scholarly journals Imaging Somatosensory Cortex Responses Measured by OPM-MEG: Variational Free Energy-based Spatial Smoothing Estimation Approach

iScience ◽  
2022 ◽  
pp. 103752
Author(s):  
Nan An ◽  
Fuzhi Cao ◽  
Wen Li ◽  
Wenli Wang ◽  
Weinan Xu ◽  
...  
Keyword(s):  
Free Energy ◽  
Somatosensory Cortex ◽  
Spatial Smoothing ◽  
Variational Free Energy

scholarly journals Active user detection and channel estimation of uplink grant-free SCMA system based on variational free energy

2020 ◽  
Vol 50 (7) ◽  
pp. 964-970
Author(s):  
Quan YUAN ◽  
ZhenYong WANG ◽  
DeZhi LI ◽  
Qing GUO ◽  
ZhenBang WANG
Keyword(s):  
Free Energy ◽  
Channel Estimation ◽  
Variational Free Energy ◽  
Active User

VARIATIONAL ELECTROSTATICS FOR CHARGE SOLVATION

2008 ◽  
Vol 07 (03) ◽  
pp. 397-419 ◽  
Author(s):  
ZHEN-GANG WANG
Keyword(s):  
Free Energy ◽  
Energy Functional ◽  
Dielectric Medium ◽  
Constitutive Relationship ◽  
Coulomb Interactions ◽  
Unified Framework ◽  
Variational Free Energy ◽  
The Poisson Equation ◽  
Charge Solvation ◽  
Electrostatic Problems

We show that the equations of continuum electrostatics can be obtained entirely and simply from a variational free energy comprising the Coulomb interactions among all charged species and a spring-like term for the polarization of the dielectric medium. In this formulation, the Poisson equation, the constitutive relationship between polarization and the electric field, as well as the boundary conditions across discontinuous dielectric boundaries, are all natural consequences of the extremization of the free energy functional. This formulation thus treats the electrostatic equations and the energetics within a single unified framework, avoiding some of the pitfalls in the study of electrostatic problems. Application of this formalism to the nonequilbrium solvation free energy in electron transfer is illustrated. Our calculation reaffirms the well-known result of Marcus. We address the recent criticisms by Li and coworkers who claim that the Marcus result is incorrect, and expose some key mistakes in their approach.

scholarly journals Knowing one's place: a free-energy approach to pattern regulation

2015 ◽  
Vol 12 (105) ◽  
pp. 20141383 ◽  
Author(s):  
Karl Friston ◽  
Michael Levin ◽  
Biswa Sengupta ◽  
Giovanni Pezzulo
Keyword(s):  
Free Energy ◽  
Self Assembly ◽  
Theoretical Formulation ◽  
Genetic Codes ◽  
Variational Free Energy ◽  
Action And Perception ◽  
Cell Groups ◽  
Biological Pattern Formation ◽  
Pattern Regulation ◽  
Epigenetic Processes

Understanding how organisms establish their form during embryogenesis and regeneration represents a major knowledge gap in biological pattern formation. It has been recently suggested that morphogenesis could be understood in terms of cellular information processing and the ability of cell groups to model shape. Here, we offer a proof of principle that self-assembly is an emergent property of cells that share a common (genetic and epigenetic) model of organismal form. This behaviour is formulated in terms of variational free-energy minimization—of the sort that has been used to explain action and perception in neuroscience. In brief, casting the minimization of thermodynamic free energy in terms of variational free energy allows one to interpret (the dynamics of) a system as inferring the causes of its inputs—and acting to resolve uncertainty about those causes. This novel perspective on the coordination of migration and differentiation of cells suggests an interpretation of genetic codes as parametrizing a generative model—predicting the signals sensed by cells in the target morphology—and epigenetic processes as the subsequent inversion of that model. This theoretical formulation may complement bottom-up strategies—that currently focus on molecular pathways—with (constructivist) top-down approaches that have proved themselves in neuroscience and cybernetics.

scholarly journals In vitro neural networks minimise variational free energy

2018 ◽  
Vol 8 (1) ◽  
Author(s):  
Takuya Isomura ◽  
Karl Friston
Keyword(s):  
Neural Networks ◽  
Free Energy ◽  
Variational Free Energy

Active Inference, Belief Propagation, and the Bethe Approximation

2018 ◽  
Vol 30 (9) ◽  
pp. 2530-2567 ◽  
Author(s):  
Sarah Schwöbel ◽  
Stefan Kiebel ◽  
Dimitrije Marković
Keyword(s):  
Free Energy ◽  
Belief Propagation ◽  
Mean Field ◽  
Field Approximation ◽  
Mean Field Approximation ◽  
Active Inference ◽  
Inference Process ◽  
Bethe Approximation ◽  
Variational Free Energy ◽  
The Mean

When modeling goal-directed behavior in the presence of various sources of uncertainty, planning can be described as an inference process. A solution to the problem of planning as inference was previously proposed in the active inference framework in the form of an approximate inference scheme based on variational free energy. However, this approximate scheme was based on the mean-field approximation, which assumes statistical independence of hidden variables and is known to show overconfidence and may converge to local minima of the free energy. To better capture the spatiotemporal properties of an environment, we reformulated the approximate inference process using the so-called Bethe approximation. Importantly, the Bethe approximation allows for representation of pairwise statistical dependencies. Under these assumptions, the minimizer of the variational free energy corresponds to the belief propagation algorithm, commonly used in machine learning. To illustrate the differences between the mean-field approximation and the Bethe approximation, we have simulated agent behavior in a simple goal-reaching task with different types of uncertainties. Overall, the Bethe agent achieves higher success rates in reaching goal states. We relate the better performance of the Bethe agent to more accurate predictions about the consequences of its own actions. Consequently, active inference based on the Bethe approximation extends the application range of active inference to more complex behavioral tasks.

