Cortical circuits for perceptual inference

1This paper looks at the functional architecture of cortical circuits from the point of view of perception; namely, the fitting or inversion of internal models of sensory data by the brain. Critically, the nature of this inversion lends itself to a relatively simple neural network implementation that shares many formal similarities with real cortical hierarchies in the brain. The basic idea that the brain uses hierarchical inference has been described in a series of papers. These papers suggest that the brain uses empirical Bayes for inference about its sensory input, given the hierarchical organisation of cortical systems. Here, we focus on how neural networks could be configured to invert these models and deconvolve sensory causes from sensory input.

2This paper comprises three sections. In the first, we introduce a free-energy formulation of model inversion or perception, which is then applied to a specific class of models that we assume the brain uses — hierarchical dynamic models. An important aspect of these models is their formulation in generalised coordinates of motion. This lends them a hierarchical form in both structure and dynamics, which can be exploited during inversion. In the second section, we show how inversion can be formulated as a simple gradient descent using neuronal networks and relate these to cortical circuits in the brain. In the final section, we consider how evoked brain responses might be understood in terms of perceptual inference and categorisation, using the schemes of the preceding section.

3This section considers the problem of inverting generative models of sensory data and provides a summary of the material in. This problem is addressed using ensemble learning or Variational Bayes. These are generic approaches to model inversion that provide an approximation to the conditional density on some causes of generalised sensory input,. Generalised input (e.g., the intensity of photoreceptor stimulation) includes the input, its velocity, acceleration, jerk, etc. Causes are quantities in the environment that generate sensory input (e.g., the orientation of an object in the visual field). The approximation of the conditional density (i.e., the probability of a particular set of causes given sensory input) is achieved by optimising a recognition density with respect to a bound on the surprise or negative log-evidence of the sensory input, as we will see next. This bound is called free-energy The free-energy comprises a cross-entropy or divergence term and surprise. By Gibb’s inequality, the divergence is greater than zero, with equality when; i.e., when the recognition density is the posterior or conditional density on the causes of sensory input. The recognition density can be optimised to minimise this bound and implicitly minimise the divergence between the recognition density and the conditional density we seek. In summary, the recognition density, induces a free-energy bound, which converts a difficult integration problem (inherent in computing the exact conditional density) into an easier optimisation problem.

4The bound can be evaluated easily because it is a function of and some generative model entailed by the brain Here, we have expressed the free-energy in terms of the negentropy of and an expected Gibb’s energy —. This energy is usually specified in terms of a likelihood and prior;, which define a generative model. This is important because it shows that we need a generative model in order to evaluate free-energy. The likelihood model just quantifies the probability of any sensations, given their cause; while the prior model encodes prior beliefs about the probability of those causes being present. It is fairly easy to show that minimising free-energy corresponds to finding a recognition density that predicts sensory input accurately, while suppressing its complexity.

5If we assume that the recognition density is Gaussian (the Laplace assumption), then we can express free-energy in terms of its sufficient statistics (i.e., its mean and covariance: ) Here is the number of unknown causes. We can now minimise free-energy w.r.t. the conditional covariances by finding the value that renders its gradient zero where a subscript means differentiation; i.e., is the free-energy gradient w.r.t. the conditional covariance, Here, the conditional precision is the inverse covariance. Critically, the conditional precision is just a function of the mean and does not have to be encoded explicitly. This means we can simplify the expression for free-energy by eliminating the curvatures of Gibb’s energy Now, the only unknown quantities are the conditional means of the causes, which only have to minimise Gibb’s energy because this is the only term that depends on them. In this paper, we will focus on time-varying causes or states of the environment:. The values we seek are the solutions to the following differential equations. This solution (which is stationary in a frame of reference that moves with its generalised motion), minimises free-energy This construction ensures that when Gibb’s energy is minimised and, the mean of the motion is the motion of the mean; i.e.,. Here is a derivative matrix operator with identity matrices along the first leading diagonal.

