherit the accuracy and stability properties of their generating backward differentiation formulas (BDFs). Discretizing the system with these CBDFs and applying consistency conditions results in reductions of the parameter ranges. We show that these reductions are better than can be obtained simply using BDFs. In addition CBDFs inherit any monotonicity properties with respect to the parameters that the vector field possesses, and we exploit these properties to make the consistency checking more efficient. We illustrate with several examples, analyze some of the behavior of our range reduction method, and discuss how the method could be extended and improved.

This paper presents an algorithm for parameter range reduction in systems of ordinary differential equations. Parameter values are assumed only to be known to lie in potentially large regions of parameter space. Interval arithmetic and a family of monotonic discretizations are used to prune regions of parameter space that are inconsistent with given time series data. The algorithm is tested on two ordinary differential equation models and the reduced ranges are shown to significantly improve the performance of traditional parameter estimation methods.

A compartmental model of the human circulatory system that illustrates Cheyne–Stokes respiration (CSR) is presented. Clinical evidence suggests that patient body position can influence the likelihood of experiencing CSR, and this model is analyzed to see if blood fluid shifts associated with body position could be the means of this influence. It is shown that lying down causes a shift in the location of the Hopf bifurcation curve associated with the onset of CSR, making it more likely.

The problem of estimating parameters from time series data is considered. A parameter range reduction scheme is employed to quickly reduce a priori ranges of parameters. The effectiveness of the scheme is tested in the presence of partial data sets using an SIR model test case. The algorithm is shown to make substantial reductions of parameter ranges when limited time series data is available. Such reductions are shown to be of benefit to traditional parameter estimation techniques.

I introduce publicly available software for accurate fitting of Hodgkin-Huxley models to voltage-clamp data. I describe the model and non-linear fitting procedure employed by the software and compare its results with the usual method of fitting such models using potassium A-current data from a pyloric dilator cell of the lobster Panulirus interruptus and sodium current data from an electrocyte cell of the electric fish Sternopygus macrurus. The set of parameter values for the model determined by this software yield current traces that are substantially closer to the observed data than those determined from the usual fitting method. This improvement is due to the fact that the software fits all of the parameters simultaneously utilizing all of the data rather than fitting steady-state and time constant parameters disjointly using peak currents and portions of the rising and falling phases. I analyze the convergence properties of the software's fitting algorithm using simulated data showing that accurate parameter values are obtained for most of the parameters using any reasonable initial values. The software also incorporates a linear pre-estimation procedure to help in determining reasonable initial values for the full non-linear algorithm. I illustrate and discuss some of the inadequacies of voltage-clamp data.

To understand the functioning of living cells, it is often helpful or even necessary to exploit inherent timescale disparities and focus on long-term dynamic behaviour. In the present study, we explore this type of behaviour for the biochemical network of the phosphotransferase system. We show that, during the slow phase that follows a fast initial transient, the network reaction rates are partitioned into clusters corresponding to connected parts of the reaction network. Rates within any of these clusters assume essentially the same value: differences within each cluster are vastly smaller than that from one cluster to another. This rate clustering induces an analogous clustering of the reactive compounds: only the molecular concentrations on the interface between these clusters are produced and consumed at substantially different rates and hence change considerably during the slow phase. The remaining concentrations essentially assume their steady-state values already by the end of the transient phase. Further, we find that this clustering phenomenon occurs for a large number of parameter values and also for models with different topologies; to each of these models, there corresponds a particular network partitioning. Our results show that, in spite of its complexity, the phosphotransferase system tends to behave in a rather simple (yet versatile) way. The persistence of clustering for the perturbed models we examined suggests that it is likely to be encountered in various environmental conditions, as well as in other signal transduction pathways with network structures similar to that of the phosphotransferase system. © 2013 FEBS.

This paper discusses the mathematical analysis of a codimension two bifurcation determined by the coincidence of a subcritical Hopf bifurcation with a homoclinic orbit of the Hopf equilibrium. Our work is motivated by our previous analysis of a Hodgkin–Huxley neuron model which possesses a subcritical Hopf bifurcation (J. Guckenheimer, R. Harris-Warrick, J. Peck, A. Willms, J. Comput. Neurosci. 4 (1997) 257–277). In this model, the Hopf bifurcation has the additional feature that trajectories beginning near the unstable manifold of the equilibrium point return to pass through a small neighborhood of the equilibrium, that is, the Hopf bifurcation appears to be close to a homoclinic bifurcation as well. This model of the lateral pyloric (LP) cell of the lobster stomatogastric ganglion was analyzed for its ability to explain the phenomenon of spike-frequency adaptation, in which the time intervals between successive spikes grow longer until the cell eventually becomes quiescent. The presence of a subcritical Hopf bifurcation in this model was the one identified mechanism for oscillatory trajectories to increase their period and finally collapse to a non-oscillatory solution. The analysis presented here explains the apparent proximity of homoclinic and Hopf bifurcations. We also develop an asymptotic theory for the scaling properties of the interspike intervals in a singularly perturbed system undergoing subcritical Hopf bifurcation that may be close to a codimension two subcritical Hopf–homoclinic bifurcation.

