Dynamics from Seconds to Hours in Hodgkin Huxley Model

Dynamics from Seconds to Hours in Hodgkin Huxley Model with Time Dependent Ion Concentrations and Buffer Reservoirs Niklas Hübel Technische Universität Berlin July 9, 214

OUTLINE Closed Models model review dynamics bistability bifurcation analysis insufficiency of ion pumps Open Models with External Reservoirs dynamics CSD time scales slow fast analysis Oscillatory Dynamics seizure like activity (SLA) and SD bifurcation analysis assign specific bifurcations to SLA and SD

INTRODUCTION Trying to find and analyze the simplest possible model of local ion dynamics that...... can be biophysically interpreted.... shows spreading depression dynamics. What has been done? A lot! An incomplete list... Hodgkin Huxley cardiac models (DiFrancesco, Noble, 198s) cortical ion dynamics: Kager, Wadman, Somjen Barreto, Cressman Schiff, Ullah Bazhenov, Fröhlich Zandt What do we do? investigate entire repertoire of ion dynamics in simple model bifurcation analysis of ion dynamics slow fast interpretation of ion dynamics in SD phase space interpretation of ion dynamics

HODGKIN HUXLEY MODEL (HH) Developed for the description of action potentials. equivalent electrical circuit lipid bilayer is a capacitor channel is a conductor in series with a battery energy from batteries not dissipated

HODGKIN HUXLEY MODEL (HH) Four rate equations of HH Model parameters dv dt dx dt = 1 C m (I Na + I K + I Cl I app ) = x (V) x τ x (V) for x {n, m, h} capitance C m leak conductances g l ion max. gated conductances g g ion ion concentrations ion i/e Three ion currents I Na = (g l Na + gg Na m3 h)(v E Na ) I K = (g l K + gg K n4 )(V E K ) I Cl = g l Cl (V E Cl) Nernst potentials E ion = 26.6mV z ln(ion i /ion e ) for ion {Na +, K +, Cl }

HH APPROXIMATIONS V / mv a) membrane potential b) sodium inactivation c) sodium activation 4 1. 1. 2.8.8 dynamic 2.6.6 adiabatic 4.4.4 6 8.2.2 1.. 5 1 15 2 25 3..2.4.6.8 1. 5 1 15 2 25 3 t / msec. h t / msec. m t / msec. The action potential dynamics can be approximated by an adiabatic approximation for the sodium activation: m = m (V) assuming a functional dependence between sodium inactivation and potassium activation: h = f (n) Two dimensional HH model rate eqations: V = 1 C m ion ṅ = n n τ n I ion gating constraints: m = m (V) h = 1 1 + exp( 6.5(n.35))

ION BASED MODEL surroundings Na + K + I P Na + I Na Na + Na + ECS Na + ICS + K K K + K + The ion based model contains intracellular space (ICS) extracellular space (ECS) + I K K + Na + Note: The membrane separates ICS and ECS. Effects from surroundings are not included here closed system Ion dynamics The flux of ions across the membrane is induced by the transmembrane currents. The novel effects include: Nernst potentials are dynamic: E ion = 26.6mV z ( ) ln ioni ion e Ion pumps are needed to maintain the resting state. ( ( 25 Nai I p = ρ 1 + exp 3 ( 1 + exp (5.5 K e ) )) 1 ) 1

ION BASED MODEL Rate equations V = 1 C m (I Na + I K + I Cl I p ) ṅ = n n τ n Na i = γ ω i (I Na + 3I p ) K i = γ ω i (I K 2I p ) Cl i = + γ ω i I Cl Note: Na i + K i Cl i Cmγ ω i V = conservation law four dimensional dynamics Constraints Gating constraints: m = m (V) h = h sig (n) Mass conservation: Na e = Na e + ω i ω e (Na i Na i ) K e = K e + ω i ω e (K i K i ) Cl e = Cl e + ω i ω e (Cl i Cl i ) Parameters: volumes ω i/e conversion factor γ