6Eq. (7) prescribes recognition dynamics that track time-varying causes or states of the world and can be thought of as a gradient descent in a moving frame of reference. The recognition dynamics for time-invariant causes (i.e., parameters, like rate constants) have a different form, because we know a priori their generalised motion is zero. In this paper, we will assume the parameters have already been learnt and focus on recognising hidden states of the environment. In summary, we have derived recognition dynamics for expected environmental states, which cause sensations. The solution to these equations minimise Gibb’s energy and (under the Laplace assumption) free-energy, which is an upper bound on their surprise. Finding these solutions corresponds to perceptual inference. The precise form of Eq. (7) depends on the generative model that defines Gibb’s energy. Next, we examine forms associated with hierarchical dynamic models.

7This section introduces a general class of generative models that the brain may use for perception. We will start with simple dynamic models and then deal with hierarchical cases later. Consider a state-space model that describes the evolution of states in the world and how they map to sensory input Here, the functions and are parameterised by (which are omitted from the following expressions for clarity). These functions correspond to equations of motion and an observer function, respectively. The states are variously referred to as sources or causal states. The hidden states mediate the influence of causal states on sensory data and endow the system with memory. We assume the random fluctuations are analytic, such that the covariance of is well defined; similarly for state noise,, which represents random fluctuations on the motion of the hidden states. Under local linearity assumptions, the generalised motion of the data or response is given by We can write this generalised state-space model more compactly as where the predicted response and motion are Gaussian assumptions about the fluctuations provide the form of the likelihood,. Similarly, Gaussian assumptions about state noise specify empirical priors, in terms of predicted motion The covariances and or precisions and are functions of precision parameters,, which control the amplitude and smoothness of random fluctuations. Generally, these covariances factorise; into a covariance proper and a matrix of correlations among generalised motion that encodes an autocorrelation function.

8Hierarchical dynamic models with levels have the following form, which generalises the model above Again, and are continuous nonlinear functions of the states. The innovations and are conditionally independent fluctuations that enter each level of the hierarchy. These play the role of observation error or noise at the first level and induce random fluctuations in the states at higher levels. The causal states link levels, whereas the hidden states link dynamics over time. In hierarchical form, the output of one level acts as an input to the next. Inputs from higher levels can enter nonlinearly into the state equations and can be regarded as changing its control parameters to produce quite complicated generalised convolutions with deep (i.e., hierarchical) structure.

9In summary, hierarchical dynamic models are about as complicated as one could imagine; they comprise causal and hidden states, whose dynamics can be coupled with arbitrary (analytic) nonlinear functions. Furthermore, these states can have random fluctuations with unknown amplitude and arbitrary (analytic) autocorrelation functions. A key aspect of these models is their hierarchical form, which induces empirical priors on the causal states. See for a discussion of approximate Bayesian inference in conditionally independent hierarchical models of static data.

10We can now write down Gibb’s energy for these generative models, which has a simple quadratic form (ignoring constants) The auxiliary variables comprise prediction errors for the generalised response and motion of hidden states, where and are the respective predictions, whose precision is encoded by. These prediction errors provide a compact way to express Gibb’s energy and, as we will see below, lead to very simple recognition schemes. For hierarchical models, the prediction error on the response is supplemented with prediction errors on the causal states Note that the data enter the prediction error at the lowest level. At intermediate levels, the prediction errors, mediate empirical priors on the causal states.

11In this section, we have seen how the inversion of dynamic models can be formulated as an optimisation of free-energy. By assuming a Gaussian (Laplace) approximation to the conditional density, one can reduce optimisation to finding the conditional means of the unknown causes of sensory data. This can be formulated as a gradient ascent in a frame of reference that moves along the path encoded in generalised coordinates (Eq. (6)). The only thing needed to implement this recognition scheme is Gibb’s energy, which is specified by a generative model. We have looked at hierarchical dynamic models, whose form provides empirical priors or constraints on inference at both a structural and dynamic level (Eq. (14)). The structural priors arise from coupling different levels of the hierarchy with causal states and the dynamic priors emerge by coupling different levels of generalised motion of the hidden states. We can now look at the recognition dynamics entailed by these models, in the context of neuronal processes in the brain.

