% ### EXhhFoundation.m ### 2018.03.19 C. Bergevin % [IN PROGRESS; see "To DO" below] % GOAL: Illustrate rationale behind Hodgkin-Huxley (1952e) bits (re their Fig.2): % O "If g_k is used as a variable the end of the record can be fitted by a first-order % equation but a third- or fourth-order equation is needed to describe the % beginning" % o "A useful simplification is achieved by supposing that g_k is % proportional to the fourth power of a variable which obeys a first-order equation. % [...] The rise in conductancetherefore shows a marked inflexion, while the fall is a simple % exponential." % o [from Nelson] % "Hodgkin and Huxley realized that a better fit could be obtained if they considered the % conductance to be proportional to a higher power of n. Figure 3 shows the results of fitting the % conductance data using a form [eqn] with powers of j ranging from 1 to 4. Using this sort of % fitting procedure, Hodgkin and Huxley determined that a reasonable fit to the K+ conductance data % could be obtained using an exponent of j=4. Thus they arrived at a description for the K+ % conductance under voltage clamp conditions" % --- Equations % o Consider the generic 1st order ODE from H&H 1952e (eqns 7-10) % [using x as a dummy variable name for n, m, or h] % + \dot{x} = \alpha_x*(1-x) - beta_x*n % OR AS % + \tau*\dot{x} + x = x_{infty} % o where \tau = 1/(\alpha_x + \beta_x) and x_{infty} = \alpha_x/(\alpha_x + \beta_x) % or \alpha_x= x_{infty}/\tau and \beta_x= (1-x_{infty})/\tau % o Solutions are of form: % + x= x_{infty}+ (x_o - x_{infty})* exp(-t/\tau) % where x_o is the initial condition, i.e., x_o= x(t=0) % --- To DO % o something seems fishy re Weiss: I can't get a decent fit to HH data % (g_k for Vm= 88 mV, see Weiss 1996 vol.2 Fig.4.23) using % o do actual nonlinear regression for different models to those data and % determine best fit params. % --> better motivates Fig.1 for comparing n^j for various j % --> possible to determine appropriate values of n_infty and tauN?? (and % compare to analytic expressions) % o repeat both items above for the different membrane polarizations and % determine the functional form of the how they depend upon Vm % --> presumably should match/reproduce Weiss 1996 vol.2 Fig.4.25 and % eqns.4.14-4.20 % --- REFs % o H&H 1952e = J. Physiol. (1952) 117, 500-544 % o Nelson, M.E. (2004) "Electrophysiological Models In: Databasing the % Brain: From Data to Knowledge" clear % ============================================================== % --- generic variable "x" params. (Fig.1) tau= 4; % [ms] xINFTY= 1; x0= 0; % initial condition % --- K (potassium) params. if (1==0) % Weiss eqns.4.18 and 4.19 (pg.191) GkBAR= 36; aK= @(X) -0.01*(X+50)./(exp(-0.1*(X+50))-1); bK= @(X) 0.125* exp(-0.0125*(X+60)); Vm= 28; % membrane potential for Gk vals below Vmx= linspace(-100,75,500); % functional form (Weiss uses "membrane potential") else % H&H 1952 eqns. 12 & 13 (and table 1 caption) GkBAR= 24.31; % val. listed (as "figure") in Table 1 caption(?) %GkBAR= 21.0; % better fit value! aK= @(X) 0.01*(X+10)./(exp((X+10)/10)-1); bK= @(X) 0.125* exp(X/80); Vm= -88; % membrane potential for Gk vals below Vmx= -linspace(-40,135,500); % func. form (HH use disp. re rest. pot.) end % Gk conductance values extracted (via deplot) for Vf-Vi= 88 mV % (presumably Vi=-60 mV such that Vm=28 mV) % (from Weiss 1996 vol.2 Fig.4.23) Gk= [0.1358 0.0000 0.3387 0.2182 0.5838 1.0731 0.7695 2.0969 1.1035 4.3834 1.5119 7.0410 2.0068 9.3710 2.9435 12.5883 4.2046 15.5825 6.4358 16.7832 8.8070 16.8921]; % ============================================================== % --- t= linspace(0,20,1000); % [ms] % --- Fig.1: Simple visual comparison of powers of 1st order kinetics x= xINFTY+ (x0 - xINFTY)* exp(-t/tau); figure(1); clf; h1= plot(t,x,'LineWidth',2); hold on; grid on; h2= plot(t,x.^2,'LineWidth',2); h3= plot(t,x.^3,'LineWidth',2); h4= plot(t,x.^4,'LineWidth',2); xlabel('Time [ms]'); ylabel('x^j'); legend([h1 h2 h3 h4],'x','x^2','x^3','x^4','Location','SouthEast'); title('First order kinetics (x) to different powers') % --- Fig.2: Comparison to actual HH data tauN= 1/(aK(Vm)+bK(Vm)); % [ms] nINFTY= aK(Vm)/(aK(Vm)+bK(Vm)); %n= nINFTY*(1- exp(-t/tauN)); n0=0.24; n= nINFTY+ (n0 - nINFTY)* exp(-t/tauN); GkMOD= GkBAR* n.^4; figure(2); clf; k1= plot(Gk(:,1),Gk(:,2),'ko','MarkerSize',6,'LineWidth',2); hold on; grid on; k2= plot(t,GkMOD,'r--','LineWidth',2); xlim([0 1.1*max(Gk(:,1))]) xlabel('Time [ms]'); ylabel('G_k [mS/cm^2]'); title('G_k conductance data (Vm= 88 mV) from Weiss vol.2 Fig.4.23 (from Hodgkin 1964, Fig.29)') % --- Fig.3: Potassium params. re their voltage depend. alpha= aK(Vmx); beta= bK(Vmx); tauNf= 1./(alpha+beta); nINFTYf= alpha./(alpha+beta); figure(3); clf; % NOTE: what precisely the x-axis represents here depends upon whether you % use the HH convention or that of Weiss subplot(221); plot(Vmx,alpha); hold on; grid on; xlabel('pot. diff [V]'); ylabel('\alpha_n') subplot(222); plot(Vmx,beta); hold on; grid on; xlabel('pot. diff [V]'); ylabel('\beta_n') subplot(223); plot(Vmx,tauNf); hold on; grid on; xlabel('pot. diff [V]'); ylabel('\tau_n') subplot(224); plot(Vmx,nINFTYf); hold on; grid on; xlabel('pot. diff [V]'); ylabel('n_{\infty}')