The first assumption is the free energy principle, which leads to active inference in the embodied context of action. The second assumption follows from need to minimise uncertainty about hidden states in the world. The third assumption is motivated easily by noting that the world is dynamic and nonlinear and that hierarchical causal structure emerges inevitably from a separation of temporal scales (Ginzburg and Landau 1950; Haken 1983). Finally, the fourth assumption is the Laplace assumption that—in terms of neural codes—leads to the Laplace code that is arguably the simplest and most flexible of all neural codes (Friston 2009).

Given these assumptions, one can simulate many neuronal processes by specifying a particular generative model. The resulting perception and action are specified completely by the above assumptions and can be implemented in a biologically plausible way; as described in many previous applications—see Table 1. In brief, the simulations in Table 1 use differential equations that minimise the free energy of sensory input using a generalised descent—see Fig. 1 and (Friston et al. 2010).

These coupled differential equations describe perception and action, respectively, and say that internal brain states—posterior expectations about hidden states—and action change in the direction that reduces free energy. The first is known as (generalised) predictive coding and has the same form as Bayesian (Kalman-Bucy) filters used in time series analysis; see also (Rao and Ballard 1999). The first term in Eq. (1) is a prediction based upon a time derivative operator. The second term—usually expressed as a mixture of prediction errors—ensures the changes in posterior expectations are Bayes-optimal predictions about hidden states of the world. The second differential equation says that action also minimises free energy—noting that free energy depends on action through sensory states. The differential equations in (1) are coupled because sensory input depends upon action, which depends upon perception through the posterior expectations. This circular dependency leads to a sampling of sensory input that is both predicted and predictable, thereby minimising free energy and surprise. To perform neuronal simulations it is only necessary to integrate or solve Eq. (1) to simulate the neuronal dynamics that encode posterior expectations and ensuing action. Figure 3 presents a simulation of saccadic eye movements, using prior expectations that lead to salient sampling. This is similar to the handwriting example in Fig. 2; however, eye movements are attracted not to points encoded by a central pattern generator but to locations that have the greatest salience. Here, salience is a function of location in visual space and reports the posterior confidence in current beliefs about the cause of sensory input that would be afforded by fictive sampling from that location.

The ensuing active inference can be regarded as a formal example of active vision (Wurtz et al. 2011)—sometimes described in enactivist terms as visual palpation (O’Regan and Noë 2001) and illustrates a number of key points. First, it discloses the nature of evidence accumulation in selecting a hypothesis or percept the best explains sensory data. Figure 3 shows that this proceeds over two timescales—within and between saccades. Within-saccade accumulation is evident even during the initial fixation, with further stepwise decreases in uncertainty as salient information is sampled. The within-saccade accumulation is formally related to evidence accumulation as described in models of perceptual discrimination (Gold and Shadlen 2003; Churchland et al. 2011). The transient changes in posterior expectations, shortly after each saccade, reflect the fact that new data are being generated as the eye sweeps towards its new target location. It is important to note that the agent is not just predicting visual input, but also how input changes with eye movements—this induces an increase in posterior uncertainty during the fast phase of the saccade. However, due to the veracity of the posterior beliefs, the posterior confidence shrinks again when the saccade reaches its target location. This shrinkage is usually to a smaller level than in the preceding saccade.