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Hierarchical models in the brain.

Friston K - PLoS Comput. Biol. (2008)

Bottom Line: This means that a single model and optimisation scheme can be used to invert a wide range of models.We present the model and a brief review of its inversion to disclose the relationships among, apparently, diverse generative models of empirical data.We then show that this inversion can be formulated as a simple neural network and may provide a useful metaphor for inference and learning in the brain.

View Article: PubMed Central - PubMed

Affiliation: The Wellcome Trust Centre of Neuroimaging, University College London, London, United Kingdom. k.friston@fil.ion.ucl.ac.uk

ABSTRACT
This paper describes a general model that subsumes many parametric models for continuous data. The model comprises hidden layers of state-space or dynamic causal models, arranged so that the output of one provides input to another. The ensuing hierarchy furnishes a model for many types of data, of arbitrary complexity. Special cases range from the general linear model for static data to generalised convolution models, with system noise, for nonlinear time-series analysis. Crucially, all of these models can be inverted using exactly the same scheme, namely, dynamic expectation maximization. This means that a single model and optimisation scheme can be used to invert a wide range of models. We present the model and a brief review of its inversion to disclose the relationships among, apparently, diverse generative models of empirical data. We then show that this inversion can be formulated as a simple neural network and may provide a useful metaphor for inference and learning in the brain.

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The ensemble density and its mean-field partition.q(ϑ) is the ensembledensity and is encoded in terms of the sufficient statistics of itsmarginals. These statistics or variational parameters (e.g., mean orexpectation) change to extremise free-energy to render the ensembledensity an approximate conditional density on the causes of sensoryinput. The mean-field partition corresponds to a factorization overthe sets comprising the partition. Here, we have used three sets(neural activity, modulation and connectivity). Critically, theoptimisation of the parameters of any one set depends on theparameters of the other sets. In this figure, we have focused onmeans or expectations µi ofthe marginal densities,q(ϑi) = N(ϑi:µi,Ci).
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pcbi-1000211-g010: The ensemble density and its mean-field partition.q(ϑ) is the ensembledensity and is encoded in terms of the sufficient statistics of itsmarginals. These statistics or variational parameters (e.g., mean orexpectation) change to extremise free-energy to render the ensembledensity an approximate conditional density on the causes of sensoryinput. The mean-field partition corresponds to a factorization overthe sets comprising the partition. Here, we have used three sets(neural activity, modulation and connectivity). Critically, theoptimisation of the parameters of any one set depends on theparameters of the other sets. In this figure, we have focused onmeans or expectations µi ofthe marginal densities,q(ϑi) = N(ϑi:µi,Ci).

Mentions: The mean-field approximationq(ϑ) = q(u(t))q(θ)q(λ)enables inference about perceptual states, causal regularities and context,without representing the joint distribution explicitly; c.f., [64].However, the optimisation of one set of sufficient statistics is a functionof the others. This has a fundamental implication for optimisation in thebrain (see Figure 10).For example, ‘activity-dependent plasticity’ and‘functional segregation’ speak to reciprocal influencesbetween changes in states and connections; in that changes in connectionsdepend upon activity and changes in activity depend upon connections. Thingsget more interesting when we consider three sets, because quantitiesencoding precision must be affected by and affect neuronal activity andplasticity. This places strong constraints on the neurobiological candidatesfor these hyperparameters. Happily, the ascending neuromodulatoryneurotransmitter systems, such as dopaminergic and cholinergic projections,have exactly the right characteristics: they are driven by activity inpresynaptic connections and can affect activity though classicalneuromodulatory effects at the post-synaptic membrane [65], while alsoenabling potentiation of connection strengths [66],[67].Furthermore, it is exactly these systems that have been implicated invalue-learning [68]–[70], attention andthe encoding of uncertainty [63],[71].


Hierarchical models in the brain.

Friston K - PLoS Comput. Biol. (2008)

The ensemble density and its mean-field partition.q(ϑ) is the ensembledensity and is encoded in terms of the sufficient statistics of itsmarginals. These statistics or variational parameters (e.g., mean orexpectation) change to extremise free-energy to render the ensembledensity an approximate conditional density on the causes of sensoryinput. The mean-field partition corresponds to a factorization overthe sets comprising the partition. Here, we have used three sets(neural activity, modulation and connectivity). Critically, theoptimisation of the parameters of any one set depends on theparameters of the other sets. In this figure, we have focused onmeans or expectations µi ofthe marginal densities,q(ϑi) = N(ϑi:µi,Ci).
© Copyright Policy
Related In: Results  -  Collection

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Show All Figures
getmorefigures.php?uid=PMC2570625&req=5

pcbi-1000211-g010: The ensemble density and its mean-field partition.q(ϑ) is the ensembledensity and is encoded in terms of the sufficient statistics of itsmarginals. These statistics or variational parameters (e.g., mean orexpectation) change to extremise free-energy to render the ensembledensity an approximate conditional density on the causes of sensoryinput. The mean-field partition corresponds to a factorization overthe sets comprising the partition. Here, we have used three sets(neural activity, modulation and connectivity). Critically, theoptimisation of the parameters of any one set depends on theparameters of the other sets. In this figure, we have focused onmeans or expectations µi ofthe marginal densities,q(ϑi) = N(ϑi:µi,Ci).
Mentions: The mean-field approximationq(ϑ) = q(u(t))q(θ)q(λ)enables inference about perceptual states, causal regularities and context,without representing the joint distribution explicitly; c.f., [64].However, the optimisation of one set of sufficient statistics is a functionof the others. This has a fundamental implication for optimisation in thebrain (see Figure 10).For example, ‘activity-dependent plasticity’ and‘functional segregation’ speak to reciprocal influencesbetween changes in states and connections; in that changes in connectionsdepend upon activity and changes in activity depend upon connections. Thingsget more interesting when we consider three sets, because quantitiesencoding precision must be affected by and affect neuronal activity andplasticity. This places strong constraints on the neurobiological candidatesfor these hyperparameters. Happily, the ascending neuromodulatoryneurotransmitter systems, such as dopaminergic and cholinergic projections,have exactly the right characteristics: they are driven by activity inpresynaptic connections and can affect activity though classicalneuromodulatory effects at the post-synaptic membrane [65], while alsoenabling potentiation of connection strengths [66],[67].Furthermore, it is exactly these systems that have been implicated invalue-learning [68]–[70], attention andthe encoding of uncertainty [63],[71].

Bottom Line: This means that a single model and optimisation scheme can be used to invert a wide range of models.We present the model and a brief review of its inversion to disclose the relationships among, apparently, diverse generative models of empirical data.We then show that this inversion can be formulated as a simple neural network and may provide a useful metaphor for inference and learning in the brain.

View Article: PubMed Central - PubMed

Affiliation: The Wellcome Trust Centre of Neuroimaging, University College London, London, United Kingdom. k.friston@fil.ion.ucl.ac.uk

ABSTRACT
This paper describes a general model that subsumes many parametric models for continuous data. The model comprises hidden layers of state-space or dynamic causal models, arranged so that the output of one provides input to another. The ensuing hierarchy furnishes a model for many types of data, of arbitrary complexity. Special cases range from the general linear model for static data to generalised convolution models, with system noise, for nonlinear time-series analysis. Crucially, all of these models can be inverted using exactly the same scheme, namely, dynamic expectation maximization. This means that a single model and optimisation scheme can be used to invert a wide range of models. We present the model and a brief review of its inversion to disclose the relationships among, apparently, diverse generative models of empirical data. We then show that this inversion can be formulated as a simple neural network and may provide a useful metaphor for inference and learning in the brain.

Show MeSH