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cont_ca_sampler.m
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cont_ca_sampler.m
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function SAMPLES = cont_ca_sampler(Y,params)
% Continuous time sampler
% Y data (normalized in [0,1])
% P intialization parameters (discrete time constant P.g required)
% params parameters structure
% params.g discrete time constant(s) (estimated if not provided)
% params.sn initializer for noise (estimated if not provided)
% params.b initializer for baseline (estimated if not provided)
% params.c1 initializer for initial concentration (estimated if not provided)
% params.Nsamples number of samples after burn in (default 500)
% params.B number of burn in samples (default 200)
% params.marg flag for marginalized sampler (default 0)
% params.upd_gam flag for updating gamma (default 0)
% params.gam_step number of samples after which gamma is updated (default 50)
% params.std_move standard deviation of shifting kernel (default 3*Dt)
% params.add_move number of add moves per iteration (default T/100)
% params.init initial sample
% params.f imaging rate (default 1)
% params.p order of AR model (p == 1 or p == 2, default 1)
% params.defg default discrete time constants in case constrained_foopsi cannot find stable estimates
% params.TauStd standard deviation for time constants in continuous time (default [0.2,2])
% output struct SAMPLES
% spikes T x Nsamples matrix with spikes samples
% bp Nsamples x 1 vector with samples for spiking prior probability
% Am Nsamples x 1 vector with samples for spike amplitude
% If marginalized sampler is used
% Cb posterior mean and sd for baseline
% Cin posterior mean and sd for initial condition
% else
% Cb Nsamples x 1 vector with samples for baseline
% Cin Nsamples x 1 vector with samples for initial concentration
% sn Nsamples x 1 vector with samples for noise variance
% If gamma is updated
% g Nsamples x p vector with the gamma updates
% Author: Eftychios A. Pnevmatikakis and Josh Merel
Y = Y(:);
T = length(Y);
% define default parameters
defparams.g = [];
defparams.sn = [];
defparams.b = [];
defparams.c1 = [];
defparams.Nsamples = 400;
defparams.B = 200;
defparams.marg = 0;
defparams.upd_gam = 1;
defparams.gam_step = 1;
defparams.A_lb = 0.1*range(Y);
defparams.b_lb = quantile(Y,0.01);
defparams.c1_lb = 0;
defparams.std_move = 3;
defparams.add_move = ceil(T/100);
defparams.init = [];
defparams.f = 1;
defparams.p = 1;
defparams.defg = [0.6,0.95];
defparams.TauStd = [.1,1];
defparams.print_flag = 0;
if nargin < 2
params = defparams;
else
if ~isfield(params,'g'); params.g = defparams.g; end
if ~isfield(params,'sn'); params.sn = defparams.sn; end
if ~isfield(params,'b'); params.b = defparams.b; end
if ~isfield(params,'c1'); params.c1 = defparams.c1; end
if ~isfield(params,'Nsamples'); params.Nsamples = defparams.Nsamples; end
