Activity dependent changes in dendritic spine density and spine structure (Crook et al. 2007)

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Accession:114342
"... In this work, we extend previous modeling studies [27] by combining a model for activity-dependent spine density with one for calcium-mediated spine stem restructuring. ... Additional equations characterize the change in spine density along the dendrite, the current balance equation for an individual spine head, the change in calcium concentration in the spine head, and the dynamics of spine stem resistance. We use computational studies to investigate the changes in spine density and structure for differing synaptic inputs and demonstrate the effects of these changes on the input-output properties of the dendritic branch. ... "
Reference:
1 . Crook SM, Dur-E-Ahmad M, Baer SM (2007) A model of activity-dependent changes in dendritic spine density and spine structure. Math Biosci Eng 4:617-31 [PubMed]
Model Information (Click on a link to find other models with that property)
Model Type: Dendrite;
Brain Region(s)/Organism:
Cell Type(s):
Channel(s): I Na,t; I K;
Gap Junctions:
Receptor(s): AMPA;
Gene(s):
Transmitter(s):
Simulation Environment: MATLAB;
Model Concept(s): Synaptic Plasticity; Synaptic Integration; Calcium dynamics;
Implementer(s):
Search NeuronDB for information about:  AMPA; I Na,t; I K;
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mbe
readme.html
dmc.m
fig8a.jpg
fig8c.jpg
gsyn.m
MBEv4n4.pdf
run_sp.m
sbf_sp.m
                            
% In this program we re-write the system of PDE's in terms of ODE's by 
% using the spectral method. We estimate the second derivative term 
% in the the first equation for V_d using the spectral method. 


function yp=sbf_sp(t,y,D2,xc)
sy=length(y);                   % Length of the initial vector
N=sy/8;                         % number of compartments
L=3;                            

        % Define vectors Vd, Vsh, Ca, Rss, nbar, n, m, and h
Vd=  y(1:N);                    % Dendrite Voltage
Vsh= y(N+1:2*N);                % Spine head Voltage
Ca=  y(2*N+1:3*N);              % Calcium concentration
Rss= y(3*N+1:4*N);              % Resistance of spine stem
nbar=y(4*N+1:5*N);              % Spine Density

        % n, m and n are the variable used in the Hodgkin huxley model.
n=   y(5*N+1:6*N);              
m=   y(6*N+1:7*N);
h=   y(7*N+1:8*N); 

                    % Parameter List
                    
A_sh = 1.31e-8;      % Surface area of each spine head cm^2
C_crit = 300;        % Critical intraspine Calcium level (nM)
C_m = 10^(-3);       % Specific membrane Capacitance (mF/cm^2)
C_min = 5;           % Calcium Lower bound (nM)
d = 0.000036;        % Dendritic Cable diameter (cm)
eps_1 = 3e-3;        % Rate of Change in Ca equation (ms^-1)
eps_2 = 7.5e-5;      % Rate of Change in Rss equation (ms^-1)
eps_3 = 1e-5;        % Rate change in nbar equation
gamma = 2.5;         % Channel Density scale factor 
L = 3;               % Dimensionless length of the cable
gbar_Na = .120;      % Maximal Sodium conductance (S/cm^2)
gbar_K = .036;       % Maximal potassium concuctance (S/cm^2);
gbar_L = .0003;      % Maximal Leackage Conductance (S/cm^2)

kappa_c = 1e-9;      % Scale Factor (Calcium Model), (mA*ms/nM)

