Leech Heart Interneuron model (Sharma et al 2020)

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Accession:264594
Fractional order Leech heart interneuron model is investigated. Different firing properties are explored. In this article, we investigate the alternation of spiking and bursting phenomena of an uncoupled and coupled fractional Leech-Heart (L-H) neurons. We show that a complete graph of heterogeneous de-synchronized neurons in the backdrop of diverse memory settings (a mixture of integer and fractional exponents) can eventually lead to bursting with the formation of cluster synchronization over a certain threshold of coupling strength, however, the uncoupled L-H neurons cannot reveal bursting dynamics. Using the stability analysis in fractional domain, we demarcate the parameter space where the quiescent or steady-state emerges in uncoupled L-H neuron. Finally, a reduced-order model is introduced to capture the activities of the large network of fractional-order model neurons.
Reference:
1 . Sharma SK, Mondal A, Mondal A, Upadhyay RK, Hens C (2020) Emergence of bursting in a network of memory dependent excitable and spiking leech-heart neurons. J R Soc Interface 17:20190859 [PubMed]
Model Information (Click on a link to find other models with that property)
Model Type: Realistic Network;
Brain Region(s)/Organism: Leech;
Cell Type(s): Leech heart interneuron;
Channel(s):
Gap Junctions:
Receptor(s):
Gene(s):
Transmitter(s):
Simulation Environment: MATLAB;
Model Concept(s): Oscillations; Bifurcation;
Implementer(s):
      %Fractional code for time series (See fig.2 in the manuscript) of Leech's heart interneuron model.

%%%% Here I refers to V^{shift}_{K2}  and u refers to the membrane voltage(v) of the manuscript.

%%% Title- Emergence of bursting in a network of memory dependent excitable and spiking Leech-Heart neurons
%%% Authors-Sanjeev Kumar Sharma; Argha Monda; Arnab Mondal; Ranjit Kumar Upadhyay and Chittaranjan Hens


clear;
 alpha=1;
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Set I
 I=-0.021;
%Set II
%  I=-0.015;
 %Set III
%  I=0.001;
 %Set IV
%  I=0.003;

 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 dt = 0.001; tspan = 0:dt:30;
  %%%%%%%% Initial conditions %%%%%%%%%%%%%%%%%%%%%%%%%
 u=rand/10;
 V=rand/10;
 w=rand/10;

  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
uu=zeros(length(tspan),1); VV=zeros(length(tspan),1); ww=zeros(length(tspan),1);
uu(1,1)=u;
VV(1,1)=V;
ww(1,1)=w;

T1=tspan(end)/10;
T2=T1+60;
NN=length(tspan);
nn=1:NN-1;

WCoet=(NN+1-nn).^(1-alpha)-(NN-nn).^(1-alpha);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
for j=1:length(tspan)-1    

    if j<2
  u=u+dt*(-2*(30*(w.^2)*(u+0.07)+8*(u+0.046)+200*(1/(1+exp(-150*(0.0305+u))).^3)*V*(u-0.045)));
   V=V+dt*(24.69*(1/(1+exp(500*(0.0333+u)))-V));
  w=w+dt*(4*(1/(1+exp(-83*(0.018+I+u)))-w));
  
    else
  %%%%% Fractional derivative starts  here%%%%%%%%%%%%%%%%%
  
       WCoe=WCoet(end-j+2:end);  % The weight   of the fractional drivative  at each  tiime t 
       kr = dt^alpha*gamma(2-alpha);     %  the kernel   from the fractional derivative and  weighted  the markovian term
        
  %%%%% Fractional derivative for u starts  here 
        
        d2dM=uu(1:j,:); % to call all past values of voltage u  for fractioanl integration
       
        TeDi=d2dM(2:j,:)-d2dM(1:j-1,:); % Delta uu (using all past values of u)  of  the  voltage memory trace of the fractional drivative  at each  tiime t 
        fraccalcu=WCoe*TeDi-d2dM(j,:);  %  The fraction derivative 
   
   %%%%% Fractional derivative of u ends   here 
        
    %%%%%   Fractional derivative for V starts  here 
    
        d2dMV=VV(1:j,1); % to call all past values  VV  for fractioanl integration
        TeDiV=d2dMV(2:j,1)-d2dMV(1:j-1,1); % Delta V (using all past values of VV) 
        fraccalcuV=WCoe*TeDiV-d2dMV(j,1);  %  The fraction derivative 
        
    %%%%% Fractional derivative ends V  here 
    %%%% ==w==  Fractional derivative for w starts  here 
      
        d2dMw=ww(1:j,1); % to call all past values  ww  for fractioanl integration
        TeDiw=d2dMw(2:j,1)-d2dMw(1:j-1,1); % Delta w (using all past values of ww) 
        fraccalcuw=WCoe*TeDiw-d2dMw(j,1);  %  The fraction derivative 
        %%%%% Fractional derivative ends w  here 
       
         u =kr*(-2*(30*(w.^2)*(u+0.07)+8*(u+0.046)+200*(1/(1+exp(-150*(0.0305+u))).^3)*V*(u-0.045)))- fraccalcu; 
         V =kr*(24.69*(1/(1+exp(500*(0.0333+u)))-V))-fraccalcuV;
         w=kr*(4*(1/(1+exp(-83*(0.018+I+u)))-w))-fraccalcuw;
        
        Memo2(j,:)=fraccalcu+uu(j,:); % to save memory over time
        Memo2V(j,:)=fraccalcuV+VV(j,:); % to save memory over time
        Memo2w(j,:)=fraccalcuw+ww(j,:);
    end
    
     uu(j+1,1)=u;
     VV(j+1,1)=V;
     ww(j+1,1)=w;
end
% figure

 plot(tspan,uu,'b');
 set(gca,'FontName','Times New Roman','FontSize',24)
 xlabel('{\bf{\it t}}','FontName', 'Times New Roman','FontSize',30,'Color','k', 'Interpreter', 'tex')
ylabel('{\bf{\it v}}','FontName', 'Times New Roman','FontSize',30,'Color','k', 'Interpreter', 'tex')
% figure;
%  plot(tspan,ww,'b');
% xlabel('t')
% ylabel('w')
%   figure
%   set(gca,'FontName','Times New Roman','FontSize',22)
%   plot(uu,VV);
%   xlabel('{\bf{\it u}}','Interpreter','tex','FontName','Times New Roman','FontSize',28)
% ylabel('{\bf{\it v}}','Interpreter','tex','FontName','Times New Roman','FontSize',28)
%   figure
%   set(gca,'FontName','Times New Roman','FontSize',22)
%   plot3(uu,VV,ww);
% xlabel('{\bf{\it u}}','Interpreter','tex','FontName','Times New Roman','FontSize',28)
% ylabel('{\bf{\it v}}','Interpreter','tex','FontName','Times New Roman','FontSize',28)
% zlabel('{\bf{\it w}}','Interpreter','tex','FontName','Times New Roman','FontSize',28)

% figure
% plot(tspan,uu,'b');
% xlabel('t')
% ylabel('u')
        
        
        
        

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