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To make a pattern, a 2-D object which will have one of the 10 crystallographic point groups assigned to it is repeated along a 1-D or 2-D lattice. Also important is invariance under a fourth kind of transformation: scaling. Concentric circles of geometrically progressing diameter are invariant under scaling. Live Science.

Basics of GROUP THEORY (Part-1) - Understanding Symmetry Operations

An isosceles triangle and a butterfly are examples of objects that exhibit reflective symmetry. Objects in 2-D have a line of symmetry; objects in 3-D have a plane of symmetry. They are invariant under reflection. So what does gauge symmetry bring to the table? Here, we develop the idea that distinct classes of inference or behaviour are equivalent to pattern formation in the nervous system. For example, the perception of a visual object involves highly organised patterns of neuronal responses throughout the visual cortical hierarchy, reflecting a functional specialisation for colour and motion [ 15 ]—right down to the architecture of classical receptive fields [ 16 , 17 ].

Specifically, the gauge perspective enables us to specify an ontology of patterns or responses that is independent of the generative model. This does not mean that the generative model is irrelevant, but rather a great deal could be learnt by knowing about the symmetries invariances of that model.

All models with a given symmetry explore the same range of pattern-forming behaviour—without reference to the underlying model [ 18 ]. For example, could experience-dependent or Hebbian plasticity [ 19 ] be an example of a symmetry that minimises variational free energy [ 20 ], and yet the products of experience lead to very different—but equally good—neuronal patterns and architectures.

In other words, could we understand phenomena like bistable perception [ 21 ] and intersubject variability [ 22 ] in terms of a gauge theory. In the animal kingdom, many aspects of behaviour are similar, yet the neural mechanisms that mediate such behaviours can be very different. For example, bees may forage using different neuronal mechanisms [ 19 ] from those used by a rat to explore a maze [ 23 ], or those we employ while foraging for information with visual saccades [ 24 ].

The difference emerges from the heterogeneity of evolutionary pressures from the ecological niche. A gauge theory that describes behaviour and establishes a mathematical framework to understand the gauge transformations that render behaviour invariant, within and between species, may be invaluable. Particularly because such an approach paves the way for a computational neuroethology, enabling us to study what properties of neuronal dynamics are conserved over species, and what constraints have caused certain neural network properties to diverge over evolution.

For example, the minimisation of variational free energy may provide an explanation for foraging that transcends the ecological context: see [ 25 ] for an example of epistemic foraging that is formally equivalent to saccadic sampling of the visual field [ 26 ].

In the study of dynamical attractors and pattern formation, it generally makes sense to study model-independent generalities as a first step, adding the details later. For example, the Lorenz attractor has found a useful role in modelling a variety of systems; from catalytic reactions [ 27 ] to meteorology [ 28 ] and immunology [ 29 ]. A technical issue here is the distinction between phase and physical space [ 18 ].

Any system evolves in a physical space, wherein transformations to a phase-space involve an unspecified change of coordinates. Such coordinate transformations can disconnect variables of the dynamical system and the variables observed in physical space e. See also S1 Appendix. The gauge-symmetry perspective alleviates the potentially problematic disconnect between physical and phase spaces: a symmetry property in phase space translates into a symmetry property in the physical space, and vice-versa, because this is the defining property of properly formulated gauge symmetries.

A related benefit of understanding the symmetries of a dynamical system is that it enables one to map from one dynamical system to another, allowing us to identify a range of solutions with identical dynamics [ 32 , 33 ]. The second important question posed by framing a gauge theory is "how are symmetries related to gauge fields? In other words, one can imagine them as fields that ensure the Lagrangian is invariant to transformations. In the next section, we use these notions to formulate the variational free energy formalism as a gauge theory for the nervous system.

The variational free energy formalism uses the fact that biological systems must resist the second law of thermodynamics i.

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In a similar vein to Maxwell's demon, an organism reduces its entropy through sampling the environment—to actively minimise the self information or surprise of each successive sensory sample this surprise is upper bounded by free energy. By doing so, it places a bound on the entropy of attributes of the environment in which it is immersed. Variational free energy operationalises this bound by ensuring internal states of the system become a replica i.

