It turns out that the problems of inferring the causes of sensory input (perceptual inference) and learning the relationship between input and cause (perceptual learning) can be resolved using exactly the same principle. Specifically, both inference and learning rest on minimizing the brain's free energy, as defined in statistical physics. Furthermore, inference and learning can proceed in a biologically plausible fashion. Cortical responses can be seen as the brain’s attempt to minimize the free energy induced by a stimulus and thereby encode the most likely cause of that stimulus. Similarly, learning emerges from changes in synaptic efficacy that minimize the free energy, averaged over all stimuli encountered. The underlying scheme rests on empirical Bayes and hierarchical models of how sensory input is caused. The use of hierarchical models enables the brain to construct prior expectations in a dynamic and context-sensitive fashion. This scheme provides a principled way to understand many aspects of cortical organization and responses. The aim of this article is to encompass many apparently unrelated anatomical, physiological and psychophysical attributes of the brain within a single theoretical perspective.

In terms of cortical architectures, the theoretical treatment predicts that sensory cortex should be arranged hierarchically, that connections should be reciprocal and that forward and backward connections should show a functional asymmetry (forward connections are driving, whereas backward connections are both driving and modulatory). In terms of synaptic physiology, it predicts associative plasticity and, for dynamic models, spike-timing-dependent plasticity. In terms of electrophysiology, it accounts for classical and extra classical receptive field effects and long-latency or endogenous components of evoked cortical responses. It predicts the attenuation of responses encoding prediction error with perceptual learning and explains many phenomena such as repetition suppression, mismatch negativity (MMN) and the P300 in electroencephalography. In psychophysical terms, it accounts for the behavioural correlates of these physiological phenomena, for example, priming and global precedence. The final focus of this article is on perceptual learning as measured with the MMN and the implications for empirical studies of coupling among cortical areas using evoked sensory responses.

This article represents an attempt to understand evoked cortical responses in terms of models of perceptual inference and learning. The specific model considered here rests on empirical Bayes, in the context of generative models that are embodied in cortical hierarchies. This model can be regarded as a mathematical formulation of the longstanding notion that ‘our minds should often change the idea of its sensation into that of its judgment, and make one serve only to excite the other’ (Locke 1690). In a similar vein, Helmholtz (1860) distinguishes between perception and sensation. ‘It may often be rather hard to say how much from perceptions as derived from the sense of sight is due directly to sensation, and how much of them, on the other hand, is due to experience and training’ (see Pollen 1999). In short, there is a distinction between percepts, which are the products of recognizing the causes of sensory input and sensation per se. Recognition (i.e. inferring causes from sensation) is the inverse of generating sensory data from their causes. It follows that recognition rests on models, learned through experience, of how sensations are caused. In this article, we will consider hierarchical generative models and how evoked cortical responses can be understood as part of the recognition process. The particular recognition scheme we will focus on is based on empirical Bayes, where prior expectations are abstracted from the sensory data, using a hierarchical model of how those data were caused. The particular implementation of empirical Bayes we consider is predictive coding, where prediction error is used to adjust the state of the generative model until prediction error is minimized and the most likely causes of sensory input have been identified.

Conceptually, empirical Bayes and generative models are related to ‘analysis-by-synthesis’ (Neisser 1967). This approach to perception, drawn from cognitive psychology, involves adapting an internal model of the world to match sensory input and was suggested by Mumford (1992) as a way of understanding hierarchical neuronal processing. The idea is reminiscent of Mackay's epistemological automata (MacKay 1956) which perceive by comparing expected and actual sensory input (Rao 1999). These models emphasize the role of backward connections in mediating predictions of lower level input, based on the activity of higher cortical levels.

Recognition is simply the process of solving an inverse problem by jointly minimizing prediction error at all levels of the cortical hierarchy. The main point of this article is that evoked cortical responses can be understood as transient expressions of prediction error, which index some recognition process. This perspective accommodates many physiological and behavioural phenomena, for example, extra classical RF effects and repetition suppression in unit recordings, the MMN and P300 in ERPs, priming and global precedence effects in psychophysics. Critically, many of these emerge from the same basic principles governing inference with hierarchical generative models.

In a series of previous papers (Friston 2002, 2003), we have described how the brain might use empirical Bayes for perceptual inference. These papers considered other approaches to representational learning as special cases of generative models, starting with supervised learning and ending with empirical Bayes. The latter predicts many architectural features, such as a hierarchical cortical system, prevalent top-down backward influences and functional asymmetries between forward and backward connections seen in the real brain. The focus of previous work was on functional cortical architectures. This paper looks at evoked responses and the relevant empirical findings, in relation to predictions and theoretical constraints afforded by the same theory. This is probably more relevant for experimental studies. We will therefore take a little time to describe recent advances in modelling evoked responses in human cortical systems to show the detailed levels of characterization it is now possible to attain.

