Projection Filtering With Observed State Increments With Applications in Continuous-Time Circular Filtering

Kutschireiter, Anna; Rast, Luke; Drugowitsch, Jan

doi:10.1109/tsp.2022.3143471

Cited by 9 publications

(19 citation statements)

References 30 publications

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Section: Resultsmentioning

confidence: 99%

“…As these two posterior, or belief, parameters fully specify the HD belief, updates of the belief in light of sensory evidence simplify to updating these two parameters. We derived the parameter update dynamics by a technique called projection filtering 20,21 , resulting in Here, f ( κ t ) is a monotonically increasing nonlinear function that controls the speed of decay of one’s certainty κ t (see Eq. (10) in Methods).…”

Section: Resultsmentioning

confidence: 99%

“…Note that as dt → 0, the precision of both angular velocity and absolute HD observations approach 0, in line with the intuition that reducing a time step size dt results in more observations per unit time, which should be accounted for by less precision per observation to avoid “oversampling”. More formally, the square-root scaling of the absolute HD observation precision with ensures that the Fisher information of the observations about the true HD scales linearly in time and κ z in the continuum limit dt → 0 (ref 21 , Theorem 2). The same applies to the dt -1 scaling of the Gaussian variance of the angular velocity observations, again achieving a Fisher information that scales linearly in time.…”

Section: Methodsmentioning

confidence: 99%

“…In contrast, including angular velocity observations (Eq. (5)) is mathematically intractable, such that the circKF relies on an approximation method called projection filtering 20 to find closed- form dynamic expressions for posterior mean and certainty (see ref 21 for technical details, and the SI for a more accessible description of the circKF).…”

Section: Methodsmentioning

confidence: 99%

“…We defined this weight as the change in HD estimate, normalized by the value of the maximum possible change, . Here, α ( κ z dt ) denotes a function that ensures a linear scaling of the Fisher information with sampling time step (see ref 21 , Theorem 2, for details about this function). Thus, by design of the observation model, the Fisher information of a single observation with reliability κ z during a time interval Δ t is given by .…”

Section: Methodsmentioning

confidence: 99%

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Bayesian inference in ring attractor networks

Kutschireiter

Basnak

Wilson

et al. 2021

Preprint

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Efficient navigation requires animals to track their position, velocity and heading direction (HD). Bayesian inference provides a principled framework for estimating these quantities from unreliable sensory observations, yet little is known about how and where Bayesian algorithms could be implemented in the brain's neural networks. Here, we propose a class of recurrent neural networks that track both a dynamic HD estimate and its associated uncertainty. They do so according to a circular Kalman filter, a statistically optimal algorithm for circular estimation. Our network generalizes standard ring attractor models by encoding uncertainty in the amplitude of a bump of neural activity. More generally, we show that near-Bayesian integration is inherent in ring attractor networks, as long as their connectivity strength allows them to sufficiently deviate from the attractor state. Furthermore, we identified the basic network motifs that are required to implement Bayesian inference, and show that these motifs are present in the Drosophila HD system connectome. Overall, our work demonstrates that the Drosophila HD system can in principle implement a dynamic Bayesian inference algorithm in a biologically plausible manner, consistent with recent findings that suggest ring-attractor dynamics underlie the Drosophila HD system.

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Section: Resultsmentioning

confidence: 99%

Section: Resultsmentioning

confidence: 99%

Section: Methodsmentioning

confidence: 99%

Section: Methodsmentioning

confidence: 99%

Section: Methodsmentioning

confidence: 99%

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Bayesian inference in ring attractor networks

Kutschireiter

Basnak

Wilson

et al. 2021

Preprint

Self Cite

View full text Add to dashboard Cite

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Optimal projection filters with information geometry

2023

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We review the introduction of several types of projection filters. Projection structures coming from information geometry are used to obtain a finite dimensional filter in the form of a stochastic differential equation (SDE), starting from the exact infinite-dimensional stochastic partial differential equation (SPDE) for the optimal filter. We start with the Stratonovich projection filters based on the Hellinger distance as introduced and developed in Brigo et al. (IEEE Trans Autom Control 43(2):247–252, 1998, Bernoulli 5(3):495–534, 1999), where the SPDE is put in Stratonovich form before projection, hence the term “Stratonovich projection”. The correction step of the filtering algorithm can be made exact by choosing a suitable exponential family as manifold, there is equivalence with assumed density filters and numerical examples have been studied. Other authors further developed these projection filters and we present a brief literature review. A second type of Stratonovich projection filters was introduced in Armstrong and Brigo (Math Control Signals Syst 28(1):1–33, 2016) where a direct $$L^2$$ L 2 metric is used for projection. Projecting on mixtures of densities as a manifold coincides with Galerkin methods. All the above projection filters lack optimality, as the single vector fields of the Stratonovich SPDE are projected optimally but the SPDE solution as a whole is not approximated optimally by the projected SDE solution according to a clear criterion. This led to the optimal projection filters in Armstrong et al. (Proc Lond Math Soc 119(1):176–213, 2019, Projection of SDEs onto submanifolds. “Information Geometry”, 2023 special issue on half a century of information geometry, 2018), based on the Ito vector and Ito jet projections, where several types of mean square distances between the optimal filter SPDE solution and the sought finite dimensional SDE approximations are minimized, with numerical examples. After reviewing the above developments, we conclude with the remaining challenges.

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