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Self-organisation  ·  Article

Transient versus asymptotic memory in bistable systems

Memory in a dynamical system can be transient or asymptotic. Transient memory is held for a time and then forgotten. Asymptotic memory persists because the system has more than one stable state, so where it ends up depends on where it started1. A bistable switch is the plain case, recording its last switching event2. The distinction lets authors compare how much of its past a cell retains3.

Earliest held
2008, Wolf DM, Fontaine-Bodin L…
Most discussed in
Memory in microbes: quantifying…, 2008
In the library
27 passages in 8 works
Rewritten
2026-10-12
18501875190019251950197520002025Wolf DM, Fontaine-Bodin L, Bischofs I…, 20081Wolf DM, Fontaine-Bodin L, Bischofs I…, 20082Wolf DM, Fontaine-Bodin L, Bischofs I…, 20083Fernando CT, Liekens AM, Bingle LE…, 20094Lyon P, 20155Law R, Levin M, 20156Power DA, Watson RA, Szathmáry E, Mills…, 20157Power DA, Watson RA, Szathmáry E, Mills…, 20158Levin M, 20199
01

Two kinds of memory

Wolf and colleagues (2008) described history-dependent behavior as spanning several time scales. Short-term, or transient, memory is stored for some time and then forgotten. A system may have effective long-term memory when its transient dynamics are long compared with environmental fluctuations. It has true asymptotic memory when its stationary state depends on initial conditions, as in nonlinear systems with multiple attractors1. Their example of the latter is the bistable switch, whose state encodes the last switching event2.

02

Measured in bacteria

Wolf and colleagues applied the distinction to B. subtilis. Different cellular systems remember different aspects of prior history, so the pathways together store more about the past than any one pathway. The total asymptotic memory was still somewhat less than the total transient memory, at 1.57 against 1.96 bits3. Fernando and colleagues (2009) analysed a circuit model of associative learning in single cells and found a bistable switch with an off and an on state. Its conditioned response lasted about 10.5 hours, then was lost as w2 fell below the threshold4.

03

Bistable switches in cells

Lyon (2015) notes that memory storage in animals with nervous systems was long thought to rely on a kinase and phosphatase pair acting as a bistable switch5. Law and Levin (2015) modelled resting potential bistability and concluded that bistable switches may be built by overexpressing particular channels in oocytes and mammalian cells6. Levin (2019) argues that voltage-gated channels readily form loops that amplify small signals or give robustness, without needing gene-regulatory mechanisms7.

04

Attractors and history

Power and colleagues (2015) take up the same vocabulary for ecosystems. In reply to commentary they agreed that multiple attractors are a generic property of dynamical systems. What they claim as interesting is the location of the attractors with respect to the history of the system8. They describe a system that converts alternate ecosystem states into alternate community states, turning patterns from past environmental states into ecological memories9.

SourcesEach quotation was checked word for word against the passage it opens.
  1. ‘true’ asymptotic memory if the stationary state of the system depends on initial conditions, as occurs in nonlinear systems with multiple attractorsWolf DM, Fontaine-Bodin L, Bischofs I, Price G, Keasling J…, 2008 · Memory in microbes: quantifying history-dependent behavior in a bacterium · open at passage 12
  2. the state of a bistable switch encodes an asymptotic memory of the last switching event.Wolf DM, Fontaine-Bodin L, Bischofs I, Price G, Keasling J…, 2008 · Memory in microbes: quantifying history-dependent behavior in a bacterium · open at passage 12
  3. the total asymptotic memory is still somewhat less than the total transient memory (1.57 vs. 1.96 bits).Wolf DM, Fontaine-Bodin L, Bischofs I, Price G, Keasling J…, 2008 · Memory in microbes: quantifying history-dependent behavior in a bacterium · open at passage 40
  4. the conditioned response continues for approximately 10.5 hours, by which time [w2]=22.8 nM. Any longer and it is lost, as the level of w2 goes below the threshold.Fernando CT, Liekens AM, Bingle LE, Beck C, Lenser T…, 2009 · Molecular circuits for associative learning in single-celled organisms · open at passage 17
  5. Memory storage in animals with nervous systems was long thought to involve a mechanism involving an autophosphorylating protein kinase and paired phosphatase that operate together as a bistable switchLyon P, 2015 · The cognitive cell: bacterial behavior reconsidered · open at passage 76
  6. Bistable switches may be constructed by overexpressing Kir2.1 in amphibian oocytes and Nav1.6 in mammalian cells, and by limiting expression of many of the non-inward-rectifying forms of voltage-gated potassium channels.Law R, Levin M, 2015 · Bioelectric memory: modeling resting potential bistability in amphibian embryos… · open at passage 41
  7. it is not the attractors per se that we claim are interesting – it is their location with respect to the history of the system.Power DA, Watson RA, Szathmáry E, Mills R, Powers ST…, 2015 · What can ecosystems learn? Expanding evolutionary ecology with learning theory · open at passage 58
  8. Figure 6 shows a system that converts alternate “ecosystem states” into alternate “community states”; thus converting patterns from past environmental states into ecological memories.Power DA, Watson RA, Szathmáry E, Mills R, Powers ST…, 2015 · What can ecosystems learn? Expanding evolutionary ecology with learning theory · open at passage 43
  9. Such channels readily enable loops which either amplify small signals (positive feedback) or implement robustness (negative feedback)Levin M, 2019 · The Computational Boundary of a "Self": Developmental Bioelectricity Drives… · open at passage 70
Linked ideas
The bacterial case, including the quantification of transient and asymptotic memory in B. subtilis, sits within this wider topic.
Asks what memory is for; the transient and asymptotic split describes how long and how stably a past state is kept.
Resting potential bistability supplies the cellular voltage states that can act as asymptotic memory.
Channel expression decides whether a cell has bistable voltage states and where its memory loci lie.
Another route by which bioelectric networks hold and update information about past states.
Basal cognitionrelated to
Cognitive vocabulary such as learning and memory is applied to single cells and circuits that show history-dependent behavior.
Where it is discussedPassages matching bistable switch, multiple attractors, asymptotic memory, transient memory
2008Memory in microbes: quantifying history-dependent behavior in a bacterium · Wolf DM, Fontaine-Bodin L, Bischofs I…17
2015What can ecosystems learn? Expanding evolutionary ecology with learning theory · Power DA, Watson RA, Szathmáry E, Mills…4
2016Evolutionary Connectionism: Algorithmic Principles Underlying the Evolution of… · Watson RA, Mills R, Buckley CL…1
2015The cognitive cell: bacterial behavior reconsidered · Lyon P1
2019The Computational Boundary of a "Self": Developmental Bioelectricity Drives… · Levin M1
2015Bioelectric memory: modeling resting potential bistability in amphibian embryos… · Law R, Levin M1
2022Probabilistic Inference with Polymerizing Biochemical Circuits · Katz Y, Fontana W1
2009Molecular circuits for associative learning in single-celled organisms · Fernando CT, Liekens AM, Bingle LE…1