Dynamical Systems for Biology - Excitability

Dynamical Systems for Biology - Excitability J. P. Keener Mathematics Department Dynamical Systems for Biology p.1/25

Examples of Excitable Media B-Z reagent CICR (Calcium Induced Calcium Release) Nerve cells cardiac cells, muscle cells Slime mold (dictystelium discoideum) Forest Fires Features of Excitability Threshold Behavior Refractoriness Recovery Resting Excited Refractory Recovering What about flush toilets? p.2/25

Intracellular Calcium What about flush toilets? p.3/25

Calcium Handling J L JNCX Extracellular Space v dc dt = J RY R J SERCA + J L J NCX Cytosol (Intracellular Space) J RYR JSERCA Sarcoplasmic Reticulum (Calcium stores) Dynamical Systems for Biology p.4/25

Calcium Handling J L JNCX Extracellular Space v dc dt = J RY R J SERCA + J L J NCX Cytosol (Intracellular Space) Sarcoplasmic Reticulum J RYR JSERCA (Calcium stores) with J RY R Ryanodine Receptor - calcium regulated calcium channel, J SERCA Sarco- and Endoplasmic Reticulum Calcium ATPase, Dynamical Systems for Biology p.4/25

the Imagine Ryanodine Receptors Ca ++ Ca ++ ++ Ca ++ Ca Dynamical Systems for Biology p.5/25

Ryanodine Receptors Flux through ryanodine receptor is diffusive, J RY R = g max P o (c c sr ) where P o = S10 3 is the open probability. To determine P o, we must find S 10 : and so on. ds 10 dt = k 1 cs 00 + k 2 S 11 k 1 S 10 k 2 cs 10 Ca ++ S 10 00 "fast" S ++ Ca "slow" "slow" Ca ++ S 01 Ca ++ "fast" S 11 Dynamical Systems for Biology p.6/25

Observation If the binding sites are independent (no cooperativity), then S 10 = mh, S 00 = (1 m)h, S 11 = m(1 h), S 01 = (1 m)(1 h), where dm dt = φ m (c)(1 m) ψ m (c)m, dh dt = φ h(c)(1 h) ψ(c)h. Furthermore, m is a fast variable, so can be taken to be in qss, m = m (c). Consequently,... Dynamical Systems for Biology p.7/25

Calcium Dynamics dc dt = (g maxp o + J er )(c e c) J SERCA, dh dt = φ h(c)(1 h) ψ h (c)h, 1 0.8 0.6 where J SERCA = V max c 2 K 2 s +c 2, P o = h 3 f(c) h 0.4 0.2 0 0 5 10 15 20 25 30 35 40 45 50 1 0.8 0.6 0.4 time (s) 0.2 0 10 2 10 1 10 0 c (µ M) Dynamical Systems for Biology p.8/25

Membrane Electrical Activity Dynamical Systems for Biology p.9/25

Membrane Electrical Activity Transmembrane potential φ is regulated by transmembrane ionic currents and capacitive currents: dφ C m dt + I ion(φ, w) = I in where dw dt = g(φ, w), w (Reminder: This is a conservation equation.) Rn Dynamical Systems for Biology p.9/25

Examples Examples include: Neuron - Hodgkin-Huxley model Purkinje fiber - Noble Cardiac cells - Beeler-Reuter, Luo-Rudy, Winslow-Jafri, Bers Two Variable Models - reduced HH, FitzHugh-Nagumo, Mitchell-Schaeffer, Morris-Lecar, McKean, etc.) Dynamical Systems for Biology p.10/25

The Hodgkin-Huxley Equations I K I l I Na Extracellular Space φ e C m I ion φ = φ φ i e Intracellular Space C m dv dt + I Na + I K + I l = 0, φ i Dynamical Systems for Biology p.11/25

The Hodgkin-Huxley Equations I K I l I Na Extracellular Space φ e C m I ion φ = φ φ i e with sodium current I Na, Intracellular Space C m dv dt + I Na + I K + I l = 0, φ i Dynamical Systems for Biology p.11/25

The Hodgkin-Huxley Equations I K I l I Na Extracellular Space φ e C m I ion φ = φ φ i e Intracellular Space C m dv dt + I Na + I K + I l = 0, with sodium current I Na, potassium current I K, φ i Dynamical Systems for Biology p.11/25

The Hodgkin-Huxley Equations I K I l I Na Extracellular Space φ e C m I ion φ = φ φ i e Intracellular Space C m dv dt + I Na + I K + I l = 0, with sodium current I Na, potassium current I K, and leak current I l. φ i Dynamical Systems for Biology p.11/25

Ionic Currents Ionic currents are typically of the form I = g(φ, t) Φ(φ) Dynamical Systems for Biology p.12/25

Ionic Currents Ionic currents are typically of the form I = g(φ, t) Φ(φ) where g(φ, t) is the total number of open channels, Dynamical Systems for Biology p.12/25

Ionic Currents Ionic currents are typically of the form I = g(φ, t) Φ(φ) where g(φ, t) is the total number of open channels, and Φ(φ) is the I-φ relationship for a single channel. Dynamical Systems for Biology p.12/25

