Stephen Wirkus

The Dynamics of Two Coupled van der Pol Oscillators with Delay Coupling

We investigate the dynamics of a system of twovan der Pol oscillators with delayed velocity coupling.We use the method of averaging to reduce the problem to the studyof a slow-flow in three dimensions.We study the steady state solutions of this slow-flow, with specialattention given to the bifurcations accompanying their change innumber and stability. We compare these stability results with numericalintegration of the original equations and show that the two sets of resultsare in excellent agreement under certain parameter restrictions.Our interest in this system is due to its relevance to coupled laseroscillators.

How to Pump a Swing

Stephen Wirkus (swirkus@cam.cornell.edu) ran on the track and cross-country teams for the University of Missouri at Kansas City, while earning B.S. degrees in mathematics and physics. He is now working toward his Ph.D. in applied mathematics at Cornell University, focusing on oscillators with time delay. With Richard Rand he also explores appli? cations of nonlinear differential equations. An avid runner, Stephen likes the outdoors and most sporting activities.

A mathematical model for photoreceptor interactions.

The interactions between rods and cones in the retina have been the focus of innumerable experimental and theoretical biological studies in previous decades yet the understanding of these interactions is still incomplete primarily due to the lack of a unified concept of cone photoreceptor organization and its role in retinal diseases. The low abundance of cones in many of the non-primate mammalian models that have been studied make conclusions about the human retina difficult. A more complete knowledge of the human retina is crucial for counteracting the events that lead to certain degenerative diseases, in particular those associated with photoreceptor cell death (e.g., retinitis pigmentosa). In an attempt to gain important insight into the role and interactions of the rods and the cones we develop and analyze a set of mathematical equations that model a system of photoreceptors and incorporate a direct rod-cone interaction. Our results show that the system can exhibit stable oscillations, which correspond to the rhythmic renewal and shedding of the photoreceptors. In addition, our results show the mathematical necessity of this rod-cone direct interaction for survival of both and gives insight into this mechanism.

Optimal Control in the Treatment of Retinitis Pigmentosa

Numerous therapies have been implemented in an effort to minimize the debilitating effects of the degenerative eye disease Retinitis Pigmentosa (RP), yet none have provided satisfactory long-term solution. To date there is no treatment that can halt the degeneration of photoreceptors. The recent discovery of the RdCVF protein has provided researchers with a potential therapy that could slow the secondary wave of cone death. In this work, we build on an existing mathematical model of photoreceptor interactions in the presence of RP and incorporate various treatment regiments via RdCVF. Our results show that an optimal control exists for the administration of RdCVF. In addition, our numerical solutions show the experimentally observed rescue effect that the RdCVF has on the cones.

Quantifying the metabolic contribution to photoreceptor death in retinitis pigmentosa via a mathematical model

Retinitis pigmentosa (RP) is a family of inherited retinal degenerative diseases that leads to blindness. In many cases the disease-causing allele encodes for a gene exclusively expressed in the night active rod photoreceptors. However, because rod death always leads to cone death affected individuals eventually lose their sight. Many theories have been proposed to explain the secondary loss of cones in RP; however, most fail to fully explain the different pathological transition stages seen in humans. Incorporating experimental data of rod and cone death kinetics from two mouse models of RP, we use a mathematical model to investigate the interplay and role of energy consumption and uptake of the photoreceptors as well as nutrient availability supplied through the retinal pigment epithelium (RPE) throughout the progression of RP. Our data driven mathematical model predicts that the system requires a total reduction of approximately 27-31% in nutrients available to result in the complete demise of all cones. Simulations utilizing retinal degeneration 1 (rd1) mouse cell count data in which cone death was delayed by altering cell metabolism in cones show that preventing a 1-2% decrease in nutrients available can permanently halt cone death even when 90% have already died. Our results also indicate that the ratio of energy consumption to uptake of cones, Dc, is mainly disrupted during the death wave of the rods with negligible changes thereafter and that the subsequent nutrient decrease is mainly responsible for the demise of the cones. The change in this ratio Dc highlights the compensation that the cones must undergo during rod death to meet the high metabolic demands of the entire photoreceptor population. Global sensitivity analysis confirms the results and suggests areas of focus for halting RP, even at later stages of the disease, through feasible therapeutic interventions.

Mathematical Model of the Role of RdCVF in the Coexistence of Rods and Cones in a Healthy Eye.

Understanding the essential components and processes for coexistence of rods and cones is at the forefront of retinal research. The recent discovery on RdCVF’s mechanism and mode of action for enhancing cone survival brings us a step closer to unraveling key questions of coexistence and codependence of these neurons. In this work, we build from ecological and enzyme kinetic work on functional response kinetics and present a mathematical model that allows us to investigate the role of RdCVF and its contribution to glucose intake. Our model results and analysis predict a dual role of RdCVF for enhancing and repressing the healthy coexistence of the rods and cones. Our results show that maintaining RdCVF above a threshold value allows for coexistence. However, a significant increase above this value threatens the existence of rods as the cones become extremely efficient at uptaking glucose and begin to take most of it for themselves. We investigate the role of natural glucose intake and that due to RdCVF in both high and low nutrient levels. Our analysis reveals that under low nutrient levels coexistence is not possible regardless of the amount of RdCVF present. With high nutrient levels coexistence can be achieved with a relative small increase in glucose uptake. By understanding the contributions of rods to cones survival via RdCVF in a non-diseased retina, we hope to shed light on degenerative diseases such as retinitis pigmentosa.

The mathematical and theoretical biology institute--a model of mentorship through research.

This article details the history, logistical operations, and design philosophy of the Mathematical and Theoretical Biology Institute (MTBI), a nationally recognized research program with an 18-year history of mentoring researchers at every level from high school through university faculty, increasing the number of researchers from historically underrepresented minorities, and motivating them to pursue research careers by allowing them to work on problems of interest to them and supporting them in this endeavor. This mosaic profile highlights how MTBI provides a replicable multi-level model for research mentorship.

Mathematical modeling of fungal infection in immune compromised individuals: Implications for drug treatment

We present a mathematical model that describes treatment of a fungal infection in an immune compromised patient in which both susceptible and resistant strains are present. The resulting nonlinear differential equations model the biological outcome, in terms of strain growth and cell number, when an individual, who has both a susceptible and a resistant population of fungus, is treated with a fungicidal or fungistatic drug. The model demonstrates that when the drug is only successful at treating the susceptible strain, low levels of the drug cause both strains to be in stable co-existence and high levels eradicate the susceptible strain while allowing the resistant strain to persist or to multiply unchecked. A modified model is then described in which the drug is changed to one in which both strains are susceptible, and subsequently, at the appropriate level of treatment, complete eradication of both fungal strains ensues. We discuss the model and implications for treatment options within the context of an immune compromised patient.

Mathematical modeling of fungal infection in immune compromised individuals: The effect of back mutation on drug treatment

We present a mathematical model that describes treatment of a fungal infection in an immune compromised patient in which both susceptible and resistant strains are present with a mutation allowing the susceptible strain to become resistant as well as a back mutation allowing resistant fungus to again become susceptible. The resulting nonlinear differential equations model the biological outcome, in terms of strain growth and cell number, when an individual is treated with a fungicidal or fungistatic drug. The model demonstrates that under any levels of the drug both strains will be in stable co-existence and high levels of treatment will never completely eradicate the susceptible strain. A modified model is then described in which the drug is changed to one in which both strains are susceptible, and subsequently, at the appropriate level of treatment, complete eradication of both fungal strains ensues. We discuss the model and implications for treatment options within the context of an immune compromised patient.