Castellani, Bazzani and Cooper (PNAS 2009) proposed that a 2-step phospho/dephosphorylation cycle on AMPA receptors (AMPARs), catalyzed by three enzymes, is bistable for biologically realistic parameters and — with multiplicative noise in the Michaelis–Menten velocities — accounts for both long-term potentiation (LTP) and long-term depression (LTD) at a single synapse. The stochastic layer of that model is, however, a mesoscopic approximation, and the paper's Discussion explicitly lists what remains unresolved: a truly microscopic, single-molecule description based on the chemical master equation (CME) that justifies the multiplicative-noise assumptions and captures molecular discreteness in small synapses; the incorporation of CaMKII autophosphorylation, expected to add a further stable state; the generalization to an (n+1)(n+1)-state cycle with nn inputs and its comparison with Bienenstock–Cooper–Munro (BCM) theory, including whether nn stable states and a sliding threshold emerge; experimental tests of the predicted LTD-induction dynamics and noise-driven spontaneous-depression time scales; and the characterization of a constructive role of noise in memory storage. The problem remains open.

Background

Long-term potentiation (LTP) and long-term depression (LTD) — the persistent activity-dependent strengthening and weakening of synaptic transmission — are widely treated as cellular substrates of memory. At glutamatergic synapses, bidirectional plasticity is strongly correlated with the phosphorylation state of the GluA1 (formerly GluR1) subunit of AMPA receptors (AMPARs), the receptors that mediate most fast excitatory transmission: phosphorylation of one subunit site by protein kinase A (PKA) and of another by Ca2+^{2+}/calmodulin-dependent protein kinase II (CaMKII) rises with LTP, whereas dephosphorylation accompanies LTD (Lee et al., 2000; Kameyama et al., 1998; Whitlock et al., 2006). Lisman (1989) had proposed that autophosphorylation of CaMKII — the ability of activated CaMKII subunits to phosphorylate each other, keeping the kinase active after calcium falls — could implement a bistable molecular switch (a system with two coexisting stable steady states separated by an unstable one) that stores memory at one synapse, and Castellani and co-workers had built calcium-dependent biophysical models of this signaling pathway (Castellani et al., 2001, 2005).

The source paper (Castellani, Bazzani and Cooper, 2009) proposed a switch of a different kind: a 2-step phospho/dephosphorylation cycle acting on the AMPAR, catalyzed by three enzymes (two kinases acting on distinct phosphorylation sites and one shared phosphatase), so that the receptor moves through three states — unphosphorylated AA, singly phosphorylated APA_P, and doubly phosphorylated APPA_{PP}. In such futile cycles, when enzymes operate near saturation (the zero-order-ultrasensitivity regime), the deterministic Michaelis–Menten rate laws — the standard velocity laws for enzyme-catalyzed reactions — can be bistable (with structural conditions analyzed by Ortega et al., 2006). The authors showed that with parameter values from the biological literature the cycle is bistable over a wide window, mapping the two stable states onto LTP-like and LTD-like phosphorylation levels, and reproducing the AMPAR phosphorylation changes observed after in vivo learning. To represent enzyme-concentration fluctuations they introduced multiplicative noise in the Michaelis–Menten velocities, obtaining a stochastic differential equation whose probability density obeys a Fokker–Planck equation in the Stratonovich sense (the Fokker–Planck equation describes the time evolution of the probability density of a stochastic process; the Stratonovich convention prescribes how fluctuating, state-dependent rates are averaged). In this mesoscopic description, noise drives diffusion between the wells of the bistable dynamics, which yields LTD-like behavior and predicts transition rates between synaptic states.

