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The evolution of epidemiological parameters, such as instantaneous reproduction number Rt, is important for understanding the transmission dynamics of infectious diseases. Current estimates of time-varying epidemiological parameters often face problems such as lagging observations, averaging inference, and improper quantification of uncertainties. To address these problems, we propose a Bayesian data assimilation framework for time-varying parameter estimation. Specifically, this framework is applied to Rt estimation, resulting in the state-of-the-art DARt system. With DARt, time misalignment caused by lagging observations is tackled by incorporating observation delays into the joint inference of infections and Rt; the drawback of averaging is overcome by instantaneously updating upon new observations and developing a model selection mechanism that captures abrupt changes; the uncertainty is quantified and reduced by employing Bayesian smoothing. We validate the performance of DARt and demonstrate its power in revealing the transmission dynamics of COVID-19. The proposed approach provides a promising solution for accurate and timely estimating transmission dynamics from reported data.
This paper presents a heuristic proof (and simulations of a primordial soup) suggesting that life-or biological self-organization-is an inevitable and emergent property of any (ergodic) random dynamical system that possesses a Markov blanket. This conclusion is based on the following arguments: if the coupling among an ensemble of dynamical systems is mediated by short-range forces, then the states of remote systems must be conditionally independent. These independencies induce a Markov blanket that separates internal and external states in a statistical sense. The existence of a Markov blanket means that internal states will appear to minimize a free energy functional of the states of their Markov blanket. Crucially, this is the same quantity that is optimized in Bayesian inference. Therefore, the internal states (and their blanket) will appear to engage in active Bayesian inference. In other words, they will appear to model-and act on-their world to preserve their functional and structural integrity, leading to homoeostasis and a simple form of autopoiesis.
•Predictive coding explains the recursive hierarchical structure of cortical processes.•Granular layer 4, which relays ascending cortical pathways, is absent from motor cortex.•Perceptual inference results if ascending sensory data modify sensory predictions action, if spinal reflexes enact descending motor and/or proprioceptive predictions.•Motor layer 4 regresses as motor predictions inherently require less modification. The agranular architecture of motor cortex lacks a functional interpretation. Here, we consider a 'predictive coding' account of this unique feature based on asymmetries in hierarchical cortical connections. In sensory cortex, layer 4 (the granular layer) is the target of ascending pathways. We theorise that the operation of predictive coding in the motor system (a process termed 'active inference') provides a principled rationale for the apparent recession of the ascending pathway in motor cortex. The extension of this theory to interlaminar circuitry also accounts for a sub-class of 'mirror neuron' in motor cortex – whose activity is suppressed when observing an action –explaining how predictive coding can gate hierarchical processing to switch between perception and action. The agranular architecture of motor cortex lacks a functional interpretation. Here, we consider a 'predictive coding' account of this unique feature based on asymmetries in hierarchical cortical connections. In sensory cortex, layer 4 (the granular layer) is the target of ascending pathways. We theorise that the operation of predictive coding in the motor system (a process termed 'active inference') provides a principled rationale for the apparent recession of the ascending pathway in motor cortex. The extension of this theory to interlaminar circuitry also accounts for a sub-class of 'mirror neuron' in motor cortex – whose activity is suppressed when observing an action –explaining how predictive coding can gate hierarchical processing to switch between perception and action. Motor cortex was localised to the precentral gyrus of apes by Sherrington in 1901 [1Sherrington C.S. Grunbaum A.S.F. An address on localisation in the 'motor' cerebral cortex as exemplified in the anthropoid apes.BMJ. 1901; ii: 1857-1859Crossref Scopus (13) Google Scholar], and was first identified histologically the following year by Campbell, using the brains of Sherrington's subjects [2Macmillan M. Alfred Walter Campbell and the visual functions of the occipital cortex.Cortex. 