DONNAN EQUILIBRIUM IN ION BASED MODEL The conservation law implies electroneutrality: = Na i + K i Cl i C mγ V ω i Q i = (Na i + K i Cl i ) = C mγ ω }{{ i } O(1 4 mm mv ) V ω i 2,16µm 3 ω e 72µm 3 F 96485C/mol A m 922µm 2 γ 9.556e 3 µm2 mol C The equilibrium without pumps... = dion i dt = ± γ ω i (g l ion +...)(V E ion) E Na = E K = E Cl E Na = E K = E Cl Q i }... is the Donnan equilibrium! Note: No impermeant anions included!

DONNAN EQUILIBRIUM IN ION BASED MODEL V/ mv 8 6 4 2 2 4 6 8 1 Potentials: pump switch-off V E K E N a 5 1 7 E C l 5 6 7 2 4 6 8 1 t/ sec ion/ mm 16 14 12 1 8 6 4 2 Concentrations: pump switch-off Na i K i Cl i Na e K e Cl e 14 9 4 2 4 6 8 1 t/ sec 5 6 7 The pump is switched off after 5sec. The transition from the physiological resting state to the Donnan equilibrium follows. ion fluxes until spiking begin spiking until depolarization block is reached final asymptotic phase until Donnan equilibrium is attained What if we turn the pumps on again?

V/ mv ion/ mm 8 6 4 2 2 4 6 8 1 16 14 12 1 8 6 4 2 Potentials: pump switch-off V E K E N a E C l 2 4 6 8 1 t/ sec Concentrations: pump switch-off Na i K i Cl i Na e K e Cl e 5 1 7 14 9 4 2 4 6 8 1 t/ sec 5 6 7 5 6 7 V/ mv ion/ mm Potentials: pump interruption 8 6 5 4 1 2 7 2 5 6 7 4 6 8 1 2 4 6 8 1 16 14 12 1 8 6 4 2 t/ sec Concentrations: pump interruption 14 9 4 2 4 6 8 1 t/ sec 5 6 7 Another stable state shows up!

FREE ENERGY STARVATION (FES) Symbol Physiological Donnan FES Units V 68 24.6 24.7 mv n.65.611.69 1 Na i 27 59.2 58.1 mm Na e 12 23.5 26.6 mm K i 131 116.9 117.9 mm K e 4 46.4 43.4 mm Cl i 9.7 27.7 27.7 mm Cl e 124 7. 7. mm E Na 39.7 24.6 2.8 mv E K 92.9 24.6 26.6 mv E Cl 68 24.6 24.7 mv Despite normal pump activity a stable state exists which...... has largely reduced ion gradients (dissipated energy).... is depolarized and cannot spike. We frame the term free energy starvation (FES) for this condition.

ion/ mm 16 14 12 1 8 6 4 2 Concentrations: pump interruption 14 9 4 2 4 6 8 1 t/ sec 5 6 7 V/ mv Potentials: pump interruption 8 6 5 4 1 2 7 2 5 6 7 4 6 8 1 2 4 6 8 1 t/ sec Phys. resting state Pumps compensate for leak currents. FES Pumps compensate for gated currents. They cannot re establish physiological conditions. Symbol phys. FES Units I l Na 1.89.7 µa/cm 2 I g Na.1 15.68 µa/cm 2 I l K 1.25.9 µa/cm 2 I g K.2 1.41 µa/cm 2 I p.63 5.25 µa/cm 2 Note: This only holds for the closed model.