12A key architectural principle of the brain is its hierarchical organisation. This has been established most thoroughly in the visual system, where lower (primary) areas receive sensory input and higher areas adopt a multimodal or associational role. The neurobiological notion of a hierarchy rests upon the distinction between forward and backward connections. This distinction is based upon the specificity of cortical layers that are the predominant sources and origins of extrinsic connections. Forward connections arise largely in superficial pyramidal cells, in supra-granular layers and terminate on spiny stellate cells of layer four in higher cortical areas. Conversely, backward connections arise largely from deep pyramidal cells in infra-granular layers and target cells in the infra- and supra-granular layers of lower cortical areas. Intrinsic connections mediate lateral interactions between neurons that are a few millimetres away. There is a key functional asymmetry between forward and backward connections that renders backward connections more modulatory or nonlinear in their effects on neuronal responses (; see also ). This is consistent with the deployment of voltage-sensitive NMDA receptors in supra-granular layers that are targeted by backward connections. Typically, the synaptic dynamics of backward connections have slower time constants. This has led to the notion that forward connections are driving and illicit an obligatory response in higher levels, whereas backward connections have both driving and modulatory effects and operate over larger spatial and temporal scales. This hierarchical aspect of the brain’s functional anatomy speaks to hierarchical models of sensory input. We now consider how this functional architecture can be understood under the inversion of hierarchical models by the brain.

13If we assume that the activity of neurons encode the conditional mean of external states causing sensory data, then Eq. (6) specifies the neuronal dynamics entailed by recognising states of the world from sensory data. Using Gibb’s energy in Eq. (14) we have Eq. (16) describes how neuronal states self-organise, when exposed to sensory input. Its form is quite revealing and suggests two distinct populations of neurons; causal or hidden state-units whose activity encodes and error-units encoding precision-weighted prediction error, with one error-unit for each state. Furthermore, the activities of error-units are a function of the states and the dynamics of state-units are a function of prediction error. This means the two populations pass messages to each other and to themselves. The messages passed within the states, mediate empirical priors on their motion, while mediates precision-dependent modulation of prediction errors. The matrix can be thought of encoding self-inhibition, which is modulated by precision (where precision might be encoded by neuromodulatory neurotransmitters like dopamine and acetylcholine).

14If we unpack these equations, we can see the hierarchical nature of the implicit message-passing This shows that error-units receive messages from the states in the same level and the level above, whereas states are driven by error-units in the same level and the level below (see Fig. 1). Critically, inference requires only the prediction error from the lower level and the level in question,. These provide bottom-up and lateral messages that drive conditional expectations towards a better prediction, to explain away the prediction error in the level below. These top-down and lateral predictions correspond to and. This is the essence of recurrent message-passing between hierarchical levels to optimise free-energy or suppress prediction error; i.e., recognition dynamics. In summary, all connections between error- and state-units are reciprocal but the only connections that link levels are forward connections conveying prediction error to state-units and reciprocal backward connections that mediate predictions. This sort of scheme is referred to as predictive coding.

15We can identify error-units with superficial pyramidal cells, because the only messages that pass up the hierarchy are prediction errors and superficial pyramidal cells originate forward connections in the brain. This is useful because it is these cells that are primarily responsible for electroencephalographic (EEG) signals that can be measured non-invasively. Similarly, the only messages that are passed down the hierarchy are the predictions from state-units that are necessary to form prediction errors in lower levels. The sources of extrinsic backward connections are deep pyramidal cells; suggesting that these encode the expected causes of sensory states (see and Fig. 1). Critically, the motion of each state-unit is a linear mixture of bottom-up prediction error (see Eq. (17)). This is exactly what is observed physiologically; bottom-up driving inputs elicit obligatory responses that do not depend on other bottom-up inputs. The prediction error itself is formed by predictions conveyed by backward and lateral connections. These influences embody the nonlinearities implicit in and. Again, this is entirely consistent with the nonlinear or modulatory characteristics of backward connections.

16It has been shown recently that hierarchical architectures (cf, Fig. 1) can be reformulated as a specific type of biased competition, where state-units receive messages from lower-level error-units and direct inputs from higher-level state-units (replacing lateral inputs from error-units in the original predictive coding scheme based on Kalman filtering; ). It has been argued that this architecture provides a more realistic model of backward connections in cortex and usefully connects predictive coding, Kalman filtering and biased competition.

