Effect of ionic diffusion on extracellular potentials (Halnes et al 2016)

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"Recorded potentials in the extracellular space (ECS) of the brain is a standard measure of population activity in neural tissue. Computational models that simulate the relationship between the ECS potential and its underlying neurophysiological processes are commonly used in the interpretation of such measurements. Standard methods, such as volume-conductor theory and current-source density theory, assume that diffusion has a negligible effect on the ECS potential, at least in the range of frequencies picked up by most recording systems. This assumption remains to be verified. We here present a hybrid simulation framework that accounts for diffusive effects on the ECS potential. ..."
1 . Halnes G, Mäki-Marttunen T, Keller D, Pettersen KH, Andreassen OA, Einevoll GT (2016) Effect of Ionic Diffusion on Extracellular Potentials in Neural Tissue PLoS Comput Biol 12:e1005193 [PubMed]
Model Information (Click on a link to find other models with that property)
Model Type: Extracellular; Neuron or other electrically excitable cell;
Brain Region(s)/Organism:
Cell Type(s): Neocortex U1 L6 pyramidal corticalthalamic cell;
Gap Junctions:
Simulation Environment: MATLAB; NEURON;
Model Concept(s): Extracellular Fields;
Implementer(s): Halnes, Geir [geir.halnes at nmbu.no]; Maki-Marttunen, Tuomo [tuomo.maki-marttunen at tut.fi];
Search NeuronDB for information about:  Neocortex U1 L6 pyramidal corticalthalamic cell;
function runit(times, jna, jk, jca, jx, icap, imemb, deltax, Avox, VECS, N, Nnrns, constsigma, diffon, nrnon, Datafile, Filename);
% Run simulation
% Prepares variables for simulation
% Runs simulation by calling function NVOXELS
% Collects data in structure S and saves it to file.

global tcounter
global Vout
global Iout

% Allocate memory. 
Nentries = length(times)*0.6
% Note: This is more or less the number of time points that Vout is
% estimated at in the RK-based numerical scheme.
tcounter = 1;
Vout = zeros(Nentries+50000,N+3); % N for nrns, 2 edges, 1 for time
Iout = zeros(Nentries+50000,N+3); % Added 50.000 elements just to make sure we have enough

% Diffusion constants
% Nano and Molecular Electronics Handbook: Sergey Edward Lyshevski. CRC Press, Taylor & Francis group 2007
lambda_o = 1.6; % ECS tortuousity, Chen & Nicholson 2000;
D_K = 1.96e-9/lambda_o^2; % Diffusion coefficients (m^2/s);
D_Ca=0.71e-9/lambda_o^2;  % 0,8 used by Gardner2015, 0.6 by Lewin2012, 0.71 in Lyshevski
D_X = 2.0e-9/lambda_o^2; % Using diffusion constant for Cl-
diffconsts = [lambda_o, D_K, D_Na, D_Ca, D_X];

% Initial conditions
Ns = 5; % 4 ion species +  V
cK0 = 3;
cNa0 = 150;
cCa0 = 1.4;
cX0 = cK0 + cNa0 + 2*cCa0;
V0 = 0;
basal = [cK0; cNa0; cCa0; cX0; V0];

[t,Y] = nvoxels(times, jna, jk, jca, jx, icap, imemb, deltax, Avox, VECS, N, Ns, basal, diffconsts, constsigma, diffon, nrnon);
%[t,Y] = nvoxels_nodiff(times, jna, jk, jca, jx, icap, deltax, Avox, VECS, N, Ns, basal);

% Unwrap variables:
Y = reshape(Y, [length(Y) N Ns]);
cKedge = cK0*ones(length(t),1);
cNaedge = cNa0*ones(length(t),1);
cCaedge = cCa0*ones(length(t),1);
cXedge = cX0*ones(length(t),1);
cK = [cKedge, Y(:,:,1), cKedge]; % mol/m^3 (or mM)
cNa = [cNaedge, Y(:,:,2), cNaedge]; % mol/m^3 (or mM)
cCa = [cCaedge, Y(:,:,3), cCaedge]; % mol/m^3 (or mM)
cX = [cXedge, Y(:,:,4), cXedge]; % mol/m^3 (or mM)
clear Y

% Interpolate to get V at same time points as cK:
% Note: With the numerical scheme used here, tv & V comes out at 1.5 times 
% as many interpolation points as t. 

tvr = Vout(:,1);
Vr = Vout(:,2:end);
tir = Iout(:,1);
Ir = Iout(:,2:end);
clear Vout;
clear Iout;

keepind = find(diff(tvr)); % some time steps are zero. Remove these
tvr = tvr(keepind); % "raw" data for global variables
Vr = Vr(keepind,:); % "raw" data
keepind = find(diff(tir)); % some time steps are zero. Remove these
tir = tir(keepind); % "raw" data
Ir = Ir(keepind,:); % "raw" data

V = interp1(tvr,Vr,t); % same #indices as concentration data
Im = interp1(tir,Ir,t);

% Prepare structure of output data
S.Datafile = Datafile;
S.Nionspecies = Ns;
S.Nvox = N+2;
S.diffconstnames = 'lambda_0 D_K D_Na D_Ca D_X';
S.diffconsts = [lambda_o, D_K, D_Na, D_Ca, D_X];
S.basalnames = 'cK0 cNa0 cCa0 cX0 V0';
S.basal = [cK0 cNa0 cCa0 cX0 V0];
S.diffon = diffon;
S.nrnon = nrnon;
S.constsigma = constsigma; 

S.geometry.deltax = deltax;
S.geometry.ECSfrac = 0.2;
S.geometry.Nnrns = Nnrns; 
S.geometry.Avox = Avox;
S.geometry.Vvox = VECS;

S.Neurondata.Nnrns = Nnrns;
S.Neurondata.jk = jk; 
S.Neurondata.jna = jna; 
S.Neurondata.jca = jca; 
S.Neurondata.jx = jx; 
S.Neurondata.icap = icap; 
S.Neurondata.imemb = imemb; 
S.Neurondata.times = times;

S.Simdata.t = t;
S.Simdata.cK = cK; 
S.Simdata.cNa = cNa; 
S.Simdata.cCa = cCa; 
S.Simdata.cX = cX;

S.Simdata.V = V; 
S.Simdata.Im = Im;

S.Simdata.tvr = tvr; % Also save the "raw" global data
S.Simdata.tir = tir;
S.Simdata.Vr = Vr;
S.Simdata.Ir = Ir;

save(Filename, 'S');

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