Initially, functional localization per se was not easy to demonstrate. For example, a meeting that took place on 4 August 1881 addressed the difficulties of attributing function to a cortical area, given the dependence of cerebral activity on underlying connections (Phillips et al. 1984). This meeting was entitled Localization of function in the cortex cerebri. Although accepting the results of electrical stimulation in dog and monkey cortex, Goltz considered that the excitation method was inconclusive because the behaviours elicited might have originated in related pathways or current could have spread to distant centres. In short, the excitation method could not be used to infer functional localization because localizationism discounted interactions or functional integration among different brain areas. It was proposed that lesion studies could supplement excitation experiments. Ironically, it was observations on patients with brain lesions some years later (see Absher & Benson 1993) that led to the concept of ‘disconnection syndromes’ and the refutation of localizationism as a complete or sufficient explanation of cortical organization. The cortical infrastructure supporting a single function may involve many specialized areas whose union is mediated by functional integration. Functional specialization and integration are not exclusive; they are complementary. Functional specialization is only meaningful in the context of functional integration and vice versa.

Figure 1 Schematic illustrating hierarchical structures in the brain and the distinction between forward, backward and lateral connections. This schematic is inspired by Mesulam's (1998) notion of sensory-fugal processing over ‘a core synaptic hierarchy, (more ...)

The notion that forward connections are concerned with the promulgation and segregation of sensory information is consistent with: (i) their sparse axonal bifurcation; (ii) patchy axonal terminations; and (iii) topographic projections. In contrast, backward connections are considered to have a role in mediating contextual effects and in the co-ordination of processing channels. This is consistent with: (i) their frequent bifurcation; (ii) diffuse axonal terminations; and (iii) more divergent topography (Salin & Bullier 1995; Crick & Koch 1998). Forward connections mediate their post-synaptic effects through fast AMPA (1.3–2.4ms decay) and GABA_A (6ms decay) receptors. Modulatory effects can be mediated by NMDA receptors. NMDA receptors are voltage-sensitive, showing nonlinear and slow dynamics (approximately 50ms decay). They are found predominantly in supragranular layers where backward connections terminate (Salin & Bullier 1995). These slow time constants again point to a role in mediating contextual effects that are more enduring than phasic sensory-evoked responses. The clearest evidence for the modulatory role of backward connections (that is mediated by ‘slow’ glutamate receptors) comes from corticogeniculate connections. In the cat LGN, cortical feedback is partly mediated by type 1 metabotropic glutamate receptors, which are located exclusively on distal segments of the relay-cell dendrites. Rivadulla et al. (2002) have shown that these backward afferents enhance the excitatory centre of the thalamic RF. ‘Therefore, cortex, by closing this corticofugal loop, is able to increase the gain of its thalamic input within a focal spatial window, selecting key features of the incoming signal.’

Angelucci et al. (2002a,b) used a combination of anatomical and physiological recording methods to determine the spatial scale of intra-areal V1 horizontal connections and inter-areal backward connections to V1. ‘Contrary to common beliefs, these (monosynaptic horizontal) connections cannot fully account for the dimensions of the surround field (of macaque V1 neurons). The spatial scale of feedback circuits from extrastriate cortex to V1 is, instead, commensurate with the full spatial range of centre-surround interactions. Thus these connections could represent an anatomical substrate for contextual modulation and global-to-local integration of visual signals.’

It should be noted that the hierarchical ordering of areas is a matter of debate and may be indeterminate. Based on computational neuroanatomic studies Hilgetag et al. (2000) conclude that the laminar hierarchical constraints presently available in the anatomical literature are ‘insufficient to constrain a unique ordering’ for any of the sensory systems analysed. However, basic hierarchical principles were evident. Indeed, the authors note, ‘All the cortical systems we studied displayed a significant degree of hierarchical organization’ with the visual and somato-motor systems showing an organization that was ‘surprisingly strictly hierarchical’.

In the post-developmental period, synaptic plasticity is an important functional attribute of connections in the brain and is thought to subserve perceptual and procedural learning and memory. This is a large and fascinating field that ranges from molecules to maps (e.g. Buonomano & Merzenich 1998; Martin et al. 2000). Changing the strength of connections between neurons is widely assumed to be the mechanism by which memory traces are encoded and stored in the central nervous system. In its most general form, the synaptic plasticity and memory hypothesis states that, ‘Activity-dependent synaptic plasticity is induced at appropriate synapses during memory formation and is both necessary and sufficient for the information storage underlying the type of memory mediated by the brain area in which that plasticity is observed’ (see Martin et al. 2000 for an evaluation of this hypothesis). A key aspect of this plasticity is that it is generally associative.