scholarly journals Reverse engineering neural networks to characterise their cost functions

2019 ◽  
Author(s):  
Takuya Isomura ◽  
Karl Friston
Keyword(s):  
Neural Networks ◽  
Free Energy ◽  
Cost Function ◽  
Neural Activity ◽  
Generative Models ◽  
Cost Functions ◽  
Prior Beliefs ◽  
Variational Free Energy ◽  
Markov Decision ◽  
Model Evidence

AbstractThis work considers a class of biologically plausible cost functions for neural networks, where the same cost function is minimised by both neural activity and plasticity. We show that such cost functions can be cast as a variational bound on model evidence under an implicit generative model. Using generative models based on Markov decision processes (MDP), we show, analytically, that neural activity and plasticity perform Bayesian inference and learning, respectively, by maximising model evidence. Using mathematical and numerical analyses, we then confirm that biologically plausible cost functions—used in neural networks—correspond to variational free energy under some prior beliefs about the prevalence of latent states that generate inputs. These prior beliefs are determined by particular constants (i.e., thresholds) that define the cost function. This means that the Bayes optimal encoding of latent or hidden states is achieved when, and only when, the network’s implicit priors match the process that generates the inputs. Our results suggest that when a neural network minimises its cost function, it is implicitly minimising variational free energy under optimal or sub-optimal prior beliefs. This insight is potentially important because it suggests that any free parameter of a neural network’s cost function can itself be optimised—by minimisation with respect to variational free energy.

scholarly journals A Measure of Information Available for Inference

2018 ◽  
Author(s):  
Takuya Isomura
Keyword(s):  
Neural Network ◽  
Free Energy ◽  
Sensory Input ◽  
External World ◽  
The State ◽  
Energy Principle ◽  
Free Energy Principle ◽  
Amount Of Information ◽  
Variational Free Energy ◽  
The Neural Network

The mutual information between the state of a neural network and the state of the external world represents the amount of information stored in the neural network that is associated with the external world. In contrast, the surprise of the sensory input indicates the unpredictability of the current input. In other words, this is a measure of inference ability, and an upper bound of the surprise is known as the variational free energy. According to the free-energy principle (FEP), a neural network continuously minimizes the free energy to perceive the external world. For the survival of animals, inference ability is considered to be more important than simply memorized information. In this study, the free energy is shown to represent the gap between the amount of information stored in the neural network and that available for inference. This concept involves both the FEP and the infomax principle, and will be a useful measure for quantifying the amount of information available for inference.

scholarly journals Sophisticated Inference

2021 ◽  
Vol 33 (3) ◽  
pp. 713-763
Author(s):  
Karl Friston ◽  
Lancelot Da Costa ◽  
Danijar Hafner ◽  
Casper Hesp ◽  
Thomas Parr
Keyword(s):  
Free Energy ◽  
Information Gain ◽  
Energy Functional ◽  
Decision Problems ◽  
Tree Search ◽  
Value Functions ◽  
Active Inference ◽  
States Of Affairs ◽  
Variational Free Energy ◽  
Belief States

Active inference offers a first principle account of sentient behavior, from which special and important cases—for example, reinforcement learning, active learning, Bayes optimal inference, Bayes optimal design—can be derived. Active inference finesses the exploitation-exploration dilemma in relation to prior preferences by placing information gain on the same footing as reward or value. In brief, active inference replaces value functions with functionals of (Bayesian) beliefs, in the form of an expected (variational) free energy. In this letter, we consider a sophisticated kind of active inference using a recursive form of expected free energy. Sophistication describes the degree to which an agent has beliefs about beliefs. We consider agents with beliefs about the counterfactual consequences of action for states of affairs and beliefs about those latent states. In other words, we move from simply considering beliefs about “what would happen if I did that” to “what I would believe about what would happen if I did that.” The recursive form of the free energy functional effectively implements a deep tree search over actions and outcomes in the future. Crucially, this search is over sequences of belief states as opposed to states per se. We illustrate the competence of this scheme using numerical simulations of deep decision problems.

scholarly journals Active Inference: A Process Theory

2017 ◽  
Vol 29 (1) ◽  
pp. 1-49 ◽  
Author(s):  
Karl Friston ◽  
Thomas FitzGerald ◽  
Francesco Rigoli ◽  
Philipp Schwartenbeck ◽  
Giovanni Pezzulo
Keyword(s):  
Free Energy ◽  
Gradient Descent ◽  
Cell Activity ◽  
Face Validity ◽  
Process Theory ◽  
Active Inference ◽  
Neuronal Dynamics ◽  
Context Learning ◽  
Repetition Suppression ◽  
Variational Free Energy

This article describes a process theory based on active inference and belief propagation. Starting from the premise that all neuronal processing (and action selection) can be explained by maximizing Bayesian model evidence—or minimizing variational free energy—we ask whether neuronal responses can be described as a gradient descent on variational free energy. Using a standard (Markov decision process) generative model, we derive the neuronal dynamics implicit in this description and reproduce a remarkable range of well-characterized neuronal phenomena. These include repetition suppression, mismatch negativity, violation responses, place-cell activity, phase precession, theta sequences, theta-gamma coupling, evidence accumulation, race-to-bound dynamics, and transfer of dopamine responses. Furthermore, the (approximately Bayes’ optimal) behavior prescribed by these dynamics has a degree of face validity, providing a formal explanation for reward seeking, context learning, and epistemic foraging. Technically, the fact that a gradient descent appears to be a valid description of neuronal activity means that variational free energy is a Lyapunov function for neuronal dynamics, which therefore conform to Hamilton’s principle of least action.

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