Symmetry in Science: An Introduction to the General Theory

This can be regarded as a formulation of the good regulator hypothesis [ 34 ], which states that every good regulator of a system must be a model of that system. We know that a gauge theory would leave the Lagrangian invariant under continuous symmetry transformations. Therefore, a gauge theory of the brain requires the interaction among three ingredients: a system equipped with symmetry, some local forces applied to the system, and one or more gauge fields to compensate for the local perturbations that are introduced. The first ingredient is a system equipped with symmetry: for the purposes of our argument, the system is the nervous system and the Lagrangian is the entropy of sensory samples which is upper-bounded by variational free energy, averaged over time.

The local forces are mediated by the external states of the world i.

Symmetry in Science : An Introduction to the General Theory by J. Rosen -

The gauge fields can then be identified by considering the fact that variational free energy is a scalar quantity based on probability measures. Let us see how. From differential geometry treatments of probability measures S3 Text , it is known that the manifold traced out by sufficient statistics of a probability measure is curved in nature [ 35 , 36 ]; more specifically, it has negative curvature and is therefore endowed with a hyperbolic geometry. Barring technical details, it suffices to understand that a functional e. Moving along such a curved manifold requires a measure of distance that corresponds to the distance between two distributions.

Technically, the Fisher information represents the curvature of the relative entropy Kullback-Leibler divergence. Put simply, distances in the curved geometry of sufficient statistics—that define variational free energy—correspond to the relative entropy in going from one point on the free energy manifold to another.

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Instead of using classical results from differential geometry [ 35 — 39 ], we will motivate the gauge formulation of variational free energy by asking an empirical question: how does neuronal activity follow the steepest descent direction to attain its free energy minimum? In other words, how does it find the shortest path to the nearest minimum?

As the free energy manifold is curved, there are no orthonormal linear coordinates to describe it. This means the distance between two points on the manifold can only be determined with the help of the Fisher information metric that accounts for the curvature. Algebraic derivations S3 Text tell us that, in such free energy landscapes, a Euclidean gradient descent is replaced by a Riemann gradient, which simply weights the Euclidean gradient by its asymptotic variance.

In the free energy framework, when the posterior probability is approximated with a Gaussian distribution the Laplace approximation; S4 Text , perception and action simply become gradient flows driven by precision-weighted prediction errors. Here, prediction errors are simply the difference between sensory input local perturbations and predictions of those inputs based upon the systems internal states that encode probability distributions or Bayesian beliefs about external states that cause sensory input.

Mathematically, precision-weighted prediction errors emerge when one computes the Euclidean gradient of the free energy with respect to the sufficient statistics. In a curvilinear space, the precision-weighted prediction errors are replaced by dispersion- and precision-weighted prediction errors. This says something quite fundamental—perception cannot be any more optimal than the asymptotic dispersion inverse Fisher information regardless of the generative model. In other words, the well-known bound upper limit on the precision of any unbiased estimate of a model parameter in statistics emerges here as a natural consequence of applying information geometry.

In the context of the Bayesian brain, this means there is a necessary limit to the certainty with which we can estimate things. We will see next, that attaining this limit translates into attention. See Box 1 for an overview. This essay considers the principle of free energy minimization as a candidate gauge theory that prescribes neuronal dynamics in terms of a Lagrangian. Here, the Lagrangian is the variational free energy, which is a functional a function of a function of a probability distribution encoded by neuronal activity.

This probabilistic encoding means that neuronal activity can be described by a path or trajectory on a manifold in the space of sufficient statistics variables that are sufficient to describe a probability distribution. In other words, if one interprets the brain as making inferences, the underlying beliefs must be induced by biophysical representations that play the role of sufficient statistics.

This is important because it takes us into the realm of differential geometry S2 Text , where the metric space—on which the geometry is defined—is constituted by sufficient statistics like the mean and variance of a Gaussian distribution. Crucially, the gauge theoretic perspective provides a rigorous way of measuring distance on a manifold, such that the neuronal dynamics transporting one distribution of neuronal activity to another is given by the shortest path. Such a free energy manifold is curvilinear, and finding the shortest path is a nontrivial problem—a problem that living organisms appear to have solved.

