Imagine the «§ Mathematical Biology University of Utah Introduction to Physiology V - Coupling and Propagation J. P. Keener Mathematics Department University of Utah Coupling and Propagation - p. 1/33 Spatially Extended Excitable Media MOTOR NEURON MITRAL CELL FROM PYRAMIDAL CELL PURKINJE CELL Neurons and axons Coupling and Propagation - p.2/33 Spatially Extended Excitable Media Mechanically stimulated Calcium waves Coupling and Propagation - p.3/33 Mathematical Biology University of Utah Conduction system of the heart % li i striae excitation atrial systole I atria; diastole ( fw irk Liar ex citation ventricular systole ventricular diastole Coupling and Propagation - p.4/33 Possibilities Mathematical Biology University of Utah Conduction system of the heart % li i striae excitation atrial systole I atria; diastole ( fw irk Liar ex citation ventricular systole ventricular diastole Electrical signal originates in the SA node. Coupling and Propagation - p.4/33 Mathematical Biology University of Utah Conduction system of the heart É Ii i striae excitation atrial systole I atria; diastole ( fw irk Liar ex citation ventricular systole ventricular diastole Electrical signal originates in the SA node. The signal propagates across the atria (2D sheet), through the AV node, along Purkinje fibers (1D cables), and throughout the ventricles (3D tissue). Coupling and Propagation - p.4/33 Spatially Extended Excitable Media i i i CL Ca Na Extracellular =B=t Intracellular Space A K Na * = ^ - \ I ■> \ / \ \ ■ \ _1_^_1_ 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 Who could have guessed? - p.8/33 Mathematical Biology University of Utah Coupled Cells Normal cell and cell with slightly elevated potassium - coupled normal cell "ischemic" cell 1 0.8 0.6 0.4 0.2 0 0 500 1000 1500 500 1000 1500 1 0.8 0.6 0.4 0.2 0 ■ \ . Í .\ \ 1 J / 1 >\ I ■> / \ \ \ \ _1_^_1_ 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Who could have guessed? - p.8/33 aagine the Possibilities Mathematical Biology University of Utah Coupled Cells Normal cell and cell with moderately elevated potassium -uncoupled i 0.8 0.6 0.4 0.2 0 normal cell "ischemic" cell 0 500 1000 1500 500 1000 1500 _ 0 0.2 0.4 0.6 0.8 1 02 04 06 08 Who could have guessed? - p.8/33 University of Utah Normal cell and cell with moderately elevated potassium coupled normal cell 0 500 1000 1500 1000 1500 0 0.2 0.4 0.6 0 1 0.8 0.6 0.4 0.2 0 ■ \ \ \ \ \ \ /"l----- / 1 1 \ 1 ■ INc1— __. _1__1_ 0 0.2 0.4 0.6 0.8 wfio could have guessed? - p.8/33 Mathematical Biology University of Utah Coupled Cells Normal cell and cell with greatly elevated potassium - uncoupled normal cell "ischemic" cell 500 1000 1500 0 500 1000 1500 1 0.8 0.6 0.4 0.2 0 ■ . Í .\ \ 1 J / 1 >\ I ■> / \ \ \ \ _1_^_1_ 0 0.2 0.4 0.6 0.8 1 1 0.8 0.6 0.4 0.2 0 ■ \ \ \ \ \ \ \ \ \ ■ / \ Y \ _i_^_i_ 0 0.2 0.4 0.6 0.8 1 Who could have guessed? - p.8/33 Imagine the Possibilities Mathematical Biology University of Utah Coupled Cells Normal cell and cell with greatly elevated potassium - coupled normal cell "ischemic" cell 500 1000 1500 500 1000 1500 0 0.2 0.4 0.6 0 0.2 0.4 0.6 0.8 1 Who could have guessed? - p.8/33 Mathematical Biology University of Utah Axons and Fibers Ve(x) redx Ve(x+dx) Extracellular space ltdx\ Cmdx -vw- lion dx Cmdx Itdx V,(x) r(- dx Cell membrane V, (x+dx) Intracellular space From Ohm's law Vi(x+dx)—Vi(x) — —Ii(x)ridx1 Ve(x+dx)—Ve(x) = —Ie(x)redx, In the limit as dx —► 0. L =-- 1 dV, Ti dx L —-- 1 dVP re dx Coupling and Propagation - p.9/33 Imagine the Possibilities Mathematical Biology University of Utah The Cable Equation Ve(x) redx Ve(x+dx) Extracellular space ltdx\ Cmdx -vw- lion dx Cmdx Itdx V,(x) r(- dx Cell membrane Vi (x+dx) Intracellular space From Kirchhoff's laws Ii{x) — Ii(x + dx) = Itdx = Ie(x + dx) — Ie(x) In the limit as dx —> 0, this becomes dh dl L = -—i = dx dx Coupling and Propagation - p.10/33 Mathematical Biology University of Utah The Cable Equation Combining these L = d í 1 dV dx V r i + re dx and, thus, dV n i j. — j — ^m, r~\ . \ -Lion, ±t d ( 1 dV dt dx \ r i + r e dx This equation is referred to as the cable equation. Coupling and Propagation - p.11/33 Mathematical Biology University of Utah Modelling Cardiac Tissue Cardiac Tissue - The Bidomain Model: At each point of the cardiac domain there are two comingled regions, the extracellular and the intracellular domains with potentials 0e and and transmembrane potential 4> = 4>i- cj)e. Coupling and Propagation - p.12/33 Mathematical Biology University of Utah Modelling Cardiac Tissue Cardiac Tissue - The Bidomain Model: At each point of the cardiac domain there are two comingled regions, the extracellular and the intracellular domains with potentials 0e and and transmembrane potential 4> = 4>i- cj)e. These potentials drive currents, ie = -creV0e, k = -cr^V<^, where ae and Oi are conductivity tensors. ■ Coupling and Propagation - p.12/33 Mathematical Biology University of Utah Modelling Cardiac Tissue Cardiac Tissue - The Bidomain Model: At each