2012; https://doi.org/10.1016/j.cortex.2012.10.007Abstract Full Text Full Text PDF PubMed Scopus (10) Google Scholar]. Although Campbell emphasised the prominent fibre architecture of motor cortex, it is the cytoarchitectural tag 'agranular' cortex, coined by Brodmann [3Brodmann, K., translated by Garey, L.J. (1909/1994) Brodmann's Localisation in the Cerebral Cortex, Smith-GordonGoogle Scholar] to describe his areas 4 and 6, that has proved the more enduring. Both authors used variations in cortical architecture for cartographic purposes, but these variants have rarely, if ever, been interpreted functionally; despite its 'fame', the dramatic recession of granular layer 4 in motor cortex has not attracted a single functional hypothesis. Cortical layers are identified by cytoarchitecture, and further characterised by patterns of intrinsic axonal and dendritic arborisation [4Shipp S. Structure and function of the cerebral cortex.Curr. Biol. 2007; 17: 443-449Abstract Full Text Full Text PDF Scopus (144) Google Scholar]. Laminar distribution distinguishes consistent types of extrinsic corticocortical connection, classified as ascending, descending, and lateral [5Felleman D.J. Van Essen D.C. Distributed hierarchical processing in the primate cerebral cortex.Cereb. Cortex. 1991; 1: 1-47Crossref PubMed Scopus (5507) Google Scholar]. These patterns are sufficiently conserved to identify a hierarchical organisation of areas in sensory systems (Box 1). Initial descriptions of sensorimotor hierarchies placed premotor above primary motor cortex (M1), with areas 3a and 3b (components of the primary somatosensory area, S1) at the lowest levels [5Felleman D.J. Van Essen D.C. Distributed hierarchical processing in the primate cerebral cortex.Cereb. Cortex. 1991; 1: 1-47Crossref PubMed Scopus (5507) Google Scholar]. Our own survey aimed to establish the polarity of key reciprocal connections, but not to arrange areas into discrete tiers [6Adams R.A. et al.Predictions not commands: active inference in the motor system.Brain Struct. Funct. 2013; 218: 611-643Crossref PubMed Scopus (414) Google Scholar]. The absence of a distinct granular layer in primary motor cortex calls for some modification of the laminar criteria, but the presence of a cryptic layer 4 [7Sloper J.J. et al.A qualitative and quantitative electron microscopic study of the neurons in the primate motor and somatic sensory cortices.Philos. Trans. R. Soc. Lond. B: Biol. Sci. 1979; 285: 141-171Crossref PubMed Scopus (64) Google Scholar, 8Skoglund T.S. et al.The existence of a layer IV in the rat motor cortex.Cereb. Cortex. 1997; 7: 178-180Crossref PubMed Scopus (38) Google Scholar, 9Mao T. et al.Long-range neuronal circuits underlying the interaction between sensory and motor cortex.Neuron. 2011; 72: 111-123Abstract Full Text Full Text PDF PubMed Scopus (318) Google Scholar] justifies the treatment of terminal patterns that target the layer 3/5 border zone as forward connections (or backward, if the pattern avoids this zone). Similar arguments apply to premotor cortex (Brodmann's area 6), sometimes described as 'dysgranular' [10Watanabe-Sawaguchi K. et al.Cytoarchitecture and intrafrontal connections of the frontal cortex of the brain of the hamadryas baboon (Papio hamadryas).J. Comp. Neurol. 1991; 311: 108-133Crossref PubMed Scopus (49) Google Scholar], owing to a rudimentary granular layer.Box 1Laminar specific connectivity and hierarchical distanceFigure I shows the laminar sources and distributions of ascending connections (green) and descending connections (red, violet, and blue), originating from a certain level (i) in a hierarchical chain.The basic laminar patterns distinguishing ascending and descending connections were originally established by studies of primate visual cortex [5Felleman D.J. Van Essen D.C. Distributed hierarchical processing in the primate cerebral cortex.Cereb. Cortex. 1991; 1: 1-47Crossref PubMed Scopus (5507) Google Scholar, 97Rockland K.S. Pandya D.N. Laminar origins and terminations of cortical connections of the occipital lobe in the rhesus monkey.Brain Res. 1979; 179: 3-20Crossref PubMed Scopus (693) Google Scholar, 98Maunsell J.H.R. Van Essen D.C. The connections of the middle temporal area and their relationship to a cortical hierarchy in the macaque monkey.J. Neurosci. 1983; 3: 2563-2586PubMed Google Scholar]. Systematic variations with hierarchical distance were later formulated as a 'distance rule' [37Markov N.T. Kennedy H. The importance of being hierarchical.Curr. Opin. Neurobiol. 2013; 23: 187-194Crossref PubMed Scopus (105) Google Scholar, 99Barone P. et al.Laminar distribution of neurons in extrastriate areas projecting to visual areas V1 and V4 correlates with the hierarchical rank and indicates the operation of a distance rule.J. Neurosci. 