17A related Bayesian algorithm called belief-propagation also rests on message-passing. In these schemes, the messages are not prediction errors but, like prediction errors, are defined self-consistently, in terms of likelihoods and empirical priors. Critically, the belief-propagation algorithm can be derived by minimising free-energy; for example, it can be shown that the Kalman filter is a special case of belief-propagation. This speaks to formal similarities between predictive coding, Bayesian filtering and belief-propagation, which could be implemented by recursive message-passing in the brain and understood in terms of free-energy optimisation.

18In summary, we have seen how the inversion of a generic hierarchical and dynamical model of sensory inputs can be transcribed onto neuronal quantities that optimise a free-energy bound on surprise. This optimisation corresponds, under some simplifying assumptions, to suppression of prediction error at all levels in a cortical hierarchy. This suppression rests upon a balance between bottom-up (prediction error) and top-down (empirical prior) influences. In the final section, we use this scheme to simulate neuronal responses. Specifically, we look at the electrophysiological correlates of prediction error and ask whether we can understand some common phenomena in event-related potential (ERP) research in terms of the free-energy formulation and message-passing in the brain.

19In this section, we examine a system that uses hierarchical dynamics as a generative model of sensory input. The aim of this section is to provide some face-validity for the functional deconstruction of extrinsic and intrinsic circuits in the previous section. To do this, we try to show how empirical measures of neuronal processes can be reproduced using simulations based on the theoretical analysis above. The example we use is birdsong and the empirical measures we focus on are local field potentials (LFP) or evoked (ERP) responses that can be recorded non-invasively. The material in section is based on the simulations described in.

20We first describe our model of birdsong and demonstrate the nature and form of this model through simulated lesion experiments. We then use simplified versions of this model to show how attractors can be used to categorise sequences of stimuli quickly and efficiently. Throughout this section, we will exploit the fact that superficial pyramidal cells are the major contributors to observed LFP and ERP signals. This means we can ascribe these signals to prediction error; because the superficial pyramidal cells are the source of bottom-up messages in the brain (see Fig. 1).

21The basic idea here is that the environment unfolds as an ordered sequence of spatiotemporal dynamics, whose equations of motion entail attractor manifolds that contain sensory trajectories. Critically, the shape of the manifold generating sensory data is itself changed by other dynamical systems that could have their own attractors. If we consider the brain has a generative model of these coupled dynamical systems, then we would expect to see attractors in neuronal dynamics that are trying to predict sensory input. In a hierarchical setting, the states of a high-level attractor enter the equations of motion of a low-level attractor in a nonlinear way, to change the shape of its manifold. This form of generative model has some key characteristics:

22First, any level in the model can generate and therefore encode structured sequences of events, as the states flow over different parts of the manifold. These sequences can be simple, such as the quasi-periodic attractors of central pattern generators or can exhibit complicated sequences of the sort associated with chaotic and itinerant dynamics (e.g., ). The notion of attractors as the basis of generative models extends the notion of encoding trajectories in terms of generalised motion, to families of trajectories that lie on the attractor manifold. Hierarchically deployed attractors enable the brain to generate and therefore predict or represent different categories of sequences. This is because any low-level attractor encodes a family of trajectories that correspond to a structured sequence. The neuronal activity representing the trajectory at any one time determines where the current dynamics are within the sequence, while the shape of the attractor manifold determines which sequence is currently being expressed.

23Secondly, if the states in a higher attractor change the manifold of a subordinate attractor, then the states of the higher attractor come to encode the category of the sequence represented by the lower attractor. This means it is possible to generate and represent sequences of sequences and, by induction sequences of sequences of sequences etc. This rests upon the states of neuronal attractors at any cortical level providing control parameters for attractors below. This necessarily entails a nonlinear interaction between the top-down effects of the higher attractor and the states of the lower attractor. Again, this is entirely consistent with the nonlinear effects of top-down connections in the real brain.