R_i = 70;            % Specific cytoplasmic resistivity (Ohm-cm)
R_m = 2500;          % Passive membrane resistence (Ohm-cm^2)
R_max = 1000000000;  % Stem resistence upper bound (Ohm)
R_min = 30000000;    % Stem resistence lower bound (Ohm)
R_sh = 1.02e11;      % Resistence of each spine head (Ohm)
V_Na = 115;          % Sodium reversal potential (mV)
V_K = -12;           % Potassium reversal potential (mV)
V_L = 10.5989;       % Leakage reversal potential (mV)
V_syn = 100;         % synaptic reversal potential (mV)

temp=22;             % Temperature parameters
phi=3^((temp-6.3)/10);

nbar_max = 100;     % Spine density upper bound (5-6 spine/10 micro m)
nbar_min = 16;      % Spine density lower bound
Ca_1 = 30;          % Lower bound of Ca concentration where LTD changes to LTP
Ca_2 = 300;         % Upper bound of Ca concentration 

% Define I1 and I2 appearing in boundary conditions
I_1 = 0;            % Injected current in the dendrite
I_2 = 0;            % Released current on the opposite side of dendrite

% Define tau_m, lambda, R_inf, and C_sh
tau_m=R_m*C_m;                          % Membrane time constant (ms)
lambda=sqrt((R_m*d)/(4*R_i));           % Space constant (cm)
R_inf=R_m/(pi*lambda*d);                % Specific input resistance (Ohm)
C_sh=A_sh*C_m;                          % Compartment specific capacitance (mF)

% Initialize vector yp
yp=zeros(8*N,1);

% Define yp(1:N): the equation for V_d  (using spectral method)
x=xc(2:N+1);
dx=xc(2)-xc(1);
D2n=D2(2:N+1,2:N+1);
D2n(:,1)=D2n(:,1)+D2(2:N+1,1);
D2n(:,N)=D2n(:,N)+D2(2:N+1,N+2);
C=dx*R_inf*(I_1*D2(2:N+1,1)+I_2*D2(2:N+1,N+2));
Iss=(Vsh-Vd)./Rss;
yp(1:N)=D2n*Vd+R_inf*nbar.*Iss+C;
yp(1:N)=yp(1:N)/tau_m;

% Function to determine the stimulus along the cable

vsyn=zeros(N,1);
for k=1:N
   vsyn(k)=gsyn(x(k),t);  
end
Isyn=vsyn.*(Vsh-V_syn);

% Define yp(N+1:2*N): the equation for V_sh.

Iion=gamma*A_sh*(gbar_Na*(Vsh-V_Na).*(m.^3).*h + ...
            gbar_K*(Vsh-V_K).*(n.^4)+gbar_L*(Vsh-V_L));
yp(N+1:2*N)=(-Iion-Isyn-Iss)/C_sh;

% Define yp(2*N+1:3*N): the equation for Ca.

yp(2*N+1:3*N)=-eps_1*(Ca-C_min)+abs(Iss)/kappa_c;

% Define yp(3*N+1:4*N): the equation for Rss.

yp(3*N+1:4*N)=-eps_2*(Rss-R_min).*(1-Rss/R_max).* ...
                     (Ca/Ca_1 - 1).*(Ca/Ca_2-1).*(Ca/C_min - 1);

% Define yp(4*N+1:5*N): the equation for nbar.

yp(4*N+1:5*N) =-eps_3*(Ca/Ca_1-1).*(Ca/Ca_2-1).*(Ca/C_min - 1).*(1-nbar/nbar_max).*(nbar-nbar_min);

% Define yp(5*N+1:6*N): the equation for n.

alfn=phi*0.01*(-Vsh+10)./(exp((-Vsh+10)/10)-1);
betn=phi*0.125*exp(-Vsh/80);
yp(5*N+1:6*N)=alfn.*(1-n)-betn.*n;

% Define yp(6*N+1:7*N): the equation for m.

alfm=phi*0.1*(-Vsh+25)./(exp((-Vsh+25)/10)-1);
betm=phi*4*exp(-Vsh/18);
yp(6*N+1:7*N)=alfm.*(1-m)-betm.*m;

% Define yp(7*N+1:8*N): the equation for h.

alfh=phi*0.07*exp(-Vsh/20);
beth=phi*1./(exp((-Vsh+30)/10)+1);
yp(7*N+1:8*N)=alfh.*(1-h)-beth.*h;


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