function [Uc,C,Ugo,Go,IGo_DA_Ach,Unogo,NoGo,INoGo_DA_Ach,Ugpe,Gpe,Ugpi,Gpi,Ut,T,Ustn,STN,E,t,Wgc_post,Wgs_post,Wnc_post,Wns_post,r,k_reward,ChI] = BG_model_function_Ach(S,Wgc,Wgs,Wnc,Wns,Correct_winner,Dop_tonic)
%% function used to simulate the network during training
% BG_model_function_Ach returns dynamical behaviour of Basal Ganglia structures and cortex
% S stimulus. Column vector of 4 elements Ei, 0<=Ei<=1 for i=1,2,3,4
% Wgc,Wgs,Wnc,Wns sets corresponding synaptic weights
% C_CORRECT_WINNER sets the desired correct response, if possible, depending on the stimulus
% Uc,Ugo,Unogo,Ugpe,Ugpi,Ut,Ustn returns the input to the sigmoidal function within time of the corresponding brain structures
% C,Go,NoGo,Gpe,Gpi,T,STN,E returns activity within time of the corresponding brain structures and energy in the cortex
% IGo_DA_Ach,INoGo_DA_Ach returns the input due to Dopa and Ach to Go and NoGo units
% t returns time
% Wgc_post,Wgs_post,Wnc_post,Wns_post returns corresponding synaptic weights after Hebbian learning
% r returns +1 for reward, -1 for punishment, NaN for no feedback
% r_story retyrn the historical record of rewards and punishments
% k_reward returns position of feedback, NaN for no feedback
% ChI returns activity within time of the cholinegic interneuron
%% tempi
tau = 15; %basal time constant
tauL = 5*tau; %time constant of lateral inhibition
%tauS = tau/5;
dt = 0.1; %step
t = (0:dt:800)'; %time [ms]
D = length(t); %number of samples
%% initialization of the structure
%C: cortex
Nc = 4; %neurons in the cortex
C = zeros(Nc,D); % activity of neurons
Uc = zeros(Nc,D); %input to the sigmoidal function
Ul = zeros(Nc,D); %contribution from the lateral inhibition
%Go: striatum, Go
Go = zeros(Nc,D);
Ugo = zeros(Nc,D);
%NoGo: striatum, No-Go
NoGo = zeros(Nc,D);
Unogo = zeros(Nc,D);
%Gpe: globus pallidus pars externa
Gpe = zeros(Nc,D);
Ugpe = zeros(Nc,D);
%Gpi: globus pallidus pars interna
Gpi = zeros(Nc,D);
Ugpi = zeros(Nc,D);
%T: thalamus
T = zeros(Nc,D);
Ut = zeros(Nc,D);
%STN: sub-thalamic nucleus
STN = zeros(1,D);
Ustn = zeros(1,D);
%E: energy (it is an index of conflict in the cortex)
E = zeros(1,D);
%ChI: cholinergic interneuron
ChI = zeros(1,D);
Uchi = zeros(1,D);
%input DA+ACh to Go and NoGo
IGo_DA_Ach = zeros(Nc,D);
INoGo_DA_Ach = zeros(Nc,D);
%% initialization of synapses
%weights from stimulus to cortex
Wcs = 1*ones(4,4); %0.2 extradiag, 1.1 su diag
%lateral inhibition
L = -1.2*ones(Nc,Nc)+diag(1.2*ones(Nc,1)); %tolto autoanello
%weights from thalamus to cortex
Wct = 4*diag(ones(Nc,1)); %diagonale
%%%%%%%%%%%% the following weights are passed as input parameters since subject to learning %%%%%%%%%%%%%%%%%%%%%
% %weights from cortex to Go: Wgc (diagonal)
% %weights from stimulus to Go: Wgs
%
% %weights from cortex to NoGo: Wnc (diagonal)
