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Hack P, Gottwald S, Braun DA. Jarzyski's Equality and Crooks' Fluctuation Theorem for General Markov Chains with Application to Decision-Making Systems. ENTROPY (BASEL, SWITZERLAND) 2022; 24:1731. [PMID: 36554136 PMCID: PMC9777588 DOI: 10.3390/e24121731] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 10/07/2022] [Revised: 11/19/2022] [Accepted: 11/24/2022] [Indexed: 06/17/2023]
Abstract
We define common thermodynamic concepts purely within the framework of general Markov chains and derive Jarzynski's equality and Crooks' fluctuation theorem in this setup. In particular, we regard the discrete-time case, which leads to an asymmetry in the definition of work that appears in the usual formulation of Crooks' fluctuation theorem. We show how this asymmetry can be avoided with an additional condition regarding the energy protocol. The general formulation in terms of Markov chains allows transferring the results to other application areas outside of physics. Here, we discuss how this framework can be applied in the context of decision-making. This involves the definition of the relevant quantities, the assumptions that need to be made for the different fluctuation theorems to hold, as well as the consideration of discrete trajectories instead of the continuous trajectories, which are relevant in physics.
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Seoane LF. Fate of Duplicated Neural Structures. ENTROPY (BASEL, SWITZERLAND) 2020; 22:E928. [PMID: 33286697 PMCID: PMC7597184 DOI: 10.3390/e22090928] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 07/31/2020] [Revised: 08/18/2020] [Accepted: 08/20/2020] [Indexed: 01/25/2023]
Abstract
Statistical physics determines the abundance of different arrangements of matter depending on cost-benefit balances. Its formalism and phenomenology percolate throughout biological processes and set limits to effective computation. Under specific conditions, self-replicating and computationally complex patterns become favored, yielding life, cognition, and Darwinian evolution. Neurons and neural circuits sit at a crossroads between statistical physics, computation, and (through their role in cognition) natural selection. Can we establish a statistical physics of neural circuits? Such theory would tell what kinds of brains to expect under set energetic, evolutionary, and computational conditions. With this big picture in mind, we focus on the fate of duplicated neural circuits. We look at examples from central nervous systems, with stress on computational thresholds that might prompt this redundancy. We also study a naive cost-benefit balance for duplicated circuits implementing complex phenotypes. From this, we derive phase diagrams and (phase-like) transitions between single and duplicated circuits, which constrain evolutionary paths to complex cognition. Back to the big picture, similar phase diagrams and transitions might constrain I/O and internal connectivity patterns of neural circuits at large. The formalism of statistical physics seems to be a natural framework for this worthy line of research.
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Affiliation(s)
- Luís F. Seoane
- Departamento de Biología de Sistemas, Centro Nacional de Biotecnología (CNB), CSIC, C/Darwin 3, 28049 Madrid, Spain;
- Instituto de Física Interdisciplinar y Sistemas Complejos (IFISC), CSIC-UIB, 07122 Palma de Mallorca, Spain
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Abstract
In a multiverse context, determining the probability of being in our particular universe depends on estimating its overall habitability compared to other universes with different values of the fundamental constants. One of the most important factors in determining this is the fraction of planets that actually develop life, and how this depends on planetary conditions. Many proposed possibilities for this are incompatible with the multiverse: if the emergence of life depends on the lifetime of its host star, the size of the habitable planet, or the amount of material processed, the chances of being in our universe would be very low. If the emergence of life depends on the entropy absorbed by the planet, however, our position in this universe is very natural. Several proposed models for the subsequent development of life, including the hard step model and several planetary oxygenation models, are also shown to be incompatible with the multiverse. If any of these are observed to play a large role in determining the distribution of life throughout our universe, the multiverse hypothesis will be ruled out to high significance.
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Wolpert DH, Kolchinsky A, Owen JA. A space-time tradeoff for implementing a function with master equation dynamics. Nat Commun 2019; 10:1727. [PMID: 30988296 PMCID: PMC6465315 DOI: 10.1038/s41467-019-09542-x] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/04/2018] [Accepted: 03/15/2019] [Indexed: 11/09/2022] Open
Abstract
Master equations are commonly used to model the dynamics of physical systems, including systems that implement single-valued functions like a computer’s update step. However, many such functions cannot be implemented by any master equation, even approximately, which raises the question of how they can occur in the real world. Here we show how any function over some “visible” states can be implemented with master equation dynamics—if the dynamics exploits additional, “hidden” states at intermediate times. We also show that any master equation implementing a function can be decomposed into a sequence of “hidden” timesteps, demarcated by changes in what state-to-state transitions have nonzero probability. In many real-world situations there is a cost both for more hidden states and for more hidden timesteps. Accordingly, we derive a “space–time” tradeoff between the number of hidden states and the number of hidden timesteps needed to implement any given function. Deterministic maps from initial to final states can always be modelled using the master equation formalism, provided additional “hidden” states are available. Here, the authors demonstrate a tradeoff between the required number of such states and the number of required, suitably defined “hidden time steps”.
