Chantal McMahon MATH723: Final Project June 8, 2009

1. The influence of sodium and potassium dynamics on excitability, seizures, and the stability of persistent states: I. Single neuron dynamics Cressman, JR et al Chantal McMahon MATH723: Final Project June 8, 2009

2. Overview • Introduce motivations behind model • Local ionic fluctuation modulating behavior • Seizure and Stability • Full Model • Reduced Model • Results • Bifurcation Diagrams • Region of Oscillation • Discussion and conclusions

3. Introduction • Explain how interrelated dynamics of Na+ and K+ affect the excitability of neurons, the occurrence of seizures and stability of persistent states of activity • Hodgkin-Huxley Equations • Assumption: intra and extra cellular concentrations of Na+ and K+ are constant. • Understanding the role that cellular control of microenvironmental factors has in modulating the neuronal response

4. Importance • Investigate the role of local fluctuations in ion concentrations in modulating behavior of a single neuron • Intrinsic excitability of neuronal networks  reversal potentials for ion species  intra and extracellular concentrations of those ions • Focus on how changes in extracellular potassium can influence excitability • Small extracellular space and weak gNa at Vrest can cause transient changes in [K]owhich have been shown to have a greater effect over neuronal behavior than changes in [Na]i and can increase neuronal excitability and cause spontaneous activity and seizure like behavior

5. Full Model • One single compartment conductance based neuron containing sodium, potassium, calcium-gated potassium and leak currents. • Each current includes dynamic variables which represent [K]o and [Na]i and are included in the reversal potential for each ionic species Supporting Rate Equations

6. Full Model Continued Extracellular Potassium Intracellular Sodium

7. Reduced Model • Eliminate fast-time-scale spiking behavior in favor of slower ion concentration dynamics • Done by replacing HH mechanism with empirical fits to time-averaged ion currents • Fixed internal and external Na and K concentration ratios and allowed cell to reach dynamical state (rest or spiking) currents were then time averaged over one second and fit to sigmoidal functions INa(inf) and IK(inf)

8. Reduced Model Cont’d

9. Bifurcations in Reduced Model • Depending on the various parameters, ion concentrations in the reduced model approach either stable equilibria (resting state) or stable periodic orbits (spiking- oscillatory behavior) • Altering parameters alters the stability of ion concentrations through bifurcations • Hopf bifurcation • Saddle-node bifurcation of periodic orbits

10. Results • In order to induce seizure like behavior, a higher concentration of extracellular potassium is considered based off of experimental data which exhibit spontaneously occurring, prolonged periods of rapid firing. • Correspondingly, the full model contains dynamics on at least two distinct time scales that are separated by four orders of magnitude: a fast spiking HH mechanism and slow overall modulation

11. Comparison Reduced to Full Model Fast Spiking Slow Modulation of Periodic Behavior

12. Bifurcation Diagrams • As compared to experimental data, low values of external potassium concentration allow [K]o to settle in a stable equilibrium. • As the steady state value increases, [K]o also increases as it reaches an unstable subcritical Hopfbifucation as [K]o is attracted to a coexisting large-amplitude periodic orbit. • Around 2.15 the periodic orbit disappears in a saddle node bifurcation. • This oscillatory behavior is the basis for the rest of the paper which is studying oscillatory behavior associated with seizures.

13. Varying Parameters And (b) or the pump strength and diffusion variable have similar behavior where an unstable, periodic orbit appears via a subcritical Hopf bifurcation and growth of stable, periodic oscillation disappears at a saddle node bifurcation of periodic orbits. (c) No saddle node bifurcation is seen in LHS of Gglia Rho = bistable region on the RHS is wide representing hysteric behavior with changes in pump strength

14. Region of Oscillation [K]o-high and constant, tonic firing Various bursting behaviors [K]o elevated, Vm = -62mV

15. Reduced Model Bifurcation Plots Gglia = 1 Gglia = 1.75 Amplitude of oscillation of [K]o is significantly smaller for the larger Gglia, also, e is now supercritical for a value of 3.2 the amplitude of [K]o decays to zero as diffusion increases.