if ~isfield(params,'B'); params.B = defparams.B; end
if ~isfield(params,'marg'); params.marg = defparams.marg; end
if ~isfield(params,'upd_gam'); params.upd_gam = defparams.upd_gam; end
if ~isfield(params,'gam_step'); params.gam_step = defparams.gam_step; end
if ~isfield(params,'std_move'); params.std_move = defparams.std_move; end
if ~isfield(params,'add_move'); params.add_move = defparams.add_move; end
if ~isfield(params,'init'); params.init = defparams.init; end
if ~isfield(params,'f'); params.f = defparams.f; end
if ~isfield(params,'p'); params.p = defparams.p; end
if ~isfield(params,'defg'); params.defg = defparams.defg; end
if ~isfield(params,'TauStd'); params.TauStd = defparams.TauStd; end
if ~isfield(params,'A_lb'); params.A_lb = defparams.A_lb; end
if ~isfield(params,'b_lb'); params.b_lb = defparams.b_lb; end
if ~isfield(params,'c1_lb'); params.c1_lb = defparams.c1_lb; end
if ~isfield(params,'print_flag'); params.print_flag = defparams.print_flag; end
end
Dt = 1; % length of time bin
marg_flag = params.marg;
gam_flag = params.upd_gam;
gam_step = params.gam_step;
std_move = params.std_move;
add_move = params.add_move;
if isempty(params.g)
p = params.p;
else
p = length(params.g); % order of autoregressive process
end
if isempty(params.init)
fprintf('Initializing using noise constrained FOOPSI... ');
params.init = get_initial_sample(Y,params);
fprintf('done. \n');
end
SAM = params.init;
g = SAM.g(:)';
if g == 0
gr = [0.9,0.1];
pl = poly(gr);
g = -pl(2:end);
p = 2;
end
gr = sort(roots([1,-g(:)']));
if p == 1; gr = [0,gr]; end
if any(gr<0) || any(~isreal(gr)) || length(gr)>2 || max(gr)>0.998
gr = params.defg;
end
tau = -Dt./log(gr);
tau1_std = max(tau(1)/100,params.TauStd(1));
tau2_std = min(tau(2)/5,params.TauStd(2));
ge = max(gr).^(0:T-1)';
if p == 1
G1 = sparse(1:T,1:T,Inf*ones(T,1));
elseif p == 2
G1 = spdiags(ones(T,1)*[-min(gr),1],[-1:0],T,T);
else
error('This order of the AR process is currently not supported');
end
G2 = spdiags(ones(T,1)*[-max(gr),1],[-1:0],T,T);
sg = SAM.sg;
SAM = params.init;
spiketimes_ = SAM.spiketimes_;
lam_ = SAM.lam_;
A_ = SAM.A_*diff(gr);
b_ = SAM.b_;
C_in = SAM.C_in;
s_1 = sparse(ceil(spiketimes_/Dt),1,exp((spiketimes_ - Dt*ceil(spiketimes_/Dt))/tau(1)),T,1);
s_2 = sparse(ceil(spiketimes_/Dt),1,exp((spiketimes_ - Dt*ceil(spiketimes_/Dt))/tau(2)),T,1);
if ~isfield(params,'prec') % FN % (from Eftychios: prec specifies to what extent you want to discard the long slowly decaying tales of the ca response. Try setting it e.g., to 5e-2 instead of 1e-2 to speed things up.)
prec = 1e-2; % precision
else
prec = params.prec; %5e-2; % FN
end
ef_d = exp(-(0:T)/tau(2));
if p == 1
h_max = 1; % max value of transient
ef_h = [0,0];
e_support = find(ef_d<prec*h_max,1,'first');
if isempty(e_support);
e_support = T;
end
e_support = min(e_support,T);
else
t_max = (tau(1)*tau(2))/(tau(2)-tau(1))*log(tau(2)/tau(1)); %time of maximum