In summary, the anatomy and physiology of cortico-cortical connections suggest that forward connections are driving and commit cells to a prespecified response given the appropriate pattern of inputs. Backward connections, on the other hand, are less topographic and are in a position to modulate the responses of lower areas. Modulatory effects imply the postsynaptic response evoked by presynaptic input is modulated by, or interacts in a nonlinear way with, another input. This interaction depends on nonlinear synaptic or dendritic mechanisms. Finally, brain connections are not static but are changing at the synaptic level all the time. In many instances, this plasticity is associative. In §3, we describe a theoretical perspective, provided by generative models, that highlights the functional importance of hierarchies, backward connections, nonlinear coupling and associative plasticity.

This section introduces learning and inference based on empirical Bayes. A more detailed discussion can be found in Friston (2002, 2003). We will introduce the notion of generative models and a generic scheme for their estimation. This scheme uses expectation maximization (EM; an iterative scheme that estimates conditional expectations and maximum likelihoods of model parameters, in an E- and M-step, respectively). We show that predictive coding can be used to implement EM and, in the context of hierarchical generative models, is sufficient to implement empirical Bayesian inference.

(3.1)

The problem the brain has to contend with is to find a function of the inputs that recognizes the underlying causes. To do this, the brain must effectively undo the interactions to disclose contextually invariant causes. In other words, the brain must perform a nonlinear unmixing of causes and context. The key point here is that the nonlinear mixing may not be invertible and that the estimation of causes from input may be fundamentally ill-posed. For example, no amount of unmixing can discern the parts of an object that are occluded by another. The corresponding indeterminacy in probabilistic learning rests on the combinatorial explosion of ways in which stochastic generative models can generate input patterns (Dayan et al. 1995). In what follows, we consider the implications of this problem. Put simply, recognition of causes from sensory data is the inverse of generating data from causes. If the generative model is not invertible then recognition can only proceed if there is an explicit generative model in the brain. This speaks to the importance of backward connections that may embody this model.

(3.2)

The conditional density of the causes, given the inputs, are given by the recognition model, which is defined in terms of the recognition distribution

(3.3)

However, as considered above, the generative model may not be inverted easily and it may not be possible to parameterize this recognition distribution. This is crucial because the endpoint of learning is the acquisition of a useful recognition model that can be applied to sensory inputs by the brain. One solution is to posit an approximate recognition or conditional density q(v;u) that is consistent with the generative model and that can be parameterized. Estimating the moments (e.g. expectation) of this density corresponds to inference. Estimating the parameters of the underlying generative model corresponds to learning. This distinction maps directly onto the two steps of EM.

(3.4)

This objective function has two terms. The first is the likelihood of the inputs under the generative model. The second term is the Kullback–Leibler divergence1 between the approximate and true recognition densities. Critically, the second term is always positive, rendering F a lower bound on the expected log likelihood of the inputs. This means maximizing the objective function (i.e. minimizing the free energy) is simply minimizing our surprise about the data. The E-step increases F with respect to the expected cause, ensuring a good approximation to the recognition distribution implied by the parameters θ. This is inference. The M-step changes θ, enabling the generative model to match the input density and corresponds to learning.

(3.5)

(3.6)

(3.7)

The first term in equation (3.7) is the prediction error that is minimized in predictive coding. The second corresponds to a prior term that constrains or regularizes conditional estimates of the causes. The need for this term stems from the ambiguous or ill-posed nature of recognition discussed above and is a ubiquitous component of inverse solutions.

Predictive coding schemes can be regarded as arising from the distinction between forward and inverse models adopted in machine vision (Ballard et al. 1983; Kawato et al. 1993). Forward models generate inputs from causes (cf. generative models), whereas inverse models approximate the reverse transformation of inputs to causes (cf. recognition models). This distinction embraces the noninvertibility of generating processes and the ill-posed nature of inverse problems. As with all underdetermined inverse problems, the role of constraints is central. In the inverse literature, a priori constraints usually enter in terms of regularized solutions. For example, ‘Descriptions of physical properties of visible surfaces, such as their distance and the presence of edges, must be recovered from the primary image data. Computational vision aims to understand how such descriptions can be obtained from inherently ambiguous and noisy data. A recent development in this field sees early vision as a set of ill-posed problems, which can be solved by the use of regularization methods’ (Poggio et al. 1985). The architectures that emerge from these schemes suggest that ‘Feedforward connections from the lower visual cortical area to the higher visual cortical area provides an approximated inverse model of the imaging process (optics)’. Conversely, ‘…the backprojection connection from the higher area to the lower area provides a forward model of the optics’ (Kawato et al. 1993; see also Harth et al. 1987). This perspective highlights the importance of backward connections and the role of priors in enabling predictive coding schemes.