16. [K]o is high enough that the cell is depolarized beyond firing threshold • -F) Decreasing [K]o oscillation amplitude- change in bust envelope shape

17. MATLAB Model Gglia = 66 e = 1.2 (Normal Value) Gglia = 60 e = 2

18. Discussion • Model Assumptions • Insatiable glial buffering • [K]i can be held for times longer than the order of individual spikes or bursts For longer times, the saturation of glia and decoupling of [K]i from sodium dynamics will lead to calculated errors. Limitation of model lies in predicting long time scale behavior.

19. Conclusions • Understand the role of ion concentration dynamics in neuronal functioning • Reduced model to describe bifurcation analysis • Understand what a broad range of bath [K] can yield seizure like activity in a single neuron • Values that elicit seizures between 4 to 8mM concentrations have been shown experimentally. • The basic mechanism that can give rise to seizure like events are Hopf bifurcations that lead to slow oscillations in the ionic concentrations.

20. References • Cressman, JR Jr. Ullah, G, Ziburkus, J, Schiff, SJ, Barreto, E. The influence of sodium and potassium dynamics on excitability, seizures, and the stability of persistent states: I. Single neuron dynamics. J ComputNeurosci26 159-170 2009 • Hodgkin, AL, Huxley, AF, A Quantitative Description of Membrane Current and its Application to Conduction and Excitation in Nerve. Bulletin of Mathematical Biology. 52(1/2) 25-71, 1990 (1952)

21. Appendix I:Model Variables

22. Appendix II: Model Parameters

23. Ina = -gna*(minf.^3).*h.*(V-Vna)-gnal*(V-Vna); Ik = -(gk*n.^4 + ((gahp*Cai)/(1+Cai))).*(V-Vk) - gkl*(V-Vk); Icl = -gcil*(V-Vcl); an = .01*(V+34)/(1-exp(-.1*(V+34))); bn = .125*exp(-(V+44)/80); ah = .07*exp(-(V+44)/20); bh = 1/(1+exp(-.1*(V+4))); dKo = -.33*Ik - 14*Ipump - Iglia - Idiff;%(2) dNai = (.33*Ina/7)-3*Ipump; dn = phi.*(an.*(1-n)-bn.*n); dh = phi.*(ah.*(1-h)-bh.*h); dCai = (-.002*gca*(V-Vca)./(1+exp(-(V+25)/2.5))) - Cai/80; dV = (1/C)*(Ina + Ik + Icl); dg = [dKo; dNai; dV; dn; dh; dCai]; Appendix III: MATLAB Full Model function dg = FullModel(t,x) Ko = x(1); Nai = x(2); V = x(3); n = x(4); h = x(5); Cai = x(6); C = 1; gna = 100; gk = 40; gahp = .01; gkl = .05; gnal= .0175; gcil = .05; phi = 3; Vcl = -81.93; gca= .1; Vca = 120; rho = 1.25;Gglia = 66; e = .8; koinf= 50; Ipump = (rho./(1+exp((25-Nai)/3))).*(1/(1+exp(5.5-Ko))); Iglia = Gglia/(1+exp((18-Ko)/2.5)); Idiff = e*(Ko-koinf); Ki = 140+18-Nai; Nao = 144-7*(Nai-18); Vna = 26.64*log(Nao/Nai); Vk = 26.64*log(Ko/Ki); am = .1*(V+30)./(1-exp(-.1*(V+30))); bm = 4*exp(-(V+55)/18); minf = am/(am+bm);

24. Appendix IV: MATLAB figure (1) C = 1; gna = 100; gk = 40; gahp = .01; gkl = .05; gnal = .0175; gcil = .05; phi = 3; Vcl = -81.93; gca = .1; gca= 120; rho = 1.25;Gglia = 66; e = .8; koinf = 50; tspan = [0 30]; Kinit = 7; Nainit = 17; Vinit =-62; ninit = .5; hinit = .5; Caiinit = 10; x0 = [KinitNainitVinitninithinitCaiinit]; [T,Y]=ode45('FullModel',tspan,x0); figure(1) subplot(3,1,1) plot(T,Y(:,3),'r') title('Spiking Behavior over time') ylabel('Vm(mV)') subplot(3,1,2) plot(T,Y(:,1),'k-') ylabel('[K]o (mM)') subplot(3,1,3) plot(T,Y(:,2)) ylabel('[Na]i (mM)') xlabel('Time(sec)')