h_max = exp(-t_max/tau(2)) - exp(-t_max/tau(1)); % max value of transient
ef_h = -exp(-(0:T)/tau(1));
e_support = find(ef_d-ef_h<prec*h_max,1,'first');
if isempty(e_support);
e_support = T;
end
e_support = min(e_support,T);
end
ef_h = ef_h(1:min(e_support,length(ef_h)))/diff(gr);
ef_d = ef_d(1:e_support)/diff(gr);
ef = [{ef_h ef_d};{cumsum(ef_h.^2) cumsum(ef_d.^2)}];
B = params.B;
N = params.Nsamples + B;
if p == 1; G1sp = zeros(T,1); else G1sp = G1\s_1(:); end
Gs = (-G1sp(:)+G2\s_2(:))/diff(gr);
ss = cell(N,1);
lam = zeros(N,1);
Am = zeros(N,1);
ns = zeros(N,1);
Gam = zeros(N,2);
if ~marg_flag
Cb = zeros(N,1);
Cin = zeros(N,1);
SG = zeros(N,1);
end
Sp = .1*range(Y)*eye(3); % prior covariance
Ld = inv(Sp);
lb = [params.A_lb/h_max*diff(gr),params.b_lb,params.c1_lb]'; % lower bound for [A,Cb,Cin]
A_ = max(A_,1.1*lb(1));
mu = [A_;b_;C_in]; % prior mean
Ns = 15; % Number of HMC samples
Ym = Y - ones(T,1)*mu(2) - ge*mu(3);
mub = zeros(2,1);
Sigb = zeros(2,2);
%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Extra tau-related params
%%%%%%%%%%%%%%%%%%%%%%%%%%%
tauMoves = [0 0];
tau_min = 0;
tau_max = 500;
%%%%%%%%%%%%%%%%%%%%%%%%%%%
for i = 1:N
if gam_flag
Gam(i,:) = tau;
end
sg_ = sg;
rate = @(t) lambda_rate(t,lam_);
[spiketimes, ~] = get_next_spikes(spiketimes_(:)',A_*Gs',Ym',ef,tau,sg_^2, rate, std_move, add_move, Dt, A_);
spiketimes_ = spiketimes;
spiketimes(spiketimes<0) = -spiketimes(spiketimes<0);
spiketimes(spiketimes>T*Dt) = 2*T*Dt - spiketimes(spiketimes>T*Dt);
trunc_spikes = ceil(spiketimes/Dt);
trunc_spikes(trunc_spikes == 0) = 1;
s_1 = sparse(trunc_spikes,1,exp((spiketimes_ - Dt*trunc_spikes)/tau(1)),T,1);
s_2 = sparse(trunc_spikes,1,exp((spiketimes_ - Dt*trunc_spikes)/tau(2)),T,1);
if p == 1; G1sp = zeros(T,1); else G1sp = G1\s_1(:); end
Gs = (-G1sp+G2\s_2(:))/diff(gr);
ss{i} = spiketimes;
nsp = length(spiketimes);
ns(i) = nsp;
lam(i) = nsp/(T*Dt);
lam_ = lam(i);
AM = [Gs,ones(T,1),ge];
L = inv(Ld + AM'*AM/sg^2);
mu_post = (Ld + AM'*AM/sg^2)\(AM'*Y/sg^2 + Sp\mu);
if ~marg_flag
x_in = [A_;b_;C_in];
if any(x_in < lb)
x_in = max(x_in,1.1*lb);
end
if all(isnan(L(:))) % FN added to avoid error in R = chol(L) in HMC_exact2 due to L not being positive definite. It happens when isnan(det(Ld + AM'*AM/sg^2)), ie when Ld + AM'*AM/sg^2 is singular (not invertible).
Am(i) = NaN;
Cb(i) = NaN;
Cin(i) = NaN';
else
[temp,~] = HMC_exact2(eye(3), -lb, L, mu_post, 1, Ns, x_in);
Am(i) = temp(1,Ns);
Cb(i) = temp(2,Ns);
Cin(i) = temp(3,Ns)';
end
A_ = Am(i);
b_ = Cb(i);
C_in = Cin(i);
Ym = Y - b_ - ge*C_in;
res = Ym - A_*Gs;
sg = 1./sqrt(gamrnd(1+T/2,1/(0.1 + sum((res.^2)/2))));
SG(i) = sg;
else
repeat = 1;
while repeat
A_ = mu_post(1) + sqrt(L(1,1))*randn;
repeat = (A_<0);
end
Am(i) = A_;
if i > B
mub = mub + mu_post(2+(0:p));
Sigb = Sigb + L(2+(0:p),2+(0:p));
end
end
if gam_flag
if mod(i-B,gam_step) == 0 % update time constants