It is at this point that the utility of a gauge theoretic approach appears; suggesting particular solutions to the problem of finding the shortest path on curved manifolds. The nature of the solution prescribes a normative theory for self-organised neuronal dynamics. In other words, solving the fundamental problem of minimising free energy—in terms of its path integrals—may illuminate not only how the brain works but may provide efficient schemes in statistics and machine learning.

Variational or Monte Carlo formulations of the Bayesian brain require the brain to invert a generative model of the latent unknown or hidden causes of sensations S3 Text. The implicit normative theory means that neuronal activity and connectivity maximises Bayesian model evidence or minimises variational free energy the Lagrangian —effectively fitting a generative model to sensory samples.

This entails an encoding of beliefs probability distributions about the latent causes, in terms of biophysical variables whose trajectories trace out a manifold.

In deterministic variational schemes, the coordinates on this manifold are the sufficient statistics like the mean and covariance of the distribution or belief, while for a stochastic Monte Carlo formulation, the coordinates are the latent causes themselves S3 Text. The inevitable habitat of these sufficient statistics e. This curvature and associated information geometry may have profound implications for neuronal dynamics and plasticity. It may be the case that neuronal dynamics—or motion in some vast neuronal frame of reference—is as simple as the pendulum S1 Appendix.

However, because the Lagrangian is a function of beliefs probabilities , the manifold that contains this motion is necessarily curved. This means neuronal dynamics, in a local frame of reference, will appear to be subject to forces and drives i. For example, the motion of synaptic connection strengths sufficient statistics of the parameters of generative models depends upon the motion of neural activity sufficient statistics of beliefs about latent causes , leading to experience-dependent plasticity.

A more interesting manifestation highlighted in the main text may be attention that couples the motion of different neuronal states in a way that depends explicitly on the curvature of the manifold as measured by things like Fisher information. In brief, a properly formulated gauge theory should, in principle, provide the exact form of neuronal dynamics and plasticity. These forms may reveal the underlying simplicity of many phenomena that we are already familiar with, such as event-related brain responses, associative plasticity, attentional gating, adaptive learning rates, and so on.

Notice that the definition of a system immersed in its environment can be extended hierarchically, wherein the gauge theory can be applied at a variety of nested levels. At every step, as the Lagrangian is disturbed e. In the setting of predictive coding formulations of variational free energy minimisation, the bottom-up or forward messages are assumed to convey prediction error from a lower hierarchical level to a higher level, while the backward messages comprise predictions of sufficient statistics in the level below.

These predictions are produced to explain away prediction errors in the lower level. From the perspective of a gauge theory, one can think of the local forces as prediction errors that increase variational free energy, thereby activating the gauge fields to explain away local forces [ 40 ]. In this geometrical interpretation, perception and action are educed to form cogent predictions, whereby minimization of prediction errors is an inevitable consequence of the nervous system minimising its Lagrangian.

Crucially, the cognitive homologue of precision-weighting is attention, which suggests gauge fields are intimately related to exogenous attention. In other words, attention is a force that manifests from the curvature of information geometry, in exactly the same way that gravity is manifest when the space—time continuum is curved by massive bodies.

In summary, gauge theoretic arguments suggest that attention and its neurophysiological underpinnings constitutes a necessary weighting of prediction errors or sensory evidence that arises because the manifolds traced out by the path of least free energy or least surprise are inherently curved. The importance of the reciprocal interactions among different scales and between the nervous system and the world has been repeatedly emphasised here and elsewhere. A quantitative formulation of this holistic aspect of information processing in the brain is clearly needed and has been framed in terms of variational free energy minimisation or, more simply, as a suppression of prediction errors encountered by an agent that is actively sampling its environment [ 12 , 14 ].

In this essay, we have reformulated the resolution of prediction error or surprise as a gauge theory for the nervous system. The free energy formulation enables us or our brains to compare a variety of hypotheses about the environment given our sensory samples.