2000; 20: 3263-32681PubMed Google Scholar]: with regard to origins, the proportion of superficial (layer 3A) neurons forming a backward projection decreases with greater distance spanned by the projection [99Barone P. et al.Laminar distribution of neurons in extrastriate areas projecting to visual areas V1 and V4 correlates with the hierarchical rank and indicates the operation of a distance rule.J. Neurosci. 2000; 20: 3263-32681PubMed Google Scholar, 100Sousa A.P.B. et al.Topographic organization of cortical input to striate cortex in the Cebus monkey: a fluorescent tracer study.J. Comp. Neurol. 1991; 308: 665-682Crossref PubMed Scopus (80) Google Scholar, 101Rockland K.S. Van Hoesen G.W. Direct temporal-occipital feedback connections to striate cortex (V1) in the macaque monkey.Cereb. Cortex. 1994; 4: 300-313Crossref PubMed Scopus (154) Google Scholar, 102Perkel D.J. et al.Topography of the afferent connectivity of area 17 in the macaque monkey: a double-labelling study.J. Comp. Neurol. 1986; 253: 374-402Crossref PubMed Scopus (137) Google Scholar], illustrated in Figure I by the 'red' terminals failing to contact level (i-2). The backward projection originating from deep layers reaches further, but these descending terminations (blue) show a progressive shift of focus upon layers 1 and 6 [51Rockland K.S. et al.Divergent feedback connections from areas V4 and TEO in the macaque.Vis. Neurosci. 1994; 11: 579-600Crossref PubMed Scopus (117) Google Scholar]. In the opposite direction, levels (i+1) and (i+2) show a progressive shift of focus of ascending terminations (green) upon layer 4 [97Rockland K.S. Pandya D.N. Laminar origins and terminations of cortical connections of the occipital lobe in the rhesus monkey.Brain Res. 1979; 179: 3-20Crossref PubMed Scopus (693) Google Scholar].The differential contribution of superficial and deep sources to superficial and deep terminations in descending projections is not well established, because few studies have used tracers with subtotal layer deposition to study interareal connectivity. At minimum, the rule may be that like connects with like, laminar-wise. Layer 6, for instance, receives the densest input when the source of the descending projection includes layer 6 of the higher area [103Coogan T.A. Burkhalter A. Hierarchical organization of areas in rat visual cortex.J. Neurosci. 1993; 13: 3749-3772PubMed Google Scholar, 104Henry G.H. et al.Projections from areas 18 and 19 to cat striate cortex: divergence and laminar specificity.Eur. J. Neurosci. 1991; 3: 186-200Crossref PubMed Scopus (40) Google Scholar]. However, layer 1 can receive descending input from deep layers in systems as diverse as primate visual and rodent somatomotor cortex [105Cauller L.J. et al.Backward cortical projections to primary somatosensory cortex in rats extend long horizontal axons in layer I.J. Comp. Neurol. 1998; 390: 297-310Crossref PubMed Scopus (156) Google Scholar, 106Angelucci A. et al.Circuits for local and global signal integration in primary visual cortex.J. Neurosci. 2002; 22: 8633-8646PubMed Google Scholar], and layer 5 can receive descending input from superficial sources, at least in cat and rat area V1 [103Coogan T.A. Burkhalter A. Hierarchical organization of areas in rat visual cortex.J. Neurosci. 1993; 13: 3749-3772PubMed Google Scholar, 104Henry G.H. et al.Projections from areas 18 and 19 to cat striate cortex: divergence and laminar specificity.Eur. J. Neurosci. 1991; 3: 186-200Crossref PubMed Scopus (40) Google Scholar]. These patterns are summarised in Figure I by the violet tone of descending terminations to layer 5 and superficial layers in level (i-1), indicating a mix of superficial (red) and deep (blue) sources at level (i). The blueing of terminals in deeper layers of level (i-1), and all layers in level (i-2), indicates a progressive domination of deep layer sources from level (i). Figure I shows the laminar sources and distributions of ascending connections (green) and descending connections (red, violet, and blue), originating from a certain level (i) in a hierarchical chain. The basic laminar patterns distinguishing ascending and descending connections were originally established by studies of primate visual cortex [5Felleman D.J. Van Essen D.C. Distributed hierarchical processing in the primate cerebral cortex.Cereb. Cortex. 1991; 1: 1-47Crossref PubMed Scopus (5507) Google Scholar, 97Rockland K.S. Pandya D.N. Laminar origins and terminations of cortical connections of the occipital lobe in the rhesus monkey.Brain Res. 1979; 179: 3-20Crossref PubMed Scopus (693) Google Scholar, 98Maunsell J.H.R. Van Essen D.C. The connections of the middle temporal area and their relationship to a cortical hierarchy in the macaque monkey.J. Neurosci. 