if p >= 2 % update rise time constant
logC = -norm(Y(:) - AM*[A_;b_;C_in])^2;
tau_ = tau;
tau_temp = tau_(1)+(tau1_std*randn);
while tau_temp >tau(2) || tau_temp<tau_min
tau_temp = tau_(1)+(tau1_std*randn);
end
tau_(1) = tau_temp;
gr_ = exp(Dt*(-1./tau_));
s_1_ = sparse(trunc_spikes,1,exp((spiketimes_ - Dt*trunc_spikes)/tau_(1)),T,1);
G1_ = spdiags(ones(T,1)*[-min(gr_),1],[-1:0],T,T);
Gs_ = (-G1_\s_1_(:)+G2\s_2(:))/diff(gr_);
logC_ = -norm(Y(:)-A_*Gs-b_-C_in*ge)^2;
%accept or reject
prior_ratio = 1;
% prior_ratio = gampdf(tau_(2),12,1)/gampdf(tau(2),12,1);
ratio = exp((logC_-logC)/(2*sg^2))*prior_ratio;
if rand < ratio %accept
tau = tau_;
G1 = G1_; %c = c_;
Gs = Gs_; gr = gr_; s_1 = s_1_;
tauMoves = tauMoves + [1 1];
else
tauMoves = tauMoves + [0 1];
end
end
%%%%%%%%%%%%%%%%%%%%%%%
% next update decay time constant
%%%%%%%%%%%%%%%%%%%%%%%
%initial logC
logC = -norm(Y(:)-A_*Gs-b_-C_in*ge)^2;
tau_ = tau;
tau_temp = tau_(2)+(tau2_std*randn);
while tau_temp>tau_max || tau_temp<tau_(1)
tau_temp = tau_(2)+(tau2_std*randn);
end
tau_(2) = tau_temp;
s_2_ = sparse(trunc_spikes,1,exp((spiketimes_ - Dt*trunc_spikes)/tau_(2)),T,1);
gr_ = exp(Dt*(-1./tau_));
ge_ = max(gr_).^(0:T-1)';
G2_ = spdiags(ones(T,1)*[-max(gr_),1],[-1:0],T,T);
if p == 1; G1sp = zeros(T,1); else G1sp = G1\s_1(:); end
Gs_ = (-G1sp+G2_\s_2_(:))/diff(gr_);
logC_ = -norm(Y(:)-A_*Gs_-b_-C_in*ge_)^2;
%accept or reject
prior_ratio = 1;
% prior_ratio = gampdf(tau_(2),12,1)/gampdf(tau(2),12,1);
ratio = exp((1./(2*sg^2)).*(logC_-logC))*prior_ratio;
if rand<ratio %accept
tau = tau_;
%c = c_;
ge = ge_; Gs = Gs_; G2 = G2_; gr = gr_; s_2 = s_2_;
tauMoves = tauMoves + [1 1];
else
tauMoves = tauMoves + [0 1];
end
ef_d = exp(-(0:T)/tau(2));
if p == 1
h_max = 1; % max value of transient
ef_h = [0,0];
e_support = find(ef_d<prec*h_max,1,'first');
if isempty(e_support);
e_support = T;
end
e_support = min(e_support,T);
else
t_max = (tau(1)*tau(2))/(tau(2)-tau(1))*log(tau(2)/tau(1)); %time of maximum
h_max = exp(-t_max/tau(2)) - exp(-t_max/tau(1)); % max value of transient
ef_h = -exp(-(0:T)/tau(1));
e_support = find(ef_d-ef_h<prec*h_max,1,'first');
if isempty(e_support);
e_support = T;
end
e_support = min(e_support,T);
end
ef_h = ef_h(1:min(e_support,length(ef_h)))/diff(gr);
ef_d = ef_d(1:e_support)/diff(gr);
ef = [{ef_h ef_d};{cumsum(ef_h.^2) cumsum(ef_d.^2)}];
end
end
if params.print_flag && mod(i,100)==0
fprintf('%i out of total %i samples drawn \n', i, N);
end
end
if marg_flag
mub = mub/(N-B);
Sigb = Sigb/(N-B)^2;
end
if marg_flag
SAMPLES.Cb = [mub(1),sqrt(Sigb(1,1))];
SAMPLES.Cin = [mub(1+(1:p)),sqrt(diag(Sigb(1+(1:p),1+(1:p))))];
else
SAMPLES.Cb = Cb(B+1:N);
SAMPLES.Cin = Cin(B+1:N,:);
SAMPLES.sn2 = SG(B+1:N).^2;
end
SAMPLES.ns = ns(B+1:N);
SAMPLES.ss = ss(B+1:N);
SAMPLES.ld = lam(B+1:N);
SAMPLES.Am = Am(B+1:N);
if gam_flag
SAMPLES.g = Gam(B+1:N,:);
end
SAMPLES.params = params.init;