Next, we introduce hierarchical models and extend the parameterization of the ensuing generative model to cover priors. This means that the constraints, required by predictive coding and regularized solutions to inverse problems, are now absorbed into the learning scheme and are estimated in exactly the same way as the parameters. These extra parameters encode the variability or precision of the causes and are referred to as hyperparameters in the classical covariance component literature. Hyperparameters are updated in the M-step and are treated in exactly the same way as the parameters.

(3.8)

(3.9)

This likelihood also plays the role of a prior on v_i at the level below, where it is jointly maximized with the likelihood p(v_i−1|v_i;θ). This is the key to understanding the utility of hierarchical models. By learning the parameters of the generative distribution of level i, one is implicitly learning the parameters of the prior distribution for level i−1. This enables the learning of prior densities.

The hierarchical nature of these models lends an important context-sensitivity to recognition densities not found in single-level models. The key point here is that high-level causes v_i+1 determine the prior expectation of causes v_i in the subordinate level. This can completely change the distributions p(v_i|v_i+1;θ), upon which inference in based, in an input and context-dependent way.

For simplicity, we will again assume deterministic recognition. In this setting, with conditional independence, the objective function is

(3.10)

The estimators ϕ_i and parameters perform a gradient ascent on the objective function

(3.11)

Inferences mediated by the E-step rest on changes in the representational units, mediated by forward connections from error units in the level below and lateral interaction with error units within the same level. Similarly, prediction error is constructed by comparing the activity of representational units, within the same level, to their predicted activity conveyed by backward connections.

This is the simplest version of a very general learning algorithm. It is general in the sense that it does not require the parameters of either the generative or the prior distributions. It can learn noninvertible, nonlinear generative models and encompasses complicated hierarchical processes. Furthermore, each of the learning components has a relatively simple neuronal interpretation (see below).

Figure 2 Upper panel: schematic depicting a hierarchical predictive coding architecture. Here, hierarchical arrangements within the model serve to provide predictions or priors to representations in the level below. The upper circles represent error units and (more ...)

(4.1)

Functionally, forward and lateral connections are reciprocated, where backward connections generate predictions of lower-level responses. Forward connections allow prediction error to drive representational units in supraordinate levels. Within each level, lateral connections mediate the influence of error units on the predicting units and intrinsic connections λ_i among the error units decorrelate them, allowing competition among prior expectations with different precisions (precision is the inverse of variance). In short, lateral, forwards and backward connections are all reciprocal, consistent with anatomical observations.

(4.2)

This is simply Hebbian or associative plasticity, where the connection strengths change in proportion to the product of pre- and postsynaptic activity, for example, . An intuition about equation (4.2) is obtained by considering the conditions under which the expected change in parameters is zero (i.e. after learning). For the backward connections, this implies there is no component of prediction error that can be explained by estimates at the higher level . The lateral connections stop changing when the prediction error has been whitened .

It is evident that the predictions of the theoretical analysis coincide almost exactly with the empirical aspects of functional architectures in visual cortices highlighted in the §2(c) (hierarchical organization, reciprocity, functional asymmetry and associative plasticity). Although somewhat contrived, it is pleasing that purely theoretical considerations and neurobiological empiricism converge so precisely.

The overall scheme implied by equation (3.11) sits comfortably with the hypothesis (Mumford 1992) that:

We have tried to show that this sort of hierarchical prediction can be implemented in brain-like architectures using mechanisms that are biologically plausible.

(4.3)

In a subsequent paper, describing DEM for hierarchical dynamic models, we will show that it is necessary to minimize the prediction errors and their temporal derivatives. However, the form of the objective function and the ensuing E- and M-steps remains the same. This means the conclusions above hold in a dynamic setting, with some interesting generalizations.