Currents Hodgkin and Huxley found that I k = g k n 4 (φ φ K ), I Na = g Na m 3 h(φ φ Na ), where τ u (φ) du dt = u (φ) u, u = m, n, h 1.0 10 0.8 h (v) n (v) 8 τ h (v) 0.6 6 0.4 m (v) ms 4 τ n (v) 0.2 0.0 2 0 τ m (v) 0 20 40 60 Potential (mv) 80 100 0 40 Potential (mv) 80 Dynamical Systems for Biology p.13/25

Action Potential Dynamics Potential (mv) 120 100 80 60 40 20 0 Gating variables 1.0 0.8 0.6 0.4 0.2 m(t) h(t) n(t) -20 0.0 0 5 10 time (ms) 15 20 0 5 10 time (ms) 15 20 35 0.8 Conductance (mmho/cm 2 ) 30 25 20 15 10 5 0 g Na g K h 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 5 10 time (ms) 15 20 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 n Dynamical Systems for Biology p.14/25

Two Variable Reduction of HH Eqns Set m = m (φ), and set h + n N = 0.85. This reduces to a two variable system C dφ dt τ n (φ) dn dt = ḡ K n 4 (φ φ K ) + ḡ Na m 3 (φ)(n n)(φ φ Na ) + ḡ l (φ φ L ), = n (φ) n. 1.0 0.8 dn/dt = 0 0.6 n 0.4 dv/dt = 0 0.2 0.0 0 40 v 80 120 Dynamical Systems for Biology p.15/25

Two Variable Models Following is a brief summary of two variable models of excitable media. The models described here are all of the form dv dt dw dt = f(v, w) + I = g(v, w) Typically, v is a fast variable, while w is a slow variable. w g(v,w) = 0 f(u,w) = 0 W* V - (w) V 0 (w) V + (w) W * v Dynamical Systems for Biology p.16/25

Cubic FitzHugh-Nagumo The model that started the whole business uses a cubic polynomial (a variant of the van der Pol equation). F (v, w) = Av(v α)(1 v) w, G(v, w) = ɛ(v γw). with 0 < α < 1 2, and ɛ small. (Remark: This model is used these days only by mathematicians.) Dynamical Systems for Biology p.17/25

FitzHugh-Nagumo Equations 0.20 dw/dt = 0 1.0 w 0.15 0.10 0.05 0.00-0.05-0.4 0.0 dv/dt = 0 0.4 0.8 v 1.2 0.8 0.6 0.4 0.2 0.0-0.2 0.0 v(t) 0.2 w(t) 0.4 0.6 time 0.8 1.0 0.20 0.15 dw/dt = 0 0.8 v(t) 0.10 dv/dt = 0 0.4 w 0.05 0.0 w(t) 0.00-0.05-0.4 0.0 0.4 v 0.8-0.4 0.0 0.5 1.0 1.5 time 2.0 2.5 3.0 Dynamical Systems for Biology p.18/25

Mitchell-Schaeffer-Karma where m(v) = dv C m dt dh τ h dt 0, v < 0 v, 0 < v < 1 1, v > 1 = g Na hm 2 (V Na v) + g K (V K v), = h (v) h, h = 1 f(v), τ h = τ open + (τ close τ open )f(v) f(v) = 1 2 (1 + tanh(κ(v v gate)), Dynamical Systems for Biology p.19/25

MSK Phase Portrait 1 0.8 φ, h 0.6 0.4 0.2 0 0 100 200 300 400 500 600 700 800 900 1000 time 1 0.8 h 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 φ Dynamical Systems for Biology p.20/25

Morris-Lecar This model was devised for barnacle muscle fiber. F (v, w) = g ca m (v)(v v ca ) g k w(v v k ) g l (v v l ) + I app G(v, w) = φ cosh( 1 2 v v 3 v 4 )(w (v) w), m (v) = 1 2 + 1 2 tanh( v v 1 v 2 ), w (v) = (1 + tanh( v v 3) 2v 4 )). 40 20 0 V 20 40 60 0 50 100 150 200 250 300 350 400 450 500 time 1 0.8 0.6 W 0.4 0.2 0 0.2 60 40 20 0 20 40 60 V Dynamical Systems for Biology p.21/25

McKean McKean suggested two piecewise linear models with F (v, w) = f(v) w and G(v, w) = ɛ(v γw). For the first, f(v) = 8 >< >: v v < α 2 α v α 2 < v < 1+α 2 1 v v > 1+α 2 f(v) 0.6 0.5 0.4 0.3 0.2 0.1 0 0.1 0.2 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 v where 0 < α < 1 2. The second model suggested by McKean had f(v) = 8 < : v v < α 1 v v > α f(v) 1 0.8 0.6 0.4 0.2 0 (-8) 0.2 0.4 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 v and γ = 0. Dynamical Systems for Biology p.22/25

Features of Excitable Systems Threshold Behavior, Refractoriness Alternans Wenckebach Patterns Dynamical Systems for Biology p.23/25

Summary From last time: Changing quantities are tracked by following the production/destruction rates and their influx/efflux rates; In case you forgot. p.24/25

Summary From last time: Changing quantities are tracked by following the production/destruction rates and their influx/efflux rates; Because reactions can occur on many different time scales, quasi-steady state approximations are often quite useful; In case you forgot. p.24/25

Summary For excitable systems: Changing quantities are tracked by following their influx/efflux rates; Pretty much the same as before! p.25/25

Summary For excitable systems: Changing quantities are tracked by following their influx/efflux rates; Because channel dynamics can occur on many different time scales, quasi-steady state approximations are often quite useful; Pretty much the same as before! p.25/25