The paper's Discussion states what remains unresolved. First, the stochastic layer is a mesoscopic approximation: a more fundamental microscopic description should be based on the chemical master equation (CME) — the continuous-time Markov chain tracking the exact copy number of every molecular species, appropriate when only tens of molecules are present, as for AMPARs in a single synapse (see Qian and Elson, 2002 for stochastic single-enzyme kinetics) — both to justify the assumptions of the multiplicative-noise model and to capture molecular discreteness. Second, the model does not include CaMKII autophosphorylation, which the authors believe can lead to a further stable state (cf. Pi and Lisman, 2008, who showed that coupled kinase and phosphatase switches produce the tristability required for LTP, a basal state, and LTD). Third, mathematically the 3-state AMPA model appears equivalent to a 2-input instance of the Bienenstock–Cooper–Munro (BCM) theory of synaptic plasticity (Bienenstock, Cooper and Munro, 1982) — a learning rule in which the sign of synaptic change depends on postsynaptic activity relative to a modification threshold that slides with the history of average activity — but the comparison for n+1n+1 states and nn inputs, and in particular whether nn stable states exist and how the model's parametric bistability windows (its 'history-dependent threshold') relate to the BCM sliding threshold, remains to be worked out. Fourth, the model's dynamical predictions — the induction of LTD and the transition rates, i.e., the time scale of noise-driven spontaneous depression — await experimental tests, for which the paper argues fluorescence measurements of phosphorylation state and electrophysiological recordings at multiple scales are suitable. Fifth, the suggested 'constructive' role of noise — fluctuations enhancing or maintaining memory storage rather than degrading it, as in noise-induced bistability of enzymatic futile cycles (Samoilov et al., 2005) — requires theoretical and experimental characterization.

Problem Statement

Within the biochemical and modeling framework of Castellani, Bazzani and Cooper (2009) — the three-enzyme, 2-step AMPAR phospho/dephosphorylation cycle whose deterministic Michaelis–Menten dynamics are bistable and whose mesoscopic stochastic extension (multiplicative enzyme-concentration noise, Fokker–Planck equation in the Stratonovich sense) reproduces LTP and LTD — establish, and where possible test experimentally, the microscopic single-molecule theory of this bidirectional synaptic switch at a single synapse. The source question comprises five linked components that jointly serve this one objective:

(1) Chemical-master-equation formulation: build the CME for the AMPAR phosphorylation cycle with discrete molecule copy numbers, including the enzymes (not only the substrate) at single-synapse abundances; derive the conditions under which the mesoscopic multiplicative-noise (Fokker–Planck/Stratonovich) approximation of the source paper is valid, and determine whether bistability and the predicted LTP/LTD transition rates between the phosphorylation states survive molecular discreteness at those copy numbers.

(2) CaMKII-autophosphorylation extension: incorporate CaMKII autophosphorylation into this model and determine whether it produces additional stable states (tristability or more complex multistable behavior), and what this implies for LTP, LTD, and depotentiation.

(3) BCM correspondence: generalize the cycle to n+1n+1 phosphorylation states with nn enzymatic inputs and either prove or refute the existence of nn stable states, and establish how the parametric multistability windows of this model class (its 'history-dependent threshold') relate to the sliding modification threshold of nn-input BCM theory.

(4) Experimental tests: test the model's dynamical predictions — the induction of LTD and the time scale of noise-driven spontaneous depression (spontaneous transitions from the potentiated to the depressed phosphorylation state) — against time-resolved measurements, such as dynamic fluorescence readouts of AMPAR phosphorylation state and electrophysiological recordings at molecular, synaptic, and network scales.

(5) Constructive noise: characterize, theoretically and experimentally, whether and how stochastic fluctuations enhance or maintain rather than degrade memory storage in this switch (for example, noise-stabilized or noise-induced phosphorylation states and their lifetimes).

The components are related as follows: (1)–(3) build the microscopic model and its mathematical structure; (4) subjects the resulting predictions to experiment; and (5) is the functional question about noise that the microscopic model (1) makes precise and that the experiments (4) can probe. The theory is to be developed within this AMPAR phosphorylation-cycle framework as stated, without replacing the components with a different model class (for example, a purely calcium-based or purely phenomenological plasticity rule).