MINIMAL PHYSIOLOGICAL AND RECOVERY PUMP RATE If we increase the pump rate ρ of ( ( )) 25 1 Nai I p = ρ 1 + exp 3 (1 + exp (5.5 K e )) 1 drastically (normally ρ = 5.25µA/cm 2 ), recovery from FES after pump interruption is possible. V / mv 4 2 2 4 6 8 HOM LP1 (4,) Bifurcations of Ion-Based Model 26 stable FP HB2 (,4) unstable FP 3 unstable LC (1,3) LP2 34 HOM HOM 33.6 34.1 34.6 HB3 (2,2) HB2 (3,1) (1,3) LP2 HB1 (4,) 1 1 2 3 4 5 ρ / (µa/cm 2 ) Two stable FP branches physiological (lower) FES (upper) Two critical pump rates minimal phys. pump rate:.89µa/cm 2 (LP1) recovery pump rate: 24.63µA/cm 2 (HB3)

SUMMARY surroundings Na + K + I P Na + I Na Na + Na + Na + + K + K Na + K + K + ECS ICS I K K + ion/ mm 16 14 12 1 8 6 4 2 Concentrations: pump interruption 14 9 4 2 4 6 8 1 t/ sec 5 6 7 V/ mv Potentials: pump interruption 8 6 5 4 1 2 7 2 5 6 7 4 6 8 1 2 4 6 8 1 t/ sec The closed neuron system can be driven into FES by pump interruption and long/strong stimulation with applied currents (not shown). The transition is permanent. The ion pumps would have to be five times stronger to recover the physiological state. Robustness?

ROBUSTNESS? Model variants We tested the effect of: gating leak currents pump model GHK V / mv V / mv a) Ip Cre & Nernst model b) 1 5 4 5 7 HB3 HB2 1 2 3 4 5 HB3 LP2 LP1 LP2 HB2 5 LP1 HB1 HB1 LP1 5 LP1 stable FP (excl. Cl ) 1 unstable FP (excl. Cl ) 1 5 5 1 2 4 6 8 1 ρ / (µa/cm 2 ) HB3 LP1 HB2 LP1 HB1 LP2 HB1 LP2 HB2 V / mv c) Ip Cre & GHK model d) 1 4 7 1 2 3 4 5 V / mv 1 Ip Kag & Nernst model HB2 LP2 HB1 1 4 7 HB2 5 1 15 2 25 3 ρ / (µa/cm 2 ) 1 5 4 7 HB3 LP1 LP2 5 LP1 HB1 HB2 HB1 HB1 LP2 stable FP (incl. Cl ) stable FP (incl. Cl ) Ip Kag & GHK model 1 2 3 4 5 1 2 3 4 5 HB2 LP2 1 2 3 4 5 ρ / (µa/cm 2 ) 2 4 6 8 1 12 14 ρ / (µa/cm 2 ) Result Model variants with voltage gated ion channels are bistable. Variants without voltage gated ion channels are not.

EVEN KAGER WADMAN SOMJEN Bifurcations of model by Kager et al. Also for the (single compartment) Kager Wadman Somjen model we find a minimal physiological pump rate 9.8µA/cm 2 and a recovery pump rate 17µA/cm 2 that is large compared to the normal value (13µA/cm 2 ). V / mv 5 5 1 LP3 lc LP HOM LP4 lc LP1 lc 5 1 15 2 HB LP2 lc ρ / (µa/cm 2 ) stable FP, leak-only stable FP unstable FP unstable LC stable LC Bistability of FES and physiological conditions apparently a generic feature of closed neuron models.

REMARK ON FIXED LEAK CURRENTS Many models contain fixed leak currents : V = 1 C m (I Na + I K + I Cl I p ). Cl i = Such a current with a fixed Nernst potential changes the dynamics dramatically! V/mV a) Potentials: fixed leak, long stimulation b) Concentrations: fixed leak, long stimulation 8 6 4 2 2 4 6 8 1 2 4 6 8 1 t/sec V E K E Na E Cl ion/mm 16 Na i 14 K i 12 Cl i 1 Na e 8 K e 6 Cl 4 e 2 2 4 6 8 1 t/sec