point of the cardiac domain there are two comingled regions, the extracellular and the intracellular domains with potentials 0e and and transmembrane potential 4> = 4>i- cj)e. These potentials drive currents, ie = -creV0e, k = -cr^V<^, where ae and Oi are conductivity tensors. Total current is ÍT = Íe + H = -creV0e - i- Coupling and Propagation - p.12/33 Mathematical Biology University of Utah Kirchhoff's laws: Total current is conserved: V • (cxíV^í + aeV #* « 0.25 (i.e. if /i is too c large, 0 is too large, or C is too small.) Coupling and Propagation - p.23/33 Mathematical Biology University of Utah With Recovery Including recovery variables dt Solitary Pulse Periodic Waves Skipped Beats dv ^d2v = D^+f(v,w) dx' dw ~dt Coupling and Propagation - p.24/33 University of Utah Coupling and Propagation - p.25/33 The APD Instability in 1D Coupling and Propagation - p.26/33 The APD Instability in 1D University of Utah Coupling and Propagation - p.26/33 Mathematical Biology University of Utah Dimension 2: Spirals 1 Atrial Flutter Coupling and Propagation - p.27/33 (íagine the Possibilities Mathematical Biology University of Utah Dimension 2: Spirals 4. Atrial Flutter Spiral instability - Meander: Torsade de Pointe Coupling and Propagation - p.27/33 Mathematical Biology University of Utah Dimension 2: Spirals Atrial Flutter Spiral instability - Meander: Torsahd duh Pwahn ■ Coupling and Propagation - p.27/33 The APD Instability in 2D Spiral Breakup Coupling and Propagation - p.28/33 Dimension 3: Ventricular Reentrant Activity Coupling and Propagation - p.29/33 Dimension 3: Cardiac Scroll Wave 3 D structure of a single scroll wave Coupling and Propagation - p.30/33 Mathematical Biology University of Utah Ventricular Fibrillation Ventricular Fibrillation L Surface View Movie 3D View Movie Still unresolved: What is the mechanism for maintenance of fibril- lation? (APD instability? Mother rotor hypothesis?) Coupling and Propagation - p.31/33 Mathematical Biology University of Utah Hypothesized Mechanisms for initiation of Reentrant Activity How is a dynamical system moved from one state (the normal heartbeat) to another (reentry)? Remark: This is a spatio-temporal system; Single cell explanations are not sufficient. Coupling and Propagation - p.32/33 Mathematical Biology University of Utah Hypothesized Mechanisms for initiation of Reentrant Activity How is a dynamical system moved from one state (the normal heartbeat) to another (reentry)? Remark: This is a spatio-temporal system; Single cell explanations are not sufficient. Anatomical - One way block on a closed 1D loop, (movie) one-way block Coupling and Propagation - p.32/33 Mathematical Biology University of Utah Hypothesized Mechanisms for initiation of Reentrant Activity How is a dynamical system moved from one state (the normal heartbeat) to another (reentry)? Remark: This is a spatio-temporal system; Single cell explanations are not sufficient. • Anatomical - One way block on a closed 1D loop, (movie) one-way block Vulnerable Period - Winfree (S1-S2) mechanism (id) (20) Coupling and Propagation - p.32/33 Mathematical Biology University of Utah Hypothesized Mechanisms for initiation of Reentrant Activity How is a dynamical system moved from one state (the normal heartbeat) to another (reentry)? Remark: This is a spatio-temporal system; Single cell explanations are not sufficient. • Anatomical - One way block on a closed 1D loop, (movie) one-way block Vulnerable Period - Winfree (S1-S2) mechanism (id) (20) Early After Depolarizations during Vulnerable Period. Coupling and Propagation - p.32/33 Mathematical Biology University of Utah Hypothesized Mechanisms for Initiation of Reentrant Activity How is a dynamical system moved from one state (the normal heartbeat) to another (reentry)? Remark: This is a spatio-temporal system; Single cell explanations are not sufficient. • Anatomical - One way block on a closed 1D loop, (movie) one-way black Vulnerable Period - Winfree (S1-S2) mechanism (id) (20) Early After Depolarizations during Vulnerable Period. Dispersion (i.e. spatial/temporal inhomogeneity) of refractoriness. Coupling and Propagation - p.32/33 Mathematical Biology University of Utah More Unresolved Issues Why is calcium overload arrhythmogenic? Coupling and Propagation - p.33/33 Mathematical Biology University of Utah More Unresolved Issues Why is calcium overload arrhythmogenic? Why is long QT syndrome arrhythmogenic? Coupling and Propagation - p.33/33 Mathematical Biology University of Utah More Unresolved Issues Why is calcium overload arrhythmogenic? Why is long QT syndrome arrhythmogenic? Why are most anti-arrhythmic drugs actually proarrhythmic? Coupling and Propagation - p.33/33 Mathematical Biology University of Utah More Unresolved Issues Why is calcium overload arrhythmogenic? Why is long QT syndrome arrhythmogenic? Why are most anti-arrhythmic drugs actually proarrhythmic? What is the mechanism of EAD's and are they truly proarrhythmic? Coupling and Propagation - p.33/33