Phase oscillator models for lamprey central pattern generators (Varkonyi et al. 2008)

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Accession:118392
In our paper, Varkonyi et al. 2008, we derive phase oscillator models for the lamprey central pattern generator from two biophysically based segmental models. We study intersegmental coordination and show how these models can provide stable intersegmental phase lags observed in real animals.
Reference:
1 . Várkonyi PL, Kiemel T, Hoffman K, Cohen AH, Holmes P (2008) On the derivation and tuning of phase oscillator models for lamprey central pattern generators. J Comput Neurosci 25:245-61 [PubMed]
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Model Information (Click on a link to find other models with that property)
Model Type: Realistic Network; Connectionist Network;
Brain Region(s)/Organism:
Cell Type(s):
Channel(s):
Gap Junctions:
Receptor(s):
Gene(s):
Transmitter(s):
Simulation Environment: MATLAB;
Model Concept(s): Temporal Pattern Generation; Oscillations; Spatio-temporal Activity Patterns; Parameter Fitting; Parameter sensitivity; Phase Response Curves;
Implementer(s): Varkonyi, Peter [vpeter at mit.bme.hu];
%[shift H fi PRC T]=couplingfunction(ve,i,j)
%
%this function determines: 
%
%-the length of the period T of a network based oscillator; 
%-the phase response curves PRC(fi) 
%-the coupling function H(shift)
%
%if tonic drive to E cells is ve; i,j determine the index of the subplot
%where the results are displayed and the total number of subplots.
%
%the dynamics of an oscillator (Fig 2) is also plotted 


function [shift H fi PRC T]=couplingfunction(ve,i,j)



%%%%%%%%%%%%%%%%%%%%% PRC
[fi0 y0 fi PRC T]=PRCdirect(@williams,[.1 0 0 0 0 0 [ve .01 .1]]);
y0=y0(:,1:6);
fi=fi';
%%%%%%%%%%%%%%%%%%%%% plotting the dynamics of an oscillator
figure(2);
if i==1
    title ('************FIGURE 2*************');
end
subplot(2,j,i);

hold on;
plot(fi0,y0(:,1:3));
hold off;
subplot(2,j,j+i);
plot(fi,PRC);
title({'FIG. 2, tonic drive =' ve});
hold off;


%%%%%%%%%%%%%%%%%%%%
dfi=fi(2)-fi(1);
dt=dfi/(2*pi/T);
y=interp1(fi0,y0,fi,'spline');
shift=-pi:pi/20:pi;%phase-shifts at which the coupling function is determined

%connectivity matrix of segmental oscillators. Rows 1,2,...,6 mean
%E,L,C,C,L,E cells, repectively; +1=excitatory connection; -1=inhibitory
%connection
conn=[0 0 0 -1 0 0; 1 0 0 -1 0 0;1 -1 0 -1 0 0];  
conn=[conn;0 0 0 0 0 0;1 0 0 0 0 0 ; 0 0 0 -1 0 0];
conn=[conn;flipud(fliplr(conn))];

% numerical integration of coupling function
pert=[];
H=[];
for k=1:length(shift) %phase shift between coupled oscillators
    for i=1:3 %the effect of incoming connections of cell i
        for j=1:length(fi)-1 %integrating over a period
            w=abs(conn(i,:));%outgoing connection strengths of i^th cell 
            
            fr=interp1(fi',y,mod(fi(j)+shift(k),2*pi),'spline'); %calculating firing rates
            fr=max(fr,0);
            
            va=sign(conn(i,:))-y(j,i); %this term takes inzto account that connections have reversion potential (+1 or -1)
            pert(j,i)=sum(fr.*w.*va*dt); %perturbations from incoming connections
        end
    end

    %The coupling function is determined from the integral of pert*PRC over
    %a period summed for all elements of the oscillator.
    h=pert.*PRC(1:end-1,:);
    H(k)=2*sum(sum(h)); %'2*' because only perturbations on half of the segment were integrated
end