% %weights from stimulus to NoGo: Wns
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%weights from No-Go to Gpe (inhibitory)
Wen = -2.2*diag(ones(Nc,1)); %diagonal
%weights from Gpe to Gpi (inhibitory)
Wie = -3*diag(ones(Nc,1)); %diagonal
%weights from Go to Gpi (inhibitory)
Wig = -36*diag(ones(Nc,1)); %diagonal
%weights from cortex to thalamus (excitatory)
Wtc = 3*diag(ones(Nc,1)); %diagonale
%weights from Gpi to thalamus (inibitoriy)
Wti = -3*diag(ones(Nc,1)); %diagonal
%%%%%%%%%%%%%%%%%%% weights from-to STN %%%%%%%%%%%%%%%%%%%
%weight from energy to STN (excitatory)
Ke = 7;
%weight from Gpe to STN (inibitory)
Kgpe = -1;
%weight from STN to Gpe (sinapsi excitatory)
Westn = 1;
%weight from STN to Gpi (excitatory)
Wistn = 30; %14
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%% weight from ChI to Go-NoGo %%%%%%%%%%%%%%%%%%%
%weight from ChI to Go(inhibitory)
wgchi = -1;
%weight from ChI to NoGo (excitatory)
wnchi = 1;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%% gains %%%%%%%%%%%%%%%%%%%
%gain from DA to Go (excitation)
Ugo_trigger = 1.074;
alpha = 0.75;
%gain from DA to No-Go (inhibition)
beta = -1;
%gain from DA a Chi (inibition)
gamma = -0.5; %-1
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% computation of the dynamics: low-pass filter + sigmoid
%parameters of the sigmoid
a = 4;
U0 = 1.0;
%tonic activity Gpe
Igpe = 1.0;
%tonic activity Gpi
Igpi = 3;
%tonic activity ChI
Ichi = 1.00;
%reward o punishment
%r = NaN no reward o initialization
%r = 1 reward
%r = -1 punishment
r = NaN;
%time of reward o punishment
%k_reward = NaN no reward o initialization
k_reward = NaN;
%time pattern of phasic dopamine
latency = 100; %ms, physiological values 50-110 ms (Schultz 1998)
klatency = ceil(latency/dt);
duration = 50; %ms, physiological values <200 ms (Schultz 1998)
kduration = ceil(duration/dt);
%% initial conditions
Ugpe(:,1) = Igpe;
Ugpi(:,1) = Igpi;
Uchi(1) = Ichi+gamma*Dop_tonic;
Ns = 4;
r_story = [];
noise1 = 0.2*randn(Nc,1);
for k = 1:D
%PRODUCTION OF PHASIC DOPAMINE
if isnan(r); %nothing
dop_phasic = 0;
else %k_reward initialization
if k>=k_reward+klatency && k<=k_reward+klatency+kduration %time interval of dopamine change
if r==1 %reward
delta_Dop = Dop_tonic;
elseif r==-1 %punishment
delta_Dop = -Dop_tonic;
end
dop_phasic = delta_Dop;
else %time interval without dopamine change
dop_phasic = 0;
end
end
DA = Dop_tonic+dop_phasic;
%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%% sigmoids
C(:,k) = 1./(1+exp(-a*(Uc(:,k)-U0)));
Go(:,k) = 1./(1+exp(-a*(Ugo(:,k)-U0)));
NoGo(:,k) = 1./(1+exp(-a*(Unogo(:,k)-U0)));
Gpe(:,k) = 1./(1+exp(-a*(Ugpe(:,k)-U0)));
Gpi(:,k) = 1./(1+exp(-a*(Ugpi(:,k)-U0)));
ChI(k) = 1./(1+exp(-a*(Uchi(k)-U0)));
T(:,k) = 1./(1+exp(-a*(Ut(:,k)-U0)));