When the generative model is dynamic (i.e. is effectively a convolution operator) the E-step, subtending inference, is more complicated and rests on a generalization of equation (3.11) to cover dynamic systems

(4.4)

In DEM, the aim is not to find the most likely cause of the sensory input but to encode the evolution of causes in terms of conditional trajectories. For static models, equation (4.4) reduces to equation (3.11), which can be regarded as a special case when there are no dynamics and the trajectory becomes a single point. The dynamic version of the E-step is based on the objective function evaluated prospectively at some point τ in the future and can be understood as a gradient ascent on the objective function and all its higher derivatives (see the second line of equation (4.4)). This prospective aspect of DEM lends it some interesting properties. Among these is the nature of the plasticity. Because the associative terms involve prospective prediction errors, synaptic changes occur when presynaptic activity is high and post-synaptic activity is increasing. This has an interesting connection with STDP, where increases in efficacy rely on postsynaptic responses occurring shortly after presynaptic inputs. In this instance, at peak presynaptic input, the postsynaptic response will be rising, to peak a short time later. The DEM scheme offers a principled explanation for this aspect of plasticity that can be related to other perspectives on its functional role. For example, Kepecs et al. (2002) note that the temporal asymmetry implicit in STDP may underlie learning and review some of the common themes from a range of findings in the framework of predictive coding (see also Fuhrmann et al. 2002).

Generative models of a dynamic sort confer a temporal continuity and prospective aspect on representational inference that is evident in empirical studies. As noted by Mehta (2001) ‘a critical task of the nervous system is to learn causal relationships between stimuli to anticipate events in the future’. Both the inference and learning about the states of the environment, in terms of trajectories, enables this anticipatory aspect. Mehta (2001) reviews findings from hippocampal electrophysiology, in which spatial RFs can show large and rapid anticipatory changes in their firing characteristics, which are discussed in the context of predictive coding (see also Rainer et al. 1999 for a discussion of prospective coding for objects in the context delayed paired associate tasks).

It would be premature to go into the details of DEM here. We anticipate communicating several articles covering the above themes in the near future. However, there are a number of complementary approaches to learning in the context of dynamic models that are already in the literature. For example, the seminal paper of Rao & Ballard (1999) uses Kalman filtering and a hierarchical hidden Markov model to provide a functional interpretation of many extra classical RF effects (see below). Particularly relevant here is the discussion of hierarchical Bayesian inference in the visual cortex by Lee & Mumford (2003). These authors consider particle filtering and Bayesian-belief propagation (algorithms from machine learning) that might model the cortical computations implicit in hierarchical Bayesian inference and review the neurophysiological evidence that supports their plausibility. Irrespective of the particular algorithm employed by the brain for empirical (i.e. hierarchical) Bayes, they each provide predictions about the physiology of cortical responses. These predictions are the subject of §5 below.

The empirical Bayes perspective on perceptual inference suggests that the role of backward connections is to provide contextual guidance to lower levels through a prediction of the lower level's inputs. When this prediction is incomplete or incompatible with the lower areas input, a prediction error is generated that engenders changes in the area above until reconciliation. When (and only when) the bottom-up driving inputs are in accord with top-down predictions, error is suppressed and a consensus between the prediction and the actual input is established. Given this conceptual model, a stimulus-related response can be decomposed into two components corresponding to the transients evoked in two functional subpopulations of units. The first representational subpopulation encodes the conditional expectation of perceptual causes ϕ. The second encodes prediction error ξ. Responses will be evoked in both, with the error units of one level exciting appropriate representational units through forward connections and the representational unit suppressing error units through backward connections (see figure 2). As inference converges, high-level representations are expressed as the late component of evoked responses with a concomitant suppression of error signal in lower areas.

In short, within the model, activity in the cortical hierarchy self-organizes to minimize its free energy though minimizing prediction error. Is this sufficient to account for classical RFs and functional segregation seen in cortical hierarchies, such as the visual system?

There are many examples where minimizing the free energy produces very realistic RFs, a very compelling example can be found in Olshausen & Field (1996). An example from our own work, which goes beyond single RFs, concerns the selectivity profile of units in V2. In Friston (2000), we used the infomax principle (Linsker 1990) to optimize the spatio temporal RFs of simple integrate and fire units exposed to moving natural scenes. We examined the response profiles in terms of selectivity to orientation, speed, direction and wavelength. The units showed two principal axes of selectivity. The first partitioned cells into those with wavelength selectivity and those without. Within the latter, the main axis was between units with direction selectivity and those without. This pattern of selectivity fits nicely with the characteristic response profiles of units in the thin, thick and inter stripes of V2. See figure 3 for examples of the ensuing RFs, shown in the context of the functional architecture of visual processing pathways described in Zeki (1993).

Figure 3 Schematic adapted from Zeki (1993) summarizing the functional segregation of processing pathways and the relationship of simulated RFs to the stripe structures in V2. LGN, lateral geniculate nucleus; P, parvocellular pathway; M, magnocellular pathway. (more ...)

