Fig. 2: Embedded Chimera state from network output using the FORCE method.
A In the FORCE method an initial sparse matrix ω0 initializes the neurons to chaotic firing rates. B A secondary set of weights ηd⊤ is learned online with the Recursive Least Squares technique. These learned weights are added to the initial sparse matrix, leading to the firing rates r(t) which are used to compute the network output. For both matrices in panels A and B blue and red edges (dark and light gray for b/w printing) indicate respectively excitatory (positive) or inhibitory (negative) connections. Edge thickness represents the weight of each connection. C The network output is given by d⊤r(t), which is a s x nt matrix (s is the total number of supervisors). Each network output column i is nt time units long and is a linear combination of the firing rates r1(t), ⋯, rN(t) with diN as coefficients. D Network output \(\cos {{{\hat{{{{{\boldsymbol{\phi }}}}}}}}}(t)\) (purple or dark gray for b/w printing) and \(\cos {{{\hat{{{{{\boldsymbol{\theta }}}}}}}}}(t)\) (orange or light gray for b/w printing) and the correspondent supervisors (black). E Embedded chimera from network output: \({{{\hat{{{{{\boldsymbol{\phi }}}}}}}}}(t)=\arctan \left[\frac{\sin {{{\hat{{{{{\boldsymbol{\phi }}}}}}}}}(t)}{\cos {{{\hat{{{{{\boldsymbol{\phi }}}}}}}}}(t)}\right]\) and \(\hat{{{{{{{{\boldsymbol{\theta }}}}}}}}}(t)=\arctan \left[\frac{\sin {{{\hat{{{{{\boldsymbol{\theta }}}}}}}}}(t)}{\cos {{{\hat{{{{{\boldsymbol{\theta }}}}}}}}}(t)}\right]\). F Mean phase velocity profile for the embedded chimera (identical profiles as in Fig. 1F). G Trajectories of the order parameter for the embedded chimera (identical trajectories as in Fig. 1G).