STN(k) = 1./(1+exp(-a*(Ustn(k)-U0)));
%%%%
%REWARD, PUNISHMENT OR NOTHING
%I look for a winner; in case, I must produce reward or punishment
if isnan(Correct_winner)==0;
if isnan(r); %I still have no result
C_winner_list = find(C(:,k)>=0.9);
lost = isempty(C_winner_list);
if lost==0 %there is al least one winner
n_C_winner = length(C_winner_list);
if n_C_winner==1 %there is just one winner, I give reward or punishment
k_reward = k; %the time of victory
if C_winner_list==Correct_winner
r = 1; %reward
else
r = -1; %punishment
end
r_story = [r_story; r];
end
end
end
end
%energy in the cortex (index of a conflict)
for i = 1:Nc
for j = i:Nc
E(k) = E(k)+C(i,k)*C(j,k);
end
end
E(k) = E(k)-(sum(C(:,k).^2));
%%%% differential equations with Euler method
Ul(:,k+1) = Ul(:,k)+dt/tauL*(-Ul(:,k)+L*C(:,k));
Uc(:,k+1) = Uc(:,k)+dt/tau*(-Uc(:,k)+Wcs*S+Ul(:,k)+Wct*T(:,k)+noise1);
IGo_DA_Ach(:,k) = alpha*DA*(Go(:,k)-0.35)+wgchi*ChI(k);
INoGo_DA_Ach(:,k) = (beta*DA+wnchi*ChI(k))*ones(Nc,1);
Ugo(:,k+1) = Ugo(:,k)+dt/tau*(-Ugo(:,k)+Wgs*S+Wgc*C(:,k)+IGo_DA_Ach(:,k));
Unogo(:,k+1) = Unogo(:,k)+dt/tau*(-Unogo(:,k)+Wns*S+Wnc*C(:,k)+INoGo_DA_Ach(:,k));
Ugpe(:,k+1) = Ugpe(:,k)+dt/tau*(-Ugpe(:,k)+Wen*NoGo(:,k)+Westn*STN(k)+Igpe);
Ugpi(:,k+1) = Ugpi(:,k)+dt/tau*(-Ugpi(:,k)+Wig*Go(:,k)+Wie*Gpe(:,k)+Wistn*STN(k)+Igpi);
Uchi(k+1) = Uchi(k)+dt/tau*(-Uchi(k)+Ichi+gamma*DA);
Ut(:,k+1) = Ut(:,k)+dt/tau*(-Ut(:,k)+Wti*Gpi(:,k)+Wtc*C(:,k));
Ustn(k+1) = Ustn(k)+dt/tau*(-Ustn(k)+Ke*E(k)+Kgpe*sum(Gpe(:,k)));
%%%%
end
%% HEBB RULE
%inizialization
S_th = 0.5*ones(length(S),1);
C_th = 0.5*ones(Nc,1);
Go_th = 0.5*ones(Nc,1);
NoGo_th = 0.5*ones(Nc,1);
gain = 0.01;
delta_Wgc = zeros(Nc,Nc);
delta_Wgs = zeros(Nc,Nc);
delta_Wnc = zeros(Nc,Nc);
delta_Wns = zeros(Nc,Nc);
if isnan(r)==0 % we had reward o punishment
if ((k_reward+klatency)>D)==0 &&((k_reward+klatency+kduration)>D)==0 %I do not train the net if still varying
%% cambio
if r==1 %reward
%maximum of the winner Go and minimum of the loosing Go
%minimum of all NoGO
val_Go(1) = min(Go(1,(k_reward+klatency):(k_reward+klatency+kduration)));
val_Go(2) = min(Go(2,(k_reward+klatency):(k_reward+klatency+kduration)));
val_Go(3) = min(Go(3,(k_reward+klatency):(k_reward+klatency+kduration)));
val_Go(4) = min(Go(4,(k_reward+klatency):(k_reward+klatency+kduration)));
val_Go(C_winner_list) = max(Go(C_winner_list,(k_reward+klatency):(k_reward+klatency+kduration)));
val_NoGo(1) = min(NoGo(1,(k_reward+klatency):(k_reward+klatency+kduration)));
val_NoGo(2) = min(NoGo(2,(k_reward+klatency):(k_reward+klatency+kduration)));
val_NoGo(3) = min(NoGo(3,(k_reward+klatency):(k_reward+klatency+kduration)));
val_NoGo(4) = min(NoGo(4,(k_reward+klatency):(k_reward+klatency+kduration)));
else %punishment
%minimum of all GO
%maximum of all NoGo