OPEN MODELS system surroundings ion reservoirs energy source extracellular ATPase channel intracellular system surroundings ion reservoirs energy source extracellular ATPase channel intracellular system surroundings ion reservoirs energy source extracellular ATPase channel intracellular Coupling to a reservoir will resolve the bistability! So far we have considered isolated Donnan closed bistability isolated boundary open isolated boundary closed isolated boundary isolated Potassium exchange with a reservoir Instead of potassium conservation we have: K e = K e + ω i ω e (K i K i ) + K e K e measures the potassium gain or loss. Dynamics of K e diffusion to ECS bath or vasculature glial buffering

CSD IN BUFFERED MODELS Name k1 k 1 B Value & unit 5e 5/sec/mM 5e 5/sec 5mM With buffering...... bistability becomes ionic excitability in both KWS and reduced ion base model! This is CSD! K e + B k 2 k1 K b V/mV k1 1+exp( (K e 15)/1.9) k 2 = B = K b + B a) Reduced model: potential b) 4 2 2 4 6 8 EK ENa ECl 1 2 4 6 8 1 12 14 16 t/sec c) Reduced: ion conc. d) ion/mm 14 12 1 8 6 4 2 2 4 6 8 1 12 14 16 d K e dt t/sec V V/mV ion/mm 5 5 1 14 12 1 = k 2 K e (B K b ) + k 1 K b 8 6 4 2 Kager et al.: potential 2 4 6 8 1 12 14 16 t/sec Kager: ion conc. 2 4 6 8 1 12 14 16 t/sec Ki Nai Cl i Ke Nae Cl e

TIME SCALES IN BUFFERED MODEL Time scale for ion dynamics from GHK equation with dimensionless potential ξ, permeability P ion : dion i dt = A m P ion z ξ ion e exp( ξ) ion i ω } i exp( ξ) 1 {{} 1/τ ion (with m p h q.1 for gated channels P ion 5µm/sec, leak P ion.5µm/sec) Forward and backward buffering time scale: τ fw buff = 1 k1 B τ bw buff = 1 k 1 Time scales τ V.5msec τ n 1msec.5sec τ ion τ fw buff τ bw buff 5sec 5h For CSD dynamics in this model particle exchange with reservoirs is by far the slowest process! slow fast analysis

POTASSIUM GAIN/LOSS AS BIFURCATION PARAMETER Slow fast analysis a) Bifurcations (reduced model): membrane potential use the slowest variable K e as a bifurcation parameter superimpose full dynamics on bifurcation diagram Phase space explanation for observed dynamics? Two stable fixed points physiological branch B phys free energy starved B FES V / mv b) K e / mm 4 2 2 4 6 8 6 5 4 3 2 HB3 LP2 HB2 LP1lc HB4 B phys B F E S 6 4 2 2 4 K e / mm HB4 HB3 B F E S HB1 LP3lc Bifurcations (reduced model): extrac. potassium LP1 LP2lc LP4lc stable FP unstable FP stable LC unstable LC LP2lc LP2 LP1lc m r LP3lc 1 HB2 LP1 B phys LP4lc HB1 6 4 2 2 4 K e / mm

IMPLICATIONS OF BIFURCATION DIAGRAM Bifurcations (reduced model): extrac. potassium Critical K e values 6.7mM HB1 1.2mM LP3 lc 17.8mM LP2 lc 21.1mM LP1 lc Implications K e / mm 6 5 4 3 2 HB3 49 46 43 HB4 HB3 LP1lc 22 17 12 B F E S 18 LP3lc 12 LP1 6 27 29 31 stable FP unstable FP stable LC unstable LC LP2lc LP2 LP1lc m r LP3lc 1 HB2 LP1 B phys LP4lc HB1 6 4 2 2 4 K e / mm maximal physiological potassium content (end of B phys at 28.7mM) potassium reduction for recovery from FES (end of B FES at 44mM) well defined levels of stable ECS potassium concentration (limit cycle have almost constant ion concentrations)