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Affiliation(s)
- David H Wolpert
- Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM, 87501, USA. .,Arizona State University, Tempe, 85281, AZ, USA.
| | | | - Jeremy A Owen
- Physics of Living Systems Group, Department of Physics, Massachusetts Institute of Technology, 400 Tech Square, Cambridge, MA, 02139, USA
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Laubichler MD, Prohaska SJ, Stadler PF. Toward a mechanistic explanation of phenotypic evolution: The need for a theory of theory integration. JOURNAL OF EXPERIMENTAL ZOOLOGY PART B-MOLECULAR AND DEVELOPMENTAL EVOLUTION 2018; 330:5-14. [DOI: 10.1002/jez.b.22785] [Citation(s) in RCA: 12] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/26/2016] [Revised: 11/03/2017] [Accepted: 11/15/2017] [Indexed: 01/01/2023]
Affiliation(s)
- Manfred D. Laubichler
- School of Life Sciences; Arizona State University; Tempe Arizona
- Marine Biological Laboratory; Woods Hole; Massachusetts
- Santa Fe Institute; Santa Fe New Mexico
| | - Sonja J. Prohaska
- Santa Fe Institute; Santa Fe New Mexico
- Computational EvoDevo Group; Department of Computer Science; Leipzig Germany
- Interdisciplinary Center of Bioinformatics; University of Leipzig; Leipzig Germany
| | - Peter F. Stadler
- Santa Fe Institute; Santa Fe New Mexico
- Interdisciplinary Center of Bioinformatics; University of Leipzig; Leipzig Germany
- Bioinformatics Group, Department of Computer Science; University of Leipzig; Leipzig Germany
- Max-Planck Institute for Mathematics in the Sciences; Leipzig Germany
- Fraunhofer Institut für Zelltherapie und Immunologie-IZI; Leipzig Germany. Department of Theoretical Chemistry; University of Vienna; Wien Austria. Center for Non-Coding RNA in Technology and Health; University of Copenhagen; Frederiksberg Denmark
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Kempes CP, Wolpert D, Cohen Z, Pérez-Mercader J. The thermodynamic efficiency of computations made in cells across the range of life. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2017; 375:20160343. [PMID: 29133443 PMCID: PMC5686401 DOI: 10.1098/rsta.2016.0343] [Citation(s) in RCA: 21] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Accepted: 07/31/2017] [Indexed: 06/01/2023]
Abstract
Biological organisms must perform computation as they grow, reproduce and evolve. Moreover, ever since Landauer's bound was proposed, it has been known that all computation has some thermodynamic cost-and that the same computation can be achieved with greater or smaller thermodynamic cost depending on how it is implemented. Accordingly an important issue concerning the evolution of life is assessing the thermodynamic efficiency of the computations performed by organisms. This issue is interesting both from the perspective of how close life has come to maximally efficient computation (presumably under the pressure of natural selection), and from the practical perspective of what efficiencies we might hope that engineered biological computers might achieve, especially in comparison with current computational systems. Here we show that the computational efficiency of translation, defined as free energy expended per amino acid operation, outperforms the best supercomputers by several orders of magnitude, and is only about an order of magnitude worse than the Landauer bound. However, this efficiency depends strongly on the size and architecture of the cell in question. In particular, we show that the useful efficiency of an amino acid operation, defined as the bulk energy per amino acid polymerization, decreases for increasing bacterial size and converges to the polymerization cost of the ribosome. This cost of the largest bacteria does not change in cells as we progress through the major evolutionary shifts to both single- and multicellular eukaryotes. However, the rates of total computation per unit mass are non-monotonic in bacteria with increasing cell size, and also change across different biological architectures, including the shift from unicellular to multicellular eukaryotes.This article is part of the themed issue 'Reconceptualizing the origins of life'.