We consider whole-cell voltage-clamp data of isolated currents characterized by the Hodgkin-Huxley paradigm. We examine the errors associated with the typical parameter estimation method for these data and show them to be unsatisfactorally large especially if the time constants of activation and inactivation are not sufficiently separated. The size of these errors is due to the fact that the steady-state and kinetic properties of the current are estimated disjointly. We present an improved parameter estimation method that utilizes all of the information in the voltage-clamp conductance data to estimate steady-state and kinetic properties simultaneously and illustrate its success compared to the standard method using simulated data and data from P. interruptus shal channels expressed in oocytes.

DifferentShakerfamily α-subunit genes generate distinct voltage-dependent K+currents when expressed in heterologous expression systems. Thus it generally is believed that diverse neuronal K+current phenotypes arise, in part, from differences inShakerfamily gene expression among neurons. It is difficult to evaluate the extent to which differentialShakerfamily gene expression contributes to endogenous K+current diversity, because the specificShakerfamily gene or genes responsible for a given K+current are still unknown for nearly all adult neurons. In this paper we explore the role of differentialShakerfamily gene expression in creating transient K+current (IA) diversity in the 14-neuron pyloric network of the spiny lobster,Panulirus interruptus. We used two-electrode voltage clamp to characterize the somaticIAin each of the six different cell types of the pyloric network. The size, voltage-dependent properties, and kinetic properties of the somaticIAvary significantly among pyloric neurons such that the somaticIAis unique in each pyloric cell type. Comparing these currents with theIAs obtained from oocytes injected withPanulirus shakerandshalcRNA (lobsterIshakerand lobsterIshal, respectively) reveals that the pyloric cellIAs more closely resemble lobsterIshalthan lobsterIshaker. Using a novel, quantitative single-cell-reverse transcription-PCR method to count the number ofshaltranscripts in individual identified pyloric neurons, we found that the size of the somaticIAvaries linearly with the number of endogenousshaltranscripts. These data suggest that theshalgene contributes substantially to the peak somaticIAin all neurons of the pyloric network.

Many neural systems display adaptive properties that occur on time scales that are slower than the time scales associated with repetitive firing of action potentials or bursting oscillations. Spike frequency adaptation is the name given to processes that reduce the frequency of rhythmic tonic firing of action potentials, sometimes leading to the termination of spiking and the cell becoming quiescent. This article examines these processes mathematically, within the context of singularly perturbed dynamical systems. We place emphasis on the lengths of successive interspike intervals during adaptation. Two different bifurcation mechanisms in singularly perturbed systems that correspond to the termination of firing are distinguished by the rate at which interspike intervals slow near the termination of firing. We compare theoretical predictions to measurement of spike frequency adaptation in a model of the LP cell of the lobster stomatogastric ganglion.

Exact solutions are derived for a nonlinear heat equation where the conductivity is a linear fractional function of (i) the temperature gradient or (ii) the product of the radial distance and the radial component of the temperature gradient for problems expressed in cylindrical coordinates. It is shown that equations of this form satisfy the same maximum principle as the linear heat equation, and a uniqueness theorem for an associated boundary value problem is given. The exact solutions are additively separable, isolating the nonlinear component from the remaining independent variables. The asymptotic behaviour of these solutions is studied, and a boundary value problem that is satisfied by these solutions is presented.

This paper analyzes the effectiveness of various monotonic discretizations of an ODE in a parameter range reduction algorithm by considering the resulting ranges for the parameter λ obtained from the simple model x′ = λx.

An algorithm that constructs a continuous piecewise linear representation, subject to a certain slope/segment length constraint, of a given set of data is described. The algorithm yields an optimal or near optimal representation, subject to this constraint, in the $L_{\infty}$ norm and does so in at worst $O(nK)$ time, where $n$ is the number of data points and $K$ the number of segments. The constraint is determined by a user-specified parameter, $t_{\min}$, which dictates a lower bound for the distance between opposite-sign slope discontinuities. For reasonable $t_{\min}$ values, the resulting representation of the data captures the signal and both smooths the noise and provides a measure of it. This representation is useful for applications that require a specified interval for the data values and also allows easy and continuous interpolation of ranges between recorded time points. The algorithm is described, some of its properties are proven, and its capabilities demonstrated with several examples. Comparisons are made with alternative techniques. Natural Sciences and Engineering Research Council of Canada, Alexander Graham Bell Canada Graduate Scholarship