Extra classical effects are commonplace and are generally understood in terms of the modulation of RF properties by backward and lateral afferents. There is clear evidence that horizontal connections in visual cortex are modulatory in nature (Hirsch & Gilbert 1991), speaking to an interaction between the functional segregation implicit in the columnar architecture of V1 and activity in remote populations. These observations suggest that lateral and backwards interactions may convey contextual information that shapes the responses of any neuron to its inputs (e.g. Kay & Phillips 1996; Phillips & Singer 1997) to confer the ability to make conditional inferences about sensory input.

The most detailed and compelling analysis of extra classical effects in the context of hierarchical models and predictive coding is presented in Rao & Ballard (1999). These authors exposed a hierarchical network of model neurons using a predictive coding scheme to natural images. The neurons developed simple-cell-like RFs. In addition, a subpopulation of error units showed a variety of extra classical RF effects suggesting that ‘non-classical surround effects in the visual cortex may also result from cortico-cortical feedback as a consequence of the visual system using an efficient hierarchical strategy for encoding natural images.’ One nonclassical feature on which the authors focus is end stopping. Visual neurons that respond optimally to line segments of a particular length are abundant in supragranular layers and have the curious property of end stopping or end inhibition. Vigorous responses to optimally oriented line segments are attenuated or eliminated when the line extends beyond the classical RF. The explanation for this effect is simple, because the hierarchy was trained on natural images, containing long line segments, the input caused by short segments could not be predicted and error responses could not be suppressed. This example makes a fundamental point, which we will take up further below. The selective response of these units does not mean they have learned to encode short line segments. Their responses reflect the fact that short line segments have not been encountered before and represent an unexpected visual input, given the context established by input beyond the classical RF. In short, their response signals a violation of statistical regularities that have been learned.

If these models are right, interruption of backward connections should disinhibit the response of supragranular error units that are normally suppressed by extra classical stimuli. Rao & Ballard (1999) cite inactivation studies of high-level visual cortex in anaesthetized monkeys, in which disinhibition of responses to surround stimuli is observed in lower areas (Hupe et al. 1998). Furthermore, removal of feedback from V1 and V2 to the LGN reduces the end stopping of LGN cells (Murphy & Sillito 1987).

The prevalence of long-latency responses in unit recordings is mirrored in similar late components of ERPs recorded noninvasively. The cortical hierarchy in figure 2 comprises a chain of coupled oscillators. The response of these systems to sensory perturbation conforms to a damped oscillation, emulating a succession of late components. Functionally, the activity of error units at any one level reflect states that have yet to be explained by higher-level representations and will wax and wane as higher-level causes are selected and refined. The ensuing transient provides a compelling model for the form of ERPs, which look very much like damped oscillation in the alpha (10Hz) range. In some instances, specific components of ERPs can be identified with specific causes, For example, the N170, a negative wave about 170ms after stimulus onset, is elicited by face stimuli relative to non-face stimuli. In the following, we focus on examples of late components. We will not ascribe these components to representational or error subpopulations because their respective dynamics are tightly coupled. The theme we highlight is that late components reflect inference about supraordinate or global causes at higher levels in the hierarchy.

A similar late emergence of selectivity is seen in motion processing. A critical aspect of visual processing is the integration of local motion signals generated by moving objects. This process is complicated by the fact that local velocity measurements can differ depending on contour orientation and spatial position. Specifically, any local motion detector can measure only the component of motion perpendicular to a contour that extends beyond its field of view (Pack & Born 2001). This aperture problem is particularly relevant to direction-selective neurons early in the visual pathways, where small RFs permit only a limited view of a moving object. Pack & Born (2001) have shown ‘that neurons in the middle temporal visual area (known as MT or V5) of the macaque brain reveal a dynamic solution to the aperture problem. MT neurons initially respond primarily to the component of motion perpendicular to a contour's orientation, but over a period of approximately 60ms the responses gradually shift to encode the true stimulus direction, regardless of orientation’.

Finally, it is interesting to note that extra classical RF effects in supragranular V1 units are often manifest 80–100ms after stimulus onset, ‘suggesting that feedback from higher areas may be involved in mediating these effects’ (Rao & Ballard 1999).

Our own work in this area uses coherent motion subtended by sparse dots that cannot fall within the same classical RF of V1 neurons. As predicted, the V1 response to coherent, relative to incoherent, stimuli was significantly reduced (Harrison et al. 2004). This is a clear indication that error suppression is mediated by backward connections because only higher cortical areas have RFs that were sufficiently large to encompass more than one dot.

In §6, we turn to the implication of error suppression for responses evoked during perceptual learning and their electrophysiological correlates.