1983; 3: 2563-2586PubMed Google Scholar]. Systematic variations with hierarchical distance were later formulated as a 'distance rule' [37Markov N.T. Kennedy H. The importance of being hierarchical.Curr. Opin. Neurobiol. 2013; 23: 187-194Crossref PubMed Scopus (105) Google Scholar, 99Barone P. et al.Laminar distribution of neurons in extrastriate areas projecting to visual areas V1 and V4 correlates with the hierarchical rank and indicates the operation of a distance rule.J. Neurosci. 2000; 20: 3263-32681PubMed Google Scholar]: with regard to origins, the proportion of superficial (layer 3A) neurons forming a backward projection decreases with greater distance spanned by the projection [99Barone P. et al.Laminar distribution of neurons in extrastriate areas projecting to visual areas V1 and V4 correlates with the hierarchical rank and indicates the operation of a distance rule.J. Neurosci. 2000; 20: 3263-32681PubMed Google Scholar, 100Sousa A.P.B. et al.Topographic organization of cortical input to striate cortex in the Cebus monkey: a fluorescent tracer study.J. Comp. Neurol. 1991; 308: 665-682Crossref PubMed Scopus (80) Google Scholar, 101Rockland K.S. Van Hoesen G.W. Direct temporal-occipital feedback connections to striate cortex (V1) in the macaque monkey.Cereb. Cortex. 1994; 4: 300-313Crossref PubMed Scopus (154) Google Scholar, 102Perkel D.J. et al.Topography of the afferent connectivity of area 17 in the macaque monkey: a double-labelling study.J. Comp. Neurol. 1986; 253: 374-402Crossref PubMed Scopus (137) Google Scholar], illustrated in Figure I by the 'red' terminals failing to contact level (i-2). The backward projection originating from deep layers reaches further, but these descending terminations (blue) show a progressive shift of focus upon layers 1 and 6 [51Rockland K.S. et al.Divergent feedback connections from areas V4 and TEO in the macaque.Vis. Neurosci. 1994; 11: 579-600Crossref PubMed Scopus (117) Google Scholar]. In the opposite direction, levels (i+1) and (i+2) show a progressive shift of focus of ascending terminations (green) upon layer 4 [97Rockland K.S. Pandya D.N. Laminar origins and terminations of cortical connections of the occipital lobe in the rhesus monkey.Brain Res. 1979; 179: 3-20Crossref PubMed Scopus (693) Google Scholar]. The differential contribution of superficial and deep sources to superficial and deep terminations in descending projections is not well established, because few studies have used tracers with subtotal layer deposition to study interareal connectivity. At minimum, the rule may be that like connects with like, laminar-wise. Layer 6, for instance, receives the densest input when the source of the descending projection includes layer 6 of the higher area [103Coogan T.A. Burkhalter A. Hierarchical organization of areas in rat visual cortex.J. Neurosci. 1993; 13: 3749-3772PubMed Google Scholar, 104Henry G.H. et al.Projections from areas 18 and 19 to cat striate cortex: divergence and laminar specificity.Eur. J. Neurosci. 1991; 3: 186-200Crossref PubMed Scopus (40) Google Scholar]. However, layer 1 can receive descending input from deep layers in systems as diverse as primate visual and rodent somatomotor cortex [105Cauller L.J. et al.Backward cortical projections to primary somatosensory cortex in rats extend long horizontal axons in layer I.J. Comp. Neurol. 1998; 390: 297-310Crossref PubMed Scopus (156) Google Scholar, 106Angelucci A. et al.Circuits for local and global signal integration in primary visual cortex.J. Neurosci. 2002; 22: 8633-8646PubMed Google Scholar], and layer 5 can receive descending input from superficial sources, at least in cat and rat area V1 [103Coogan T.A. Burkhalter A. Hierarchical organization of areas in rat visual cortex.J. Neurosci. 1993; 13: 3749-3772PubMed Google Scholar, 104Henry G.H. et al.Projections from areas 18 and 19 to cat striate cortex: divergence and laminar specificity.Eur. J. Neurosci. 1991; 3: 186-200Crossref PubMed Scopus (40) Google Scholar]. These patterns are summarised in Figure I by the violet tone of descending terminations to layer 5 and superficial layers in level (i-1), indicating a mix of superficial (red) and deep (blue) sources at level (i). The blueing of terminals in deeper layers of level (i-1), and all layers in level (i-2), indicates a progressive domination of deep layer sources from level (i). There is a consistent asymmetry between forward connections from sensory to motor areas and the reverse backward connections (e.g., between M1 and area 3a) [11Kunzle H. Cortico-cortical efferents of primary motor and somatosensory regions of the cerebral cortex in Macaca fascicularis.Neuroscience. 