You are an expert in writing code in many different languages. Your goal is to perform code completion for the following code, keeping in mind the rest of the code and the file meta data.
    Metadata details:
    filename: spm_int_ode.m
    path of file: ./repos/spm
    
    the code of the file until where you have to start completion: function [y] = spm_int_ode(P,M,U)
% Integrate a MIMO nonlinear system (using classical ODE solvers)
% FORMAT [y] = spm_int_ode(P,M,U)
% P   - model parameters
% M   - model structure
% U   - input structure or matrix
%
% y   - (v x l)  response y = g(x,u,P)
%__________________________________________________________________________
% Integrates the MIMO system described by
%
%    dx/dt = f(x,u,P,M)
%    y     = g(x,u,P,M)
%
% using an Runge-Kutta(4,5) scheme over the times implicit in the input.
% ode45 is based on an explicit Runge-Kutta (4,5) formula, the Dormand-
% Prince pair. It is a one-step solver - in computing y(tn), it needs only 
% the solution at the immediately preceding time point y(tn-1). In general,
% ode45 is the best function to apply as a "first try" for most problems.
%
% ode113 is a variable order Adams-Bashforth-Moulton PECE solver. It may be
% more efficient than ode45 at stringent tolerances and when the ODE file 
% function is particularly expensive to evaluate. ode113 is a multi-step 
% solver - it normally needs the solutions at several preceding time points
% to compute the current solution
%
% see also ode45; ode113
%--------------------------------------------------------------------------
%
% SPM solvers or integrators
%
% spm_int_ode:  uses ode45 (or ode113) which are one and multi-step solvers
% respectively.  They can be used for any ODEs, where the Jacobian is
% unknown or difficult to compute; however, they may be slow.
%

    
function [y] = spm_int_ode(P,M,U)
% Integrate a MIMO nonlinear system (using classical ODE solvers)
% FORMAT [y] = spm_int_ode(P,M,U)
% P   - model parameters
% M   - model structure
% U   - input structure or matrix
%
% y   - (v x l)  response y = g(x,u,P)
%__________________________________________________________________________
% Integrates the MIMO system described by
%
%    dx/dt = f(x,u,P,M)
%    y     = g(x,u,P,M)
%
% using an Runge-Kutta(4,5) scheme over the times implicit in the input.
% ode45 is based on an explicit Runge-Kutta (4,5) formula, the Dormand-
% Prince pair. It is a one-step solver - in computing y(tn), it needs only 
% the solution at the immediately preceding time point y(tn-1). In general,
% ode45 is the best function to apply as a "first try" for most problems.
%
% ode113 is a variable order Adams-Bashforth-Moulton PECE solver. It may be
% more efficient than ode45 at stringent tolerances and when the ODE file 
% function is particularly expensive to evaluate. ode113 is a multi-step 
% solver - it normally needs the solutions at several preceding time points
% to compute the current solution
%
% see also ode45; ode113
%--------------------------------------------------------------------------
%
% SPM solvers or integrators
%
% spm_int_ode:  uses ode45 (or ode113) which are one and multi-step solvers
% respectively.  They can be used for any ODEs, where the Jacobian is
% unknown or difficult to compute; however, they may be slow.
%
% spm_int_J: uses an explicit Jacobian-based update scheme that preserves
% nonlinearities in the ODE: dx = (expm(dt*J) - I)*inv(J)*f.  If the
% equations of motion return J = df/dx, it will be used; otherwise it is
% evaluated numerically, using spm_diff at each time point.  This scheme is
% infallible but potentially slow, if the Jacobian is not available (calls
% spm_dx).
%
% spm_int_E: As for spm_int_J but uses the eigensystem of J(x(0)) to eschew
% matrix exponentials and inversion during the integration. It is probably
% the best compromise, if the Jacobian is not available explicitly.
%
% spm_int_B: As for spm_int_J but uses a first-order approximation to J
% based on J(x(t)) = J(x(0)) + dJdx*x(t).
%
% spm_int_U: like spm_int_J but only evaluates J when the input changes.
% This can be useful if input changes are sparse (e.g., boxcar functions).
% spm_int_U also has the facility to integrate delay differential equations
% if a delay operator is returned [f J D] = f(x,u,P,M)
%
% spm_int:   Fast integrator that uses a bilinear approximation to the 
% Jacobian evaluated using spm_bireduce. This routine will also allow for
% sparse sampling of the solution and delays in observing outputs
%
%__________________________________________________________________________

% Karl Friston
% Copyright (C) 2008-2022 Wellcome Centre for Human Neuroimaging


% convert U to U.u if necessary
%--------------------------------------------------------------------------
if ~isstruct(U),  U.u  = U; end