In §5, we introduced the notion that evoked response components in sensory cortex encode a transient prediction error that is rapidly suppressed by predictions mediated by backward connections. If the stimulus is novel or inconsistent with its context, then this suppression is compromised. An example of this might be extra classical effects expressed 100ms or so after stimulus onset. In the following, we consider responses to novel or deviant stimuli as measured with ERPs. This may be important for empirical studies because ERPs can be acquired noninvasively and can be used to study humans in both a basic and clinical context.

Figure 4 Repetition suppression as measured with fMRI in normal subjects. Top panel: estimated hemodynamic responses to the presentation of faces that were (red), and were not (blue), seen previously during the scanning session. These estimates were based on a (more ...)

In §5, we presented empirical examples where coherence or congruency was used to control the predictability of stimuli. Below, we look at examples where rapid sensory and perceptual learning renders some stimuli more familiar than others do. The behavioural correlate of repetition effects is priming (cf. global precedence for congruency effects). The example we focus on is the MMN elicited with simple stimuli. However, there are many other examples in electrophysiology, such as the P300.

Neither the change-specific neuron nor the intrinsic adaptation hypotheses are consistent with the theoretical framework established above. The empirical Bayes scheme would suggest that a component of the N1 response, corresponding to prediction error, is suppressed more efficiently after learning-related plasticity in backward and lateral connections. This suppression would be specific for the repeated aspects of the stimuli and would be a selective suppression of prediction error. Recall that error suppression (i.e. minimization of free energy) is the motivation for plasticity in the M-step. This repetition suppression hypothesis suggests the MMN is simply the attenuation of the N1. Fortunately, the apparent dislocation of N1 and MMN sources has been resolved recently: Jääskeläinen et al. (2004) show that the MMN results from differential suppression of anterior and posterior auditory N1 sources by preceding stimuli. This alters the centre of gravity of the N1 source, creating a specious difference between N1 and MMN loci when estimated using dipole equivalents.

In summary, both the E-step and M-step try to minimize free energy; the E-step does so during perceptual synthesis on a time-scale of milliseconds and the M-step does so during perceptual learning over seconds or longer. If the N1 is an index of prediction error (i.e. free energy), then the N1, evoked by the first in a train of repeated stimuli, will decrease with each subsequent stimulus. This decrease discloses the MMN evoked by a new (oddball) stimulus. In this view, the MMN is subtended by a positivity that increases with the number of standard. Recent advances in the MMN paradigm that use a ‘roving’ paradigm show exactly this (see Baldeweg et al. 2004; see also figure 5).

Figure 5 Schematic using empirical results reported in Baldeweg et al. (2004) relating the MMN to a predictive error suppression during perceptual learning. The idea is that perceptual synthesis (E-step) minimizes prediction error ‘online’ to terminate (more ...)

The mechanistic link between plasticity and the MMN provided by perceptual learning based on empirical Bayes is important because a number of neuropsychiatric disorders are thought to arise from aberrant cortical plasticity. Two interesting examples are dyslexia and schizophrenia. In dyslexia, the MMN to pitch is attenuated in adults and the degree of attenuation correlates with reading disability. In contrast, in schizophrenia, the MMN to duration is more affected and has been shown to correlate with the expression of negative symptoms (Näätänen et al. 2004). This is potentially important because the disconnection hypothesis for schizophrenia (Friston 1998) rests on abnormal experience-dependent plasticity. From the point of view of this paper, the key pathophysiology of schizophrenia reduces to aberrant representational learning as modelled by the M-step. If the MMN could be used as a quantitative index of perceptual learning, then it would be very useful in schizophrenia research (see also Baldeweg et al. 2002). In §7, we show that the MMN can indeed be explained by changes in connectivity.

Although it is pleasing to have a principled explanation for many anatomical and physiological aspects of neuronal systems, the question can be asked, is this explanation empirically useful? We conclude by showing that recent advances in the DCM of evoked responses now afford measures of connectivity among cortical sources that can be used to quantify theoretical predictions about perceptual learning. These measures may provide mechanistic insights into putative functional disconnection syndromes, such as dyslexia and schizophrenia. The advances in data analysis are useful because they use exactly the same EM scheme to analyse neurophysiological data as proposed here for the brain's analysis of sensory data.

Each region or source is modelled using a neural mass model decribed in David & Friston (2003), based on the model of Jansen & Rit (1995). This model emulates the activity of a cortical area using three neuronal subpopulations, assigned to granular and agranular layers. A population of excitatory pyramidal (output) cells receives inputs from inhibitory and excitatory populations of inter neurons, via intrinsic connections. Within this model, excitatory inter neurons can be regarded as spiny stellate cells found predominantly in layer 4 and in receipt of forward connections. Excitatory pyramidal cells and inhibitory inter neurons will be considered to occupy agranular layers and receive backward and lateral inputs (see figure 6).

