The free energy principle made simpler but not too simple

Friston, Karl J.; Costa, Lancelot Da; Sajid, Noor; Heins, Conor; Ueltzhöffer, Kai; Pavliotis, Grigorios A.; Parr, Thomas

doi:10.1016/j.physrep.2023.07.001

Cited by 55 publications

(13 citation statements)

References 120 publications

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“…To formulate and solve the problem of the evolutionarily optimal simplest neural network, it is convenient to use the toolkit for the analysis of stochastic equations of the form (1), developed in the context of nonequilibrium statistical physics and also previously applied to the analysis of functioning neural networks (but not their evolutionary optimization). 18,27,28,32,33,35-42…”

Section: Methodsmentioning

confidence: 99%

Simplest Model of Nervous System. I. Formalism

Sinitskiy

2023

Preprint

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This study presents a novel, highly simplified model of the nervous system, inspired by one hypothetical scenario of its origin. The model is designed to accommodate both mathematical derivations and numerical simulations, offering a template for studying generalized principles and dynamics beyond the specifics of the referenced origin scenario. The model offers a holistic perspective by treating the nervous system and the environment (in their simplest forms) as parts of one system and, together with a companion paper, notes the key role of evolutionary factors (in this model, predator evasion) in shaping the properties of the nervous system. To emphasize these fundamental principles, some aspects, such as the highly dimensional nature of the networks or detailed molecular mechanisms of their functioning, are omitted in the current version. Analytically, the model facilitates insights into the stationary distribution as a solution to the Fokker-Planck equation and the corresponding effective potential and rotation (solenoidal) terms. Numerically, it generates biologically plausible (given its high abstraction) solutions and supports comprehensive sampling with limited computational resources. Noteworthy findings from the study include limitations of the commonly used weak noise approximation and the significance of rigorous mathematical analysis over heuristic interpretations of the potential. We hope that this abstract model will serve as a fruitful tool for better understanding a complete set of principles for modeling nervous systems.

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Section: Methodsmentioning

confidence: 99%

Simplest Model of Nervous System. I. Formalism

Sinitskiy

2023

Preprint

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“…Recently, a powerful technique to analyze such dynamic systems has been proposed, inspired by methods from nonequilibrium statistical physics. [4][5][6][7][8][9][10][11][12][13][14][15][16] Instead of considering individual stochastic trajectories that can be obtained from equations ( 1)-( 4) by numerical integration, consider an ensemble of such systems characterized by a probability distribution function. In general, this function should be time-dependent; however, in many cases, a stationary probability distribution function Pstat(x,r) exists.…”

Section: Model Definitionmentioning

confidence: 99%

“…The formalism outlined in the previous subsection is commonly used in the literature to analyze neural networks dynamics, given that functions fi (and thus, u and Q) are predetermined. [4][5][6][7][8][9][10][11][12][13][14][15][16] In this work, we use this formalism -for the first time, to the best of our knowledge -to address a new problem, namely, the problem of evolutionary optimization of neural networks. A traditional approach to this problem would be to define the specific forms for f1 and f2, for example, as in equations ( 2) and (3), integrate them to get an ensemble of trajectories, and then calculate the evolutionary fitness as defined in a given model as a temporal average over all trajectories.…”

Section: Model Definitionmentioning

confidence: 99%

“…Within the framework of this model, we will consider this value R0 as constant. The appearance of predators leads to the death of some of the "Dickinsonia", thus reducing the growth rate of the population by the amount of Pdeath, which can be found from equation (14). Furthermore, the functioning of the nervous system and effectors, while reducing Pdeath, also requires the expenditure of "Dickinsonia" resources, which can no longer be directed towards population growth, thus reducing the growth rate by amounts proportional to I1 and I2.…”

Section: Probability Of An Organism's Death As Demonstrated Inmentioning

confidence: 99%

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Simplest Model of Nervous System. II. Evolutionary Optimization

Sinitskiy

2023

Preprint

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In this work, we build upon a simple model of a primitive nervous system presented in a prior companion paper. Within this model, we formulate and solve an optimization problem, aiming to mirror the process of evolutionary optimization of the nervous system. The formally derived predictions include the emergence of sharp peaks of neural activity (‘spikes’), an increasing sensory sensitivity to external signals and a dramatic reduction in the cost of the functioning of the nervous system due to evolutionary optimization. Our work implies that we may be able to make general predictions about the behavior and characteristics of the nervous system irrespective of specific molecular mechanisms or evolutionary trajectories. It also underscores the potential utility of evolutionary optimization as a key principle in mathematical modeling of the nervous system and offers examples of analytical derivations possible in this field. Though grounded in a simple model, our findings offer a novel perspective, merging theoretical frameworks from nonequilibrium statistical physics with evolutionary principles. This perspective may guide more comprehensive inquiries into the intricate nature of neural networks.

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“…The notion that the brain is a predictive or generative machine has been formalized in neuroscience as predictive coding and active inference (or the free-energy principle) [8][9][10][11][12] .…”

Section: Introductionmentioning

confidence: 99%

Future shapes present: autonomous goal-directed and sensory-focused mode switching in a Bayesian allostatic network model

Idei,

Tani,

Ogata

et al. 2024

Preprint

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Trade-offs between moving to achieve goals and perceiving the surrounding environment highlight the complexity of continually adapting behaviors. The need to switch between goal-directed and sensory-focused modes, along with the goal emergence phenomenon, challenges conventional optimization frameworks, necessitating heuristic solutions. In this study, we propose a Bayesian recurrent neural network framework for homeostatic behavior adaptation via hierarchical multimodal integration. In it, the meta-goal of “minimizing predicted future sensory entropy” underpins the dynamic self-organization of future sensorimotor goals and their precision regarding the increasing sensory uncertainty due to unusual physiological conditions. We demonstrated that after learning a hierarchical predictive model of a dynamic environment through random exploration, our Bayesian agent autonomously switched self-organized behavior between goal-directed feeding and sensory-focused resting. It increased feeding before anticipated food shortages, explaining predictive energy regulation (allostasis) in animals. Our modeling framework opens new avenues for studying brain information processing and anchoring continual behavioral adaptations.

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The free energy principle made simpler but not too simple

Cited by 55 publications

References 120 publications

Simplest Model of Nervous System. I. Formalism

Simplest Model of Nervous System. I. Formalism

Simplest Model of Nervous System. II. Evolutionary Optimization

Future shapes present: autonomous goal-directed and sensory-focused mode switching in a Bayesian allostatic network model

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