References

  1. Castellani GC, Bazzani A, Cooper LN (2009) Toward a microscopic model of bidirectional synaptic plasticity. Proc Natl Acad Sci USA 106(33):14091-14095. doi:10.1073/pnas.0905988106, https://doi.org/10.1073/pnas.0905988106
  2. Bazzani A, Castellani GC, Giampieri E, Remondini D, Cooper LN (2012) Bistability in the chemical master equation for dual phosphorylation cycles. Journal of Chemical Physics 136: 235102. doi:10.1063/1.4725180, https://doi.org/10.1063/1.4725180 (preprint arXiv:1106.4445, https://arxiv.org/abs/1106.4445)
  3. Agarwal A, Adams R, Castellani GC, Shouval HZ (2012) On the precision of quasi steady state assumptions in stochastic dynamics. Journal of Chemical Physics 137: 044105. doi:10.1063/1.4731754, https://doi.org/10.1063/1.4731754
  4. Remondini D, Giampieri E, Bazzani A, Castellani G, Maritan A (2013) Analysis of noise-induced bimodality in a Michaelis-Menten single-step enzymatic cycle. Physica A 392: 336-342. doi:10.1016/j.physa.2012.09.005, https://doi.org/10.1016/j.physa.2012.09.005
  5. de Oliveira LR, Bazzani A, Giampieri E, Castellani GC (2014) The role of non-equilibrium fluxes in the relaxation processes of the linear chemical master equation. Journal of Chemical Physics 141: 065102. doi:10.1063/1.4891515, https://doi.org/10.1063/1.4891515
  6. Borri A, Carravetta F, Mavelli G, Palumbo P (2013) Chemical master equations: a mathematical scheme for the multi-site phosphorylation case. Proceedings of the 3rd International Conference on Simulation and Modeling Methodologies, Technologies and Applications (MATES 2013). doi:10.5220/0004632306810688, https://doi.org/10.5220/0004632306810688
  7. Castellani G, Cooper LN, De Oliveira LR, Blais BS (2021) Energy Consumption and Entropy Production in a Stochastic Formulation of BCM Learning. Journal of Computational Biology 28: 257-268. doi:10.1089/cmb.2020.0118, https://doi.org/10.1089/cmb.2020.0118
  8. Pi HJ, Lisman JE (2008) Coupled phosphatase and kinase switches produce the tristability required for long-term potentiation and long-term depression. Journal of Neuroscience 28(49):13132-13138. doi:10.1523/JNEUROSCI.2348-08.2008, https://doi.org/10.1523/JNEUROSCI.2348-08.2008
  9. Chen M, Ren W, Wang X (2016) Depotentiation from potentiated synaptic strength in a tristable system of coupled phosphatase and kinase. Frontiers in Computational Neuroscience 10: 104. doi:10.3389/fncom.2016.00104, https://doi.org/10.3389/fncom.2016.00104
  10. Singh D, Bhalla US (2018) Subunit exchange enhances information retention by CaMKII in dendritic spines. eLife 7: e41412. doi:10.7554/eLife.41412, https://doi.org/10.7554/eLife.41412
  11. Bartol TM, Ordyan M, Sejnowski TJ, Rangamani P, Kennedy MB (2025) A spatial model of autophosphorylation of CaMKII predicts that the lifetime of phospho-CaMKII after induction of synaptic plasticity is greatly prolonged by CaM-trapping. Frontiers in Synaptic Neuroscience 17: 1547948. doi:10.3389/fnsyn.2025.1547948, https://doi.org/10.3389/fnsyn.2025.1547948
  12. Higgins D, Graupner M, Brunel N (2014) Memory maintenance in synapses with calcium-based plasticity in the presence of background activity. PLoS Computational Biology 10(7):e1003834. doi:10.1371/journal.pcbi.1003834, https://doi.org/10.1371/journal.pcbi.1003834