Figure 6 This schematic shows the state equations describing the dynamics of one source. Each source is modelled with three subpopulations (pyramidal, spiny stellate and inhibitory inter neurons) as described in Jansen & Rit (1995) and David & (more ...)

To model ERPs, the network receives inputs via input connections. These connections are exactly the same as forward connections and deliver inputs w to the spiny stellate cells in layer 4. In the present context, these are subcortical auditory inputs. The vector C controls the influence of the ith input on each source. The matrices A^F, A^B, A^L encode forward, backward and lateral connections, respectively. The DCM here is specified in terms of the state equations shown in figure 6 and a linear output equation,

(7.1)

(7.2)

Within each subpopulation, the evolution of neuronal states rests on two operators. The first transforms the average density of presynaptic inputs into the average postsynaptic membrane potential. This is modelled by a linear transformation with excitatory and inhibitory kernels parameterized by H and τ. H controls the maximum postsynaptic potential and τ represents a lumped rate constant. The second operator S transforms the average potential of each subpopulation into an average firing rate. This is assumed to be instantaneous and is a sigmoid function. Interactions, among the subpopulations, depend on constants γ_1,2,3,4, which control the strength of intrinsic connections and reflect the total number of synapses expressed by each subpopulation. In equation (7.2), the top line expresses the rate of change of voltage as a function of current. The second line specifies how current changes as a function of voltage, current and presynaptic input from extrinsic and intrinsic sources. Having specified the DCM, one can estimate the coupling parameters from empirical data using EM (Friston et al. 2003), using exactly the same minimization of free energy proposed for perceptual learning.

Six sources were identified using conventional procedures (David et al. 2005) and used to construct DCMs (see figure 7). To establish evidence for changes in backward and lateral connections beyond changes in forward connections, we employed a Bayesian model selection procedure. This entailed specifying four models that allowed for changes in forward, backward, forward and backward and changes in all connections. These changes in extrinsic connectivity may explain the differences in ERPs elicited by standard relative to oddball stimuli. The models were compared using the negative free energy as an approximation to the log evidence for each model. From equation (3.4), if we assume the approximating conditional density is sufficiently good, then the free energy reduces to the log evidence. In Bayesian model selection of a difference in log evidence of three or more can be considered as very strong evidence for the model with the greater evidence, relative to the one with less. The log evidence for the four models is shown in figure 7. The model with the highest evidence (by a margin of 27.9) is the DCM that allows for learning-related changes in forward, backwards and lateral connections. These results provide clear evidence that changes in backward and lateral connections are needed to explain the observed differences in cortical responses.

Figure 7 Upper right: transparent views of the cortical surface showing localized sources that entered the DCM. A bilateral extrinsic input acts on primary auditory cortices (red), which project reciprocally to orbito-frontal regions (green). In the right hemisphere, (more ...)

These differences can be seen in figure 8 in terms of the responses seen and those predicted by the fourth DCM. The underlying response in each of the six sources is shown in the lower panel. The measured responses over channels have been reduced to the first three (spatial) modes or eigenvectors. The lower panel also shows where changes in connectivity occurred. The numbers by each connection represent the relative strength of the connections during the oddball stimuli relative to the standards. The percentages in brackets are the conditional confidence that these differences are greater than zero (see David et al. 2005 for details of this study). Note that we have not attempted to assign a functional role to the three populations in terms of a predictive coding model, nor have we attempted to interpret the sign of the connection changes. This represents the next step in creating theoretically informed DCMs. At present, all we are demonstrating is that exuberant responses to rare stimuli, which may present a failure to suppress prediction error, can be explained quantitatively by changes in the coupling among cortical sources, which may represent perceptual learning with empirical Bayes.

Figure 8 Auditory oddball paradigm: DCM results for the FBL model of the previous figure. Upper panel: the data are the projection of the original scalp time-series onto the three first spatial modes or eigenvectors. Note the correspondence between the measured (more ...)

In summary, we estimated differences in the strength of connections for rare and frequent stimuli. As expected, we could account for detailed differences in the ERPs by changes in connectivity. These changes were expressed in forward, backward and lateral connections. If this model is a sufficient approximation to the real sources, then these changes are a non-invasive measure of plasticity, mediating perceptual learning in the human brain.

The Wellcome Trust funded this work. I would like to thank my colleagues for help in writing this paper and developing the ideas, especially Cathy Price, Peter Dayan, Rik Henson, Olivier David, Klaas Stephan, Lee Harrison, James Kilner, Stephan Kiebel, Jeremie Mattout and Will Penny. I would also like to thank Dan Kersten and Bruno Olshausen for didactic discussions.