24Finally, this particular model has implications for the temporal structure of perception. Put simply, the dynamics of high-level representations unfold more slowly than the dynamics of lower-level representations. This is because the state of a higher attractor prescribes a manifold that guides the flow of lower states, which could change quite rapidly. We will see an example of this below when considering the perceptual categorisation of different sequences of chirps subtending birdsongs. This suggests that neuronal representations in the brain will change more slowly at higher levels (; see also ). One can turn this argument on its head and use the fact that we are able to recognise sequences of sequences (e.g., ) as an existence proof for this sort of generative model. In the examples below, we will try to show how autonomous dynamics furnish generative models of sensory input, which behave much like real brains, when measured electrophysiologically.

25The toy example used here deals with the generation and recognition of birdsongs. We imagine that birdsongs are produced by two time-varying causal states that control the frequency and amplitude of vibrations of the syrinx of a songbird (see Fig. 2). There has been an extensive modelling effort using attractor models at the biomechanical level to understand the generation of birdsong (e.g., ). Here we use the attractors at a higher level to provide time-varying control over the resulting sonograms. We drive the syrinx with two states of a Lorenz attractor, one controlling the frequency (between two to five kHz) and the other (after rectification) controlling the amplitude or volume. The parameters of the Lorenz attractor were chosen to generate a short sequence of chirps every second or so. To endow the generative model with a hierarchical structure, we placed a second Lorenz attractor, whose dynamics were an order of magnitude slower, over the first. The states of the slower attractor entered as control parameters (the Raleigh and Prandtl number) to control the dynamics exhibited by the first. These dynamics could range from a fixed-point attractor, where the states of the first are all zero; through to quasi-periodic and chaotic behaviour, when the value of the Raleigh number exceeds an appropriate threshold (about twenty four) and induces a bifurcation. Because higher states evolve more slowly, they switch the lower attractor on and off, generating distinct songs, where each song comprises a series of distinct chirps (see Fig. 3).

26This model generates spontaneous sequences of songs using autonomous dynamics. We generated a single song, corresponding roughly to a cycle of the higher attractor and then inverted the ensuing sonogram (summarised as peak amplitude and volume) using the message-passing scheme described in the previous section. The results are shown in Fig. 3 and demonstrate that, after several hundred milliseconds, the veridical hidden states and supraordinate causal states can be recovered. Interestingly, the third chirp is not perceived, in that the first-level prediction error was not sufficient to overcome the dynamical and structural priors of the model. However, once the subsequent chirp had been predicted correctly the following sequence of chirps was recognised with a high degree of conditional confidence. Note that when the second and third chirps in the sequence are not recognised, first-level prediction error is high and the conditional confidence about the causal states at the second level is low (reflected in the wide 90% confidence intervals). Heuristically, this means that the synthetic bird listening to the song did not know which song was being emitted and was unable to predict subsequent chirps.

27This example provides a nice opportunity to illustrate the relative roles of structural and dynamic priors. Structural priors are provided by the top-down inputs that reshape the manifold of the low-level attractor. However, this attractor itself contains an abundance of dynamical priors that unfold in generalised coordinates. Both provide important constraints on the evolution of sensory states, which facilitate recognition. We can selectively destroy these priors by lesioning the top-down connections to remove structural priors or by cutting the intrinsic connections that mediate dynamic priors. The latter involves cutting the self-connections in Fig. 1, among the causal and state-units. The results of these two simulated lesion experiments are shown in Fig. 4. The top panel shows the percept as in the previous panel, in terms of the predicted sonogram and prediction error at the first and second level. The subsequent two panels show exactly the same things but without structural (middle) and dynamic (lower) priors. In both cases, the synthetic bird fails to recognise the sequence with a corresponding inflation of prediction error, particularly at the sensory level. Interestingly, the removal of structural priors has a less marked effect on recognition than removing the dynamical priors. Without dynamical priors there is a failure to segment the sensory stream and, although there is a preservation of frequency tracking, the dynamics per se have completely lost their tempo. Although it is interesting to compare and contrast the relative roles of structural and dynamics priors; the important message here is that both are necessary for veridical perception and that destruction of either leads to suboptimal inference. Both of these empirical priors prescribe dynamics which enable the synthetic bird to predict what will be heard next. This leads to the question ‘what would happen if the song terminated prematurely?’