val_Go(1) = min(Go(1,(k_reward+klatency):(k_reward+klatency+kduration)));
val_Go(2) = min(Go(2,(k_reward+klatency):(k_reward+klatency+kduration)));
val_Go(3) = min(Go(3,(k_reward+klatency):(k_reward+klatency+kduration)));
val_Go(4) = min(Go(4,(k_reward+klatency):(k_reward+klatency+kduration)));
val_NoGo(1) = max(NoGo(1,(k_reward+klatency):(k_reward+klatency+kduration)));
val_NoGo(2) = max(NoGo(2,(k_reward+klatency):(k_reward+klatency+kduration)));
val_NoGo(3) = max(NoGo(3,(k_reward+klatency):(k_reward+klatency+kduration)));
val_NoGo(4) = max(NoGo(4,(k_reward+klatency):(k_reward+klatency+kduration)));
end
%%
for l=1:Nc
delta_Wgc(l,l) = gain*max(0,(C(l,k_reward)-C_th(l))).*(val_Go(l)-Go_th(l));
delta_Wnc(l,l) = gain*max(0,(C(l,k_reward)-C_th(l))).*(val_NoGo(l)-NoGo_th(l));
end
for row=1:Nc
for col=1:Nc
delta_Wgs(row,col) = gain*(val_Go(row)-Go_th(row))*max(0,(S(col)-S_th(col)));
delta_Wns(row,col) = gain*(val_NoGo(row)-NoGo_th(row))*max(0,(S(col)-S_th(col)));
end
end
end
end
Wgc_post = Wgc+delta_Wgc;
Wgs_post = Wgs+delta_Wgs;
Wnc_post = Wnc+delta_Wnc;
Wns_post = Wns+delta_Wns;
%% CONTROL ON SYNAPSES
%synapse saturation
delta_opt = 0.2;
sinapsi_ecc_max = 1.2;
max_ecc = sinapsi_ecc_max+delta_opt;
Wgc_opt_max = max_ecc*diag(ones(Nc,1)); %diagonal
Wgc_opt_min = zeros(Nc,Nc);
Wnc_opt_max = max_ecc*diag(ones(Nc,1)); %diagonal
Wnc_opt_min = zeros(Nc,Nc);
Wgs_opt_max = max_ecc*ones(Nc,Nc); %full
Wgs_opt_min = zeros(Nc,Nc);
Wns_opt_max = max_ecc*ones(Nc,Nc); %full
Wns_opt_min = zeros(Nc,Nc);
[rowgc,colgc,valgc] = find(Wgc_post>Wgc_opt_max);
if isempty(rowgc)==0
for i=1:length(rowgc)
Wgc_post(rowgc(i),colgc(i)) = Wgc_opt_max(rowgc(i),colgc(i));
end
end
[rowgc,colgc,valgc] = find(Wgc_post<Wgc_opt_min);
if isempty(rowgc)==0
for i=1:length(rowgc)
Wgc_post(rowgc(i),colgc(i)) = Wgc_opt_min(rowgc(i),colgc(i));
end
end
[rownc,colnc,valnc] = find(Wnc_post>Wnc_opt_max);
if isempty(rownc)==0
for i=1:length(rownc)
Wnc_post(rownc(i),colnc(i)) = Wnc_opt_max(rownc(i),colnc(i));
end
end
[rownc,colnc,valgnc] = find(Wnc_post<Wnc_opt_min);
if isempty(rownc)==0
for i=1:length(rownc)
Wnc_post(rownc(i),colnc(i)) = Wnc_opt_min(rownc(i),colnc(i));
end
end
[rowgs,colgs,valgs] = find(Wgs_post>Wgs_opt_max);
if isempty(rowgs)==0
for i=1:length(rowgs)
Wgs_post(rowgs(i),colgs(i)) = Wgs_opt_max(rowgs(i),colgs(i));
end
end
[rowgs,colgs,valgs] = find(Wgs_post<Wgs_opt_min);
if isempty(rowgs)==0
for i=1:length(rowgs)
Wgs_post(rowgs(i),colgs(i)) = Wgs_opt_min(rowgs(i),colgs(i));
end
end
[rowns,colns,valns] = find(Wns_post>Wns_opt_max);
if isempty(rowns)==0
for i=1:length(rowns)
Wns_post(rowns(i),colns(i)) = Wns_opt_max(rowns(i),colns(i));
end
end
[rowns,colns,valns] = find(Wns_post<Wns_opt_min);
if isempty(rowns)==0
for i=1:length(rowns)
Wns_post(rowns(i),colns(i)) = Wns_opt_min(rowns(i),colns(i));
end
end