  13. Elliott T (2011) Stability against fluctuations: scaling, bifurcations, and spontaneous symmetry breaking in stochastic models of synaptic plasticity. Neural Computation 23: 674-734. doi:10.1162/neco_a_00088, https://doi.org/10.1162/neco_a_00088
  14. Karbowski J (2021) Energetics of stochastic BCM type synaptic plasticity and storing of accurate information. Journal of Computational Neuroscience 49: 71-106. doi:10.1007/s10827-020-00775-0, https://doi.org/10.1007/s10827-020-00775-0
  15. Lee HK, Barbarosie M, Kameyama K, Bear MF, Huganir RL (2000) Regulation of distinct AMPA receptor phosphorylation sites during bidirectional synaptic plasticity. Nature 405:955-959. doi:10.1038/35016089, https://doi.org/10.1038/35016089
  16. Kameyama K, Lee HK, Bear MF, Huganir RL (1998) Involvement of a postsynaptic protein kinase A substrate in the expression of homosynaptic long-term depression. Neuron 21:1163-1175. doi:10.1016/s0896-6273(00)80633-9, https://doi.org/10.1016/s0896-6273(00)80633-9
  17. Lisman J (1989) A mechanism for the Hebb and the anti-Hebb processes underlying learning and memory. Proc Natl Acad Sci USA 86:9574-9578. doi:10.1073/pnas.86.23.9574, https://doi.org/10.1073/pnas.86.23.9574
  18. Castellani GC, Quinlan EM, Cooper LN, Shouval HZ (2001) A biophysical model of bidirectional synaptic plasticity: dependence on AMPA and NMDA receptors. Proc Natl Acad Sci USA 98(22):12772-12777. doi:10.1073/pnas.201404598, https://doi.org/10.1073/pnas.201404598
  19. Castellani GC, Quinlan EM, Bersani F, Cooper LN, Shouval HZ (2005) A model of bidirectional synaptic plasticity: from signaling network to channel conductance. Learning & Memory 12(4):423-432. doi:10.1101/lm.80705, https://doi.org/10.1101/lm.80705
  20. Bienenstock EL, Cooper LN, Munro PW (1982) Theory for the development of neuron selectivity: orientation specificity and binocular interaction in visual cortex. Journal of Neuroscience 2(1):32-48. doi:10.1523/JNEUROSCI.02-01-00032.1982, https://doi.org/10.1523/JNEUROSCI.02-01-00032.1982
  21. Samoilov M, Plyasunov S, Arkin AP (2005) Stochastic amplification and signaling in enzymatic futile cycles through noise-induced bistability with oscillations. Proc Natl Acad Sci USA 102(7):2310-2315. doi:10.1073/pnas.0406841102, https://doi.org/10.1073/pnas.0406841102
  22. Qian H, Elson EL (2002) Single-molecule enzymology: stochastic Michaelis-Menten kinetics. Biophysical Chemistry 101-102:565-576. doi:10.1016/s0301-4622(02)00145-x, https://doi.org/10.1016/s0301-4622(02)00145-x
  23. Bazzani A, Castellani GC, Cooper LN (2010) Eigenvalue distributions for a class of covariance matrices with application to Bienenstock-Cooper-Munro neurons under noisy conditions. Physical Review E 81:051917. doi:10.1103/physreve.81.051917, https://doi.org/10.1103/PhysRevE.81.051917
  24. Whitlock JR, Heynen AJ, Shuler MG, Bear MF (2006) Learning induces long-term potentiation in the hippocampus. Science 313:1093-1097. doi:10.1126/science.1128134, https://doi.org/10.1126/science.1128134
  25. Ortega F, Garces JL, Mas F, Kholodenko BN, Cascante M (2006) Bistability from double phosphorylation in signal transduction: kinetic and structural requirements. FEBS Journal 273:3915-3926. doi:10.1111/j.1742-4658.2006.05394.x, https://doi.org/10.1111/j.1742-4658.2006.05394.x