SLOW CHLORIDE Chloride is slower than sodium and potassium (τ Cl 5sec) vary chloride as a prameter V / mv a) reduced model: potential b) Kager et al.: potential 1 3 5 7 1 6 2 2 Ke / mm 2 4 6 8 1 12 8 4 Ke / mm c) reduced: extrac. potassium d) Kager: extrac. potassium V / mv Result: family of topologically equivalent FP curves Ke / mm 6 stable FP for dyn. Cl unstable FP for dyn. Cl 4 2 1 6 2 2 Ke / mm Ke / mm 6 stable FP for imperm. Cl unstable FP for imperm.cl 4 Hopf line for imperm.cl 2 12 8 4 Ke / mm Recovery threshold The recovery threshold is then the line of Hopf bifurcations. Arrows indicate K e changes due to (m) flux across membrane (r) exchange with reservoir

CSD IN PHASE SPACE V/mV a) Reduced model: potential b) Kager et al.: potential 4 EK 2 ENa ECl 2 V 4 6 8 1 2 4 6 8 1 12 14 16 5 5 1 t/sec c) Reduced: ion conc. d) ion/mm 14 12 1 8 6 4 2 2 4 6 8 1 12 14 16 t/sec V/mV ion/mm 14 12 1 8 6 4 2 2 4 6 8 1 12 14 16 t/sec Kager: ion conc. Ki Nai Cl i Ke Nae Cl e 2 4 6 8 1 12 14 16 t/sec V / mv Ke / mm a) reduced model: potential b) Kager et al.: potential 1 3 5 7 1 6 2 2 Ke / mm c) reduced: extrac. potassium d) Kager: extrac. potassium SD for pump interruption 6 4 2 1 6 2 2 Ke / mm V / mv SD for Ke elevation 2 4 6 8 1 12 8 4 Ke / mm Ke / mm 6 4 2 12 8 4 Ke / mm SD in reduced ion based model and KWS. Ignition by potassium elevation and pump interruption. Course of events after stimulation 1. vertical transition from B phys to B FES 2. diagonal transition along B FES until threshold 3. abrupt vertical depolarization from B FES to B phys 4. slow asymptotic recovery

SCHEMATIC VIEW ON CSD extrac. potassium / mm 7 6 5 4 3 2 1 transmembrane through reservoirs Phase space scheme for SD C FES D E A B phys. 6 4 2 2 4 potassium gain or loss / mm New insights concerning ignition threshold recovery mechanism recovery threshold SD duration Note: Recovery is not due to the ion pumps! SD phases and time scales AB stimulation (instantaneous) BC ECS potassium accumulation, depolarization τ ion.5sec CD buffering, diffusion τ fw buff 5sec DE abrupt repolarization τ ion.5sec EA final recovery τbuff bw

OSCILLATORY DYNAMICS We investigate oscillatory dynamics for bath coupling with elevated potassium concentrations (λ = 3e 2/sec). J diff = λ(k bath K e ) d K e dt = J diff Bifurcation analysis Classify these pathologically important types of ion dynamics. Oscillation Types seizures for 8.5mM tonic firing for 12mM periodic SD 15mM V/mV a) Potentials b) Potentials c) Potentials 6 EK 6 6 4 ENa 4 4 2 ECl 2 2 2 V 2 4 2 4 6 4 6 8 6 8 1 8 1 1 2 3..2.4.6.8 1. 3 6 9 12 d) Concentrations e) Concentrations f) Concentrations ion/mm 14 13 12 2 1 1 2 3 t/sec V/mV ion/mm 13 12 2 15..2.4.6.8 1. t/sec V/mV ion/mm 14 Ki 12 Nai 1 Cl i 8 Ke 6 Nae 4 Cl e 2 3 6 9 12 t/sec

BIFURCATION DIAGRAM FOR K bath Result a) Bifurcations (reduced model): membrane potential seizure like activity (SLA) via supercr. torus bif. T SLA is 16 45sec periodic SD via subcrit. torus bif. T SD is 35 55sec hysteresis conclusion SLA graded SD all or none V / mv 5 5 1 15 b) 9 8 7 6 5 4 3 2 1 Ke / mm PD HB1 TR4 TR3 TR2 LP2 HB2 HB3 LP1 LP1lc 5 1 15 2 Kbath / mm stable FP unstable FP stable LC unstable LC stable torus Bifurcations (reduced model): extrac. potassium LP2 LP1 HB2 HB1 HB3 5 1 15 2 Kbath / mm LP2lc HB4 TR1 HB4