1978; 3: 25-39Crossref Scopus (90) Google Scholar, 12Stepniewska I. et al.Architectonics, somatotopic organization, and ipsilateral cortical connections of the primary motor area (M1) of owl monkeys.J. Comp. Neurol. 1993; 330: 238-271Crossref PubMed Scopus (271) Google Scholar, 13Leichnetz G.R. Afferent and efferent connections of the dorsolateral precentral gyrus (area 4, hand/arm region) in the macaque monkey, with comparisons to area 8.J. Comp. Neurol. 1986; 254: 460-492Crossref PubMed Scopus (203) Google Scholar], but the reciprocal connections among motor areas are of a distinct nature: there is a backward pattern of termination for projections from premotor areas to M1 [10Watanabe-Sawaguchi K. et al.Cytoarchitecture and intrafrontal connections of the frontal cortex of the brain of the hamadryas baboon (Papio hamadryas).J. Comp. Neurol. 1991; 311: 108-133Crossref PubMed Scopus (49) Google Scholar], yet the reverse connections (e.g., M1 to SMA, supplementary motor area) are columnar [11Kunzle H. Cortico-cortical efferents of primary motor and somatosensory regions of the cerebral cortex in Macaca fascicularis.Neuroscience. 1978; 3: 25-39Crossref Scopus (90) Google Scholar, 12Stepniewska I. et al.Architectonics, somatotopic organization, and ipsilateral cortical connections of the primary motor area (M1) of owl monkeys.J. Comp. Neurol. 1993; 330: 238-271Crossref PubMed Scopus (271) Google Scholar, 13Leichnetz G.R. Afferent and efferent connections of the dorsolateral precentral gyrus (area 4, hand/arm region) in the macaque monkey, with comparisons to area 8.J. Comp. Neurol. 1986; 254: 460-492Crossref PubMed Scopus (203) Google Scholar], of the sort normally associated with lateral connections. Hence, the premotor areas may top the hierarchy, as previously suggested [5Felleman D.J. Van Essen D.C. Distributed hierarchical processing in the primate cerebral cortex.Cereb. Cortex. 1991; 1: 1-47Crossref PubMed Scopus (5507) Google Scholar, 14Shipp S. The importance of being agranular: a comparative account of visual and motor cortex.Philos. Trans. R. Soc. Lond. B: Biol. Sci. 2005; 360: 797-814Crossref PubMed Scopus (98) Google Scholar], but there is little evidence for a classical ascending pathway through motor areas [6Adams R.A. et al.Predictions not commands: active inference in the motor system.Brain Struct. Funct. 2013; 218: 611-643Crossref PubMed Scopus (414) Google Scholar, 14Shipp S. The importance of being agranular: a comparative account of visual and motor cortex.Philos. Trans. R. Soc. Lond. B: Biol. Sci. 2005; 360: 797-814Crossref PubMed Scopus (98) Google Scholar]. In this review, we attempt to reconcile the laminar architecture and connectivity in both visual and sensorimotor hierarchies within a popular theoretical framework for describing cortical operations [15Friston K. The free-energy principle: a rough guide to the brain?.Trends Cogn. Sci. 2009; 13: 293-301Abstract Full Text Full Text PDF PubMed Scopus (1008) Google Scholar]. A percept can be regarded as a hypothesis that explains sensory input [16Gregory R.L. Perceptions as hypotheses.Philos. Trans. R. Soc. Lond. B: Biol. Sci. 1980; 290: 181-197Crossref PubMed Scopus (477) Google Scholar, 17Friston K. et al.Perceptions as hypotheses: saccades as experiments.Front. Psychol. 2011; 3: 151Google Scholar] – on occasion, an erroneous hypothesis, as demonstrated by classic illusions (Figure 1A) . The percept interprets sensory data, such that what we see is the inferred cause of the sensations, not merely an image of the data per se [18Helmholtz, H., translated by Southall, J.P.C., ed. (1860/1962) Handbuch der Physiologischen Optik (Vol. 3), DoverGoogle Scholar]. In Figure 1A, the facial features have an ambiguous depth structure that is resolved by our past experience of convex faces. The ability to infer the cause of visual sensations (e.g., a face) rests on an internal, generative model of how objects generate sensory data [19Friston K. A theory of cortical responses.Philos. Trans. R. Soc. Lond. B: Biol. Sci. 2005; 360: 815-836Crossref PubMed Scopus (2556) Google Scholar, 20Kersten D. et al.Object perception as Bayesian inference.Annu. Rev. Psychol. 2004; 55: 271-304Crossref PubMed Scopus (843) Google Scholar]. Generative models are required to finesse the problem of sensory indeterminacy (e.g., ambiguity) that illusions aptly illustrate. A generative model also has a temporal aspect: velocity is not a property of an instantaneous scene or 'snapshot', but an attribute that integrates sensory evidence over time. Biological motion detection implies recognition of complex motion patterns, such as a reach and grasp movement, or a repetitive action, such as walking [21Blake R. Shiffrar M. Perception of human motion.Annu. Rev. Psychol. 