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A system responding to a stochastic driving signal can be interpreted as computing, by means of its dynamics, an implicit model of the environmental variables.
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β ⟨ W diss [ x t → x t + 1 ] ⟩ = β ( ⟨ E ( s t , x t + 1 ) ⟩ p ( s t , x t + 1 ) − ⟨ E ( s t , x t ) ⟩ p ( s t , x t ) ) − β ( ⟨ F neq [ p ( s t | x t + 1 ) ] ⟩ p ( x t + 1 ) − ⟨ F neq [ p ( s t | x t ) ] ⟩ p ( x t ) ) = H [ s t | x t + 1 ] − H [ s t | x t ] = I mem ( t ) − I pred ( t ) \beta\left<W_{\rm diss}[x_{t}\shortrightarrow x_{t+1}]\right>\\ =\beta\left({\hbox{$\left⟨\vbox to1.0pt{}\right.$}}E(s_{t},x_{t+1}){\hbox{$\left⟩\vbox to1.0pt{}\right.$}}_{p(s_{t},x_{t+1})}-{\hbox{$\left⟨\vbox to1.0pt{}\right.$}}E(s_{t},x_{t}){\hbox{$\left⟩\vbox to1.0pt{}\right.$}}_{p(s_{t},x_{t})}\right)\\ -\beta\left({\hbox{$\left⟨\vbox to1.0pt{}\right.$}}F_{{\rm neq}}[p(s_{t}|x_{t+1})]{\hbox{$\left⟩\vbox to1.0pt{}\right.$}}_{p(x_{t+1})}-{\hbox{$\left⟨\vbox to1.0pt{}\right.$}}F_{{\rm neq}}[p(s_{t}|x_{t})]{\hbox{$\left⟩\vbox to1.0pt{}\right.$}}_{p(x_{t})}\right)\\ =H[s_{t}|x_{t+1}]-H[s_{t}|x_{t}]=I_{{\rm mem}}(t)-I_{{\rm pred}}(t)
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