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Affiliation(s)
| | - David Wolpert
- The Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501, USA
- Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
- Beyond Center, Arizona State University, Tempe, AZ 85287, USA
| | - Zachary Cohen
- Department of Biology, University of Illinois, Urbana Champagne, Urbana, IL 61801, USA
| | - Juan Pérez-Mercader
- Department of Earth and Planetary Sciences, Harvard University, Cambridge, MA 02138, USA
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Grau-Moya J, Krüger M, Braun DA. Non-Equilibrium Relations for Bounded Rational Decision-Making in Changing Environments. ENTROPY 2017; 20:e20010001. [PMID: 33265092 PMCID: PMC7512193 DOI: 10.3390/e20010001] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 07/30/2017] [Revised: 12/17/2017] [Accepted: 12/18/2017] [Indexed: 11/16/2022]
Abstract
Living organisms from single cells to humans need to adapt continuously to respond to changes in their environment. The process of behavioural adaptation can be thought of as improving decision-making performance according to some utility function. Here, we consider an abstract model of organisms as decision-makers with limited information-processing resources that trade off between maximization of utility and computational costs measured by a relative entropy, in a similar fashion to thermodynamic systems undergoing isothermal transformations. Such systems minimize the free energy to reach equilibrium states that balance internal energy and entropic cost. When there is a fast change in the environment, these systems evolve in a non-equilibrium fashion because they are unable to follow the path of equilibrium distributions. Here, we apply concepts from non-equilibrium thermodynamics to characterize decision-makers that adapt to changing environments under the assumption that the temporal evolution of the utility function is externally driven and does not depend on the decision-maker’s action. This allows one to quantify performance loss due to imperfect adaptation in a general manner and, additionally, to find relations for decision-making similar to Crooks’ fluctuation theorem and Jarzynski’s equality. We provide simulations of several exemplary decision and inference problems in the discrete and continuous domains to illustrate the new relations.
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Affiliation(s)
- Jordi Grau-Moya
- Max Planck Institute for Intelligent Systems, Stuttgart 70569, Germany
- Max Planck Institute for Biological Cybernetics, Tübingen 72076, Germany
- PROWLER.io, Cambridge CB2 1LA, UK
| | - Matthias Krüger
- Max Planck Institute for Intelligent Systems, Stuttgart 70569, Germany
- 4th Institute for Theoretical Physics, Universität Stuttgart, Stuttgart 70569, Germany
| | - Daniel A. Braun
- Max Planck Institute for Intelligent Systems, Stuttgart 70569, Germany
- Max Planck Institute for Biological Cybernetics, Tübingen 72076, Germany
- Institute of Neural Information Processing, Universität Ulm, Ulm 89081, Germany
- Correspondence: ; Tel.: +49-731-5024150
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Abstract
In complex environments, there are costs to both ignorance and perception. An organism needs to track fitness-relevant information about its world, but the more information it tracks, the more resources it must devote to perception. As a first step towards a general understanding of this trade-off, we use a tool from information theory, rate-distortion theory, to study large, unstructured environments with fixed, randomly drawn penalties for stimuli confusion ('distortions'). We identify two distinct regimes for organisms in these environments: a high-fidelity regime where perceptual costs grow linearly with environmental complexity, and a low-fidelity regime where perceptual costs are, remarkably, independent of the number of environmental states. This suggests that in environments of rapidly increasing complexity, well-adapted organisms will find themselves able to make, just barely, the most subtle distinctions in their environment.
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Affiliation(s)
- Sarah E Marzen
- Physics of Living Systems, Department of Physics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA
- Redwood Center for Theoretical Neuroscience, and Department of Physics, University of California at Berkeley, Berkeley, CA 94720, USA
| | - Simon DeDeo
- Department of Social and Decision Sciences, Carnegie Mellon University, 5000 Forbes Avenue, BP 208, Pittsburgh, PA 15213, USA
- Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501, USA
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Abstract
For many organisms, the number of sensory neurons is largely determined during development, before strong environmental cues are present. This is despite the fact that environments can fluctuate drastically both from generation to generation and within an organism's lifetime. How can organisms get by by hard coding the number of sensory neurons? We approach this question using rate-distortion theory. A combination of simulation and theory suggests that when environments are large, the rate-distortion function-a proxy for material costs, timing delays, and energy requirements-depends only on coarse-grained environmental statistics that are expected to change on evolutionary, rather than ontogenetic, time scales.
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Affiliation(s)
- Sarah Marzen
- Department of Physics, Redwood Center for Theoretical Neuroscience, University of California at Berkeley, Berkeley, California 94720, USA.,Physics of Living Systems, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
| | - Simon DeDeo
- Center for Complex Networks and Systems Research, Department of Informatics, Indiana University, 919 East 10th Street, Bloomington, Indiana 47408, USA.,Department of Social and Decision Sciences, 5000 Forbes Avenue, BP 208, Pittsburgh, Pennsylvania 15213, USA.,Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, New Mexico 87501, USA
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Correction: Wolpert, D.H. The Free Energy Requirements of Biological Organisms; Implications for Evolution. Entropy 2016, 18, 138. ENTROPY 2016. [DOI: 10.3390/e18060219] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/16/2022]
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