2007; 58: 47-73Crossref PubMed Scopus (681) Google Scholar]. In other words, the generative model of the brain is more like a narrative or scenario, predicting sequences of events. The scenario enables predictions about what may happen next. If a head is turning, for instance, a frontal view of a face may soon be replaced by a profile [22Perrett D.I. et al.Seeing the future: natural image sequences produce 'anticipatory' neuronal activity and bias perceptual report.Q. J. Exp. Psychol. 2009; 62: 2081-2104Crossref PubMed Scopus (55) Google Scholar]. Generative models are necessarily hierarchical (in space and time). If the visual system operates as a generative model, the percept corresponding to a particular cause is not specified at only one level, but has multiple levels of description. Take face processing, for example: a high-level face area encodes view-invariant face identity, whereas lower levels are view specific but less identity specific [23Freiwald W.A. Tsao D.Y. Functional compartmentalization and viewpoint generalization within the macaque face-processing system.Science. 2010; 330: 845-851Crossref PubMed Scopus (441) Google Scholar]. Features such as hair, eye, and skin colour are also encoded elsewhere [24Conway B.R. Tsao D.Y. Color architecture in alert macaque cortex revealed by FMRI.Cereb. Cortex. 2006; 16: 1604-1613Crossref PubMed Scopus (71) Google Scholar]. In addition, because face cells are size and position invariant [25Rolls E.T. Neurophysiological mechanisms underlying face processing within and beyond the temporal cortical visual areas.Philos. Trans. R. Soc. Lond. B: Biol. Sci. 1992; 335: 11-20Crossref PubMed Scopus (292) Google Scholar], lower areas must represent the 'filled-in surface' and 'border ownership' attributes of a percept [26Pollen D.A. Fundamental requirements for primary visual perception.Cereb. Cortex. 2008; 18: 1991-1998Crossref PubMed Scopus (24) Google Scholar, 27Poort J. et al.The role of attention in figure-ground segregation in areas V1 and V4 of the visual cortex.Neuron. 2012; 75: 143-156Abstract Full Text Full Text PDF PubMed Scopus (155) Google Scholar, 28Qiu F.T. et al.Figure-ground mechanisms provide structure for selective attention.Nat. Neurosci. 2007; 10: 1492-1499Crossref PubMed Scopus (167) Google Scholar]. In short, the gestalt of a 'face' has multiple components. In modelling terms, the high-level face area provides the highest stamp of recognition, guiding and contextualising inference about physical attributes in lower-level areas. Here, we shall use the term 'expectations' to refer to the representations of causes encoded at each level. Predictive coding schemes (e.g., [29Rao R.P. Ballard D.H. Predictive coding in the visual cortex: a functional interpretation of some extra-classical receptive-field effects.Nat. Neurosci. 1999; 2: 79-87Crossref PubMed Scopus (2864) Google Scholar]) describe the inversion of a generative model, in order to recognise causes from their sensory consequences. In global terms, the model generates predictions of sensory input from high-level representations of causes; more specifically, the expectations at any given level predict the expectations at the level below. The model is inverted using a 'guess it and try it' approach (Figure 1B): each level computes 'prediction errors' by subtracting top-down predictions from its current expectations. The requisite predictions are based on expectations from the level above and conveyed by top-down or backward connections. Bottom-up prediction error signals are then passed forwards to modify expectations in the level above. This iterative, reciprocal exchange of predictions and errors minimises prediction error at every level of the hierarchy and provides a plausible explanation for visual sensations, in terms of expectations at multiple levels. In generalised formulations of hierarchical predictive coding, there are three sorts of expectation: expected 'causes', 'states', and 'precisions' [15Friston K. The free-energy principle: a rough guide to the brain?.Trends Cogn. Sci. 2009; 13: 293-301Abstract Full Text Full Text PDF PubMed Scopus (1008) Google Scholar]. Causes are invariant aspects of the world that create regularities in sensory data, such as objects in the visual scene. Their correspondence to elements of the scene is concrete at lower levels (e.g., a colour), and increasingly abstract at higher levels of the hierarchy (e.g., a smile). Whereas causes model categorical aspects of the world, states model their dynamics; that is, the fluctuations caused by the interactions among causes (e.g., motion of an object) or between cause and context (e.g., a rotating object and its illumination). Finally, precision corresponds to the reliability (inverse amplitude of random fluctuations) of causes and states. Therefore, expected precision determines the relative confidence in descending predictions and ascending prediction error. The differential equations describing predictive coding are provided elsewhere [30Adams R.A. et al.The computational anatomy of psychosis.Front Psychiatry. 