Ordinary and Particial Differential Equations: An Introduction to Dynamical Systems

Differential equations arise in a variety of contexts, some purely theoretical and some of practical interest. As you read this textbook, you will find that the qualitative and quantitative study of differential equations incorporates an elegant blend of linear algebra and advanced calculus. This book is intended for an advanced undergraduate course in differential equations. The reader should have already completed courses in linear algebra, multivariable calculus, and introductory differential equations.https://scholarship.richmond.edu/bookshelf/1162/thumbnail.jp

Analysis of a Modified Feedback Control Technique for Suppressing Electrical Alternans in Cardiac Tissue

Alternans is an abnormal cardiac rhythm in which action potential duration alternates from beat-to-beat. In order for an implanted pacemaker to successfully seize control of the heart rhythm, its electrical stimuli have to be carefully timed relative to the firing of the heart’s specialized pacemaker cells. In this manuscript, we use mathematical techniques to analyze a novel feedback control algorithm for suppressing alternans. We model the cardiac rhythm and the effect of the controller using a system of two nonlinear difference equations. Our analysis reveals that it is often advantageous not to allow the pacemaker to intervene in every beat when attempting to control alternans

Modeling Action Potential Reversals in Tunicate Hearts

Tunicates are small invertebrates which possess a unique ability to reverse flow in their hearts. Scientists have debated various theories regarding how and why flow reversals occur. Here we explore the electrophysiological basis for reversals by simulating action potential propagation in an idealized model of the tubelike tunicate heart. Using asymptotic formulas for action potential duration and conduction velocity, we propose tunicate-specific parameters for a two-current ionic model of the action potential. Then, using a kinematic model, we derive analytical criteria for reversals to occur. These criteria inform subsequent numerical simulations of action potential propagation in a fiber paced at both ends. In particular, we explore the role that variability of pacemaker firing rates plays in generating reversals, and we identify various favorable conditions for triggering retrograde propagation. Our analytical framework extends to other species; for instance, it can be used to model competition between the sinoatrial node and abnormal ectopic foci in human heart tissue

Notes

Notes by W. D. Rollison, E. L. Hessmer, John M. Ruberto, William M. Cain, and John V. Leddy

Cellular automata simulation of topological effects on the dynamics of feed-forward motifs

<p>Abstract</p> <p>Background</p> <p>Feed-forward motifs are important functional modules in biological and other complex networks. The functionality of feed-forward motifs and other network motifs is largely dictated by the connectivity of the individual network components. While studies on the dynamics of motifs and networks are usually devoted to the temporal or spatial description of processes, this study focuses on the relationship between the specific architecture and the overall rate of the processes of the feed-forward family of motifs, including double and triple feed-forward loops. The search for the most efficient network architecture could be of particular interest for regulatory or signaling pathways in biology, as well as in computational and communication systems.</p> <p>Results</p> <p>Feed-forward motif dynamics were studied using cellular automata and compared with differential equation modeling. The number of cellular automata iterations needed for a 100% conversion of a substrate into a target product was used as an inverse measure of the transformation rate. Several basic topological patterns were identified that order the specific feed-forward constructions according to the rate of dynamics they enable. At the same number of network nodes and constant other parameters, the bi-parallel and tri-parallel motifs provide higher network efficacy than single feed-forward motifs. Additionally, a topological property of isodynamicity was identified for feed-forward motifs where different network architectures resulted in the same overall rate of the target production.</p> <p>Conclusion</p> <p>It was shown for classes of structural motifs with feed-forward architecture that network topology affects the overall rate of a process in a quantitatively predictable manner. These fundamental results can be used as a basis for simulating larger networks as combinations of smaller network modules with implications on studying synthetic gene circuits, small regulatory systems, and eventually dynamic whole-cell models.</p

Rate-dependent propagation of cardiac action potentials in a one-dimensional fiber

Action potential duration (APD) restitution, which relates APD to the preceding diastolic interval (DI), is a useful tool for predicting the onset of abnormal cardiac rhythms. However, it is known that different pacing protocols lead to different APD restitution curves (RCs). This phenomenon, known as APD rate-dependence, is a consequence of memory in the tissue. In addition to APD restitution, conduction velocity restitution also plays an important role in the spatiotemporal dynamics of cardiac tissue. We present new results concerning rate-dependent restitution in the velocity of propagating action potentials in a one-dimensional fiber. Our numerical simulations show that, independent of the amount of memory in the tissue, waveback velocity exhibits pronounced rate-dependence and the wavefront velocity does not. Moreover, the discrepancy between waveback velocity RCs is most significant for small DI. We provide an analytical explanation of these results, using a system of coupled maps to relate the wavefront and waveback velocities. Our calculations show that waveback velocity rate-dependence is due to APD restitution, not memory.Comment: 17 pages, 7 figure

The role of cation-dependent chloride transporters in neuropathic pain following spinal cord injury

<p>Abstract</p> <p>Background</p> <p>Altered Cl^-homeostasis and GABAergic function are associated with nociceptive input hypersensitivity. This study investigated the role of two major intracellular Cl^-regulatory proteins, Na⁺-K⁺-Cl^-cotransporter 1 (NKCC1) and K⁺-Cl^-cotransporter 2 (KCC2), in neuropathic pain following spinal cord injury (SCI).</p> <p>Results</p> <p>Sprague-Dawley rats underwent a contusive SCI at T9 using the MASCIS impactor. The rats developed hyperalgesia between days 21 and 42 post-SCI. Thermal hyperalgesia (TH) was determined by a decrease in hindpaw thermal withdrawal latency time (WLT) between days 21 and 42 post-SCI. Rats with TH were then treated with either vehicle (saline containing 0.25% NaOH) or NKCC1 inhibitor bumetanide (BU, 30 mg/kg, i.p.) in vehicle. TH was then re-measured at 1 h post-injection. Administration of BU significantly increased the mean WLT in rats (p < 0.05). The group administered with the vehicle alone showed no anti-hyperalgesic effects. Moreover, an increase in NKCC1 protein expression occurred in the lesion epicenter of the spinal cord during day 2–14 post-SCI and peaked on day 14 post-SCI (p < 0.05). Concurrently, a down-regulation of KCC2 protein was detected during day 2–14 post-SCI. The rats with TH exhibited a sustained loss of KCC2 protein during post-SCI days 21–42. No significant changes of these proteins were detected in the rostral region of the spinal cord.</p> <p>Conclusion</p> <p>Taken together, expression of NKCC1 and KCC2 proteins was differentially altered following SCI. The anti-hyperalgesic effect of NKCC1 inhibition suggests that normal or elevated NKCC1 function and loss of KCC2 function play a role in the development and maintenance of SCI-induced neuropathic pain.</p