28We repeated the above simulation but terminated the song after the fifth chirp. The corresponding sonograms and percepts are shown with their prediction errors in Fig. 5. The left panels show the stimulus and percept as in Fig. 4, while the right panels show the stimulus and responses to omission of the last syllables. These results illustrate two important phenomena. First, there is a vigorous expression of prediction error after the song terminates prematurely. This reflects the dynamical nature of the recognition process because, at this point, there is no sensory input to predict. In other words, the prediction error is generated entirely by the predictions afforded by the dynamic model of sensory input. It can be seen that this prediction error (with a percept but no stimulus) is almost as large as the prediction error associated with the third and fourth stimuli that are not perceived (stimulus but no percept). Second, it can be seen that there is a transient percept, when the omitted chirp should have occurred. Its frequency is slightly too low but its timing is preserved, in relation to the expected stimulus train. This is an interesting stimulation from the point of view of ERP studies of omission-related responses. These simulations and related empirical studies (e.g., ) provide clear evidence for the predictive capacity of the brain. In this example, prediction rests upon the internal construction of an attractor manifold that defines a family of trajectories, each corresponding to the realisation of a particular song. In the last simulation we look more closely at perceptual categorisation of these songs.

29In the previous simulations, we saw that a song corresponds to a sequence of chirps that are preordained by the shape of an attractor manifold that is controlled by top-down inputs. This means that for every point in the state-space of the higher attractor there is a corresponding manifold or category of song. In other words, recognising or categorising a particular song corresponds to finding a fixed location in the higher state-space. This provides a nice metaphor for perceptual categorisation; because the neuronal states of the higher attractor represent, implicitly, a category of song. Inverting the generative model means that, probabilistically, we can map from a sequence of sensory events to a point in some perceptual space; where this mapping corresponds to perceptual recognition or categorisation. This can be demonstrated in our synthetic songbird by ignoring the dynamics of the second-level attractor and exposing the bird to a song and letting the states at the second level optimise their location in perceptual space. To illustrate this, we generated three songs by fixing the Raleigh and Prandtl variables to three distinct values. We then placed uninformative priors on the second-level causal states (that were previously driven by the hidden states of the second-level attractor) and inverted the model in the usual way. Fig. 6 shows the results of this simulation for a single song. This song comprises a series of relatively low-frequency chirps emitted every 250 ms or so. The causal states of this song (song C in the next figure) are recovered after the second chirp, with relatively tight confidence intervals (the blue and green lines in the lower left panel). We then repeated this exercise for three songs. The results are shown in Fig. 7. The songs are portrayed in sonogram format in the top panels and the inferred perceptual causal states in the bottom panels. The left panel shows the evolution of the causal states for all three songs as a function of peristimulus time and the right panel shows the corresponding conditional density in the causal or perceptual space of these two states after convergence. It can be seen that for all three songs, the 90% confidence interval encompasses the true values (red dots). Furthermore, there is very little overlap between the conditional densities (grey regions), which means that the precision of the perceptual categorisation is almost 100%. This is a simple but nice example of perceptual categorisation, where sequences of sensory events with extended temporal support can be mapped to locations in perceptual space, through Bayesian deconvolution of the sort entailed by the free-energy formulation.

30This paper has suggested that the architecture of cortical circuits speaks to hierarchical generative models in the brain. The estimation or inversion of these models corresponds to a generalised deconvolution of sensory inputs to disclose their causes. This deconvolution could be implemented in a neuronally plausible fashion, where neuronal dynamics self-organise when exposed to inputs to suppress free-energy. The focus of this paper has been on the nature of the hierarchical models and, in particular, how one can understand message-passing among neuronal populations in terms of perception. We have tried to demonstrate their plausibility, in relation to empirical observations, by interpreting the prediction error, associated with model inversion, with observed electrophysiological responses.

31The ideas reviewed in this paper have a long history, starting with the notion of neuronal energy; covering ideas like efficient coding and analysis by synthesis to more recent formulations in terms of Bayesian inversion and Predictive coding (e.g., ). This work has also tried to provide support for the notion that the brain uses dynamics to represent and predict causes in the sensorium.