2013; 4: 47Crossref PubMed Scopus (482) Google Scholar], together with the theory relating predictive coding to Bayesian inference [15Friston K. The free-energy principle: a rough guide to the brain?.Trends Cogn. Sci. 2009; 13: 293-301Abstract Full Text Full Text PDF PubMed Scopus (1008) Google Scholar, 31Friston K. The free-energy principle: a unified brain theory?.Nat. Rev. Neurosci. 2010; 11: 127-138Crossref PubMed Scopus (3544) Google Scholar]. Here, we consider the computational architecture and its implementation by neuronal circuitry. Figure 2 shows five kinds of computational unit (cf. neuronal ensembles): expectation and error units for causes and states, and units signalling expected precision. To recap, expectation units encode the expected causes and states describing events (scenarios) in the environment, whereas error units report inconsistencies between expectations at different levels or, at the sensory level, the mismatch between predictions and sensory input. Units encoding expected precision modulate the gain of error units and endow them with greater or lesser weight. This cortical gain control balances the influence of prediction errors at different levels in the hierarchy. Accordingly, precision is associated with the top-down deployment of attention [32Feldman H. Friston K.J. Attention, uncertainty, and free-energy.Front. Hum. Neurosci. 2010; 4: 215Crossref PubMed Scopus (769) Google Scholar] in the sensory domain and action selection in the context of affordance. In summary, expectation and error units interact to update beliefs about causes and states in the world, with one crucial distinction: expected causes are updated by reciprocal exchanges between hierarchical levels, whereas expected states are updated within each level. Below, we suggest a neural implementation of the predictive coding model outlined above (noting that alternative formulations could specify a different neural architecture [33Spratling M.W. Reconciling predictive coding and biased competition models of cortical function.Front. Comput. Neurosci. 2008; 2: 4Crossref PubMed Scopus (95) Google Scholar]). We prefabricate the scheme in visual cortex, as a model of hierarchical processing, before transcribing it to motor cortex and illustrating its explanatory scope through the example of mirror neurons. We now attempt to marry the computational anatomy of predictive coding with cortical microcircuitry. For simplicity, we focus on updating expected causes: our aim is not to specify exactly how such computations are performed at the synaptic level, but to indicate how they might map onto the laminar architecture of extrinsic and intrinsic cortical connections. The scheme shown in Figure 3 is inferred from anatomy alone; there is no explicit physiological categorisation of the notional expectation, error and precision units, but we make the provisional assumption that all three are represented in some form by pyramidal cells (or by excitatory, spiny stellate cells in layer 4). Extrinsic and intrinsic axonal ramifications typically contact inhibitory interneurons as well as pyramidal neurons [34Isaacson J.S. Scanziani M. How inhibition shapes cortical activity.Neuron. 2011; 72: 231-243Abstract Full Text Full Text PDF PubMed Scopus (999) Google Scholar], but the former are largely exclude
This paper reviews recent developments in statistical structure learning; namely, Bayesian model reduction. Bayesian model reduction is a method for rapidly computing the evidence and parameters of probabilistic models that differ only in their priors. In the setting of variational Bayes this has an analytical solution, which finesses the problem of scoring large model spaces in model comparison or structure learning. In this technical note, we review Bayesian model reduction and provide the relevant equations for several discrete and continuous probability distributions. We provide worked examples in the context of multivariate linear regression, Gaussian mixture models and dynamical systems (dynamic causal modelling). These examples are accompanied by the Matlab scripts necessary to reproduce the results. Finally, we briefly review recent applications in the fields of neuroimaging and neuroscience. Specifically, we consider structure learning and hierarchical or empirical Bayes that can be regarded as a metaphor for neurobiological processes like abductive reasoning.
I know little about social psychology and less about cognitive consistency theory; however, I greatly enjoyed reading the deconstruction of this paradigm by Kruglanski et al. (this issue). I learne...