BIFURCATION DIAGRAM FOR K bath AND FOR K e K e / mm 6 5 4 Bifurcations (reduced model): extrac. potassium 4 HB3 22 18 stable FP LP1lc LP3lc 17 unstable FP 12 2 LP1 stable LC 12 6 unstable LC 49 46 43 27 29 31 BF E S 2 3 HB4 HB3 4 LP2lc 2 LP2 LP1lc m r 6 LP3lc 1 HB2 LP1 Bphys HB1 8 LP4lc 6 4 2 2 4 K e / mm K e / mm Bifurcations (reduced model): potassium gain/loss, bath coupling TR4 LP1 TR2 HB1 TR3 LP2lc LP2 LP1lc stable FP HB4 HB2 HB3 unstable FP stable LC TR1 unstable LC stable torus 5 1 15 2 K bath / mm Bifurcations can be related: LP1 lc TR1 LP2 lc TR2 LP3 lc TR3 LP4 lc TR4 Relevance of Close Model Phase Space Many results for parametrical K e translate almost directly to the full system.

SLA VS SD IN PHASE SPACE a) extrac. potassium: SLA b) 7 7 6 11 6 5 7 5 Ke / mm 4 3 2 1 24 29 1 8 6 4 2 2 4 K e / mm Ke / mm 4 3 2 1 extrac. potassium: SD trajectory 1 8 6 4 2 2 4 K e / mm SLA is oscillation around physiological conditions and LCs at low ECS potassium. SD is a large excursion to FES and subsequent return. SLA and SD are of fundamentally different nature!

SUMMARY AND OVERVIEW: OPEN, CLOSED AND ISOLATED extrac. potassium / mm 7 6 5 4 transmembrane through reservoirs Phase space scheme for SD C FES 3 D 2 1 B E A phys. 6 4 2 2 4 potassium gain or loss / mm system surroundings ion reservoirs isolated boundary energy source extracellular ATPase intracellular open channel system surroundings ion reservoirs isolated boundary energy source extracellular ATPase intracellular closed channel system surroundings ion reservoirs isolated boundary energy source extracellular ATPase intracellular isolated channel Open vs Closed Model physiol. resting state (stable) physiol. resting state (stable) physiol. resting state (non-exisiting) Pumps cannot recover physiological conditions from FES. In ionic excitability ion exchange with surroundings leads to recovery. time scales, thresholds... FES (metastable) CSD Suscept. FES (stable) Tissue at risk? FES (stable) Dead

SUMMARY: OSCILLATORY DYNAMICS Key results SLA and SD related to different bifuractions SD and SLA of fundamentally different nature SD is all or none SLA is graded (probably model specific) approximative values of SD and SLA thresholds can be obtained from K e bifurcation diagram Ke / mm V / mv a) Bifurcations (reduced model): membrane potential 5 PD TR4 TR3 TR2 LP1lc LP2 HB2 HB3 5 LP1 HB1 1 15 5 1 15 2 Kbath / mm b) Bifurcations (reduced model): extrac. potassium 9 stable FP 8 unstable FP 7 stable LC 6 unstable LC stable torus 5 Ke / mm 4 3 HB3 2 1 LP2 LP1 HB2 HB1 5 1 15 2 Kbath / mm a) extrac. potassium: SLA b) extrac. potassium: SD 7 7 11 6 6 trajectory 5 7 5 4 24 29 4 Ke / mm LP2lc HB4 TR1 HB4 3 3 2 2 1 1 1 1 8 6 4 2 2 4 8 6 4 2 2 4 Ke / mm Ke / mm

Thank you and... Markus Dahlem Eckehard Schöll Frederike Kneer Steven Schiff...