Cortical activity is the product of interactions among neuronal populations. Macroscopic electrophysiological phenomena are generated by these interactions. In principle, the mechanisms of these interactions afford constraints on biologically plausible models of electrophysiological responses. In other words, the macroscopic features of cortical activity can be modelled in terms of the microscopic behaviour of neurons. An evoked response potential (ERP) is the mean electrical potential measured from an electrode on the scalp, in response to some event. The purpose of this paper is to outline a population density approach to modelling ERPs. We propose a biologically plausible model of neuronal activity that enables the estimation of physiologically meaningful parameters from electrophysiological data. The model encompasses four basic characteristics of neuronal activity and organization: (i) neurons are dynamic units, (ii) driven by stochastic forces, (iii) organized into populations with similar biophysical properties and response characteristics and (iv) multiple populations interact to form functional networks. This leads to a formulation of population dynamics in terms of the Fokker–Planck equation. The solution of this equation is the temporal evolution of a probability density over state-space, representing the distribution of an ensemble of trajectories. Each trajectory corresponds to the changing state of a neuron. Measurements can be modelled by taking expectations over this density, e.g. mean membrane potential, firing rate or energy consumption per neuron. The key motivation behind our approach is that ERPs represent an average response over many neurons. This means it is sufficient to model the probability density over neurons, because this implicitly models their average state. Although the dynamics of each neuron can be highly stochastic, the dynamics of the density is not. This means we can use Bayesian inference and estimation tools that have already been established for deterministic systems. The potential importance of modelling density dynamics (as opposed to more conventional neural mass models) is that they include interactions among the moments of neuronal states (e.g. the mean depolarization may depend on the variance of synaptic currents through nonlinear mechanisms). Here, we formulate a population model, based on biologically informed model-neurons with spike-rate adaptation and synaptic dynamics. Neuronal sub-populations are coupled to form an observation model, with the aim of estimating and making inferences about coupling among sub-populations using real data. We approximate the time-dependent solution of the system using a bi-orthogonal set and first-order perturbation expansion. For didactic purposes, the model is developed first in the context of deterministic input, and then extended to include stochastic effects. The approach is demonstrated using synthetic data, where model parameters are identified using a Bayesian estimation scheme we have described previously.
Thalamotomy at the ventralis intermedius nucleus for essential tremor is known to cause changes in motor circuitry, but how a focal lesion leads to progressive changes in connectivity is not clear. In order to understand the mechanisms by which thalamotomy exerts enduring effects on motor circuitry, a quantitative analysis of directed or effective connectivity among motor-related areas is required. We characterized changes in effective connectivity of the motor system following thalamotomy using (spectral) dynamic causal modelling (spDCM) for resting state fMRI. To differentiate long-lasting treatment effects from transient effects, and to identify symptom-related changes in effective connectivity, we subject longitudinal resting-state fMRI data to spDCM, acquired 1 day prior to, and 1 day, 7 days, and 3 months after thalamotomy using a non-cranium-opening MRI-guided focused ultrasound ablation technique. For the group-level (between subject) analysis of longitudinal (between session) effects, we introduce a multi-level parametric empirical Bayes analysis (PEB) for spDCM. We found remarkably selective and consistent changes in effective connectivity from the ventrolateral nuclei and the supplementary motor area to the contralateral dentate nucleus after thalamotomy, which may be mediated via a polysynaptic thalamic-cortical-cerebellar motor loop. Crucially, changes in effective connectivity predicted changes in clinical motor-symptom scores after thalamotomy. This study speaks to the efficacy of thalamotomy in regulating the dentate nucleus in the context of treating essential tremor. Furthermore, it illustrates the utility of PEB for group-level analysis of DCM in quantifying longitudinal changes in effective connectivity; i.e., measuring long-term plasticity in human subjects noninvasively.
Abstract Natural language syntax yields an unbounded array of hierarchically structured expressions. We claim that these are used in the service of active inference in accord with the free-energy principle (FEP). While conceptual advances alongside modelling and simulation work have attempted to connect speech segmentation and linguistic communication with the FEP, we extend this program to the underlying computations responsible for generating syntactic objects. We argue that recently proposed principles of economy in language design—such as “minimal search” criteria from theoretical syntax—adhere to the FEP. This affords a greater degree of explanatory power to the FEP—with respect to higher language functions—and offers linguistics a grounding in first principles with respect to computability. While we mostly focus on building new principled conceptual relations between syntax and the FEP, we also show through a sample of preliminary examples how both tree-geometric depth and a Kolmogorov complexity estimate (recruiting a Lempel–Ziv compression algorithm) can be used to accurately predict legal operations on syntactic workspaces, directly in line with formulations of variational free energy minimization. This is used to motivate a general principle of language design that we term Turing–Chomsky Compression (TCC). We use TCC to align concerns of linguists with the normative account of self-organization furnished by the FEP, by marshalling evidence from theoretical linguistics and psycholinguistics to ground core principles of efficient syntactic computation within active inference.
Abstract This chapter shows that new methods for measuring effective connectivity allow us to characterize the interactions between brain regions which underlie the complex interactions among different processing stages of functional architectures. It reviews the basic concepts of effective connectivity in neuroimaging. The methods introduced to assess effective connectivity are multiple linear regression, covariance structural equation modelling and variable parameter regression. The first example demonstrates that non-linear interactions can be characterized using simple extensions of linear models, while in the second, structural equation modelling is introduced as a device that allows one to combine observed changes in cortical activity and anatomical models. Finally, the chapter concludes that the approach to neuroimaging data and regional interactions is an exciting endeavour, which is starting to attract more attention.
